Logic in the TractatusbyMax WeissA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Philosophy)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2013? Max Weiss 2013AbstractWhat is logic, in the Tractatus? There is a pretty good understanding what islogic in Grundgesetze, or Principia. With respect to the Tractatus no comparablydefinite answer has been received. One might think that this is because, whetherthrough incompetence or obscurantism, Wittgenstein simply does not propounda definite conception. To the contrary, I argue that the text of the Tractatus sup-ports, up to a high degree of confidence, the attribution of a philosophicallywell-motivated and mathematically definite answer to the question what is logic.iiPrefaceWarum hier pl?tzlich Worte?(5.452)It is something of an accident that I ended up writing a dissertation aboutthe Tractatus. In 2009, I was casting around for a thesis topic and somehow foundmyself in Carl Posy?s seminar on Brouwer at TheHebrewUniversity. He seemedlike a pretty good person to ask. When we met, he took out his copy of theCritique of Pure Reason and read out a passage in which he discerned insightslater conveyed in certain early 1970s lectures at Princeton on the philosophyof language. History?or, better, long conversation?brings a structure that hethought I needed. It also helped that Posy clearly enjoys talking about Brouwer.Then it did come down to early Wittgenstein.1 In my first year as an under-graduate, my favorite course had been an introduction to logic taught by San-ford Shieh. I asked him what to read next, and he said, ?well, the Grundlagen ofcourse. Then then youmight want to look at the Tractatus?. So I went out, foundthe Tractatus, and carried it around for a year or two. It ended up on my book-shelf and remained there for about ten years. Some feeling of personal failurelingers around that episode, which maybe a whole dissertation begins to redress.When I first began to read the Tractatus as an adult, then, the first thing thatstruck me was the extreme implausibility of its characterization as a kind of logi-cal atomism. It seemed just incomprehensible to me that Hume or Russell couldsay that each object has written into its nature all its possibilities of occurrence instates of affairs. But it also struck me as unpalatable to respond by arguing thatthe Tractatus is not atomist but holist, since that seems to amount to little more1Or Spinoza. So it could have been worse.iiithan consigning it to the opposing role in some Kantian antinomy. Anyway, thisinitial response was enough for a dissertation prospectus, but that was about it.Then in 2012 I had the good luck to end up in Cambridge, MA, where, as ithappened, Warren Goldfarb was giving a seminar on the Tractatus.2 A few weeksinto the seminar, he pointed out that Wittgenstein tried to develop resources foranalysis of the ancestral of a relation. Goldfarb said he was working on the ques-tion what is the complexity of the property of being a tautology in the Tractatussystem.I was intrigued by Goldfarb?s question, and spent much of that year tryingto assign it a definite mathematical content. The basic idea behind the construc-tion of propositions in the Tractatus is a process of inductive generation. Someelementary propositions are initially given. Furthermore, for any given bunchof propositions, some proposition is the joint denial of the propositions in thatbunch. This construction has some fairly obvious mathematical and philosoph-ical similarities with the cumulative hierarchy of sets. For the cumulative hier-archy of sets is described as the result of repeatedly applying, ?as much as possi-ble?, an operation of set-formation to multiplicities of items ?already? obtained.Moreover, neither construction is an ordinary inductive definition which servesto single out objects from some antecedently given class of objects: rather, theyare each inductions of the sort that Kleene (1952, 258) calls ?fundamental?, pur-porting to generate the entities of the domain in the first place. However, thisresemblance, while suggestive, should not be taken too flatfootedly. In particu-lar, it is implausible and boring to suppose that for any multiplicity of proposi-tions whatsoever, a proposition can be constructed which denies exactly them.What, by Wittgenstein?s lights, actually is the extent of the capacity to surveymultiplicities?At 5.501 Wittgenstein says that one can specify multiplicities of propositionsby ?stipulating the range of a propositional variable.? He gives ?three ways? offixing such ranges.3 The first two ways basically correspond to the syntax of first-2He and a student couldn?t quite agree about whether it had been ten or eleven years since itsprevious offering.3References to entries of the Tractatus (Wittgenstein 1989) will take this standard form, soleexception being single-digit entries which will be prefaced by the letter T. (I will not refer to itspreface.) Reference to entries of the Prototractatus will take the same form but prefixed by PT.ivorder logic. But the third way allows fixing the range of a variable by constructinga so-called ?form-series??roughly, an inductive specification of a multiplicity offormulas. Sadly, Wittgenstein says almost nothing about which operations canbe used in such inductions.Frustrated by Wittgenstein?s vagueness on this point, I decided to explore thehypothesis that one can form the joint denial of any effectively enumerable col-lection of formulas. Under this hypothesis, there then followed an analogy of theTractatus system with the hyperarithmetical hierarchy, a technical constructionwhich arose in mid 20th-century mathematical logic. Goldfarb wasn?t impressed.He urged, in particular, that according to Wittgenstein, we need to be able to?see? which things are subformulas of a given formula. So I tossed aside the arith-metization and tried just to say very carefully how Wittgenstein?s constructionseemed to work. Rather than imposing external ideas about computability ontothe collection of propositional signs, I tried instead to think about operationsthat one might think of as native to the realm of propositional signs itself. Theidea that immediately suggested itself was that of substitution of a formula for apropositional variable in a formula. It quickly became clear that such a simpleidea actually did the required technical work. That is, by forming a disjunctionof the results of iterated substitution, it is fairly straightforward to express theancestral of a given relation. In this way, I settled on what seems to me to be aplausible approximation of the system of logic developed in the Tractatus. It isan ?approximation from below? in the sense that everything expressible in it isclaimed to be expressible in the Tractatus. No converse claim is intended, but Idoubt that significant strengthenings are interpretively defensible.My original formulation of this analysis used a bizarre metalanguage: in par-ticular Wittgenstein?s talk of variable-ranges was handled with plural variables.This made the formulation pretty hard to understand (according to people whotried to read it). So I recast the whole thing as a construction inside a particularstandard model of the axiomatic set theory KPU. The result is a mathematicallydefinite explication of the property of being a tautology which Goldfarb?s origi-References to the Notebooks (Wittgenstein 1979) will always take the form: ?.?.? with the numer-als ?,?,? respectively denoting the day, month, and year. A Roman letter at the end of such areference indicates particular paragraphs of the denoted passage.vnal question addressed. This notion is definite despite avoiding any prejudgmentwhatsoever about the number and form of elementary propositions. By riskingsuch prejudgment, one can instantiate the abstract notion to more concrete ones.Anyway, all this appears in Chapter 5, which embodies the main claim of thethesis: ?here is what Wittgenstein thinks logic is?.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The general propositional form . . . . . . . . . . . . . . . . . . . . . . . . . 121.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.1 Logic and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.2 What about reality? . . . . . . . . . . . . . . . . . . . . . . . . . 261.1.3 Sign and symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.1.4 Propositions as truth-functions . . . . . . . . . . . . . . . . . 451.2 Workings of the GPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.2.1 The one-many problem . . . . . . . . . . . . . . . . . . . . . . 491.2.2 Some early-Russellian technology . . . . . . . . . . . . . . . . 521.2.3 Variables in the Tractatus . . . . . . . . . . . . . . . . . . . . . 551.2.4 Propositional variables: some details . . . . . . . . . . . . . . 601.2.5 Insufficiency of local constraints . . . . . . . . . . . . . . . . 641.2.6 The general propositional form . . . . . . . . . . . . . . . . . 671.3 The GPF in action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711.3.1 A guide to analysis . . . . . . . . . . . . . . . . . . . . . . . . . 711.3.2 Origins of the independence criterion . . . . . . . . . . . . . 812 Objectual generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.1 What is the issue? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93vii2.2 Nonelementary propositions . . . . . . . . . . . . . . . . . . . . . . . . 972.3 Fogelin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.4 Geach-Soames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.5 Wehmeier and Ricketts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.6 Pictoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.7 The picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243 Formal generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.1 Two problems with higher-order generality . . . . . . . . . . . . . . 1373.2 Wittgenstein?s alternative . . . . . . . . . . . . . . . . . . . . . . . . . . 1423.3 Form-series in analysis in the NB . . . . . . . . . . . . . . . . . . . . . 1473.4 Another role for form-series? . . . . . . . . . . . . . . . . . . . . . . . . 1563.5 T5.252 and the evolving conception of hierarchy . . . . . . . . . . . 1583.6 The riddle of T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.7 An alternative reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.1 Ways One and Two, a sketch . . . . . . . . . . . . . . . . . . . . . . . . 1814.2 Way Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.3 The ancestral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.4 More fun with root-substitution . . . . . . . . . . . . . . . . . . . . . . 1904.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935 Effing it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.1.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1975.1.2 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.1.3 Indication and denial . . . . . . . . . . . . . . . . . . . . . . . . 2015.1.4 Truth and possibility . . . . . . . . . . . . . . . . . . . . . . . . 2085.2 Realizations ofB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.2.1 Relational signatures . . . . . . . . . . . . . . . . . . . . . . . . 2125.2.2 FOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2165.3 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222viiiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226ixIntroductionThe aim of this thesis is to answer the following three questions about logic inthe Tractatus.(1) Is there such a thing?(2) What is it?(3) Why is it that?The thesis addresses the questions in reverse order. Chapter 1 addresses question(3), chapters 2-4 address question (2), and Chapter 5 addresses question (1). In thepresent introductory chapter I?ll summarize these results, and insodoing advancewhat is the main claim of the thesis: that Wittgenstein in the Tractatus develops aconception of logic which is philosophically well-motivated and mathematicallydefinite.(3) Pursuing question (3) we might ask what does the author of the Tractatuswant out of a logico-philosophical engagement with logic itself? How did thesubject, or discipline of logic capture his imagination; why, for that matter, doesit warrant his respect? Here it is amusing to compareWittgenstein?s attitude withthat of Descartes. [[cite]] Like Wittgenstein, Descartes thought that logic wasno great body of theoretical knowledge, but for Descartes, this was because hethought that logic was boring. Wittgenstein, to the contrary, felt that logic wasrather too interesting, and that the expressive and clarificatory power of logic be-witches us into delusions about truth and thought. What explains the difference?In 1879 Frege laid the foundation of the modern subject of logic in his bookthe Begriffsschrift. Frege?s preface cites a dream in Leibniz of a lingua characteris-tica, or characteristic language: a language adequate to the subject-matter, in such1a way that the marks of concepts correspond to the parts of the correspondingexpressions of the language. If a problem arises in astronomy, or chemistry orjurisprudence?then we can take out our notebooks and calculate on the expres-sions of the concepts which are constitutive of that problem. Frege remarks thatit is too much to expect all at once a universal characteristic language, a languageadequate all at once to all subjects matter, but suggests that piecemeal progresshad happened already in arithmetic, geometry, and chemistry. And he announcesin BG the completion of another?the central piece, a notation adequate to thesubject-matter of the science of logic. This piece of the puzzle is central becauselogic is what determines what counts as rational justification. The concepts oflogic constitute the evidentiary structure internal to each special science.But logic isn?t just the common portion of the special sciences but a scienceitself. Logic for Frege has a subject-matter of its own: the properly logica objectsand concepts, objects and concepts or functions whose nature we acknowledge (ifonly partially) in any achievement of knowledge. So for example the True itselfis a logical object, as is the False; negation is a logical function, is that functionwhich takes the False to the True and everything else to the False. A basic law oflogic is that the True is the negation of the negation of itself, for example. Thislaw is essential to the use of nonconstructive reductio ad absurdum argumentsin number theory, for example. Logic, for Frege, is a science, the science whosesubject-matter is constitutive of rational justification.So, logic for Frege is that science whose subject-matter constitutes rationaljustification and its truths articulate the general structure of rational justifica-tion. But, for Frege logic is also itself a rational science, consisting of a body oftruths. Being rational, its truths stand in a kind of natural order of justificationor grounding themselves, so that a given truth of logic will rest on other truths,and those on others, and at the bottom you have the basic laws of logic, the truthswhich are basic. The logician, now, makes evident, with respect to some proposi-tion of logic, that it is in fact a truth, bymaking evident this relation of groundingwhich it bears to other already recognized truths. The logician does this by re-ducing the grounding relation down to a series of inferential steps, each itselfevidently justificatory. This is the construction of a proof. What Frege himselfdoes, is codify the concept of proof itself, by giving an axiomatic system. This is2a general recipe for proofs?one might say, the general form of the logically true,or at least, the general form of the theorem.It?s this conception of logic as a science thatWittgenstein, in his book, sets outto destroy. For Wittgenstein, logic is not a science. There are no ?logical truths?.So-called ?logical truths? are really ?tautologies?, i.e., redundancies or pleonasms,like saying the same thing somebody already said; they add no information. Norare there any ?logical objects??logical vocabulary is mere punctuation. Logic hasno subject-matter to describe, so it cannot have any truths of its own. Now, withrespect to all this, it is not just that Wittgenstein wants to prove himself right intheory; he wants to effect meaningful social change, by getting people not to dowhat, under Frege?s program, they?d be doing.So far this is mostly just what we might expect from the author of the Tracta-tus, some stylish opinionation. Can all this talk be substantiated? Wittgenstein?swork had better have some bones in it if it is to disrupt a program with the coher-ence and power of Frege?s. On my understanding, Wittgenstein takes the schemeof Frege?s program and transforms it.Frege uses the word ?proposition? to mean, roughly ?theorem?, or ?state-ment to be proven?, and therefore to mean something that is logically true.Wittgenstein, amusingly, says: there is no such thing as a proposition that is log-ically true. For Wittgenstein, a proposition is essentially not logically true. It isessentially logically possibly false, and logically possibly true. For Wittgenstein,propositions are mere possibilities.Now, this of courses raises the question: what is a possibility? Is it possiblethat this dissertation not have been written? Or that when this sentence waswritten, it was 12:35pm in Vancouver? Or that it have been 12:35pm on the sun?Wittgenstein thinks that, in general, something makes something a possibility.Some possibilities Wittgenstein takes to be what I?ll called basic: their being pos-sible has no further explanation (just as Frege?s being an axiom has no furtherexplanation). Other possibilities are mere positions with respect to the obtainingand non-obtaining of the basic possibilities. Thus, the basic possibilities deter-mine all possibilities, or, all possibilities supervene on the basic ones.For Wittgenstein, this relationship of supervenience of possibilities on basicpossibilities gets made evident through analysis. Analysis reduces an instance of3such supervenience of a possibility to the repeated truth-operations. Analysisthus represents a possibility as the result of repeatedly applying infinitary con-junction, disjunction, negation, etc., on the possibilities which are basic. WhatWittgenstein himself does in the Tractatus is to codify the notion of analysis, bygiving what he calls the general form of the truth-function. So, whereas Fregecodifies the general form of the logically true and says, go find some proofs,Wittgenstein codifies the general form of the logically possible, and says go findsome analyses. Whereas for Frege, logic gives knowledge of truths, for Wittgen-stein logic gives only clarity.(2) Let me now very briefly summarize my answer to the question what logicis supposed to be according the Tractatus. Wittgenstein intends his response toFrege to transform the character of logical activity, so that it is no longer directedtoward recognition of truths of its own proper (incidentally, highest) species, butinstead only toward sharper grasp of what it is for possibilities to obtain and notobtain. The heart of this response is the articulation of the general form of theproposition as the general form of the truth-function. On this articulation, thereare some propositions, the basic or elementary ones, such that every propositionis just a truth-function of them, so that for a proposition to be true is no morethan a matter of which elementary propositions are true. But how, precisely, dothere get to be such things as truth-functions of elementary propositions?Wittgenstein aims with the general propositional form (GPF) to give a gen-eral description of the propositions in any sign-language whatsoever. This goalof generality requires that the GPF contain only what is essential to any wayof constructing truth-functions which a particular sign-language might exploit.Nonetheless, the GPF form is, like Frege?s axiomatizations, itself a kind of techni-cal construction, which requires seemingly accidental hacks and manipulations.Onmy understanding, Wittgenstein?s construction can bemade explicit by posit-ing a scheme of notations, together with a method of associating truth-functionsto notations, in such a way as to sustain the view that to be a truth-function isto be a truth-function associated to a notation in the scheme. The particular de-tails of the scheme itself are to some extent inessential; the actual content of thedescription of propositions which the GPF embodies involves rather only the4totality of associated truth-functions.The scheme of notations can be characterized inductively. Thus, in particu-lar, every basic or elementary proposition may be taken to be a notation itself.But second, suppose that we?ve constructed some bunch of propositional nota-tions, which are, by hypothesis, expressions of possibility. Then, in general wecan construct another expression of possibility by signalling agreement and dis-agreement with obtaining and nonobtaining of possibilities in the bunch. Anexpression constructed by such a signal is the result of a truth-operation on theexpressions antecedently constructed. Wittgenstein initially decides that truth-operations themselves must be of finite number, and also in some sense trans-parently simple. But then he realizes that actually, it suffices to consider a singletruth-operation, that of joint denial.Thus, in general, a nonelementary propositional notation will consist of asign of an operation N , together with an expression A which indicates someother propositional signs A1,A2, . . .. The propositional notation NA is said (inmy terminology) to deny ?directly? the notations A1,A2, . . . which its associatedpointer A indicates. A notation Ai may itself be an elementary proposition, orit may have the form NB where B in turn points to further notations B1, . . . ,Bkwhich are those directly denied by B . In this way, any nonelementary propo-sitional notation stands at the origin of a tree generated from it via the directdenial relation. How does this feature of a notation determine that the notationcorresponds to a truth-function of elementary propositions? It is naturally suffi-cient that the tree of direct denial associated to a notation must be well-founded,terminating only in notations which are elementary.Thus, rather than thinking of propositions as ?built up? from elementarypropositions, instead we can think of the general propositional form as an insis-tence that all propositions can be arranged in a notational structure on whichthe direct denial relation is wellfounded. Analysis is then just a matter of find-ing such a notation for the sentences and thoughts whose truth-conditions wealready grasp. Such a notation for the totality of sentences will induce a ?pre-dicted? consequence relation. As I argue in Chapter 1, an analysis is sound if thepredicted consequence relation agrees with the actual one.Chapter 2 addresses a puzzle in the desiderata for an understanding of logic5in the Tractatus. Wittgenstein insists that we can?t prejudge the question howmany elementary propositions there are, or what is the way in which elemen-tary propositions are built up out of names. Yet any sane explanation of first-order logic certainly prejudges this question. How, in particular, are we to renderquantification, i.e., generalizations over objects, if we don?t know the structuresof propositions? The solution to this problem lies in the 3.31s. Wittgensteindoesn?t there explain propositional functions in terms of ?swapping? of names(or objects) in the way that Frege and Russell do. Rather, he explains them interms of the curiously primitive notion of an ?expression?, the result of some-how ?turning a name into a variable?. Now, Wittgenstein?s point is this. Anexpression marks the sense of a proposition if it is the result of ?removing? aname from that proposition. A propositional function presents an expression,and it ranges over those propositions whose sense that expression marks. But,now Wittgenstein?s funny ?exclusive? understanding of the name-variables fol-lows immediately. So the notion of expression yields the desired technical conse-quences. On the other hand, it also guarantees that the propositions in the rangeof a propositional function all contain the same number of names?hence, havethe same ?mathematical multiplicity?. In this way, Wittgenstein?s distinctive un-derstanding of quantificational generality turns out to be deeply rooted in theidea of propositions as pictures. Moreover, this account requires no assumptionsabout propositions and names: the result of removing the name from that propo-sition is determined merely by the name alongside the proposition. So, Chapter2 addresses the way of stipulating the range of a propositional variable whichunderlies objectual quantification.4Chapter 3 addresses Wittgenstein?s last listed method of fixing the range ofa variable. This method is decidedly non-first-order, that of giving a form-series.Here, Wittgenstein?s ideas are much less developed. Moreover, sinceWittgensteinseems to use something like this method to give the general propositional form,the status of the method is quickly implicated by any serious consideration of dif-ficult interpretive questions about the point of the Tractatus as a whole. But therehas been very little serious study of the details of the development of this idea in4One can also stipulate the range if the range is finite by simply listing its elements?alternatively, one simply write the elements of the range instead of using a variable at all.6Wittgenstein?s thinking. I devote Chapter 3 to this task, and several important in-terpretive results emerge. Wittgenstein sees himself in the Tractatus, in giving thegeneral propositional form, as reconceptualizing the type-theoretical stratifica-tion of propositions found in Russell and Whitehead. The type of a propositionis merely aspectival, depending just on the complexity of a notational exhibitionof its truth-functionality. Thus, Wittgenstein sees the form-series of 4.1252 as anexample of movement from type to type in what becomes of the hierarchies ofRussell and Whitehead, since the formulas in that form-series have increasing no-tational complexity. In this way, Wittgenstein replaces a system of type-theoreticconstraints with a survey of notational freedoms.(1) Finally, in Chapters 4 and 5 I turn to the question whether there is such athing as that which Wittgenstein takes logic to be. The answer to this questionis certainly not settled in the foregoing chapters, since they yield a philosophicalcharacterization of logic which may for one reason or another resist any definiterealization. I show that such a definite realization exists, by actually constructingone.Chapter 4 is an informal introduction. Given that Wittgenstein never devel-oped a systematic analysis of the device of formal series, one can only indulge ininterpretively informed speculation how it might be used. I propose what mustbe the simplest possible approach which meets what is known of Wittgenstein?sexpectations, and explore its use in conjunction with the understanding of ob-jectual generality already developed in Chapter 2. In particular I show how toexpress the ancestral of a relation, and then also some related notions of interestin Wittgenstein?s analytical program.Finally, in Chapter 5, I give the systematic reconstruction of Wittgenstein?slogical system. The reconstruction mimics the scheme of 4.51: ?suppose I amgiven all elementary propositions: then I can simply ask what propositions I canconstruct out of them. And there I have all propositions, and that fixes theirlimits.? So, the reconstruction assumes as given a certain basis, B, which con-sists of the totalities of elementary propositions and of names, together with arelation which, in a sense to be made precise, determines how names characterizethe sense of elementary propositions. The structure B then determines a corre-7sponding class HF(B) of hereditarily finite sets over the totalities of elementarypropositions and names. I then explicitly construct, within HF(B), the totalityZ of propositional notations. This shows that althoughWittgenstein?s system hassome semantic affinities with subsystems of infinitary logic, nonetheless its syn-tax is (almost) perfectly finitary. The semantic notions of truth and validity areanother story. Truth becomes a relation between a formula and a subset of theclass of elementary propositions. The truth-function associated to a notation isthe class of those sets to which that notation bears the truth-relation. I concludeby suggesting that although Wittgenstein rejected full second-order quantifica-tion, nonetheless his conception of logic as elucidation of possibility commitshim to realism about the notion of truth-possibility for elementary propositions.A philosophical theme. The overarching aim of the dissertation is to try toanswer the question what is logic in the Tractatus. However, the Tractatus beingwhat it is, everything hangs together (or falls apart). Occasionally I found myselfstraying into logically inexcusable philosophical matters, sometimes even reach-ing for the same idea on multiple occasions. So in the end I decided to write someof these down, with apologies to the unexpecting reader.Let me just mention here one salient example: throughout the unfolding ofthis project, I?ve been struck more and more by how deeply the conception ofpropositions as pictures pervades Wittgenstein?s thinking about logic itself. Apriori it seemed to me almost impossible that the two topics had anything todo with each other?for one thing, at the outset I found only one of them in-teresting. Nonetheless, again and again in trying to understand Wittgenstein?sthinking about logic I found myself driven to appeal to some aspect or other ofthe idea of propositions as pictures. Moreover, in different theoretical contextsquite different aspects of the idea became salient. So it seems to me that thereis no one single way in which propositions are like pictures; rather, it is a deep,multifaceted heuristic that guides almost all of Wittgenstein?s logical thinking.In Chapter 1, the key influence of the picture theory is this: that in seeing apicture, we thereby see in the picture a possibility for the world. The proposition,in this way like a picture, ?shows its sense?: the proposition shows how the worldis if it is true. This aspect of pictoriality should not be understood psychologisti-8cally. For example, it would be a mistake of the sort that Frege inveighed againstin the Grundlagen, and which I take Wittgenstein to have appreciated deeply, tosuppose that seeing something as a picture entails any sort of a mental episode.For example, it is that sort of mistake which leads to the old complaint that thenegation of a picture can?t be a picture. (That is just like the complaint that it isimpossible to visualize zero things.) Rather, seeing something as a picture meanscommanding its representational powers. Grasping a picture means being able todiscern logical relations between it and other pictures, it means being able (at leastin appropriate cases) to bring about the world that is as the picture depicts it tobe, to hope or fear that the world is as depicted, to see the picture or its negationin the world itself, and so on. Like Frege, Wittgenstein never calls into questionthis power of grasping, never postulates some independent purchase on proposi-tional structure out of which such questioning might gather strength. Rather, byreflection on the understanding of propositions, which their pictoriality makespossible, we might come to reach a clear view of their logical order. Talk of thepictorial likeness of propositions and the world thus recapitulates the idea thatsentences of ordinary language are in logically perfect order as they are.In Chapter 2, a quite different aspect of the pictoriality of propositions emergesto guide the discussion. Wittgenstein thinks of a picture as something like amodel of reality, so that ?names? are arranged in a picture as their bearers arethereby said to be arranged in the world. Thus, a picture, or a proposition, isin some sense a ?hanging-together? of names, just as a possible fact is a hanging-together of objects. Now a fact, Wittgenstein thinks, has a definite ?mathematicalmultiplicity?; roughly speaking, this is?perhaps among other things?amatter ofhow many objects it incorporates. A proposition expressing the fact must havethe same multiplicity, i.e., must contain the same number of different names.Thus there cannot be different names of the same object. Nor does it really makesense to distinguish between different occurrences of a name in a proposition,any more than there is any natural way to talk about the number of times anobject occurs in a fact. How many times does Obama occur in the fact that heis father of both Sasha and Malia? (Of course, people are not actually objects,but set that aside.) This question makes no sense. Nor, then, is there any moresense to the question how many times Obama?s name occurs in the proposition9that he is their father. As it turns out, all this metaphysical waxing has logical sig-nificance: for the idea of mathematical multiplicity is the key to Wittgenstein?sinsight that the equality predicate is redundant. For, once we refuse to countdifferent occurrences of a name in a proposition, then the so-called ?exclusive?interpretation of the name-variables is practically forced on us.Yet another aspect of the conception of propositions as pictures operates inChapter 3. A big problem in that chapter is to try to figure out what justi-fies Wittgenstein?s appeal to this seemingly bizarre device of the form-series inhis analysis of logical structure. What, in general, licenses the analysis of someproposition into others? Here, the guiding idea is that a proposition must haveits logico-mathematical content in common with what it depicts. So, Wittgen-stein thinks, even though ?Socrates? isn?t really a name, still we must be able toreason with ?Socrates? as though it were really a name: thus, he insists, the logicof Principia Mathematica is (pretty much) applicable to ordinary, unanalyzed sen-tences just as they are. But how can this be, if such application means treating asnames items which are not really names at all? The proposition to be analyzedmust bear a quite definite resemblance to the propositions into which analysistransforms it. Wittgenstein must then have felt that a form-series variable is likea propositional function, in bearing some such quite definite resemblance to itsvalues.So there is an interpretive claim: that Wittgenstein?s thinking about logicalmatters is shaped pervasively and multifariously by the conception of proposi-tions as pictures. Soon I felt this feature of the form of Wittgenstein?s thinkingto be reflected in the content of his thinking too, in what one might call a thesisof primacy of picturing. The point is that it is a mistake to think that pictur-ing is something achieved with signs when signs are put to (appropriate?) use.Rather, as McCarty (1991, 72ff) puts it, ?understanding is like standing?. Signsare an abstraction from pictures, and come into view only when something goeswrong. Like the one-sided face of illusions, signs are epiphenomenal. So it isbackward to postulate a task of assignment of reference to names, because to bea name is to go proxy for an object in a picture. But it is also backward to thinkof facts and objects as ?popouts? of antecedently determinate inferential practice:no inference runs free of the responsibility to truth and falsehood. On a very10schematic level, one might put the point as follows: that the formal unity be-tween proposition and situation precedes both its decomposition into names andits individuation as a nexus of the inferential web. In even more schematic form:there is no horizontal falling apart without vertical unity.5That, however, is about as far as I should go in an introduction. So let mejust conclude with a remark on the structure of exposition. The material of eachchapter was prepared for standalone presentation (containing advertisements forother material). So, each chapter can be read independently of the others. Inparticular, somebody who quickly wants the ?technical core? may just read ?1 ofChapter 5, consulting Chapter 4 for motivation and examples. Chapters 1 and2 are addressed to a historically informed philosophical audience. Chapter 3 isprimarily exegetical. Chapter 1 establishes the general interpretive outlook.5No mortal interpreter could fail to find something of themselves, or of their teachers, in whatthey read. Here I should acknowledge the influence of Roberta Ballarin and Ori Simchen, tor-menting as usual the Carnap of Alan Richardson.11Chapter 1The general propositional formAll the propositions of ordinarylanguage are actually, as they are,in perfect logical order. (5.5563)1.1 AnalysisAt about the midpoint of the Tractatus, there appears an announcement:It now seems possible to give the most general propositional form:that is, to give a description of the propositions of any sign-languagewhatsoever6 in such a way that every possible sense can be expressedby a symbol satisfying the description, and every symbol satisfyingthe description can express a sense, provided that the meanings of thenames are suitably chosen.. . .The general form of the proposition is: This is how things stand.(T4.5)After slogging through a tedious and slightly confusing exposition of truth-tablesand of truth-functional validity, when the reader?s patience is running thin al-ready, this elaborate buildup arises more or less out of the blue, ascending to itsvaunted climax. The climax itself may then seem a little bit trite.7 Nonetheless,6This translation is Pears-McGuinness. Ramsey-Ogden have here instead: ?description of thepropositions of some one sign-language?. The German has ?irgend einer Zeichensprache?, withemphasis on the first two words.7As several commentators have observed.12I think that it nicely summarizes some of the early Wittgenstein?s fundamentalideas.The mention of ?things? is intended to bear no weight.8 Indeed, I thinkthat whatever information there is here, it lies inside of the demonstrative ?this?.One might as well read the entire sentence as an ?ahem?, carefully enunciatedas the speaker sweeps back a curtain to show you. What such a speaker wouldshow you, behind the curtain, is how things are. Perhaps, in the best instance,the speaker, in revealing what is behind the curtain, shows you how things are,because, behind the curtain, there they are.But, not every instance is of the best kind. For example, it might arise that,say, things are too far away, or, for that matter, that somebody will not lookat them. For example, Hamlet resorts to hiring actors, who go proxy for thereal villains and victims. Thereby the king is brought despite himself to see howthings are: it suffices that he see the actors, arranged with respect to each other inthis visceral narrative arc, and then, suddenly to realize whom they represent.Unlike the king, of course, usually people do want to see how things are.They might be, so to speak, curious. A common practice, in such a case, is toask somebody. But let?s consider this condition of wondering: what is it that?swondered? Whatever it is, one can even ask of somebody if that?s how thingsare. So, it looks as though the general form of the proposition could also be putas a question: is this how it is? Once again there is just a demonstrative in thesubject-position, with an accompanying ?ahem? now modulated differently. Butthis alternative formulation brings something out: that, a priori, things mightbe as we?ve wondered, and, they might not be. As Wittgenstein put the point,?thought can be of what is not the case?. One can hardly pose a question bypointing the worldly circumstances that would form its answer, for it is igno-rance of those worldly circumstances which animates the interrogation in thefirst place.According to Wittgenstein, rather than simply pointing out how the worldis, instead people resort to pictures: that is, to constructing a representation ofsome situation from a position outside of that situation. We can make ourselves8In German, it is Es verh?lt sich so und so.13understood in this way, because in the picture, names are arranged as things arethereby said to be arranged. Thus, asWittgenstein puts the point, a picture showsits sense; it shows how things stand if it is true. Since a picture is not what itportrays but a replica, there is room for two possibilities: that things be, and thatthey be otherwise than, the picture portrays them.So, the Tractatus conception of pictures has two sides. The picture must bedifferent from what is depicted, in order for the two to disagree. But the picturemust be the same as what is depicted, in order for the picture to show how thingsstand if it is true. The difference arises because in the picture names go proxy fortheir bearers. The similarity arises because the picture and what it depicts havein common what Wittgenstein calls pictorial form.Wittgenstein puts forward this conception of pictures as a completely generalconception of thought. Thus, for Wittgenstein, every thought is, in some suchway, a picture. The thought of things in a certain way, is very much like the sightof them that way. What is curious about this idea is that it seems, to put it mildly,like it might not be right.Tomakematters worse, Wittgenstein goes on to say that a thought is a propo-sition with sense, and that the totality of propositions with sense is language (4,4.001). It seems to me that somebody who wants to maintain that thinking is akind of language will not help their case by maintaining that thoughts are likepictures, because, for example, sentences and pictures seem so unlike each other.As Frege puts the point, ?It is striking that visible and audible things turn uphere along with things which cannot be perceived with the senses. This suggeststhat shifts of meaning have taken place? (1984, 352). How could Wittgenstein?sconception of propositions as pictures even seem to be right?As Frege points out, the problems with this conception are manifold:Is a picture considered as a mere visible and tangible thing really true,and a stone or a leaf not true?A picture is supposed to represent something. . . . . It might be sup-posed from this that truth should consist in correspondence of a pic-ture to what it depicts. [. . . ]Now a correspondence is a relative relation. But this goes against the14use of the word ?true?, which is not a relative term. . . .A correspondence, moreover, can only be perfect if the correspond-ing things coincide and so just are not different things. It is supposedto be possible to test the genuineness of a bank-note by comparingit stereoscopically with a genuine one. But it would be ridiculousto compare a gold piece stereoscopically with a twenty-mark note.(Frege 1984, 352).9In the following two sections, I?ll try to explain how the conception of propo-sitions as pictures derives from Wittgenstein?s understanding of the nature oflogic.1.1.1 Logic and analysisFrege?s 1879 Begriffsschrift opens with this remarkable passage:In apprehending a scientific truth we pass, as a rule, through vari-ous degrees of certitude. Perhaps first conjectured on the basis of aninsufficient number of particular cases, a general proposition comesto be more and more securely established by being connected withother truths through chains of inferences, whether consequences arederived from it that are confirmed in some other way or whether,conversely, it is seen to be a consequence of propositions already es-tablished. [. . . ] The most reliable way of carrying out a proof, obvi-ously, is to follow pure logic, a way that, disregarding the particularcharacteristics of objects, depends solely on those laws upon whichall knowledge rests. (Frege 1967, 5)Frege thus opens his book with a summary of the means of recognizing scientifictruths. He distinguishes between the way that we happen to come to consider aproposition to be true, and the best possible foundation which the truth of thatproposition could receive. How we happened to discover the truth may havepractically nothing to do with the content of the truth itself. But, following the9Though this passage was published in 1918, its composition dates to Wittgenstein?s boyhood(Frege 1979, 127).15accident of its discovery, we begin to see the grounds in nature upon which itrests. The kind of grounds to which this deepening grasp will lead, depends onthe particular nature of that truth.According to Frege, it is obvious that the most reliable kind of proving of ascientific truth would disregard all particular characteristics of objects, and de-pend only on laws on which all knowledge rests. It would follow only ?purelogic, a way that, disregarding the particular characteristics of objects, dependssolely on those laws upon which all knowledge rests? (5). As is well known,Frege strove to uncover such a source for the truths of number. This would yielda rigorous proof of the laws of number on the sole basis of maximally general lawsof logic. The value of such a proof would not consist in its raising our subjectiveconfidence in the truth of the laws of arithmetic. Rather, it would reveal theirplace in a natural justificatory order. Basic logical laws would thus themselvesstand at the root of a science of pure reason, which develops upward throughthe expression of further logical truths and blossoms into the science of number.Thus, in devising his ideography, Frege had in mind ?right from the start?, the?expression of a content? (Frege 1979, 12): the contents of general logical laws,of arithmetic theorems, and finally of the application of all these rational truthsthroughout the special sciences of astronomy, chemistry, biology and so on.Wittgenstein?s response is to declare that the propositions of logic are tau-tologies. Now, when he wrote the Tractatus, the word ?tautology? was not atechnical term, or not much of one. It meant something like ?redundancy?, or?pleonasm?.10 So as he immediately explains, the point is that ?the propositionsof logic say nothing?, and that ?all theories that make a proposition of logic ap-pear to have content are false?.11 Of course, by the time Wittgenstein composedthis passage, Frege had published his Begriffsschrift and Grundgesetze, and Rus-sell and Whitehead had published the Principia. There was a firmly establishedmethod of ?proving? logical propositions, i.e., of deriving them by rules of infer-ence from initially given propositions. Wittgenstein acknowledges the existenceof this method as a datum:The proof of logical propositions consists in the following process:10See Dreben and Floyd (1991) for a fascinating discussion.116.11, 6.111.16we produce them out of other logical propositions by successivelyapplying certain operations that always generate further tautologiesout of the initial given ones. (6.126)This passage encapsulates Wittgenstein?s ambivalent regard for the logicalachievements of Frege and Russell. On the one hand, for example, at 3.325Wittgenstein urges upon philosophers the use of a ?sign-language that is governedby logical grammar?by logical syntax.? And, he says, ?the conceptual notationof Frege and Russell is such a language, though, it is true, it fails to exclude allmistakes.? Wittgenstein finds that Frege and Russell correctly aim at construct-ing a sign-language which does not just ape the whims of words but respects theforms of logical syntax?even if, in pursuit of this aim, they sometimes fall short.For example, he complains that Frege and Russell ?introduce generality in asso-ciation with logical product or sum?, which ?made it difficult to understand thepropositions in which both ideas are embedded? (5.521). And, ?it is self-evidentthat identity is not a relation between objects? (5.5301); ?Russell?s definition of?=? is inadequate? (5.5302). And again, according to 4.1273, the way in whichFrege and Russell want to express the ancestral ?contains a vicious circle?. Thus,Frege and Russell sometimes falter in the matters of detail. But, Wittgenstein canonly pick nits like this because there is something about their projects which hedoes find illuminating, or seductive?the analysis of logical structure.Thus, for example, Frege?s discovery of the quantifier reveals something ofthe essence of propositions. For, as Wittgenstein puts it, ?in a proposition theremust be exactly as many distinguishable parts as in the situation that it repre-sents.? He considers various alternative notations for generality, for example, anotation which would express the universal generalization of a formula F x bysimply prefixing the letter G to it. Such a notation lacks the relevant ?multiplic-ity? achieved by Frege?s quantifier notation, and therefore must be inadequate(4.0411). One cannot, in the expression of generality, ?get away? from the kindof representational structure that the quantifier notation makes explicit. For thisreason, the quantifier notation reveals something in the nature of representationitself.12 The quantifiers embody some kind of important insight.12Here, one might wonder: in the nature of all representation? Or just in representation of gen-erality? But, Wittgenstein held that much ordinary representation is covertly general. Thus, my17So, Wittgenstein thinks that the Frege-Russell program of analysis of logi-cal structure is somehow on the right track. But I said, he regards their logicalachievements with a kind of ambivalence. The basic reason for this is that theysubordinate the analysis of logical structure to a further aim, the developmentof an ultimate deductive system which would form the organon of a new sci-ence. Frege and Russell, so it seems, devoted years of their lives to the senselesstransformation of empty formulas into empty formulas by emptiness-preservingrules?this in spite of Frege?s emphatic insistence that, ?right from the start? hehad as his aim with ideography the ?expression of a content.?13 Now, Wittgen-stein certainly concedes that an ideography like Frege?s can serve for the expres-sion of a content, and indeed, can so serve as an improved, elucidatory form ofexpression. But, at least to the extent that the underlying logical analyses are freeof defects, a sentence will have a proof in a system of Frege or Russell?s only if itfails to express a content. So there is certainly nothing to be found along the linesof a new and central rational deductive science that Frege, following Leibniz,envisioned. It now looks like Wittgenstein?s reading of Frege is that he single-handedly invented the piano, and even tuned it pretty well, but never envisionedits use to make music. Such an interpretation of Frege looks not implausible butincoherent: one cannot invent a piano but as a musical instrument. Wittgen-stein must find it puzzling how it could have been the logicists who ?broughtabout an advance in Logic comparable only to that which made Astronomy outof Astrology, and Chemistry out of Alchemy?.14There must be something?if not to how things must be if a tautology istrue?at least, to the fact that a proposition is a tautology, which explains howsuch facts could have so entranced the inventors of serious logic.The fact that the propositions of logic are tautologies shows the formal?logical?properties of language and the world.claim would be at least: quantifier notation reveals something in the nature of ordinary represen-tation. One might also argue that Wittgenstein, following Frege, finds the articulated structure ofany representation, general or not, to be coeval with the relationship of that representation to thegeneralities it instantiates. However, Wittgenstein?s understanding of this relationship is somewhatdifferent from Frege?s, so I won?t press the point here.13This irony is noted by Juliet Floyd (Floyd and Shieh 2001, 154).14Wittgenstein 1913, 351. Thanks to Goldfarb for goading me into this question.18The fact that a tautology is yielded by this particular way of connect-ing its constituents characterizes the logic of its constituents.If propositions are to yield a tautology when they are connected ina certain way, they must have certain structural properties. So theiryielding a tautology when combined in this way shows that they pos-sess these structural properties. (6.12)The propositions of logic demonstrate the logical properties of propo-sitions by combining them so as to form propositions that say noth-ing.This method could also be called a zero-method. In a logical propo-sition, propositions are brought into equilibrium with one another,and the state of the equilibrium then indicates what the logical con-stitution of these propositions must be. (6.121)15These passages present Wittgenstein?s error-theoretic interpretation of logi-cism. Since Wittgenstein holds that logical truths are not truths at all, there can-not be such a thing as a science whose proper truths are the propositions of logic.Nonetheless, this notion identified by Frege, being a proposition of logic, musthave some kind of logical significance which explains how it could have so en-tranced him. In the passages just quoted, Wittgenstein sketches a kind of inquiryin which interest in the property of being a tautology takes the form it deserves.The deserved form of this interest is the following: that a tautology results fromcombining some propositions in a certain way tells us about the structure ofthose propositions. Wittgenstein gives examples of this sort of inference not justin the not 6.12s, but also in the 4.12s and the 5.13s. That conditionalizing A onB yields a tautology tells us that B entails A; that a tautology results by negating15Note that in the Tractatus, 6.121 comes in the wake of the seagulls of 6.1203, which once gaveme the impression that ?this method? refers to the flight of the seagulls. But, that is not correct.In the Prototractatus, the antecedent of 6.121 is a commentary on the antecedent of 6.1221, whichsimply remarks that it says something about p ? q and p that p?(p ? q)? q is a tautology. ThePrototractatus contains no seagulls; Wittgenstein seems to have dredged them up from some veryold notes, perhaps in a fit of sentimentality (the editors of KS (144) cite the youthfully exuberantletter to Russell of 1903 (number 28 in McGuinness, ed., 2008), whose claim to a decision proce-dure for quantificational logic is explicitly retracted in 6.1203?s first sentence). Thus, the Tractatusnumbering of 6.121 is appropriate, since 6.121 is a commentary not on 6.1203 but on 6.12.19the conjunction of Awith B tells us that A contradicts B .16So, in the 6.12s,Wittgenstein sketches out a distinctive distribution of knownsand unknowns that characterizes a certain kind of inquiry. What we don?t know,and what we want to investigate, are the internal structures of some propositions,say A and B . On the other hand, we know how to combine A and B into a singleproposition (5.131) whose sense is a function of the senses of A and B (5.2341).And, moreover, since a tautology must itself show that it is a tautology (6.127),we can tell whether or not a tautology results by combining A and B into a sin-gle proposition. Consider a simple case, in which the analyst combines queriedpropositions A and B into the assembly ?(A?B), where, it turns out, this assem-bly is a tautology. The analyst knows that negation of A?B is true if and only ifA?B is false, that A?B is true if and only if A and B are both true, and further-more, that a tautology is always, in a degenerate sense, true (4.46ff). In this way,that ?(A?B) is a tautology shows the analyst that A and B are never both true, orin other words, that A contradicts B . So, for example, the analyst might recordthat ?Ludwig is a bachelor? contradicts ?Ludwig is unmarried?. Then, ?Ludwigis a bachelor? might be rewritten ?Ludwig is not married, and . . . ?, or ?Ludwigis married? might be rewritten ?Ludwig is not a bachelor, and . . . ? where in eachcase, the ellipses remain to be filled in by considering some concepts of gender.This is the sort of result that might be extracted from the information that a givenproposition is a tautology, and, according to Wittgenstein, such results are whatmakes the property of being a tautology worth of attention. Such results, I takeit, are the results of what Wittgenstein understands to be logical analysis.Of course, the example I just gave is implausibly simplistic. Less trivial exam-ples are readily found, like ?if there are at least three cows and at least four hens,and if no cows are hens, and if all cows and hens are animals, then there are at leastseven animals.? This and related examples might lead to a quantificational analy-sis of number-words of the sort that Frege considers in ?55 of the Grundlagen;17eventually, though, other examples involving arithmetical generalizations mightjustify Frege?s rejection of the quantificational analysis in favor of something else.In any case, more interesting analytical problems are ready to hand.164.1211; 5.13-5.131.17Frege 1953, 67.20Still, there is something rather strained in this discussion. Why is it necessaryat all to combine the queried propositions into a single one and ask whether it is atautology? Can?t we simply discern between the queried propositions themselveswhatever internal relations guarantee the tautologousness of their combination?That is, isn?t enough just to consider the relations between the constituent sen-tences ?there are at least three cows?, ?. . . at least four hens?, etc.? Wittgensteinputs the point as follows.18If the truth of one proposition follows from the truth of others, thisfinds expression in relations in which the forms of the propositionsstand to one another: nor is it necessary for us to set up these rela-tions between them, by combining themwith one another in a singleproposition; on the contrary, the relations are internal, and their ex-18Proops (2002, 288) describes the quoted remark as ?darkly metaphorical?, and spends much of(2002) ?unpacking? it. Proops? view is that 5.132 attacks, in Frege anyway, a ?proof-theoretic? ac-count of logical consequence that might be extrapolated from the account of dependence-between-truths of in ?Foundations of Geometry II? (Frege 1984, 333ff). I think, in contrast, that Wittgen-stein?s point is much more direct. Given that tautologies don?t describe the world, and so lacksense, why did Frege and Russell keep on asserting them? Being logicians, they begin with a nat-ural desire to understand logical consequence (and other internal relations between propositions).But since logical consequence finds no direct expression in their systems, they therefore sublimatethe desire to understand it into a sterile enumeration of tautologous conditionals. The reflectionof internal relations between propositions in the tautologousness of logical combinations of thosepropositions gets mistaken, by the logicist, for interesting facts which tautologies report. In thisway, the predilection for tautologies is just one among many examples of the confusion of materialrelations with formal ones ?which is very widespread among philosophers? (4.122). I see the driftof the 5.13s to be smoothly continuous with the 6.12s and diagnostic of obsession with all kindsof tautology, including disjunctions, biconditionals, negated sentences, etc., and correspondinglyfinding the actually interesting underlying facts to be not just entailments but also contradictions,equivalences, etc. So, onmy reading, it is only because Frege asserts so many conditionals, i.e., onlybecause his system contains a primitive sign for conditional rather than for disjunction, conjunc-tion, etc., that Wittgenstein focuses on logical consequence in the 5.13s. In a way, then, my viewis somewhat closer to the view of Ricketts (1985), according to which Wittgenstein attributes toFrege and Russell confusion about the concept of inference rule?though I don?t take Wittgensteinto attribute a Carroll-style confusion to Frege or Russell.Proops finds that Wittgenstein attacks a Fregean theory of entailment that a scholar might ex-trapolate from the ?Foundations II? passage and one highly compressed remark in Grundlagen (at?17). But since Frege doesn?t think the general concept of entailment is very important, and per-haps even doubts its cogency, it is not clear why he should be much bothered by criticism of his ac-count of it?so much the worse for your concept, he might reply. In contrast, I take Wittgenstein?starget to be the elephant in the room, that Frege (and Russell) keep on making senseless assertions.On my reading, 5.132 fits naturally in the thematic nexus of the 4.12s-5.13s-6.12s, which form asystematic response to logicism.21istence is an immediate result of the existence of the propositions.(5.131)So, Wittgenstein thinks that it is a theoretically important datum that a propo-sition is a tautology. Nonetheless, this datum is only important because of itsmembership in a broader family of logical phenomena. It is just as interestingthat a proposition be contradictory, or that some propositions entail or contra-dict another. The logicists? enumeration of tautologies is misguided not onlybecause of the underlying false pretense that the tautologies are the truths in theorganon of some new science. It is moreover misguided because tautologousnessis just one logically important phenomenon amongmany others. Some such phe-nomena are also just properties of single propositions, like, for example, logicalcontingency; but others are not properties but relations between propositions,like entailment, or joint contradictoriness or joint satisfiability.As I understand Wittgenstein, then, the analyst begins with an appreciationof internal relations between propositions, such as entailment, exclusion, and soon. This appreciation embodies a grasp of how the world must be, if a propo-sition is true. Analysis then seeks to clarify thoughts which are obscure andconfused, by setting propositions which express them into relations with eachother, and noting the logical properties of the results. Thus, analysis is a mat-ter of construction. But it is construction on ordinary propositions, proceedingupward from our ordinary, human appreciation of the force of what we say, ofprecisely how we, in picturing the world, merely thereby hold the world to yesor no. Rather than speaking of analysis as ?top down?, then, I?d speak of it as?top up?.In the distribution of knowns and unknowns which characterizes the predica-ment of analysis, the logical structure of propositions falls halfway in shadow andhalfway in sunlight. On the one hand, Wittgenstein seems to take for grantedthat the significance of logical combinations of already understood propositionsis completely clear. Indeed, he flamboyantly pushes this assumption to the brink,or, it?s often complained, past19 the brink of tenability, declaring, for example,that ?proof in logic, is merely a mechanical expedient to facilitate the recognition19Cf., among many, Potter 2009, 191.22of tautologies in complicated cases? (6.1262). Thus, returning to our simplisticexample, in order that the tautologousness of ?(A? B) count as evidence that Acontradicts B , Wittgenstein must take for granted a grasp of the rules for evaluat-ing negation and conjunction, a recognition of applicability of those rules to theexpression ?(A? B), and the capacity to detect logical consequences of such ap-plications of the evaluation rules. Taking all this for granted, the part of ?(A?B)which, as it were, exists over and above A and B themselves, is completely trans-parent. The presumption of transparency of logical construction is to be justifiedby appeal to the character of ordinary speaking, thinking and understanding. Inparticular, supposing two people each to have asserted propositions, then a thirdperson may deny that they are both right, by uttering the words ?you are notboth right?. These words do not just dump the speaker into some undifferen-tiated role of naysaying, but construct an intellectual position explicitly, so thatfrom the construction everybody recognizes, modulo what the first two speakerssaid, what it takes for the third speaker to be right or wrong.On the other hand, propositions are given to the analyst as only superficiallyarticulated wholes. There is nothing on the surface of ?Ludwig is a bachelor? and?Ludwig is married? to indicate why one entails the other but not conversely, foron the surface these have the same structure.20 As Wittgenstein rather quaintlyputs it, ordinary language is not designed to reveal the forms of thought under-neath it, anymore than clothing is designed to reveal the form of the body (4.002).So, one cannot infer from the surface structure of the sentence ?Ludwig is a bach-elor?, the form of the thought it expresses. Analysis therefore proceeds by almostBaconian poking and prodding of its subject matter.We?ve supposed that, given as subject-matter the two propositions A and B , alogical analyst constructs the assembly ?(A? B). In this complex assembly, theoccurrence of A and B is manifest: indeed, Wittgenstein goes so far as to speak ofA and B as occurring in ?(A?B) as ?constituents? (Bestandteile), even though thisseverely violates his logico-grammatical sensibilities.21 Moreover, it is not just the20Wittgenstein does acknowledge, for example, that ?Ludwig is a bachelor? carries something onits surface, which explains, say, how the result of conjoining it with ?Ludwig is a philosopher? en-tails ?some philosopher is a bachelor?. However, this surface structure will presumably disappearon analysis, giving way to some other structure which legitimates the same inferences and more.21 Cf. ?a proposition cannot occur ?in? another?, MN, p.116.23occurrence of A and B but the logical meaning of this occurrence which ?(A?B)makes plain: ?(A?B) is, as it were by construction, that proposition which is trueprecisely when not both of A and B are true. Wittgenstein officially introducesthis relationship between the construction and its raw materials as follows:The structures of propositions stand in internal relations to eachother. (5.2)In order to give prominence to these internal relations, we can adoptthe following mode of expression: we can represent a proposition asthe result of an operation that produces it out of other propositions(which are the bases of the operation) (5.21).In the particular case we?re considering, Wittgenstein represents a certainproposition, C , as the result of negating the conjunction of A and B , by express-ing it in the notation ?(A?B). The fact that C can be so represented shows thatC stands in a certain internal relation to A and B . Now, because the propositionC is a tautology, this internal relation of A and B to C points out an internalrelation between A and B themselves, namely that A contradicts B . Accordingto 5.21, it ought to be possible to give prominence to the contradiction of A byB , by presenting B as the result of an operation on A. The obvious move is totry representing B simply as ?A, so that ?Ludwig is a bachelor? would be writtenas ?Ludwig is not married?. But, B can be so represented only if the formulaB ??A is a tautology, i.e., only if the following is a tautology: that Ludwig is abachelor iff Ludwig is not married. But Ludwig might be divorced, or not of mar-riageable age, etc. So one cannot simply represent the situation in which Ludwigis a bachelor as the situation which obtains when he is not married. Rather, hisbachelorhood consists of his not being married together with further conditions,say that he?s of marriagable age, but has never been married; let?s write these asA?. Now, granting that B ??A?A? is indeed a tautology, then B itself can sim-ply be written as ?A?A?. This rewriting of B in terms of A and A? is anotherinstance, perhaps even a paradigm, of the method described at 5.21.Of course, following tradition, we might want to acknowledge that the pred-icate, being a bachelor, entails more than just being of marriageable age yet un-married; someone does satisfy these conditions without being a bachelor if she24is a ?she?. However, what is given for analysis is not the predicate, being a bach-elor, but rather just the whole proposition that Ludwig is a bachelor. There isnot much record of the early Wittgenstein having developed any particular viewson the metaphysics of gender. But it seems likely that he would have held that?Ludwig is male? is itself a tautology, i.e., analytic, since he probably held, forexample, that ?Ludwig is in the room? logically entails ?a male is in the room?.22Then he would likewise have held that ?Ludwig is unmarried? entails ?Ludwig ismale?. So, in the analysis of the single, whole proposition ?Ludwig is a bachelor?,the extra conjunct suggested by the traditional analysis would be redundant. Wecould have anticipated that the suggestion of tradition may not be immediatelyapplicable, because whereas the tradition aims at analyzing predicates (say, intospecies and differentia), Wittgenstein aims at analyzing whole propositions.23We can see how Wittgenstein takes analysis to illuminate logical structure ofthe propositions analyzed. For example, we began with the datum that a con-junction ?(A?B) was a tautology, which is not explained by the overt structureof A,B . The datum indicates the obtaining of internal relations between A andB , and in turn the possibility of rewriting B as the result of an operation on A(along with some auxiliary operands). Upon replacing the expression B with itsanalyzed form ?A?A? in B , we obtain a formula ?(A??A?A?)whose validity isnow completely independent of the forms of its as yet unanalyzed constituents.Thus, rewriting B as ?A?A? explains the originally given datum that ?(A? B)is a tautology. Of course, we aren?t yet done, since, for example, A? was con-structed from propositions to the effect that Ludwig has never been married, andthat he is of marriageable age; the internal relations in which these stand to theproposition that he is married, remain to be articulated.Analysis therefore begins by observing an internal relation between propo-sitions which is not explained by their overt structure. By hook or by crook,22In NB2, he certainly held, for example, that if x is a part of y, then ?y is in the room? logicallyentails ?x is in the room? (cf. e.g. 17.6.15f) Moreover, he seems to have held that propositionsabout x entail propositions about the configurations of the parts of x. The acceptance of suchentailments is clearest in the Notebooks, but it surfaces in the Tractatus, particularly at 3.24. Suchprinciples, together with some not unquestionable assumptions about the metaphysics of gender,do yield the conjectured entailment from ?Ludwig is in the room? to ?a male is in the room?.23Here, he follows the path cleared by Frege. See in particular T3.3: ?only in the context of aproposition has a name a meaning?.25the analyst finds a way to rewrite the given propositions in terms which do thenexplain the observed internal relation. Now, a typical initial datum will be thattwo propositions cannot both be true, or that one cannot be true unless anotheris, and so on. Moreover, pursuit of the desired rewriting is constantly guided byknowledge of obtaining and nonobtaining of internal connections as well: forexample, such knowledge is needed to reject the hypothesis that to be a bache-lor is simply to be unmarried. So, analysis takes as its data not what is true butwhat is possible: as Wittgenstein puts it, ?logic deals with every possibility andall possibilities are its facts? (2.0121). Thus, supposing A to be analyzable as theresult O(A1, . . . ,Ak) of applying operation O to some other propositions, thenthere are no possibilities in which A is true but O(A1, . . . ,Ak ) false, or vice versa.Conversely, onceO andA1, . . . ,Ak are chosen so that no possibilities discriminatebetween A and O(A1, . . . ,Ak ), then nothing remains to distinguish between thelogical positions of A and O(A1, . . . ,Ak ) at all, and so they are the same position.Absenting global theoretical considerations, nothing impeaches the analysis of Aas O(A1, . . . ,Ak ). In the long run, however, the overarching aim of analysis is tomake all covert logical structure explicit in the forms of propositional expression.This is a global program that addresses internal relations between all proposi-tions; it is this responsibility to the global structure of language as a whole whichdetermines what is to be analyzed in terms of what.So, Wittgenstein conceives of analysis as introducing logical structure to ex-plain antecedently grasped possibilities and necessities. Necessity, then, is thebasis of logical structure. But now, the following is clear: Wittgenstein simplyhas no room for any kind of necessity but logical necessity. Necessities unex-plained by logical structure can only indicate that analysis is incomplete. Thisresult, that ?there is only logical necessity? (6.37), is the reason for Wittgenstein?saccount of propositions as pictures. But more on that in the next section.1.1.2 What about reality?As I?ve just argued, Wittgenstein?s conception of analysis leaves no room for anybut logical necessity. But this doctrine is double-edged.24 It may look at first like24Thanks to Roberta Ballarin for this way of putting the point.26a purification of the concept of necessity, but at the same time it entangles theapplication of logic with metaphysics. For, as Wittgenstein acknowledged, theworld is full of necessity.A spatial object must be situated in infinite space. (A spatial point isan argument-place.)A speck in the visual field, though it need not be red, must have somecolor: it is, so to speak, surrounded by color-space. Notes must havesome pitch, objects of the sense of touch some degree of hardness, andso on. (2.0131)A property is internal if it is unthinkable that its object should notpossess it.(This shade of blue and that one stand, eo ipso, in the internal relationof lighter to darker. It is unthinkable that these two objects shouldnot stand in this relation.) (4.123)25[. . . ] the simultaneous presence of two colors at the same place inthe visual field is impossible, in fact logically impossible, since it isruled out by the logical structure of color. (6.3751)25The next sentence of this entry is ?(Here the shifting use of the word ?object? correspondsto the shifting use of the words ?property? and ?relation?.)? From 4.122, it?s clear that by ?shiftinguse? of ?property? and ?relation?, he means the ?confusion between internal relations and relationsproper?, as between internal and genuine properties. Since each of ?internal relation? and ?relationproper? expresses a formal concept, the slide is a confusion between two different formal concepts.Thus, the corresponding shift in use of the word ?object? ought also to be a confusion betweentwo different formal concepts. Now, Wittgenstein holds that the concept of simple object and theconcept of complex object are distinct formal concepts. A sharp statement of this point occurs in1913:Russell?s ?complexes? were to have the useful property of being compounded, andwere to combine with this the agreeable property that they could be treated like ?sim-ples?. But this alone made them unserviceable as logical types, since there would besignificance in asserting, of a simple, that it was complex. But a property cannot be alogical type? (NL 100-101).So, I conjecture that Wittgenstein has in mind at 4.123 a confusion between the use of ?object?to refer to simples and its use to refer to complexes. (Such confusion will not be unfamiliar tostudents of NB2.) The objects mentioned at 4.123 and 2.0131 presumably support the necessitiesthey do because they are not really ?objects? in the sense of 2.02 (?objects are simple?); rather theyare complexes, so that statements about them stand in internal relations to statements about theirconstituents (2.0201, 3.24). So, the last sentence of 4.123 suggests that the necessities cited in thetext are purportedly genuine, rather than just metaphorical.27We are now in a position to summarize these results in an inference. Sincethe world is full of necessity, and since there is no necessity but logical, thereforelogic fills the world.26I have been promising for some time to explain why Wittgenstein thinks thatpropositions are pictures of reality. The picture theory is largely driven by twoconvictions about the nature of the logic. First, logic illuminates internal rela-tions between propositions, internal relations which weave together all suscep-tibility to truth and falsehood. But second, because internal relations subsumethe necessities of the world, therefore the world itself must somehow be full oflogic. From these two convictions, Wittgenstein is driven to the conclusion thatinternal relations between propositions must reflect the necessities and possibil-ities of the world. It is by virtue of a likeness of propositions to the world thatthe necessities and possibilities of the world can be grasped through understand-ing of internal relations between propositions. This likeness of propositions tothe world is the pictoriality of propositions: ?sign and thing signified must bealike with respect to their total logical content? (4.9.14). Thus, Wittgenstein aimswith the doctrine of pictoriality of propositions to secure an identity betweenthe logical space of the truth and falsehood of propositions in language and theontological space of the obtaining and nonobtaining of possible situations in theworld.One route to unifying the spaces of propositions and of situations was al-ready trod by Russell:27 that is: just identify the propositions with the facts, sothat a proposition simply is the fact that would obtain were it true. Suppose that aproposition says how things are, by specifying a way for things to be, and specify-ing which things are that way. The simplest way to do this is for the propositionsimply to consist of those things, being that way. But then, a proposition cannotsay how things are unless things are that way. So, a proposition cannot consistof the things which it?s about, standing in the relation in which the propositionrepresents them to stand.26Wittgenstein does say at 5.61: ?Die Logik erf?llt die Welt?. But he surely intends more in thatpassage than I will have to offer today.27Having been blazed by Moore (1899).28Nonetheless, as the conception of analysis drives Wittgenstein to insist, theremust be something in common between propositions and situations. Wittgen-stein appears to settle for the next best thing to Russell?s identity theory. Aproposition says of some things that they are in a certain way, by being somethings, standing to each other in just that way?only in the proposition, we findnot the constituents of the fact standing for themselves, but some other itemsgoing proxy for them. These proxies, or ultimate propositional constituents, arewhat Wittgenstein calls names. So, as Wittgenstein puts it:In a proposition a situation is, as it were, constructed by way of ex-periment. [. . . ] (4.031)One name stands for one thing, another for another thing, and theyare combined with one another. In this way the whole group?like atableau vivant[lebendes Bild]?presents a state of affairs. [4.0311]Wittgenstein thus conceives of propositions as pictures or models of reality.As I claimed, what motivates this conception is its potential to secure a unity be-tween the spaces of propositions and of possibilities. For, onWittgenstein?s view,it is part of what makes something a picture that the possibility of the picture be,somehow, at one with the possibility of what it depicts. Wittgenstein?s intro-ductory remarks on the representational power of pictures make this motivationexplicit:That the elements of a picture are related to one another in a deter-minate way represents that things are related to one another in thesame way.Let us call this connection of its elements the structure of the picture,and let us call the possibility of this structure the pictorial form ofthe picture. (2.15)Pictorial form is the possibility that things are related to one anotherin the same way as the elements of the picture. (2.151)These sentences are pretty fraught with verbal ambiguities. They do have thefollowing ontological counterparts:29The determinate way in which objects are connected in a state ofaffairs is the structure of the state of affairs. (2.032)Form is the possibility of structure. (2.033)2.15b and 2.032a strike me as ambiguous between explaining ?structure? as ei-ther the relation in which the elements of the complex stand to each other, or thecircumstance that those particular elements stand in the relation in which theystand. The subsequent talk of the ?possibility of structure? slightly favors thesecond reading, since it is not really what it means for a relation to be possible,unless this is understood as the possibility of its instantiation. So I take ?struc-ture? to be tantamount to ?picturing fact?. This interpretation is also supportedby Wittgenstein?s later usage of ?structure? as for example in 4.1211, 5.13, etc.28For example, the structures of f a and ga can show that a occurs in both onlyif the structures involve a itself. But, Wittgenstein also apparently intends thatthe sort of things shown by the structures of facts (or pictures or propositions)depend only on cross-reference to those things between the facts, rather than onthe identities of the things?this is what makes them structural. So, the structureof a fact must involve elements of the fact in such a way as to make sense of thequestion whether the structures of a variety of facts have elements in common.But there is no need to suppose anything more to the involvement of elementsin structures than what is required to decide whether and how various structureshave elements in common.Following the introduction of the notion of structure, 2.15b and 2.033 intro-duce a corresponding notion of form, as the possibility of structure. Accordingly,in the case of a state of affairs, form is the possibility that the elements of a stateof affairs do stand in the relation in which they therein stand, and likewise inthe case of a picture, form is the possibility that the elements of the picture standin the relation in which they stand therein. An obvious interpretation of theseremarks is that form is what makes structure possible. This interpretation is nat-urally elaborated by example. A structure might consist of a golf ball sitting inthe hole of a donut. The golf ball and the donut are so shaped as to explain,jointly, that as it turns out, one fits the other, i.e., their two shapes then jointly28As pointed about by Ramsey (1923, 466).30explain the possibility of the structure. But, note that the possibility is explainedonly by the two shapes together. So, although the donut turns out to accommo-date the golf ball, there is nothing to the donut in itself which would presage theirgraceful union. Wittgenstein sharply rejects such an understanding of objects: ?ifa thing can occur in a state of affairs, the possibility of the state of affairs mustbe written into the thing itself? (2.012). Thus, for Wittgenstein, it is not the casethat objects have, on their own, forms which, considered alongside the forms ofother objects, only then explain possibilities of structure.Citing 2.012 and related passages, Michael Kremer remarks: ?we seem to bedriven inexorably towards a holistic reinterpretation of the picture theory, ac-cording to which it is not individual propositions but the whole system of lan-guage which ?pictures? the world? (1992, 421). Thus, although a propositionmust, in order to picture reality, share its logical form with reality, it is not tothe single proposition, but to the whole of language, that logical form should beascribed. For Kremer, language as a whole is a ?system of signs governed by rulesof application?; these rules of application of signs are what constitute the formof language as a whole. Now, according to Kremer, we need an account not justof logical form of language as a whole, but, also, derivatively, of the logical formsof individual propositions. The logical form of one proposition will be given byits characteristic place in the ?overall network of propositions?. For example, adisjunction might occupy a place as it were above each of its disjuncts, below eachof which would be the conjunction of those two disjuncts. On the other hand,reality itself also has a logical form, in which situations are arranged with respectto relations of necessitation, exclusion, etc. A sameness of form between languageand reality then induces a correspondence between propositions and situations;a proposition then depicts the situation to which it corresponds.On this account, logical form, which Kremer tends to identify with logicalmultiplicity, is still what something must share with reality in order to repre-sent reality. A system of notation, like those Wittgenstein scrutinizes at 4.0411,may ?lack the necessary multiplicity to mirror the logical structure of the world,by lacking the wherewithal to reflect certain logical interrelationships betweensituations reflected in the standard notation? (1992, 422). Kremer?s account ofmultiplicity, like that of the Hintikkas (1986), appears to presuppose that there31is such a thing as the multiplicity of a system of notation, such that, with respectto this multiplicity, we can raise the question whether it is adequate to mirrorreality. Kremer and Hintikka agree that being formally adequate to reality is acondition on language. Of course, Kremer differs from the Hintikkas over rel-ative priority of the loci of form within language: the Hintikkas think that itflows from the parts to the whole, whereas Kremer thinks it flows in the otherdirection.29Apparently in elaborating the holistic nature of logical form, Kremer goeson repeatedly to compare language as a whole with a picture (421,422,423). It isnot at all clear to me how to understand this comparison. Saying proposition isa picture entails distinguishing in the proposition the form it must share with re-ality, from the particular modification of that form which makes something thatproposition. The former is that in virtue of which a proposition is susceptible totruth and falsehood, the latter is how reality must be if the proposition is true.But then what is it to call language as a whole a picture? What are the poles oftruth and falsehood for language as a whole? What does reality have to do tosay yes? I can?t see any other answer here than: that language and reality havethe possibilities of agreement and disagreement with respect to their forms. Lan-guage as a whole depicts reality to be in a certain way?only, the kind of way thatlanguage-as-a-whole depicts for reality is a super-way, perhaps the structure of thesystem of coordinates at which the possibilities of obtaining and nonobtaining ofstates of affairs take their place.30In the context of this shift to holism, Kremer cites the following famous pas-sage:How can logic?all embracing logic, which mirrors the world?usesuch particular hooks and manipulations?31 Only because they areall connected with one another in an infinitely fine network, thegreat mirror. (5.511)3229McCarty (1991, 52-53) warns against this hazard of ?top-down? approaches to the Tractatus.30More in ?1.3 on ?pre-pictorial? structure.31Pears-McGuinness translateHaken und Manipulationen as ?crotchets and contrivances?, whichI can?t do with a straight face. It may be that Wittgenstein alludes here to a means of weaving orcrocheting together of the infinitely fine network mentioned in the next sentence.32This translation is not in Kremer.32There is a puzzle about this passage which Kremer doesn?t mention. A map, forexample, can be evaluated for its adequacy by comparison with the region it pur-ports to depict. This is because the map is a self-standing image, a structure whichexists independently of that with which it is to be compared. On the other hand,a mirror is a piece of glass, one side of which is covered with a reflective material.The glass and the underlying reflective material do not themselves constitute astructure, such that the mirror is then to be evaluated for adequacy based on theresemblance of this structure to, say, Ludwig?s face. That is rather what we wouldsay of something like a portrait. The mirror in itself is blank. The very imageof ?mirroring? undermines the conception of the relationship between languageand reality which Kremer and the Hintikkas use it to illustrate.Sullivan (2001) squares up to the problem that is bothering me. To restate:the problem is just, why do the logical properties of propositions have anythingat all to do with the material necessities of the world?let alone, in their struc-ture, reflect those material necessities completely? Or as Sullivan puts it: ?whatconnects our picturing-propensities with the space of possible worlds?? (9). Itake him to agree that part of the point of the metaphor of picturing is to ges-ture toward some connection of our saying-propensities and the space of possibleworlds. But does the metaphor of picturing lead to any insight here? Or is it justpalliative care?Sullivan helpfully emphasizes Wittgenstein?s insistence on the connection be-tween pictures and possibility. Thus he cites these passages:The picture contains the possibility of the state of affairs it repre-sents. (2.203)The thought contains the possibility of the situation of which it isthe thought. What is thinkable is also possible. (3.02)The proposition determines a place in logical space. The existenceof the logical place is guaranteed by the existence of the constituentparts alone, by the existence of the significant proposition. (3.4)Following 3.4, it becomes clear in 3.41 that by ?constituents?Wittgenstein meansconstituents of the proposition, and these are presumably names. So it is tempt-ing to fall back on the idea that type-preserving referential links between names33and objects guarantee that every sentence does pick out a place in logical space(see 4.0312). But, for Sullivan, that referential links should by themselves securecoincidence of possibility seems like ?dubiously grounded confidence? (12). Sulli-van then suggests that in relying only on reference here, we omit any role for thenotion of pictorial form. Thus, rather than accepting at face value 2.15?s appar-ent verbal stipulation that pictorial form just is the possibility of the picture, thesuggestion is that pictorial form takes on some explanatory burden of its own.This suggestion strikes me as exegetically plausible. Wittgenstein?s talk ofform alludes to Kantian terminology of ?forms of intuition?, apparently in par-ticular to the rhetoric that ?we? cannot represent objects of intuition but as subjectto the conditions which inhere in that form of intuition. This is pretty explicitat 2.0121: ?Just as we are quite unable to think spatial objects outside of space,or temporal objects outside of time, so too there is no object that we can imagineexcluded from the possibility of combining with others.? Such overtones are toostrong to ignore, although their purpose is another matter.33Sullivan suggests that we should ?think of the pictorial form of a picture asconstituted by the form of its elements? (13). The elements of the picture, be-ing significant expressions in a language, are therefore subject to distinguishingrules of use, which determine their potential for combining into propositions;this constitutive combinatory potential of an expression is its form. Any partic-ular proposition is then an ?actualization of the possibilities built into the formsof its constituents? (14).34 If, then, the combinatory potentials of names lineup with the combinatory potentials of the objects for which they stand, then aproposition, actualizing as it does the the potentials of the constituents, wouldtherefore guarantee the potential in objects for a corresponding reality. But why,Sullivan continues, should we assume any such alignment? Sullivan proposes thatthe alignment is simply a condition of picturing, and the hypothesized alignmentcannot be amystery?although he does then call it a ?demand?, and starts looking33I do not think that the Kantian rhetoric here should be taken to suggest that Wittgenstein issome kind of Kantian: Wittgenstein might well be turning Kantian rhetoric against Kant, as heclearly does at 6.233. I hope to investigate these matters elsewhere.34This formulation strikes me as both toomuch and too little. Toomuch, because the possibilityof the proposition is already built into any one of its constituents. Too little, because some relationsare neither symmetric nor asymmetric.34for motivation.Sullivan uncovers a motivation for this demand which reveals some uneasewith his account of pictorial form as constituted by the forms of pictorial ele-ments. On his official account of pictorial form, a spatial picture containing ared toy car and a blue toy car might have a different form from a spatial picturecontaining a red toy car and a red toy stop sign: stop sign and car have differentcombinatory potential. However, this official account seems to sit poorly withthe text. The text seems to imply that all spatial pictures have one and the sameform: the form of space. And Sullivan clearly registers this implication. A spatialpicture, Sullivan acknowledges, may be made up of ordinary spatial objects, spa-tially arranged. Thus, as ordinary spatial objects, these elements?say, toy cars,have forms of objects, namely space, color, mass, etc. But, the forms of the pic-ture and its elements, qua bald facts and objects, must be distinguished from thepicture?s properly pictorial form. In a spatial picture, the toy cars go proxy forother items in reality, say cars. On the other hand, other parts of the picture donot go proxy for things distinct from themselves in reality, and yet contribute tothe pictorial form: in the case of a spatial picture, the spatial arrangement of thetoy cars simply is (modulo scaling factors) the arrangement of cars depicted. Fi-nally, some aspects of the picture, say, that the toy cars are made of plastic ratherthan wood, have no pictorial significance at all. What is essential to the picture(to what Wittgenstein calls the ?symbol?) is just the role of the pictorial elementsto go proxy for objects, and those properties of the pictorial elements which thepicture asserts of the corresponding objects.35Now, the crucial point for my purposes is this: merely in seeing somethingas a picture, we already discern in it some parts which go proxy for objects, andsome properties of those parts which the picture asserts of those objects. Suchpower of discernment is prior to seeing something as a picture at all. Seeingsomething as a picture requires decomposition into constituent names and theirassertoric assembly. Spatial pictures are unified by the existence of a single power35Note that, of course, a spatial picture may also be, for example, a color-picture, if the colorsof the elements of the picture are intended to say that the corresponding objects are so colored.But, a spatio-colored picture is at the same time a (merely) spatial one, and can therefore be graspedmerely as such, by taking the colors to go proxy for themselves.35of discernment, which is necessary and sufficient for seeing all spatial pictures aspictures.36 This power is the grasp of the spatial form of picturing. Wittgensteinholds that mere grasp of the spatial form of picturing suffices to see what is shownby any spatial picture, i.e., suffices to see in the picture how things must be if it istrue. The basis of this conviction is that the spatial form of picturing just is spatialform, it is what is common to every spatial possibility, to every determinationof space. Thus, an identity between all spatial pictures and all spatial possibilitiesunderlies the capacity to see what a spatial picture shows, i.e., to see how thingsmust be if a spatial picture is true. Conversely, it also underlies the conversecapacity to see the picture in the possible fact.37 In sum, it underlies the capacityto picture spatial facts to ourselves.We?ve been struggling with the question how what can be spatially picturedis just the same as what can happen in space. The tempting answer is that some-thing about spatial pictures out to coincide with something about space. But, thisanswer is wrong. For it presupposes that there is something like a way that a pur-ported spatial picture is, or a way that all purported spatial pictures are, whicheven makes room to wonder about a coincidence of possibilities. And there issupposed to be no room to wonder here. But why not? I think it may help toswitch metaphors:A gramophone record, the musical idea, the written notes, and thesound-waves, all stand to one another in the same internal relation ofdepicting that holds between language and the world. There is a gen-eral rule by means of which the musician can obtain the symphonyfrom the score, and which makes it possible to derive the symphonyfrom the groove on the gramophone record, and using the first rule,to derive the score again. That is what constitutes the inner sim-ilarity between these things which seem to be constructed in suchentirely different ways. [. . . ] (4.014)The possibility of all imagery [Gleichnisse], of all of our pictorial36This is an oversimplification. An accurate, though less evocative statement is this: the formof a picture consists in a single power of discernment, which is necessary for seeing all pictures ofthat form as pictures.37After all, picturing is a two-way street.36modes of expression, is contained in the logic of depiction. (4.015)Recast in terms of the musical metaphor, our questions becomes something likethis. What is the guarantee, not that a performance render the score faithfully,but just that the score can be rendered at all? Couldn?t there be an intrinsicallyunplayable musical manuscript? Conversely, couldn?t there be a musical perfor-mance which could not be written down? Could there be an inexpressible mu-sical thought? These questions come into focus when we set aside the grammo-phone, and consider, say, 19th century European art music. Would it be possibleto hear aMahler symphony that could not be written down? CouldMahler writedown a symphony that could not be played? Could you play one that couldn?t beheard? But now, as should be clear, the questions are getting silly. Their answer issomething like ?maybe, sort of?. In one degenerate sense or another, there mightbe an unplayable score, but it would be a score only, as it were, by accident (hencethe degeneracy). The connections between the realms of musical composition,and performance practice, and audience appreciation, all belong to the nature of(19th-century European art) music. The general unification of these media byitself precedes, and makes possible, any particular activities in one medium orthe other. The different formats of music agree with each other, falling apart intodiscrete nodes with matching combinatory potentials, because they have a com-mon form. This common form is (19th century European art) music. Withoutmusic, a score could be at best a piece of (20th century) graphic design. It wouldbe logically possible, of course, to write a computer programwhich translates thepatterns of the piece of graphic design into sound (say, as conceptual art). But theresulting structural resemblance between the ink patterns and the sounds wouldnot be the internal relation between score and performance. In other words, tothe question, how could the existence of a musical score guarantee the possibilityof its performance, the answer is: the unity of all scores and performances pre-cedes the isolated possibility of existence of any particular score or performance.There is no such thing as a score, unless there is such a thing as (19th centuryEuropean art) music; nor is there such a thing as a performance. For otherwise:a score, or a performance, of what?37The previous paragraph is of course too picturesque to be really philosoph-ical. But the basic point can be put a little more dryly: the unity between anygiven picture and the situation it depicts is prior to the articulation of anything asthat picture. For, the possibility of the situation is already contained in the merepossibility of that picture. Or, better, these possibilities are one and the same.What makes both picture and situation possible is logical form, or the form ofreality. The form of reality itself, prior to any of its accidental modifications, al-ready underlies the unity between the attributes in which the modifications takeplace. So, the proposition is articulated, because it, like any possible thought,participates in logical form and thereby stands at one with the situation it de-picts. A proposition is articulated, while a merely typographic entity is not. Fora proposition to be articulated just is for it to participate in logical form. Andto participate in logical form is to submit to the rule whereby a situation can beread off. It is likewise by participating in logical form that a world falls apartinto facts to be read off in propositions.38 In slogan form: there is no horizontalfalling apart without vertical unity.1.1.3 Sign and symbolI now want to clear up certain ontological or nomenclatural questions aboutlinguistic expressions. Wittgenstein famously distinguishes between ?sign? and?symbol?. The subtleties of this distinction are obviously implicated in difficultinterpretive questions about the nature, or lack thereof, of nonsense, senseless-ness, and so on. I can?t do justice to these questions here, nor to the voluminousliterature they?ve engendered. For present purposes, my interest in Wittgen-stein?s account of linguistic expressions turns on the widespread attribution toWittgenstein of a conception of logical truth as somehow basically ?linguistic?in nature. I will aim to suggest, though certainly not prove, that Wittgenstein?snotion of a linguistic expression undermines that attribution.It may be helpful to distinguish two kinds of questions. First, we might askwhat is the Tractatus ontology of linguistic expressions. Second, we might ask38So, logical form is not ?imposed? on the world, but the world itself, in some sense, falls apartalong its lines.38how various terms in the Tractatus are used in different contexts to pick out con-cepts in the ontology. After all, it is entirely possible, and I think, almost cer-tainly true, that while Wittgenstein?s ontology is fairly clearly worked out, theusage of terminology is somewhat uneven.As I argued in ?1.1.2, seeing some bald circumstance as a picture presupposessome grasp of a general system or form of picturing. This claim applies in particu-lar to occurrences of sentences of a natural language. That is, what embodies suchan occurrence is, for all that?s sensibly perceptible, just a bald circumstance, andseeing it as a picture of reality requires a grasp of logical form. Similarly, a graspof logical form is just as well required to see in some piece of reality which inten-tionally articulated actions depict it. In general, then, a bald circumstance ?fallsapart? into a picture. The picture itself is a nexus or hanging-together of names; itrepresents that the objects of the names stand to each other as the names stand toeach other in a picture. This hanging together is what I?ll call the pictorially sig-nificant commonality between the picture and depicted situation, or, for short,the ?common cement?.Some kinds of picture share pictorially significant commonalities with whatthey depict which are material: for these material commonalities to be pictoriallysignificant is for the picture to represent its objects as instantiating them. Thusfor example, in a spatial picture of a cup on a table, the contiguity of the cup-image and table-image is a pictorially significant material commonality becausethe picture thereby represents the cup and image as instantiating contiguity. Incontrast, logical pictures share no pictorially significant material commonalitieswith what they depict. Their purely logical ?common cement? is a purely logicalconnection between their constituents (presumably something like instantiationof one by others).Among the various bald circumstances that turn out to obtain, some of themconstitute utterances or inscriptions. Such circumstances fall apart into articu-lated pictures of reality, and in particular, into pictures which are purely logical.The purely logical pictures into which utterance/inscription/etc.-constitutingcircumstances fall apart might be called sentences, propositions, or propositionalsigns. Now, as I?ve argued, it is essential to such a logical picture to enjoy a formal(or ?vertical?) unity with a situation which is thereby the depicted situation. But,39just as nothing counts as performing one musical score unless that score is oneamong many, likewise there can be no single isolated moment of picturing: assoon as one picture is possible, many pictures are possible too. So, one circum-stance can?t appear as a sentence unless lots of possible circumstances do so too:for example, nothing can appear as a sentence until something else can appear asthat sentence, and something else can appear as its negation, and so on.As a matter of psychological fact, however, it is not the case that endlesslymany merely possible circumstances could, in the fallible human eye, all at oncefall apart into pictures of distinct though logically related situations unless thecircumstances were constructed somehow systematically. How in practice wemanage to achieve this humanly necessary systematicity is a matter of empiricalpsychology or anthropology. Certain unsurveyably complicated understandingsare involved already in securing that different circumstances are occurrences ofthe same sentence. Of course, we need conventions not just to determine a same-sentence relation, but also to determine when circumstances belong to logicallyrelated sentences, i.e., sentences depicting logically related situations. Here iswhere the need arise for the ?hacks and manipulations? that Wittgenstein men-tions at 5.511, logical signs like ?, ?, ?, etc. These work together with repeatedexploitation of a same-sentence relation between circumstances to make psycho-logically possible the simultaneous appearance of a whole infinite multiplicity oflogical positions. The resulting systematically structured realm of possibilities ofoccurrence of sentences is what Wittgenstein calls a system of signs.Much of the problems interpretingWittgenstein?s nomenclature arise throughdifficulties navigating the complicated structure of the system of possibilities ofoccurrences of sentences (together, of course, with the unevenness of his usage).But, relying on matters of empirical fact, it is possible to distinguish variousnatural regularities in the structure. First, for example, it is to some extent ar-bitrary which circumstances should be put forward as sentences, so in practicethere arises a multiplicity of distinct systems of sentencehood?English, French,German, Chinese, etc. Consequently, some possibilities are given as sentencesaccording to English, others according to French, and so on, depending on theintentions and history of the author. Wittgenstein sometimes distinguishes be-tween sentences as sentences of one language as opposed to another and accord-40ingly requiring translations between them (4.025). But there are other distinc-tions which cut across languages. For example, languages exploit various sys-tems of material regularities and internal relations between elements of thesesystems in order to secure that one and the same picture of reality can be, say,whispered to a lover, carried around in the pocket, stored in data warehouses inColorado. Here Wittgenstein distinguishes between sentences that are written,versus spoken, etc (4.011). Yet another dimension, again somewhat orthogonalto the others, comes from the fact that the hacks and manipulations psychologi-cally required for the expression of infinitely many distinct but logically relatedpositions issue in redundancies so that structurally different patterns of hackingand manipulating converge on the same picture of reality. Here Wittgenstein isvery fond of remarking that ??p = p (4.0621, 5.41, 5.254). These three kindsof system, as I said, are at least somewhat orthogonal, so one could in principleconstruct from them several different notions of ?type?. I don?t think that muchphilosophical profundity will be achieved by insisting that Wittgenstein must al-ways mean one versus another of them. Taken flatfootedly, the passages cited inthis paragraph together yield the amusing position that sentences are different indifferent languages, different if spoken or written down, but the same if one isthe double negation of the other.In fact, the situation is even more complicated still. As an anthropologicalfact, it is possible for human beings to pretend that some circumstance falls apartfor them into a sentence, even though it does no such thing. For example, anactor in a play may sit in an armchair turning over some sheets of newsprintwith sentence-like ink markings, and thereby act out the reading of a newspaper,although, its being (let?s suppose) not really a newspaper but only a prop, he canbe doing no such thing.39 Now, such pretenses are intelligible, and so, it seems,we must have some concept not just of sentences but also of sentence-likenesses.For example, it is clear that (under our supposition) the ink markings on theactor?s prop are not really sentences, any more than the piece of plastic in his39It has been speculated that such performances sometimes transpire in coffee shops, libraries,or philosophical seminars. It has even been speculated that it is possible to carry out such a perfor-mance without realizing this oneself. I do not advance any such speculations in the course of thisthesis.41pocket is really a gun. Rather?and here we need to spell out the story a bitmore?they may be just blocks of randomly printed characters. This, of course,can be discovered on closer inspection.Thus, there are things which, in some sense appear to be sentences, yet whichturn out, after all, not to be sentences. But, this example is slightly far-fetched,since actual newspapers make good props as newspapers. The example doesn?tshow that there is an interesting kind of thing, which we might call a ?toy sen-tence? (or ?mock sentence?) in the way that there is indisputably the kind ofthing we call a toy gun. It shows only that sometimes, something seems to bea sentence yet turns out not to be one. There is no general license to supposethat, for a given kind of thing, there is another kind of thing, namely, the kind ofthing which seems like a thing of the first kind. Actually, two different posits areconceivable here. One is the posit of a kind K ? such that things in K ? merely seemlike things in kind K but actually aren?t; and the other is a posit of a kind K ?? suchthat things of kind K ?? either are, or seem to be, things of kind K . Let?s call K ?the pseudo-Ks, and K ?? the quasi-Ks. Are there such things as pseudosentences,or a quasisentences?Some commentators have supposed that the notions of pseudosentence orquasisentence plays an important role in the Tractatus. For example, MichaelKremer (1997, 98ff) argues that the very Tractatus usage of ?sign? (Zeichen) gen-erally refers to things belonging to a kind of quasisentence. More specifically,Kremer argues that the usage of ?sign? in the Tractatus should be explicated interms of a notion sign-design which he borrows from Wilfrid Sellars. Roughly,given a printed inscription of a sentence, its associated Sellarsian sign-design is theshape of the inked region (or, perhaps, the shape of the associated region of thetypeblock). Thus, a Sellarsian sign-design is a shape, and a phenomenon instanti-ates a sign-design by in some sense ?having? that shape. Now, it is clearly true thatunder certain conditions, something can seem to be a sentence if it instantiates aSellarsian sign-design. It is perhaps not true under every condition?for example,subterranean crystalization of boron, or libidinous meanderings of pond scum,might in some sense ?have? the shape that is the given sign-design, yet fail evenpossibly to seem to be sentences. Of course, it may be that the shapes of sign-designs are so sharply fitted to an actual occurrence of a sentence that the given42sentence is practically guaranteed to be the thing that instantiates it. But then thething that instantiates it is a sentence, not something that only seems like one.The force of the analysis of signs as Sellarsian sign-designs is that there exists aclass of shapes such that, to seem like a sentence just is to have a shape in that class.Now, in some sense, of course, such a class of shapes may platonistically exist.Suppose it does. The resulting notion of quasi-sentence is that it has a shape in thisclass. When, however, do two things belong to the same quasisentence? Whenthey have the same shape? But then, for example, it is impossible to transcribea quasisentence, or read it aloud, and so on. The concept of shape just doesn?tbend in the right ways. The notion of Sellarsian sign-design doesn?t illuminatethe notion of sign in the Tractatus.Wittgenstein does write, however:A sign is what can be perceived of a symbol. (3.32)So one and the same sign (written or spoken, etc.) can be commonto two different symbols?in which case they will signify in two dif-ferent ways. (3.321)A full interpretation of this passage runs beyond my purposes here. Nonetheless,3.32 identifies a concept of sign which would organize occurrences of sentencesby perceptual similarity. It is presumably this identification which inspires theSellarsian analysis. However, that analysis rests on pretty implausible psychol-ogy. Perception of language seems to be very special. For example, one can mem-orize the contents of a document without ever noticing whether it is set in serifor sans-serif.40 For that matter, it is possible to know what someone said, to theword, without knowing whether this was uttered or written. A reasonable treat-ment of ?signs? in the sense of the 3.32s should, at the very least, follow a notionof perceptual similarity which is characteristic of language processing rather thanappreciation of graphic design. Imagine, for example, a game of telephone whereinstead of speaking people transmit the message through all different kinds offormats?by typed document, flash drive, megaphone, semaphore, etc. Then, Ipropose as a first pass, occurrences of sentences are occurrences of the same ?sign?40At least, such is the ideal of typographers.43if one is liable to lead to the other in such a game. Of course, this proposal has itsdefects, since the relation is not even symmetric?for example, some things aremuch more likely to be misheard than to be mishearings. But, the right accountof ?sign? in the sense of the 3.32s may result by some such extrapolation from4.014.41The resulting notion of quasisentence, the ?sign? of the 3.32s which so ex-ercises Kremer, is linguistically epiphenomenal. As Wittgenstein insists in the4.01s, what makes a something a picture of reality is the rule, or law of projec-tion, by means of which it can be not just transmitted from the mouth of oneperson to another, or from a mouth to a pen, and so on, but also that it can beprojected into the world itself, and back out from the world into language. It islaws of projection not just within language but between language and the worldwhich distinguishes a sentence from a mere stone or a leaf. As a matter of an-thropological happenstance, these laws of projection can be traced out in a richlycomplex tissue of practices. A proper part of the tissue, or perhaps one shouldsay, an organ of the organism, may consist in, for example, a relay between dif-ferent tokens of an electronic document. Such organs can be misaligned withrespect to the patterns of sameness and difference they serve to maintain. Thenotion of ?mere sign? serves only to draw attention to the possibility of suchmisalignments?it pertains, as it were, not to life but to sickness. Mere signs, likesicknesses, are privations, and they have no nature themselves. Just as for Aristo-tle, anyway, there is nothing that it is to be a sickness, similarly for Wittgenstein,there is nothing that it is to be a ?mere sign?. Or to vary the metaphor a bit: tosay ?the sign is dead? is to say that it is a corpse. To say that a symbol is a signtogether with its use is like saying that a human being is a corpse that is alive.I?ve been arguing mainly about Wittgenstein?s ontology, and have acknowl-edged that the 3.32s point to frailties of the linguistic organism. So, then, I con-cede that in the 3.32s, the word ?sign? means ?mere sign?, provided that ourunderstanding of ?mere sign? is philosophically adequate. However: the us-age of ?sign? that predominates in the Tractatus raises no question whatsoever?indeed, forecloses the question?whether what it describes is really a sentence,41Such an account would better suit Kremer?s interpretive aims, since it, unlike his own, couldaccount for how philosophical illusions propagate from one victim to the next.44i.e., whether it really holds the world to yes or no. This is clear, for example,in the term ?propositional sign?. If propositional signs are identified merely assigns, i.e., as quasisentences, what is the difference between a quasisentence and aquasiname? To which category should we assign, e.g., ALEX BURROWS? And asFrege demanded, what distinguishes it from a mere stone or leaf?1.1.4 Propositions as truth-functionsThe term ?truth-function? has a standard usage in 20th century logic. For exam-ple, Shoenfield (1967) gives the following explanation:A noteworthy feature of the formula A? B is that in order toknowwhetherA?B is true or false, we only need to knowwhetherAis true or false and whether B is true or false; we do not have to knowwhatA and B mean. We can express this more simply by introducingsome terminology. We select two objects, T and F , which we calltruth values. It does not matter what these objects are, so long asthey are distinct from each other. We then assign a truth value toeach formula as follows: we assign T to each true formula and Fto each false formula. Then we see that the truth value of A? B isdetermined by the truth values of A and B .A truth-function is a function from the set of truth-values to theset of truth-values. We can restate our remark as follows: there is abinary truth-function H?such that if a and b are the truth valuesof A and B respectively, then H?(a, b ) is the truth-value of A? B .(Shoenfield 1967, 11)Shoenfield, like Frege, takes for granted that a sentence is true or false, and followsFrege?s procedure in identifying two objects with truth-values. Of course, Fregewould not appreciate Shoenfield?s remark that ?it does not matter what theseobjects are?. For Frege, a sentence stands to a truth-value as a name stands to anobject, and so it surely matters which object is the True, just as it matters whichobject is the planet third closest to the sun, or which object is the number ofplanets closer to the sun than it. But, having corrected this insouciance,42 Frege42And perhaps also the indifference to knowing what A and B ?mean?.45would happily recognize Shoenfield?s subsequent explanation as clearly fixing themeaning of the sign ?, by determining which function the sign ? denotes.43Wittgenstein, of course, would be shocked by all this. First of all, regardingthe suggestion that the word ?and? should be a name (of a function), he has thisto say:My fundamental thought is that the so-called ?logical constants? arenot representatives. (4.0312)To illustrate this thought, he devised a so-called ?truth-table? notation, whichlooks like this:p qT T TT FF TF FAnd, he remarks:It is clear that a complex of the signs ?F ? and ?T ? has no object (orcomplex of objects) corresponding to it, just as there is none corre-sponding to the horizontal and vertical lines or to the brackets.?There are no ?logical objects?. (4.441)Now, in response to this, Frege-Shoenfield might reply: ?well, your notationis surely pedantic, but you pay too much attention to signs. The content is whatthat matters, and a truth-table is just a name of a function. That is to say, thatnotation is an open term, in the two free variables p and q , which, under anassignment of values to p and q , denotes a truth-value.? Here,Wittgensteinmightsay, Frege-Shoenfield is misled by the use of the letters p and q . They are not ?freevariables? at all, but schematic letters which contribute to a general illustration ofvarious possible uses of truth-tables. An actual use of a truth-table is a sentence,or propositional sign (4.44); therefore it says something about the world and istrue or false depending on how the world is. For example, here is an expression43Given a stipulation for the case when not both of a and b are truth-values.46of ?it?s raining and it?s sunny?:It?s raining It?s sunnyT T TT FF TF FThis tabular expression says how the world is, by expressing agreement and dis-agreement with the various possibilities of truth and falsehood of the sentencesin the upper row. If we drop the rightmost column, i.e., the column separatedoff to the right by a double line, then the resulting truncated table (as at 4.31)simply enumerates truth-possibilities for the entries in the uppermost row. Eachof these rows ?means? (4.3) a truth-possibility for the uppermost entries, by sym-bolizing this truth-possibility in a way that, Wittgenstein says, ?can easily beunderstood? (4.31). By appending a T , after the double line, to such a row, wethen signal agreement with that truth-possibility; by omitting such a T we signaldisagreement. Thus, the truth-table for conjunction as a whole expresses agree-ment and disagreement with truth-possibilities for uppermost entries, which itsvarious rows symbolize.44As usual with Wittgenstein?s criticisms of Frege, this one certainly tells ussomething about Wittgenstein.45 For Wittgenstein, the truth-functionality of44 Wittgenstein?s understanding of the tabular notation of 4.442 might be understood with ref-erence to the seagull notation presented at 6.1203. In seagull notation, a truth-function of twoelementary propositions can be written as follows. First, write T pF followed by T qF . Now,above these inscriptions, draw a seagull whose wingtips touch the T next to the p and the T of theq , and another whose wingtips touch the T of the P and the F of the q . Underneath, draw twomore seagulls with wingtips covering the other two combinations. Finally, correlate a T with thebodies of some seagulls, and an F with the bodies of some others.In this notation, the T and F around the p and q represent two poles of truth and falsehoodwhich are intrinsic to them. These T s and F s correspond to the T s and F s in the rows representingtruth-possibilities for p and q in a truth-table. On the other hand, the T s and F s which arecorrelated with them correspond to the T s and F s which represent agreement and disagreementwith truth-possibilities. These outer T s and F s are expressions of agreement and disagreementwith the T s and F s to which they are correlated. All the T s and F s are thus ?poles? of someproposition or other. But, the innermost poles are not expressions of agreement and disagreementwith the propositions to which they are correlated.45This motif is beautifully explored in Goldfarb (2001a).47a proposition is the functional dependence of its truth on the truth or false-hood of other propositions; these other propositions have their possibilities oftruth and falsehood independently of each other. But, the possibilities of truthand falsehood of propositions simply are the possibilities of the obtaining andnonobtaining of situations. Thus, truth-functionality is a functional dependenceof the obtaining or non-obtaining of one situation on some others. The senseof ?function? in Wittgenstein?s usage of ?truth-function? is therefore not the pu-rified mathematical sense which appears in Frege, but rather the vulgate of theeconomist or engineer. Accordingly, for example, ?this surface reflects whitelight? is a truth-function of ?this surface reflects red light?, ?. . . orange light?,?. . . yellow light?, etc., because whether or not a surface reflects white light is afunction of whether or not it reflects wavelengths of light from throughout thevisible spectrum. Likewise, statements of the form ?the pressure in the tube is pp.s.i.? are truth-functions of statements of the form ?the temperature in the tubeis k degrees F? because pressure is a function of temperature.1.2 Workings of the GPFObviously, the general propositional form is supposed to express what is com-mon to all propositions. This has two halves: to be completely general, thatis, to contain nothing not common to all propositions, and yet to characterizepropositions, hence to characterize nothing other propositions. But, as we?veseen, Wittgenstein already pointed out that to give such a form is a pretty triv-ial task: it suffices to do hardly more than clear your throat. Why, then, doesWittgenstein circle back and arduously build up again toThe general form of a truth-function is [p,? ,N (? )].This is the general form of a proposition. (T6)This time, he did away with the throat-clearing. But, where we once lookedthrough the window of a demonstrative to see the image of proposition as pic-ture, now the view is obstructed by this notational gadget. Why the obtrusionof structure here, in what must be common to all propositions? The point is toshow how it might turn out that all necessary, or, in the jargon, ?internal? re-48lations between propositions are really just logical ones and hence inhere in themere possibility of representation, which is the same as the possibility of what isrepresented.The general strategy is to show that there?s some class of propositions amongstwhich no necessary connections obtain, such that all possibilities for the worldare the results of truth-operations on propositions in this class, the class of so-called elementary propositions. A proposition expressed as a denial of someothers depends, for its susceptibility to truth-and-falsehood, on the correspond-ing susceptibility of the propositions it denies. But, claims T6, similarly everyproposition depends for its susceptibility to truth-and-falsehood on the suscepti-bility to truth-and-falsehood of the propositions that are elementary. Moreover,the detailed structure of this dependence will be the explanatory source of all nec-essary connections. So, propositions which are not necessarily connected to eachother must be separable by some possibility for truth-and-falsehood amongst thepropositions which are elementary. And conversely, any distribution of truthand falsehood over the elementary propositions represents a genuine possibility,so that all the separations apparently effected by truth-possibilities for elemen-tary propositions are actually genuine.At this point, Wittgenstein?s philosophical adventure leads into some rockytechnical terrain. For he needs to show that propositions are the results of truth-operations on elementary propositions, or, in other words, that they are truth-functions of elementary propositions.1.2.1 The one-many problemTo fix an image of the sort of technical situation we?ve now encountered, let mesuggest an analogy from the very most elementary foundations of mathematics.The set-theoretic analyst of mathematics confronts as its data various mathemat-ical objects and internal relations between them. For example, a vector spacebears some internal relations to various particular vectors. These vectors standin internal relations to each other?say, that one is a scalar multiple of another.Similarly, the vector space as a whole stands in internal relations to other vec-tor spaces, and, for that matter, to other mathematical structures. The audacity49of set-theoretic foundations is to suppose that out of all the internal relations inmathematics, only one internal relation needs to be taken as fundamental, that ofset-membership. All other internal relations are complex iterations of this singlefundamental one. The question, then, is how can we understand a vector spaceas an object identifiable by the mere fact that exactly such-and-such objects bearto it the relation of set-membership, these other objects being so identifiable too?How, in such a way, can we reconstruct the immense variety of internal relationsbetween mathematical structures with which we began? All such relations willhave to be ?reduced? to complex iterations of the now-fundamental relation ofset-membership.The technical predicament of the logical analyst in the Tractatus is somewhatsimilar. In particular, the analyst has given as data certain possible intellectualpositions, say that Johnson flies, that nobody flies, that nobody flies while sleep-ing, etc. The proposition that nobody flies bears some internal relation to theproposition that Johnson flies, and another internal relation to the propositionthat nobody flies while sleeping. The audacity of logical analysis as Wittgensteinenvisages it is to suppose that out of all these internal relations only one needsto be taken as fundamental: this relation is the one which a proposition bearsto some others, in virtue of which it can be uniquely identified as the denial ofthem.Now, the data presented to the set-theoretic analyst of mathematics demandsome device for ensuring that a set be somehow connected by the relation ofset-membership to a large multiplicity of other sets. For example, a vector spacemay need to be eventually somehow connected by the set-membership relation touncountably many individual vectors. To this end, the set-theorist makes certainexistential assumptions, to the effect, say, that there exists a set to which infinitelymany objects belong, and, say, that there exists a set containing as elements all thesubsets of a given set, and so on. These assumptions are ?external? or ?transcen-dent? in the sense that they bear no immediate connection to the originally givendata, and rather, in some sense, transcend the local mathematical subject-matter.But given these existential assumptions, the set-theoretic foundationalist buildsup, using the postulated objects and constructional procedures, counterparts ofthe originally given mathematical data and their internal relations. A question50whether a mathematical structure and its set-theoretic counterpart are ?the same?might arise, though it can be set aside.The logical analyst of mental life as instructed by the Tractatus similarly en-counters data which demand some device for connecting intellectual positionsto large multiplicities of other intellectual positions. For example, the proposi-tion that nobody flies is connected by the direct denial relation to the variouspropositions about Johnson, Moore, and so on. On my understanding, Wittgen-stein?s procedure here departs somewhat that of the set-theoretic foundationalist.While the set theorist introduces as technical devices existential assumptions andconstructional procedures which are external or transcendent, the devices of theTractarian analyst are in some correspondingly vague sense, immanent. That is,the internal relation that Wittgenstein takes to be basic already appears amongthe logical data that we need to appreciate in order for there to be an analyticalproblem in the first place. Merely in virtue of the understanding of propositionswe must already appreciate the possibility of representing a proposition as thedirect denial of some others. So, rather than stepping outside, and formulatingsome ?objective? existential postulations, the logical analyst simply makes mani-fest the logical relationships already present in the data.Nonetheless, the essential point here is this. In purporting to describe therealm of mathematics solely by means of the relationship of set-membership,the set-theorist incurs a technical problem to explain how, in general, a set canbe connected by this relation to extraordinarily many other objects. Wittgen-stein similarly incurs a problem to explain how a proposition can be connectedby the direct-denial relation to many other propositions. Indeed, eventually, allpropositions must somehow or other be connected by iterations of direct denial.Moreover, in each case, the essential problem will be to find a method of repre-senting multiplicities of items, in such a way that a given item can be representedas the item which bears the chosen fundamental relation to exactly those otheritems. Let?s call this problem the one-many problem.511.2.2 Some early-Russellian technologySo, Wittgenstein attempts to construct a general form of the proposition whichwould, in virtue of its structure, show that the arbitrary proposition gets its rep-resentational nature from its particular position in the network of internal rela-tionships that inheres in the totality of propositions. For, the general proposi-tional form shows that the representational nature of an arbitrary proposition isits functional dependence for truth on the truth and falsehood of propositions ina certain distinguished class, those which are elementary. And in particular, thearbitrary proposition is shown by the general propositional form to inherit itstruth-functionality thanks to the fact that it can be grasped as the direct denialof some other propositions, whose truth-functionality would be already presup-posed.In this way, Wittgenstein?s presentation of the general propositional formdemands substantial technical resources. For, it presumes the possibility of rep-resenting a proposition as connected in some cases, via this relation of directdenial, to some possibly infinite multiplicities of propositions. Thus, Wittgen-stein incurred a burden here to demonstrate that this technical challenge of theone-many problem could be met, at least in principle. The understanding of thisphase of the book therefore requires some grasp of the technical resources fromwhich Wittgenstein could draw to meet this burden.My historical hypothesis is that Wittgenstein silently drew the basic idea forhis treatment of the one-many problem from Russell?s treatment of classes in his1903 book The Principles of Mathematics. It is believed that Wittgenstein did readthis work when in Manchester,46 as his interests began to shift from engineeringto foundations of mathematics.Nonetheless, this derivation of Tractatus technology from 1903 Russell issomewhat speculative. I do think that regardless of how Wittgenstein actually46Russell (1951) writes confusingly that in Manchester ?Wittgenstein became interested in theprinciples of mathematics?. McGuinness (2005) says only that Wittgenstein?s enthusiasm for Prin-ciples and Grundgesetze ?leads one to suppose? he had read both books. Monk (1991, 30) assertsthat Wittgenstein?s reading PoM in Manchester was a decisive event in Wittgenstein?s life, and thatWittgenstein spent two terms there mainly studying PoM and Grundgesetze. But Goldfarb justlyremarks in (2001a) that he finds Monk?s claim ?somewhat hyperbolic?. In any case, as Goldfarbpointed out to me, Wittgenstein clearly alludes to PoM at 5.5351.52got the ideas, the ideas are actually there in this early Russell.47 So even if thehistorical hypothesis is not quite correct, the marvelous lucidity of Russell?s ex-position still helps to explain important parts of the Tractatus.According to Russell, works on logic ?distinguish between two standpoints,that of extension and that of intension? (66). But, according to him, ?Symboliclogic has its lair? in some intermediate region between the two. In particular,Russell holds that classes must be understood extensionally, so that no two classeshave the same elements. Actually, as Russell understands the concept of class,one simply cannot avoid this conclusion. In particular, he holds that one candefine a class by simply listing its terms. For example, the numbers two andseven are a class. Similarly, the number two just by itself is a class, the class towhich just the number two belongs. 48 In other words, which, I?m sorry, are notgrammatical, one might want to say: ?a class to which some objects belong just isthose objects.? Here, Russell observes, the grammatical difficulty is essential: thephrase ?a class? is grammatically singular, but, a class is in general intrinsicallymany things, being defined by a phrase like ?the numbers two and seven? whichis grammatically plural. So, he concludes, the difficulty is ?not removable by abetter choice of technical vocabulary? (70).Thus, while we, nowadays, are accustomed to thinking of there being oneobject, a set, to which many objects may bear the relation of set-membership,Russell?s corresponding conception of a class is primarily a conception just ofthose many objects. Russell does introduce a notion that corresponds moreclosely to our notion of a set, for which he uses the phrase ?class as one?, in47Of course, it might be rash to rule out that actually all possible ideas (and, perhaps someimpossible ones) are to be found in early Russell.48So, Russell thinks, a class to which just one object belongs just is that object. To this ideathere arises an objection (77) which appears also in Grundgesetze (Frege 1964, 48n). Consider, forexample, the class ((2,7)), whose sole member is itself a class (2,7) two members are the numberstwo and seven. Since a class to which just one object belongs just is that object, therefore ((2,7))is just the same as (2,7). But then, it follows that whatever belongs to (2,7) must also belong to((2,7)). Hence both 2 and 7 belong to ((2,7)). But, only one object belongs to ((2,7)), namely theclass (2,7). Hence, 2 and 7 must both be identical to that object, and must therefore be identical toeach other. To this objection, Russell rejects the assumption that just one object belongs to ((2,7)).For, after all, the class (2,7) is really two objects, the numbers 2 and 7. More generally, Russellallows, when the elements of a class are themselves classes, then a class can be ?decomposed?, orrepresented as a bunch of items, in more than one way.53contrast to this prior conception of ?class as many?. But, I think, for Wittgen-stein, the conception of ?class as many? is prior. I?ll reserve ?class? and ?mul-tiplicity? for this notion, and use ?set? for Russell?s ?class-as-one?. Where it iscustomary to denote a set by enclosing the names of its elements in curly bracesas in {a, b , . . .}, I?ll denote a class-as-many by enclosing the names of its elementsin parentheses as in (a, b , . . .). In particular, for example, there are three dis-tinct sets {a, . . . , b , . . .}, {a, . . . ,{b , . . .}}, and {{a, . . . , b , . . .}}. On the other hand,(a, . . . , b , . . .), (a, . . . , (b , . . .)), ((a, . . . , b , . . .)) are all the same class, which is justthe objects a, . . . , b , . . ..So far, Russell?s account of classes has invoked only extensional resources: todefine the class containing such-and-such items, one simply forms a list of thenames of those items. But, as Russell observes, were we to try by enumerationto define an infinite class, ?Death would would cut short our laudable endeavourbefore it had attained its goal? (69). Since in logic we do deal with infinite classes,logic must ?find its lair? in the regions intermediate between intension and exten-sion. For example, when considering a proposition about all numbers, we do notthereby consider each and every number, but rather only a concept, all numbers.Nonetheless, the proposition is in some sense about all numbers because, Russellsays, the concept, all numbers, denotes exactly those things which are numbers(73). Thus, the ?inmost secret of our power to deal with infinity? (73) is that aninfinite collection can be denoted by a concept which is only finitely complex.On Russell?s account,49 classes can contain items of any kind, anything thatfalls under his most general category of being: objects, propositions, proposi-tional functions, variables, denoting concepts, and so on. But, then the notionof class has a particular special case, the notion of a ?class of propositions of con-stant form? (90). Roughly speaking, such a class results from a proposition byallowing one or more of its objectual constituents to vary freely. Russell claimsthat this special case is more fundamental than the notion of class in general, forthe general case can be defined in terms of the special one, though not conversely(89).50 In turn, a class of propositions-of-constant form is defined by the resultof replacing an object in a proposition with a variable. For Russell, of course, a49As far as I can make of it just now, through Tractatus-eyes.50He seems to have in mind a definition by appeal to the notion (?) of ?such that? (93).54proposition is not a linguistic or representational entity, but is rather, as it were,the fact that would obtain were the proposition true. So a variable, also, seemsfor Russell to be somehow nonlinguistic. A variable, in turn, presents a definingconcept, so that the items falling under the concept are the variable?s values (91).In the fundamental case, of the ?true? variable which has unrestricted range, thepresented concept is, according to Russell, said to be formal (91). But in any case,it is in general by presenting a concept that a variable ranges over a multiplicity ofitems. Most importantly, a variable does not name this or that one of its values.Nor, on the other hand, does it plurally name the multiplicity of those values.Rather, through its associated concept, the variable indeterminately indicates orranges over them. Now, when a constituent of a proposition is replaced by thisvariable, then the result is a propositional function. The propositional functiondetermines a class of propositions of constant form, namely those propositionswhich result according as the variable in turn gives way to some one or other ofits values.1.2.3 Variables in the TractatusWittgenstein?s approach to so-called one-many problem bears clear marks of Rus-sell, and I want to emphasize some commonalities, but first let me mention someimportant differences. The first difference is in the general realization of theideas. PoM is tentative and exploratory, seeming to run down every possible con-ceptual path, and Russell?s discussion is extremely complicated. In contrast, itseems to me, Wittgenstein thought these issues through to a fairly simple andsystematic conception. Second, Wittgenstein distinguishes clearly between signsand what they express, or, as he puts it, between signs and the symbols to whichthey belong. Thus, Wittgenstein distinguishes between propositional signs, nom-inal signs, operators, etc., on the one hand, and propositions, names, operations,etc., on the other. In particular, a variable, for Wittgenstein, is just a sign. Just asthe symbol to which a propositional sign belongs is a proposition, the symbol towhich a variable belongs is a formal concept.Wittgenstein proposes to represent a proposition as the joint denial of thevalues of a propositional variable (5.501). Thus, to explain how a proposition can55be logically related to infinitely many propositions, it would suffice to explainhow a variable can assume infinitely many values. Echoing Russell, Wittgensteinsays that if the range of the variable is supposed to be finite, then one can fix itsrange simply by enumerating the objects in its range; but, one cannot assume thatthe range of every variable is indeed finite (5.501). And so, one cannot assumethe possibility in every case of fixing the range of a variable by bare enumeration.Instead, according to Wittgenstein, a variable secures its range by presenting aformal concept (4.1272); in turn the formal concept specifies some multiplicityof items as those which fall under the concept; this multiplicity is the range ofthe variable (5.501).Just as in Russell, a variable in the Tractatus does not itself name any particularone of its values; nor does it plurally name the multiplicity of those values. Thus,for example, if ??? is a variable which ranges over the totality of people, then sign-arrangement ?? flies? does not express any one definite proposition. Rather, thissign-arrangement serves only to mark off what is common to the various propo-sitions to the effect that this or that element of the range of ??? flies. However,according to Wittgenstein, one can convert the indefinitely indicating expression??? into a plural expression which does stand for things: by placing a bar overthe variable ???, one forms a plural term ??? which stands at once for each andevery value of ???. Since, in our case, the range of the variable ??? is the totality ofpeople, the plural term ??? stands for that totality: it stands for that class-as-manywhich, we are tempted to say, ?just is? all of the people, i.e., the following logicalformula would be correct:?= (Moore, Johnson, Russell,. . . ).However, this example of the variable ranging over people is importantlymisleading. For Wittgenstein holds that all variables ?can be construed as propo-sitional variables, even variable names?.51 So, the ground-level explanation of51So, Russell reduces the general notion of class to the notion of class of propositions as constantform. But, this notion of class of propositions of constant form itself depends on the notion ofvariable, which takes as its range the extension of a formal concept. Such an extension (there maybe only one such, the All) includes not just objects but propositions, etc. Thus, Russell does nothold in PoM that all variables can be construed as propositional variables.56logic will appeal only to variables which range over propositions, and all otherforms of variation must be reduced to propositional variation. But, for example,how are we so to reconstrue such an apparently nominal variable as ??? whoserange is the people?Well, how does the apparent existence of nominal variables arise in the firstplace? It looks like, for example, one can understand the variable name ? to playsome essential role in expressing a generalization over people, say in expressingthe generalization that nobody flies. For example, suppose we begin with theproposition that Johnson flies, and then in this proposition regard Johnson asvariable. Thus, rather than considering the proposition to say of Johnson thatJohnson flies, we consider it to say of an arbitrary person that that person flies.This indication of an arbitary person we might effect with the variably meaning-ful sign?which, combined with the constantly meaningful sign. . . fliesyields a corresponding variably meaningful sign? flies.In just the same way, a constantly meaningful signJohnsonmight be combined with the constantly meaningful expression. . . fliesto yield the constantly meaningfulJohnson flies.57But, Wittgenstein thinks, this explanation puts the cart before the horse. Onecannot explain themeaning of an expression ?Johnson flies? by pointing out John-son and pointing out (an instance of ?) flight, because ?only in the context of aproposition has a name meaning.? For, as I?ve claimed, Wittgenstein proposes tounderstand the expressive power of a proposition by finding its place in the total-ity of propositions. So, similarly, one does not explain the function of a variablymeaningful sign ?? flies? by stipulating that ??? ranges over people and ?. . . flies?stands for flying. Rather, the complex expression ?? flies? is itself what is prior:it is a propositional variable which ranges over each and every proposition to theeffect that this or that person flies. Thus, one could perhaps derivatively explainthe range of the nominal variable ??? by suggesting that ??? stands to ?Johnson?as ?? flies? stands to ?Johnson flies?. But, this explanation applies to ??? only in-sofar as it occurs in the propositional variable ?? flies?, and so it is not of muchindependent interest.But, at this point a natural question arises. Russell?s notion of a class as manyhas two strong justifications. First, some notion of class appears to be essential tomathematics. Second, Russell?s particular fundamental realization of the conceptof class appears to be perfectly familiarly implicit in the use of ordinary Englishnoun phrases like ?Johnson and Moore? or ?the people?. That is, we can?t but takefor granted that sentences like ?people need food? somehow mean something;and the natural account is simply that ?the people? stands for all people. More-over, as we?ve seen, the expression ?the people?, can?t simply contribute to theproposition expressed all people; rather, it contributes a concept which in turndenotes the people whose organismic nature then decides the truth of the origi-nal proposition. Thus, to put the point in Tractatus notation, Russell?s account of??? and ??? is very naturally motivated. But it does not seem that Wittgenstein?sreplacement concepts, of propositional variable and its extension, are similarlynaturally motivated. In particular, while one might perhaps try to take an ex-pression like ?he or she flies? as a model for the use of expressions like ?? flies?, itseems hopeless to find any sort of English phrase which would somehow standfor the corresponding range of the propositional variable (i.e., to the extensionof the formal concept the variable presents). In other words, it is not at all clearwhat ?? flies? is supposed to mean. For, Wittgenstein holds, propositions cannot58be named. Perhaps one could get some kind of approximation by simply listing,as in ?that Johnson flies, that Moore flies,. . . .? But, if such a list works logicallyas per Wittgenstein?s intention, then it isn?t to be distinguished from the simplemultiplicity of expressions ?Johnson flies?, ?Moore flies?,. . . . So, it is probably fairto say that the result of applying a bar operation does not appear to correspondto any single naturally isolable English phrase.But if the bar notation doesn?t correspond to any natural segments of English,then what is Wittgenstein?s justification for introducing it in the basic account ofcontrol over possibly infinite multiplicities? Consider, for example, the proposi-tion that nobody flies. Wittgenstein holds that this proposition bears an internalrelation to various other propositions: in particular, say, to the propositions thatJohnson flies, that Moore flies, and so on. In particular, the proposition that no-body flies is constituted by its position as the joint denial of all such instances.Thus, Wittgenstein thinks, this position itself embodies the relevant control overthat multiplicity. That is, for Wittgenstein, there is no one ?object? or ?set?which collects together a multiplicity of propositions; rather, there is instead aproposition which bears to the multiplicity a fixed logical relation?for example,there is the joint denial of those propositions. And, this unification is generallyalready implicit in representational practice. Thus, the analyst begins with thedatum that to affirm a propositionnobody flies,is exactly to deny the propositionsMoore flies, Johnson flies, . . . .Yet, since the original proposition does not manifestly contain these many propo-sitions, etc., there must be a variable ? such that? = (Moore flies, Johnson flies, . . . )so that in turnnobody flies=N (? ).59The question remains how to fix the values of the propositional variable ?? ?.This, Wittgenstein claims, is something ?to be stipulated.? Such stipulations re-quire technical resources for the construction of variables, which Wittgensteinoutlines at 5.501. To describe those resources is the task of the next section.This section began by laying out some Russellian resources for the controlof possibly infinite multiplicities. We transposed these resources to the Tractatusframework, where it is insisted that all variables can be construed as proposi-tional variables. But this meant that fundamental to the Tractatus framework issome notion of a multiplicity of propositions, which lacks any obvious form ofunitary expression in ordinary English. The question then arose what could pos-sibly be the intuitive source of this transposed technical notion. I suggest thatjust as, for Russell, the intuitive source of the notion of a class-as-many of ob-jects is not any single representational unit but rather simply a correspondingmultiplicity of names, similarly for the Tractatus, the intuitive source of the no-tion of a class-as-many of propositions is not any single representational unit butrather a corresponding multiplicity of propositions. Such multiplicity warrantsour analytic attention precisely in case a proposition we want to understand canbe represented as the result of applying a logical operation to the elements of thatmultiplicity.1.2.4 Propositional variables: some detailsLet me now turn to the details of Wittgenstein?s technical resources of stipula-tion of the value-ranges of variables. It is at 5.501 that Wittgenstein describesthree Ways of fixing the values of a variable. Now, on Wittgenstein?s account, tofix the range of a variable is just to supply the formal concept under which falls ex-actly the items in the range, i.e., the formal concept whose extension is the range.Thus, at bottom the threeWays are really ways of fixing the use of a sign as a vari-able by specifying the formal concept to which it belongs. So in a way, the waysreally embody commitments to certain formal-conceptual resources which areavailable to the analyst of language or thought. It is a difficult interpretive ques-tion what legitimates exactly these analytical resources. Naturally a constraintarises in the form of conditions of the possibility of representing a proposition60as the result of applying a logical operation to some bunch of other propositions:what makes it possible for a bunch to appear in this way?52 At 5.501 Wittgen-stein does not give any principle but contents himself with three examples oflegitimate stipulative procedures.53Way 1. Wittgenstein says that a variable presents a formal concept, and that therange of this variable is the totality of propositions which fall under the concept.But, as I?ve suggested, the theoretical appeal to the notion of concept and hence ofvariable derives from the analytical datum that a proposition might be logicallyrelated to some multiplicity of propositions which is not a priori finite. Afterall, as Wittgenstein supposes in the Notebooks (18.6.15f) and acknowledges in theTractatus (4.2211), it might turn out that facts and states of affairs are infinitelycomplex.Nonetheless, some propositions seem to be naturally representable withoutany immediate appeal to not a priori finite multiplicities. Thus for example wehave quite naturally an analysisNeither Moore nor Johnson flies=N (Moore flies, Johnson flies).In this case, apparently, Wittgenstein holds that one can simply stipulate that theformal concept ? be that which is satisfied by just the two explicitly formulatedpropositions, so that ? = (Moore flies, Johnson flies).Way 2. The second way of stipulating the values of a variable is the naturalcase we?ve already considered, and which basically amounts to Wittgenstein?streatment of quantificational generality. This treatment is well-encapsulated byWittgeinstein?s remark that unlike Frege and Russell, he dissociates generalityfrom truth-function (5.52). As we?ve seen,Nobody flies=N (Moore flies, Johnson flies,. . . )52A natural suggestion is that one can ?mechanically check? whether a proposition belongs tothe bunch. But this presupposes a concept of computability over propositions.53To the second and third of which I devote entire chapters, and what follows is just an outlineof those results.61and so we would like to construct ? such that? = (Moore flies, Johnson flies,. . . ).To this end, note that among these propositions in ? there is a certain constantaspect and a variable aspect, and correspondingly such aspects in their expres-sions. Replacing the variable aspect with an indeterminately meaningful sign ???gives a sign-arrangement? flies.This sign-arrangement seems to articulate a commonality amongst the verbal ex-pressions of all the propositions in the desired range, because its indeterminatelymeaningful part corresponds to what is not constant to the expressions of thepropositions in that range. We might therefore try to take the sign-arrangementitself to be a propositional variable. Since the sign-arrangement appears to sig-nify the common content of the propositions in the desired range of the variable,the corresponding formal concept might be given as follows: to be marked by thecontent that ?? flies? presents. The generality of the original proposition then liesin the means by which a subordinate expression points out those propositions towhich the operation of denial is then applied.54Note, however, that the analysis just given is somewhat equivocal. For exam-ple, it is easy to imagine Frege or Russell asking: isTweetie fliesa value of ?? flies?? But if Tweetie is a bird (let us suppose), then, it seems, whetherTweetie flies cannot count against the truth of the proposition that nobody flies.Frege or Russell would take this to show that the proposition that nobody fliesactually amounts to the denial ofMoore is a person and Moore flies, Johnson is a person and Johnson flies,. . . .54Wittgenstein?s account of quantificational generality is highly unusual in ways unmentionedhere; but its eccentricities are deeply rooted. I defend this claim elsewhere.62Thus, the stipulation? = ? is a person and ? flieswould be taken to lead to the desired analysisnobody flies=N (? ).This ?Fregean improvement? requires a revision in the original datum that itwould be outright contradictory to maintain that nobody flies but that Mooreflies. Clearly, Frege or Russell would indeed regard the original argument ?no-body flies, so Moore doesn?t fly? as enthymematic. It is not really clear to mewhat to say about this from Wittgenstein?s point of view, since it is doubtful thathe could regard ?Moore is a person? as ruling out any possibilities.55Way 3. According to Wittgenstein, the third way of fixing the range of a vari-able was overlooked by Frege and Russell. Apparently, it has also been over-looked in most subsequent development of logic, and as yet its use has not beenrigorously studied in the secondary literature. But, as is often the case in the Trac-tatus the basic idea is simple if peculiar: to incorporate into logical structure theconcept ?and so on?. That is, suppose given a proposition p, and a operation Owhich generates a proposition from a proposition. Then, according to Wittgen-stein, there exists a variable whose range consists of the totality of propositionsthat can be obtained by repeatedly applying the the operation O to the originallygiven proposition p.In developing this method, Wittgenstein might have had in mind some such55The solution to this problem may lie in the near-certainty that, for Wittgenstein, ?Moore?,?Johnson?, etc., are not really names since their bearers aremortal. But then, it may be that ?Mooreflies? and ?Tweetie flies? have different logical forms. Then, the result of removing ?Moore? from?Moore flies? would not be an expression which marks the sense of ?Tweetie flies?. I am thinkingof the passage in the Notebooks where Wittgenstein observes that one can say of the rod, but not ofthe ball, that it is leaning against the wall. Actually, an even closer example comes in the hopelessEnglish translation of a similar passage to the effect that he cannot say of his watch that it is lyingon the table. That is, in English, ?the watch is lying on the table? does not, appearances (?) to thecontrary, have the same form as ?the client is lying on the table.? Thus, saying that nothing is theway the watch is said to be in the proposition that the watch is on the table, should entail nothingabout the client.63example as:None of Russell?s ancestors fly,which, so it seems, is the denial of the propositions in the seriesa parent of Russell fliesa parent of a parent of Russell fliesa parent of a parent of a parent of Russell flies...We thus seek to construct a formal concept ? whose extension contains just thesepropositions. Such a construction can be written as? = [a parent of Russell flies, ? , a parent of ? ].According to T5.2522, the notation in the definiens works roughly like this. Thewhole thing is supposed to define a series of items. The first item in the seriesis indicated by the leftmost entry in the square brackets. The second and thirdentries together give an operation for generating an item from an item?here, aproposition from a proposition. Roughly speaking, they characterize the opera-tion by means of ?before and after shots?. So in this case, the operation, appliedto a proposition, gives the result of plugging the proposition in for ? in the frag-ment: a parent of ? . Now, the rest of the series consists of just what can beobtained by repeatedly applying the operation to the repeatedly given item. Theterms of this series are thus, according to the definition, the values of ? . So wenow have once againNone of Russell?s ancestors fly=N (? ).1.2.5 Insufficiency of local constraintsAs we?ve seen, the analysis of any particular proposition has always the samepattern. Given the proposition p, we seek a formal concept ? , such that p is64the joint denial N (? ) of the propositions in the extension of ? . Of course, thisin itself is almost completely trivial. Since as we?ve seen, N (N (p)) = p, there-fore to find ? such that p = N (? ) it suffices to take ? to characterize the singleproposition N (p). More generally, any one particular proposition has severalpossible analyses. To give another example, suppose we begin with a datum thatthe proposition q is true iff both propositions r and s are true. Then, it will betempting to conclude that the analysisq =N (N (r ),N (s))must be the correct one, according to which q is somehow ?composed? from rand s and therefore more complex than them?this will be particularly so if itappears that verbal expressions of r and of s are parts of a verbal expression of q .However, the datum by itself is equally consistent with the pair of analysesr =N (N (q , t ))s =N (N (q ,N (t )))according to which actually r and s would be ?composed? of q and some otherrandomly chosen proposition t .The 5.2s observe that propositions stand in internal relations to each other,so that one proposition can be represented as the result of an operation on otherpropositions. This means, in practice, finding a propositional sign belongingto the first proposition by attaching the sign of an operation to a multiplicityof signs which belong to the other propositions. The analysis of propositionssimply applies doggedly, toward a certain end, a certain restricted form of thispractice. Thus, analysis always represents a proposition as the result of an oper-ation of joint denial. Such representation is the syntactical result of prependingthe letter ?N ? to the enclosure in parentheses of a multiplicity of propositionalsigns. Since, however, in general it will be impracticable to inscribe that multi-plicity outright, instead we construct a notation ?? ? which plays the same logicalrole as that multiplicity.So understood, analysis just applies a refined form of the ordinary practice of65expressing a proposition as the result of a truth-operation. But then, what we?veso far said doesn?t suffice to pin down how analysis might be in any sense explana-tory, elucidatory, or otherwise interesting. For, as we?ve observed, the practicecan be applied to a given proposition in infinitely many divergent ways. Onemight observe that it must be, with respect to some given proposition, that it betrue if and only if two others are true, and thus represent it as result of truth-operations on those others. But this only codifies notationally a relationshipbetween three propositions, and various relationships between various proposi-tions can be codified similarly. Nothing, then, makes a proposition in itself to be aconjunction as opposed to a disjunction, etc. We might try to insist, for example,that p is a conjunction if it has the form N (N (p),N (q)) for some propositionq , but by picking q to be a tautology, then every proposition has this form.56Truth-operational articulatedness is an artifact only of this or that particular wayof realizing a proposition in signs.What underlies this essential relationality of logical structure is the idea ofa proposition as a picture of reality. Consider, for example, a proposition withrespect to some cup and table that the first is on the second. This propositionreally depicts the cup and the table, so that as the cup and the table would be, pal-pably, combined in the world, so their proxies are combined in the proposition.Now, roughly speaking, the proposition that the cup is on the table is equivalentto the disjunction of two propositions, that the cup is on the left half of the table,and that it is on the right half. Hence, one might think, the original propositionis really the disjunctive result of gluing together two pictures of reality. Each ofthose two pictures contains a cup, and a table, and so the result of gluing themtogether contains two cups, and two tables, such that the one cup is on the lefthalf of the one table, and the other cup is on the right half of the other table. But,this is clearly not what the original proposition is intended to depict; rather, theoriginal proposition depicts just one cup, on just one table. Because there is nomore than one cup and one table in the situation depicted, similarly there is no56The general idea here is well-expressed in the Tractatus by the remark that an operation is not amark of the sense of propositions, but rather a mark of difference between senses of propositions.Thus, the operation of conjunction might be presented notationally by ?? ,? ,N (N (? ),N (? ))?whose structure marks a ?difference? or internal relation between three propositions.66multiplicity of occurrences of proxies in the picture.57In any case, since logical structure is essentially relational, a proposition isneutral with respect to logical structure when considered only intrinsically. Butthis means that we cannot understand analysis as unfolding ?intrinsic? natureof propositions; one cannot simply ?look inside? a proposition to see what it ismade of, and then look inside what it is made of to see what that is made of,etc. Thus, local constraints on the adequacy of analysis do not tell us the orderof analysis. The adequacy of analysis must depend on some further, ?global?constraint. The point of T6 is to unfold this constraint, and that?s what we nowturn to considering.1.2.6 The general propositional formIn this section I just want to analyze the formulation of the T6-variable thatWittgenstein gives at T6:[p,? ,N (? ).]This formulation obviously resembles the notation explained at T5.2522. Manycommentators have therefore supposed that the explanation at T5.2522 is in-tended to cover the use at T6. But, I think this is a mistake. The reason is that theexplanation at T5.2522 simply does not explain this use. In particular, T5.2522treats only a singular, constantly meaningful sign as its left entry, indicating thesingle initial term of the signified form-series. Similarly, it treats only a singular,variably meaningful sign as its middle entry, which together with the right-handentry marks out an operation which, applied to an arbitrary single term of theseries, yields another term. But, at T6, we have instead a left entry which is aplural constantly meaningful expression. Likewise, the middle entry at T6 is aplural, variably meaningful expression which, together with the right-hand en-try, marks out an operation which yields a term of the series when applied to amultiplicity of such terms. Thus, the explanation at T5.2522 doesn?t specify thesymbol at T6. Surprisingly, in Prototractatus Wittgenstein does explain use of the57I contend in Chapter 2 that this particular application of the thought of propositions as pic-tures is what drives Wittgenstein?s stipulation that in a generalized proposition, distinct variablenames cannot assume the same value. Thus, the redundancy of the equality predicate is a merecorollary of the pictoriality of propositions.67sort that appears at T6:We write the general term of a form-series like this:|x0, x,O?(x)|The x0 are the initial terms of the series, the x are any of its terms,and O?(x) is any term in the progression of the series which the op-eration O produces out of the x.58So, I take the summary import of the notation at T6 to be as follows. Thesquare bracket notation as a whole is a variable, such that the range of this variablecontains? Every item of the multiplicity denoted by its barred left-hand entry;? every result of applying the given operation to a value of the barred middleentry,? and nothing else.Just for an innocuous example, let?s use this method to construct a formalconcept C whose extension C is the totality of natural numbers greater than 1.For this construction, we take as given the concept k of being a prime number.And, we take as given the concept ? of: being a finite multiplicity. Finally, let?(?) be the result of the operation to compute the product of the elements of thegiven instance of the concept ?. Then, the concept C whose range is the totalityof numbers greater than 1 is[k,?,?(?)].58This is my probably pretty bad translation. The German is:Schreiben wir das allgemeine Glied der Formenreihe so:|x0, x,O?(x)|Die x0 sind die Anfangsglieder der Reihe, die x beliebige ihrer Glieder, und O?(x)dasjenige Glied welches beim Fortschreiten in der Reihe durch die Operation O?(x)aus den x ensteht.Note in particular that the German does contain the plurals.68Let?s now argue that 12 falls withinC . First note that 2 and 3 fall under the formalconcept, k, of being a prime number, and therefore belong to the extension k ofthat concept. By the explanation of generalized form-series notation, it followsthat 2 and 3 are initial terms of the presented form-series, and hence fall withinC . Now, being either 2 or 3 is a formal concept, say e1, whose extension fallsentirely within C . So, again by explanation of form-series notation, it followsthat the product ?(e1) of the elements of the extension of e1 falls within C aswell, that is, 6 falls within C . Similarly, being either 2 or 6 is a formal concept,say e2, whose extension we can now say falls entirely within C . So the product12=?(e2) falls within C .59Now, let?s turn to the particular use at T6. In that case,? ?p? works as a constant denoting the multiplicity of initial terms of theseries;? ?? ? works as a variable whose values are multiplicities of terms of the series;? ?N (? )? works as a variable ranging over results of applying the operationmarked by the letter ?N ? to the elements of an arbitrarily chosen value of?? ?.And the import is that ?[p,? ,N (? )]? works as a variable whose range consists of? the multiplicity of items denoted by ?p?;? every value assumed by ?N (? )? when the value of ?? ? is a multiplicity ofitems already in the range.It remains, then, to identify the denotation of ?p? and the range of ?? ? andthe operation signified by ?? ,N (? )?. Generally, Wittgenstein uses the letter ?p?as a variable not just over elementary propositions, but over propositions ingeneral.60 But, in the entry immediately after the introduction of T6-variable,59For brevity, I?ve carried out this explanation entirely within material mode. Carnap (1934)says that many sentences of Wittgenstein?s ?which at first appear obscure become clear when trans-lated into the formal mode of speech? (303).60Wittgenstein actually stipulates at 4.24 that p, q , r, . . . range over elementary propositions. Butby my count, there are only two entries other than T6 where p, q , r, . . . clearly range over elemen-tary propositions only: 4.31 and 5.101. I take these letters to be used as variables ranging over allpropositions in the following passages: 5.12s, 5.132, 5.1362, 5.141, 5.152, 5.151, 5.31, 6.1221.69Wittgenstein explains: ?What this says is just that every proposition is a resultof successive applications to elementary propositions of the operation N ?(? )?(T6.001). This suggests that the initial terms of the intended form-series are pre-cisely the elementary propositions. Since ?p? is intended to denote the multi-plicity of initial terms, it is therefore natural to suppose that ?p? here works as avariable whose range is the totality of elementary propositions. Then, in turn,?p? works as a constant which denotes the multiplicity of values of ?p?, i.e., thetotality of elementary propositions.Anscombe remarks that ?as in Frege, ?? ? marks an informal exposition.?While I?d say that the entirety of the Tractatus is informal exposition (if not out-right poetry), it does appear that Wittgenstein?s use of ?? ? is pretty messy anddiverse. That letter seems to serve in the Tractatus like the computer linguist?s?foo? as a catch-all for grammatical concepts too weird to deserve their own typeof variable.61 But, the usage closest to that in T6 appears in the 5.5s, and in partic-ular at 5.501, where he explains the bar notation itself. In that passage, it appearsthat ?? ? is used, somewhat sloppily, as a variable which ranges over variables;but I think the intention is for it there to range over formal concepts. As we?veseen, our aim is that ?? ? be a variable which takes as its values certain multiplic-ities of terms of the form-series. Since the terms of the form-series in questionare precisely the totality of propositions, therefore our aim is that ?? ? range overmultiplicities of propositions. However, it does not range over what we wouldcall arbitrary sets of terms, because Wittgenstein does not take the notion ofarbitrary set for granted.62 But, then: over which multiplicities? The naturalsuggestion is that 5.501 itself provides the answer: a multiplicity falls within the6152521, 55, 5501, 5502, 551, 552, 6, 6001, 601, 603, 61203. He tends for some reason to use it forbases of an operation; but some of these uses are singular and some plural. Possibly, Wittgensteinthinks of an expression like ?N (? )? as extremely incomplete, because it tries to signal an operation,but instead amounts only to a variable over all propositions (if such a variable exists), since everyproposition is the negation of some propositions. That is, because an operation doesn?t character-ize the sense of propositions it can?t be presented by a variable as in ?N (? )?. Rather, the operationmarks a difference between propositions, and is therefore presented by means of two variables, asin ?? ,N (? )?.62That is: the cardinality of the range of this variable is, I contend, not greater than the cardi-nality of the form-series itself (unless the number of elementary propositions is finite). See theappendix for further discussion.70range of ?? ? if and only if that multiplicity can be specified by some such formalconcept as those of the three kinds introduced at 5.501.We now have the following:? ?p? ranges over the elementary propositions? ?p? denotes the totality of values of ?p?? ?? ? ranges over formal propositional concepts? ?? ? ranges over the extensions of formal propositional concepts.Then, the range of the general propositional variable will include? every elementary proposition;? every joint denial of the extension of a formal propositional concept;? and nothing else.1.3 The GPF in action1.3.1 A guide to analysisSo I take it that from a technical point of view, the workings of the notationalform employed in T6 are, if a little delicate in detail, basically straightforwardin broad brushstrokes. In other words, it is fairly easy to see how that kind ofnotation should in general secure the range of a variable, as, for example, in thecase of the definition of the class of integers greater than 1. One begins with abunch of items, and with an operation for constructing items from bunches ofitems. Repeatedly apply this operation until no new items result. The values ofthe variable so constructed are precisely those items which must eventually soappear.However, while we can now understand certain uses of the pluralized squarebracket notation, it is not clear that we can make sense of the particular use atT6. For, so far we have only established, with respect to pluralized square bracketnotation, a result of the form ?intelligible in, intelligible out?. That is, when71specifying the class of numbers greater than 2, we simply took for granted thenotion of prime number, and the notion of finite product. Granted our graspof these notions, the square bracket notation makes sense too. Now, I thinkthat the operator N and indeed, at least roughly, the means of specification of itsbases, have been explained more or less adequately, and that these issues are nolonger fundamentally in question. But, it is still not at all clear that we have anyunderstanding of the use of ?p? in the T6 notation, where it is purported to varyover the totality of elementary propositions. Indeed, Wittgenstein goes out of hisway to make sure that it would seem to be difficult to try to construct a variablewith just the elementary propositions as its range:We now have to answer a priori the question about all the possibleforms of elementary propositions.Elementary propositions consist of names. Since, however, we areunable to give the number of names with different meanings, weare also unable to give the composition of elementary propositions.(5.55)If I cannot say a priori what elementary propositions there are, thenthe attempt to do so must lead to obvious nonsense. (5.5571)As BrianMcGuinness memorably said about the constituents of elementary propo-sitions, these are just not the kind of thing we encounter on the street. We justdon?t know what an elementary proposition is like: even if some thoughts arethoughts to the effect, for some elementary proposition or other, that it is thecase, still, we have no way to tell, as it were by inspection, that this is actually so.Thus, we cannot fix the values of the variable ?p? simply by listing: not onlyare there perhaps infinitely many of them, but Wittgenstein gives no suggestionhow to recognize an elementary proposition if we saw it in the street. And, be-sides the outright listing, the only other way to specify a multiplicity of proposi-tions is by the internal relationships of the elements of the multiplicity to somepropositions already at hand. But such specification would require a grasp of theforms of elementary propositions, and this is precisely what we are not supposedto have, at least not a priori. But then, it is just not clear how we can under-stand the function of ?p? in the notation for the general propositional form as a72propositional variable at all. For it is just not clear how its range is supposed tobe fixed.Wittgenstein does seem to acknowledge, in the midst of the pessimistic 5.55s,that in spite of its not a priori status, still there is something like a question whatelementary propositions there are, and, of course, therefore also an answer to thatquestion. But, he says, ?the application of logic decides what elementary propo-sitions there are? (5.557). So, to understand how the letter ?p? is supposed to takeup its function as a variable ranging over the totality of elementary propositions,we need to understand Wittgenstein?s conception of the application of logic. Inthe remainder of this section, I will spell out what I take to be this conception,and thereby propose an answer to Sullivan?s question.Of course, the problem is not just that we have no way to identify the elemen-tary propositions. To identify a proposition as elementary requires identifyingit as maximally early in the order of analysis. So if we just knew the order ofanalysis a priori, that would somehow be enough. But in fact, so far we haveidentified no principled way to determine which propositions are to be analyzedin terms of which. When any small bunch of logically interrelated propositionsis considered in isolation, there will be various ways of analyzing some in termsof the others, and thus various possible conclusions as to which propositions inthe bunch are more fundamental than the others in the order of analysis. Evensomething that looks like a conjunction might turn out to be prior, in the orderof analysis, to each of its conjuncts.At this point, I think it is essential to note that the general propositional formis supposed to secure the truth-functionality thesis. That is, the general form ofthe proposition is supposed to be the general form of truth-functions of elemen-tary propositions. So, that every proposition is a truth-function of elementarypropositions is supposed to follow from the fact that every proposition falls un-der the general form of such truth-functions. The question then arises, how is itthat the general form of the proposition actually secures the truth-functionalityof what falls under it?The answer to this question rests, I think, on three key insights. First, truth-functionality explains the logical relationships of entailment and incompatibil-ity between propositions. For according to the truth-functionality thesis, what-73ever elementary propositions turn out to be, every proposition is an expres-sion of agreement and disagreement with truth-possibilities for them. Then,one proposition entails another provided that the first agrees only with thosetruth-possibilities with which the second proposition agrees. Similarly, for exam-ple, two propositions are incompatible provided that there is no truth-possibilitywith which they both agree.The second insight characterizes the relationship between truth-operationsand truth-functionality. Applied to elementary propositions, a truth-operationyields a truth-function of those original elementary propositions. But further-more, the result of applying a truth-operation to truth-functions of elementarypropositions is itself a truth-function of elementary propositions [[5.3]].The third insight is due to Frege, that analysis discloses new forms of ex-pression. That is, we might discover, through reflection on the nature of ourthoughts, that their apparently immediate interconnections are actually mediatedby heretofore unarticulated hypotheses.63 This may amount to the discovery ofthoughts not even obviously expressible at all in the words we had to begin with(hence the exotic typography of Begriffsschrift). Thus, the domain of analysis can-not be just the thoughts which a priori we might try to bound by the languagethat is already familiar to us. Rather, it must include further means of expres-sion whose use is to begin with merely possible. Of course, such signs with amerely possible use cannot be analyzed to begin with; so to begin with they canappear only as analysantia. But, on the other hand, one should not suppose thatanalysans must itself be unanalyzable, merely say, on account of its novelty orits aura of technical sophistication. For every analysans, like every analysandum,must itself be completely integrated into the system of signs, and this may in turnrequire that it too be analyzed.Let?s now turn to the nature of application of logic, which begins with thetotality of propositions. As we?ve just seen, this totality cannot be supposedto be bounded by the totality of actually formable sentences. So, consideredconcretely, the totality will not be covered by totality of prima facie understoodactually formable sentences, but must include thoughts which can be formulated63Broadly speaking, this exposure of unarticulated hypotheses is the logicist strategy to establishthe dispensability of intuition in mathematical argument.74only upon further reflection. Nonetheless, we do begin in the middle, amongstwhat we think we understand. And the aim is to construct an analysis.I now turn to the question what is an analysis. For expository purposes, Iwant to separate the question what is an analysis from a further question whenis an analysis good. In the most schematic form, an analysis is a function f suchthat, for some propositions A, the function f associates to A a collection f (A)of other propositions. The import of this is that the analysis f says that A is thejoint denial of the elements of f (A). That is, it is in the nature of an analysis fthat for every A such that f analyzes A, an insistenceA=N ( f (A)).is a fundamental commitment of f . By such an insistence, f consigns the assert-ibility conditions of A to the assertibility conditions of f (A). For, according tothe insistence, a totally negative verdict on f (A) just is an affirmation of A, and atotal but not totally negative verdict on f (A) just is a denial of A. Commitmentsso encapsulated can be thought of as the ?local? commitments of f . Whether ornot f does well to undertake this or that such local commitment is a point to beinvestigated. Since some propositions are incompletely grasped at the outset, andindeed, since a priori we have no idea what is to be analyzed in terms of what,we are certainly not at the outset in a position to evaluate the correctness of localcommitments. We can only, at best, volunteer them tentatively, on a case-by-casebasis. But as stipulated, the question whether something is an analysis at all isseparate from the question whether, as an analysis it is good.Now, among the commitments that an analysis undertakes, merely as an anal-ysis, there are not just local commitments but also a global one. Note that it is,in general, only for some but not all propositions A that f returns a collectionf (A) of propositions such that f alleges that A is their joint denial. These propo-sitions we will say that f analyzes; the rest, that f leaves unanalyzed. Now, theglobal commitment of f as an analysis is this: to explain all necessary connec-tions between propositions in terms of its analyses of propositions. Hence, fundertakes the commitment that no necessary connections obtain between thepropositions it leaves unanalyzed. This means, however, that f must envisage75each total verdict of agreement and disagreement on unanalyzed propositions toconstitute a genuine possibility. And conversely, since propositions left unana-lyzed are supposed to be explanatorily ultimate with respect to the obtaining andnonobtaining of logical connections, f also undertakes a commitment that suchverdicts on the propositions it leaves unanalyzed exhaust all possibilities. Let?scall such a total verdict, coherent or not, an f -envisaged possibility. Of course,f might be wrong about whether what it envisages is actually possible, and con-versely whether it does envisage all the possibilities. But as already remarked, wewill not have any way to tell this immediately since we probably do not have agood immediate understanding of the propositions that f leaves unanalyzed.With this idea of an f -envisaged possibility in hand, we can now spell outwhat it means for f to secure the truth-functionality of every proposition: fmust bear witness that every proposition is a truth-function of the propositionsthat f leaves unanalyzed. More precisely, f secures the truth-functionality of theproposition A provided that for every f -envisaged possibility M , either f saysthat M affirms A, or f says that M denies A.Given that f does secure truth-functionality, there is the further questionwhether f does so correctly. Now, a priori we will probably have no under-standing at all of the propositions that f leaves unanalyzed. Thus, nothing couldimmediately count for or against some hypothesis to the effect that such-and-such envisaged possibility does affirm (or does deny) such-and-such proposition.But, the fact that f bears witness to truth-functionality of all propositions com-mits f to further conclusions about logical relationships. This is because truth-functionality generates all logical relationships, and so f , being an analysis, mustacknowledge this. Thus, the testimony of f in regard to the relationship be-tween possibilities and propositions commits f to verdicts on all questions as toentailment, consistency, and so on. For example, suppose that f envisages nopossibility to affirm A and deny B . Now, f pretends to envisage all genuine pos-sibilities. Hence, f commits to the conclusion that there is no possibility underwhich A holds and B fails. But, is this right? Or does such a possibility exist?Conversely, suppose that f envisages some possibility to affirm A and deny B .Since f pretends to envisage only those possibilities which are genuine, it followsthat f commits to the existence of some such possiblity, hence to the conclusion76that f does not entail B . Is this right? Or actually does no such possibility exist?In this way, we can see that the proposed analysis absorbs the structure ofcommitments that makes something an analysis in the first place. To be an analy-sis is to acknowledge that if propositions are such-and-such truth-functions thensuch-and-such logical relations hold. Thus, commitments about logical relation-ships between propositions some of which may be unknown require commit-ments about logical relationships between propositions all of which are known.It is because the commitments of analysis lift up into the light in this way that ananalysis is correct or incorrect. The correctness condition on f can now be statedmore precisely as follows: that the answer to any logical question is what f says itis. Thus, if f is correct, then it must be if and only if some bunch of propositionsdoes entail A that f says it does; and it must be if and only if such-and-such bunchof propositions is logically inconsistent that f says it is, etc.We?ve now spelled out what it is for an analysis f to be good: f must se-cure the truth-functionality of all propositions, and f must do so correctly. Thiselaboration appeals crucially to questions about what f says. In particular, we?veobserved that f envisages such-and-such possibilities, and that f might say that agiven possibility affirms a proposition or that it denies it. We?ve also argued thatif f says stuff of that form, then f commits also to saying things about entail-ment, consistency, and so on. However, we?ve dropped only hints about whenf must allow that a possibility M affirms a proposition A. But the answers dofollow from what has been said so far. Let M be an arbitrary possibility envis-aged by f . On the one hand, suppose that f leaves A unanalyzed. Recall that forf to envisage M is for f to acknowledge M as a total verdict of agreement anddisagreement on propositions left unanalyzed by f . Since f leaves A unanalyzed,this means that either f takes M to affirm A or f takes M to deny A. So, if fleaves A unanalyzed, we simply stipulate that what f says M does to A is justwhat f takes it to do. Now on the other hand, suppose that f analyzes A. Thismeans precisely that f lets what f says M does to A depend on what f says Mdoes to the propositions into which f analyzes A. This, after all, is what it is forf to have given an analysis of A, i.e., for f to have represented A as the result ofa truth-operation on other propositions. Specifically, f represents A as the jointdenial of the propositions f (A). Thus, f says that M affirms A if f says M de-77nies B for all B in f (A); on the other hand, f says that M denies A if there?s atleast one proposition in f (A) that f says M affirms and if for every propositionin f (A), f says either that M affirms it or that it denies it. This completes thestipulations on what f must say about M and A. We further insist that f be asquiet as possible, so that f does not say anything more about what M does to Athan what is required by those stipulations.Thus, we?ve determined that the goodness of f depends on what f says aboutaffirmation and denial of propositions by the truth-possibilities f envisages. Andwe?ve now elaborated on just when f must say anything to that effect, at leastinasmuch as f is an analysis at all. However, it does not follow from what we?vesaid so far that f , being an analysis, actually does secure the truth-functionality ofpropositions. Hence, an analysis might fail to be good even before the questionof correctness arises. Here are some examples?I?ll work the first in detail; therest are similarly routine.1. Suppose that there are three propositions A, B , and C , such that f (A) =(B), f (B) = (A), and f leavesC unanalyzed. Then, a possibility M which fenvisages is simply a verdict on C , e.g., an affirmation of C . So, suppose fsays that M affirmsC but that f is otherwise silent; it remains to verify thatf is an analysis. Clearly f says that M does to an unanalyzed propositionprecisely what M does do to it. And, if f analyzes a proposition, then fsays that M affirms it whenever f says M denies all the propositions intowhich it is analyzed, i.e, never; and similarly f says M denies a propositionwhenever f analyzes it into some propositions about all of which f saysM issues a verdict but about at least one of which f says M affirms it. So,f says nothing to the effect that M affirms A or denies A. Thus, f is ananalysis which does not secure truth-functionality of A.2. Suppose there is a proposition A such that f (A) = A, and a C which fleaves unanalyzed.3. Suppose that there is an infinite sequence of propositionsA0,A1,A2, . . . suchthat f (A0) = (A1), f (A1) = (A2), f (A2) = (A3), . . . , and that there is anotherproposition C left unanalyzed by f .784. Consider each of examples 1-3, but without the C .These examples all indicate that an analysis f can fail to secure truth-functionality.But, note that all four examples have something in common, that in each case,applied to some proposition the analysis f never halts. More precisely, let?s writeA f B provided that B belongs to f (A) for any A and B . Then, in each case, itturns out that f induces an ?infinite descending chain?A0 f A1 f A2 ? ? ? .For example, in the first case the chain is just A f B f A. . .. To rule outthese particular pathologies, it would suffice to insist that f induces no infinitedescending chain.Let?s say that analysis f is well-founded if it induces no infinite descendingchain. The following question now arises. If f is well-founded, then must fsecure the truth-functionality of all propositions? The answer is yes. To see this,suppose that f is well-founded. Further suppose, toward contradiction, that fdoes not secure truth-functionality of a proposition A. Then, f envisages someverdict M on propositions left unanalyzed by f such that f says neither that Maffirms A nor that M denies A. Now, since f envisages M as a total verdict onunanalyzed propositions, if A were left unanalyzed, then f would have to sayeither that M affirms A or that M denies A. So, f must analyze A. Generalizing,if f is silent about what M does to a proposition, then that proposition cannot beleft unanalyzed. But now, since f is an analysis, if f is silent, with respect to M ,about none of the propositions in terms of which f analyzes a proposition, thenf cannot be silent about that proposition either. It follows that if f is silent withrespect to M about a proposition, then f must also be silent about one of thepropositions in terms of which f analyzes that proposition. Hence, there mustexist an infinite descending chain of propositions, each of which f analyzes intoits successor.64 This contradicts the original hypothesis.We have thus given a sufficient condition for f to secure the truth-functionalityof all propositions: namely, that f be well-founded. In a way, this condition is64This argument seems to require a weak form of the axiom of choice, but I am not sure if thisis invariant under philosophically insignificant rearrangements of concepts.79quite natural. For example, suppose that we consider the analysis of a propositionin some sense to explain it. This explanation would itself introduce further ideas,each of which must either be understood, or be explained. An infinite descendingchain of such explanations would leave the desire for understanding unsatisfied.65The concept of well-foundedness therefore appears to be of central interestto the understanding of the general propositional form.66 Historically, this con-cept was introduced by Mirimanoff (1917).67 And there is no reason to supposeWittgenstein himself found it a clear articulation in the course of composing Trac-tatus.68 Nonetheless, it seems to me that the concept of wellfoundedness, and thestructure of the induction used at T6 to fix the general propositional variable, aretwo sides of the same coin.69 If we put logic aside for a moment, and considerpsychology instead, then, one may say that, so it happens, people who seek to un-derstand the logical relationships between propositions do begin in the middle,with propositions that await analysis into propositions that are not immediatelyfamiliar. So, not as a matter of logic, but rather, as a matter of psychology, itmight be said that the concept of well-foundedness better suits the predicamentof the analyst.65This remark is, of course, intended only as a motivational heuristic. I do think that Wittgen-stein thought of analysis as, in some sense clarificatory. But, locally speaking it is not really ?expla-nation? in any obvious sense.66So far as I know, the first mention of the concept in this connection is due to Goran Sund-holm (1997), who suggests that the kind of order induced by the general propositional form is awell-founded partial order. One might introduce a geenuine partial order relation on the totalityof propositions, induced by the general propositional form, by taking the transitive closure of ?.But I don?t know of any place in the Tractatuswhere such an ordering is mentioned. There is an un-fortunate tendency among some commentators to talk about the set of propositions as exhibitingthis or that order type without spelling out which relation on the set so orders it.67But, as Goldfarb urged in conversation, it?s plausible that Wittgenstein would have learned theconcept of wellordering from Russell (1903).68But, then again, maybe he does. As Goldfarb pointed out, 4.221 says: ?it is obvious that anal-ysis must arrive at elementary propositions, which consist of names in immediate combination?.69Indeed, there is a general result to bear this out. Now, let (? f A) be the multiplicity of propo-sitions B such that B ? f A. Let I f be the least subset of the totality of propositions such that I fcontains N (? A) for each proposition A. It can then be shown, by an argument similar to whatis given in the main text, that every proposition belongs to I f . The analogy with T6 is this. Thepropositions left unanalyzed by f are simply the propositions A such that (?A) is empty; these getput into I f automatically and correspond to the elements of p. If (? A) is not empty, then (? A)corresponds to the analysantia ? of an analyzed proposition, so that f can be understood to claimthat A is N (? ).80Obviously, some mathematics remains to be explored here, but it is time toset that aside. For, we are now in a position to justify the following remark: asit is presented in T6, the general propositional form is simple, natural, and easyto understand. What T6 says is this. There is a correct analysis of the totalityof propositions, which represents some propositions as joint denials of others; inturn, those others may also be so represented, or they may not be; but in anycase, no such chain goes on forever. And that is all T6 says. Hence, the remark isjustified. Now, givenWittgenstein?s account of analysis, it follows that an analysisof the sort described at T6 would secure that every proposition is a truth-functionof the propositions left unanalyzed. Such an analysis, being correct, thereby givesthe answer to every question as to which propositions follow from which others,as to which propositions are jointly inconsistent, and so on. So in particular, theanalysis would be a description of all logically true propositions. But this resultis unsurprising, because the answers to such questions are precisely what must beknown in order that the analysis have been discovered in the first place.1.3.2 Origins of the independence criterionThe style of my discussion in this chapter has been seemingly aprioristic. Nonethe-less, I assert a definite interpretive claim: that no proposition is elementary initself, but is elementary only in virtue of the analysis of the entirety of language.Analysis represents all propositions to be truth-operations of some fixed bunchpropositions in such a way that the resulting iterative structure explains all neces-sary connections between states of affairs. Those propositions not represented asresults of truth-operations are the elementary ones; it is a trivial corollary of thecompleteness of analysis, and the means by which analysis predicts the obtainingof necessary connections, that no necessary connections obtain between elemen-tary propositions. But did I just make all this stuff up? In this section, I surveysome discussions from Wittgenstein?s NB3, and claim to pinpoint the precise mo-ment when Wittgenstein establishes this conception of elementariness.BetweenApril 1916 until October 1917Wittgenstein produced his third wartimenotebook. Unlike its predecessors, this volume contains signs of a concerted in-81vestigation of the general form of the proposition.70 It opens at 15.4.16 with ananticipation of T5.556: ?We can only foresee what we ourselves construct?. Thisremark, I think, encapsulates a tension that pervades the Notebooks and maybethe Tractatus as well. What follows this remark is the very natural question: ?Butthen where is the concept of a simple object still to be found?? For, Wittgensteinseems to think that to have a concept of the simple objects would entail foreseeingtheir forms or combinatory possibilities. Since, however, neither simple objectsnor their forms are constructed by us, he?s therefore perplexed by the questionhow we could grasp the concept of simple object. But why does Wittgensteinthink we know what a simple object is? In the previous notebook, and almosta year before (19.6.15), Wittgenstein had concluded that the concepts of thing,relation, property and so on cannot be derived from experience. Were they todepend on experience, then they could not be used in logic. But we do makeuse of them in logic, as with variables x or function notations ?x. At 15.4.16the grounds for thinking we must grasp a concept of simple object are basicallythe same. He says: ?We must be able to construct the simple functions becausewe must be able to give each sign a meaning. For the only sign which guaran-tees its meaning is function and argument.? Here, the point must be that oneconstructs a ?simple function? by abstracting a name of a simple object from asimple proposition; hence, knowing what a simple object is would seem to berequired to confer meaning on signs.Wittgenstein?s reason for saying that the function-argument pattern guaran-tees its own meaning must be that the function-argument pattern is the pictorialform of a proposition which partakes of it. For, pictorial form would be what iscommon between the proposition and the situation depicted, so that once objectsare assigned as meanings to the constituent names, then a combination of namesunder that form says something without further ado. Now, this assignment ofmeanings to names requires seeing (?simple?) propositions as functions of namesin the first place, and thus requires the ability to ?construct the simple functions?,70Inasmuch as Wittgenstein?s question is what a proposition is, of course, everything he doesis an investigation of the general propositional form. In particular, the persistent questions inNotebooks 1 and 2 about subject-predicate form in unanalyzed propositions seem to me to involvethe fundamental issues of foreseeability which motivate conviction of the existence of g.p.f. Butnot really until Notebook 3 do we find concerted attempts to articulate the g.p.f.82i.e., the functions which result from elementary propositions by turning a nameinto a variable.The next day?s entry (16.4.16) begins, ?Every simple proposition can be broughtinto the form ?x?.71 It seems clear that ?simple proposition? is a terminologi-cal variant of ?elementary proposition?. Since 15.4.16 anticipates 5.556, and the5.55s address the question of foreseeing the number and forms of elementarypropositions, it looks like this is what?s really at stake here early in NB3. So, thefirst sentence of 16.4.16 seems to be that the form ?x should allow us to spec-ify the totality of elementary propositions. This, I think, is confirmed by nextcouple of paragraphs.That is why we may compose all simple propositions from this form[?x].Suppose that all simple propositions were given to me: then it cansimply be asked what propositions I can construct from them. Andthese are all propositions and this is how they are bounded.As the Notebooks? editors observe, the last two sentences anticipate 4.51, itself acommentary on the first mention in the Tractatus of the general propositionalform at 4.5. Thus, it is hard to resist the conclusion that at 16.4.16 Wittgen-stein identifies two, jointly determinative sources of the general propositionalform: namely first, the general form of what is not constructed, i.e., the generalform ?x of simple propositions, and second, the general form of constructionof propositions from propositions. Granted an analysis of possibilities of con-struction, a bound on the totality of propositions would derive from a boundon the elementary propositions. And so, Wittgenstein?s aim here seems to beto find a bound on the totality of propositions by identifying a common form?x from which all elementary propositions ?can be composed?. What expressesthis common form would be a variable which takes the totality of elementarypropositions as its range.71On my understanding, Wittgenstein has said here only that it must somehow be possibleto isolate a name in a proposition, but not in any way explained how this should takes place.(For example, presumably the constituents of ordinary sentences are not names.) While the merepossibility of such isolation is a purely logical result, the account of how the possibility is realizedwould be a result of the application of logic.83The details of this idea are not entirely clear to me. As a first pass one mightspeculate that Wittgenstein proposed that an elementary proposition72Quine respects Frege and Russellis analyzable as the value of a function?xas applied to( ) respects Frege and RussellandQuine.Thus, the proposal somehow relies on a decomposition of elementary propo-sitions into names and elementary propositional functions.73 This might bethought to be some kind of extension of the function-argument method of range-fixing described at T5.501.74 However, since at least in the Tractatus, Wittgensteinholds that it is not just in elementary propositions but in any proposition thata name can be turned into a variable. So it is not clear how function-argumentstructure could distinguish elementary propositions from arbitrary ones.Despite the difficulties of understanding this discussion in detail, it seemsclear that at 16.4.16 Wittgenstein tries to establish the existence of a general, dis-tinguishing form of elementary propositions. This form would find expressionin a variable, thereby subserving an articulation of the general form of all propo-sitions. Seven months later, such a general strategy reappears in the following72Of course, I don?t claim the below is an elementary proposition, let alone one envisaged byWittgenstein.73Speculation along these lines may draw some further support from the last of the April 1916Notebook entries just now considered, namely 27.4.16. But there, it?s worth noting that the decom-position is said to issue in three arguments, ?x, ?( ), and x. The extra argument ?x is presum-ably a descendant of the concept of logical form from Russell?s Theory of Knowledge manuscript,and an ancestor of the logische Urbild of 3.315.74I agree with, or will double down on?the suggestion in Ricketts (2012) that that this kindof approach does give an instance of genuinely quantificational generalization with respect tothe predicate-position, and that it probably explains Wittgenstein?s use of such generalization at(5.5261).84passage.The fact that it is possible to erect [muss sich aufstellen lassen] thegeneral form of proposition means nothing but: every possible formof proposition must be foreseeable. [21.11.16]Thus, the foreseeability of logical forms guarantees not just that some form iscommon to all propositions, but moreover that the form explains how thoseforms could be known to us a priori.75So, inNovember 1916,Wittgenstein resumes his analysis of the general propo-sitional form systematically. After declaring that the analysis ought to secure theforeseeability of all propositional forms, he writes: ?We now need a clarifica-tion of the atomic function and the concept ?and so on??. This, evidently, recallsfrom 16.4.16 the dichotomy of determinative sources of the general propositionalform. After concluding the 21.10.16 entry with discussion of the ?and so on? half,he returns at 23.11.16 with the following suggestion:What does the possibility of the operation depend on?On the general concept of structural similarity.As I conceive, e.g., the elementary propositions, there must be some-thing common to them; otherwise I could not speak of them all col-lectively as the ?elementary propositions? at all.In that case, however, they must also be capable of being developedfrom one another as the results of operations. [23.11.16]In this passage, Wittgenstein makes another attempt to construct a variable rang-ing just over the elementary propositions. The plan seems to be to specify themby means of a form-series variable. This, Wittgenstein seems to think, mustbe possible merely if elementary propositions have some commonly distinctivestructural feature.76 So, it seems we have here a clear use of an assumption I re-75Note that the Prototractatus links foreseeability of logical form with determinacy of sense.76Such a linkage of formal features with formal series reappears in somewhat attenuated formin the Prototractatus, in particular in the PT5.00534s, though there formal commonality appearsas a necessary but not sufficient condition for specifiability of some bunch of propositions by aform-series.85jected in Ricketts, namely that the instances of a universal generalization couldbe given by means of a formal series.77,78In any case, it is now evident that actually in the Notebooks, Wittgensteinmade two attempts to identify the common form of the elementary proposi-tions. Throughout these attempts, Wittgenstein?s ultimate aim was to constructthe form of all propositions. His strategy was to articulate the general propo-sitional form in two stages. First, identify that form which is common to theelementary propositions. Second, identify the general form of construction ofpropositions from propositions. The ground for Wittgenstein?s aim to articulatethe general propositional form was that the forms of all propositions must beforeseeable to us a priori. The details of Wittgenstein?s strategy for articulatingthis form demonstrate that the commitment to the foreseeability of the formsof propositions involved a commitment to the foreseeability of the forms of ele-mentary propositions.Sullivan is correct in saying that nothing remains of these attempts in the Trac-tatus. And as he observes, the 5.55s insist on their futility. Nonetheless, Wittgen-stein?s presentation of the general propositional form in the Tractatus presupposessome concept of the elementary proposition. But what, in the Tractatus, is thebasis of our grasp of this concept?Wittgenstein winds down his second, 23.11.16 attempt at articulating theforms of elementary propositions with the following remark: ?if there reallyis something common to two elementary propositions which is not common toan elementary proposition and a complex one, then this common thing must becapable of being given general expression in some way.? I think that the phrasingof the hypothesis here already evinces some doubt that some formal feature doescharacterize propositions as elementary. Indeed, I claim that a few days later,Wittgenstein rejected the hypothesis altogether. A few days later, in a single-lineentry he writes:Either a fact is contained in another one, or it is independent of it.77I suspect that Ricketts worked out his interpretation with an eye on these early passages.78There is some reason to think that Wittgenstein may at the time have conceived of the secondapproach to be a refinement of the first approach. For, at 22.5.15 (which I?ll discuss in more detailshortly) Wittgenstein suggests that quantificational generality can be analyzed by means of form-series.86[28.11.16]Rephrasing, this says that dependence of one fact on another is a sufficient con-dition for the containment of one by the other. I suggest that Wittgensteinhereby analyzes containment in terms of dependence. Granted that elementari-ness was antecedently characterized as containment of no further propositions,the remark would then reduce the concept of elementariness to the concept ofdependence. As for the notion of dependence, it?s clear that in other passages inthe Notebooks, as well as in the Prototractatus and Tractatus, Wittgenstein takesthis to be a purely logical, and indeed purely truth-functional notion.79 Thus, Iconjecture that at 28.11.16, Wittgenstein proposes a purely truth-functional char-acterization of the concept of elementariness itself. The details of such a char-acterization are not explicit there. But some such details are certainly implicit.For example, assuming Wittgenstein intended that elementary propositions donot contain each other, 28.11.16 implies outright that elementary propositionsare truth-functionally independent.A month or so later, some consequences of such a truth-functional character-ization of elementariness emerge explicitly:It is clear that the logical product of two elementary propositionscan never be a tautology.If the logical product of two propositions is a contradiction, and thepropositions appear to be elementary propositions, we can see thatin this case the appearance is deceptive. (E.g.: A is red and A is green.)[8.1.17]These remarks follow what I take to be the sharpest expression of the generalpropositional form in the Notebooks, in which for the first time Wittgensteinspeaks of the totality of propositions as generated in terms of a single operation.I take the 8.1.17 entry to endorse an alternative account of elementariness, anaccount which remains in the Tractatus. On this new approach, nothing intrinsicto a proposition makes it elementary. Rather, to be elementary is simply to beindependent of all other elementary propositions. (Cf. 4.211) Thus, the class of79It?s clear that dependence has no conceivable nonlogical meaning in the Tractatus. As fordependence in NB, see for example 10.6.15a.87elementary propositions is characterized by the mutual independence of its ele-ments. Of course, many classes are so characterized. However, not every suchclass is such that every proposition results by truth-operations from its elements,so that all necessary connections between propositions derive from the structureof the operational genesis. That, and only that, is what identifies a class of propo-sitions as the elementary ones. It is from this conception of elementariness thatthere originates the idea of states of affairs, whose mutually independent possibil-ities of obtaining and non-obtaining explains the truth-susceptibility of elemen-tary propositions and thereby, via the method of constructing propositions frompropositions, also explains the structure of the totality of propositions.80 Only inarriving at this conception of elementariness at the end of NB3 doesWittgensteinat last make room for the remarks of the 5.55s, whose earliest known ancestorsappear in the Prototractatus.8180It follows from this conception of elementary propositions that, if it is in virtue of the unsur-veyability of elementary propositions that the general propositional form is, somehow or other,transcendental, then the ultimate analysis of any one proposition is itself also transcendental. For,on this conception, what identifies a proposition as elementary is only its relationship to all otherelementary propositions, together with the role that the entire class of elementary propositionsplays within the global structure of the totality of propositions. So, distinguishing any propositionas elementary requires recognizing all elementary propositions as elementary. But some proposi-tions are distinguished as elementary by a complete analysis of any one proposition. Hence, acomplete analysis of any one proposition requires surveying the totality of elementary proposi-tions. And surveying the totality of elementary propositions was what was supposed to preventthe general propositional form from appearing as a variable in the determination of sense of anordinary proposition.81Thus, the 1s and 2s appear to present a dogmatic and aprioristic theory of the internal structureof elementary propositions, na?vely inspired by the surface structure of ordinary language. Butthese opening remarks are grounded in Wittgenstein?s conclusion at the end of NB3, that we can?thave knowledge of the sort to which they tempt us.88Chapter 2Objectual generalityDon?t get involved in partialproblems. . . . (1.11.14n)In the late 1970s Robert Fogelin sparked a little industry by defending a neg-ative answer to the following apparent question: ?is the system of the Tractatusadequate to First-Order-Logic-with-Equality?? In the early 1980s, Peter Geachand Scott Soames rebutted Fogelin?s argument. Actually, according to them, ?thesystem is adequate.? It is now a rite of passage for would-be Tractatus scholars topromise to end the ensuing debate.I would like to break with tradition, and argue that there is much more to besaid than the disputants acknowledge?indeed, that there is more to be said than Icould possibly have said here. It seems to me that the very phrasing ?is the systemadequate?? already begs the question against the Tractatus outlook on signs, onsign-systems, on what these things do for us, and what they do to us.Juliet Floyd has suggested that one of Wittgenstein?s overarching aims in theTractatus is to ?break the metaphysical idolatry of notation?, and expressed doubtthatWittgenstein had any interest in constructing a smoothly functioning systemof notation. I agree that the Tractatus aims to disrupt the sort of smooth function-ing in logic or philosophy into which notational systems can seduce us.82 Onesource of value in the Tractatus is its power to reawaken us to questions that aresilenced by the well-practiced movements of the muscles in our hands.Now, let me be clear. I presume that it is a straightforward historical questionwhether Wittgenstein grasped the elementary syntactical functioning of quan-82It seems to me that this is why Wittgenstein, as Goldfarb [[cite]] emphasizes, does not putforward any particular begriffsschrift of his own.89tificational logic, just as it is a straightforward question whether, say, Herbrandgrasped it, and just as it is straightforward, if actually questionable, whether anundergraduate student in the exam last week grasped it. Fogelin has even raiseddoubts about this. It is notoriously difficult to resolve certain varieties of skep-ticism about matters of historical fact, and I will mainly leave this problem toprofessionals like Geach and Soames.So although Fogelin?s provocation deserved its torrent of enthusiastic rebut-tals, their very enthusiasm strikes me as somehow contrary to the spirit of theTractatus. And perhaps it is discomfort with such choruses of affirmation thatprovides the main animus of this chapter. But let?s get down to some sharperpoints of doctrine and methodology.First and foremost, these rebuttals of Fogelin all aim to present some smoothlyfunctioning system of notation, a purported best possible candidate for beingwhat Wittgenstein himself simply disdained to formulate, perhaps on account ofhis literary taste, or sense of station in life.83 However, it was part of the veryidea of giving the general form of the proposition that it contain only what isessential to any system of signs whatsoever. Thus it was Wittgenstein?s aim pre-cisely to prescind from any artifacts of notation, ever so apt as they eventuallyare to be mistaken for determinations of the subject matter. Perhaps Wittgen-stein?s most characteristic criticism of Frege and Russell, after all, is that theypervasively mistake formal relations for material ones, widgets of notation forways of the world. Fogelin?s respondents therefore fuel the very march of nota-tional hegemony that Wittgenstein sought to disrupt. I will therefore seek to layout Wittgenstein?s conception of quantificational generality in a way that givesno satisfaction to the lust for smooth functioning, and yet do so clearly. And Iwill also try to illustrate Wittgenstein?s strategy of prescinding from features ofthe sort that are liable to be confused with what is essential to all articulations ofobjectual generality whatsoever?trying to see, as McCarty (1991, 61) puts it, justwhat we can get away with, and what we cannot get away without.A second, and related point, is that much of contemporary logical practicearose in a tradition that stems from Hilbert. Good logical training thus prompts83I have in mind in particular the remark that Ramsey was a ?bourgeois thinker?.90the interpreter to foist onto the Tractatus a way of thinking about representationwhich is foreign to it. Signs become mute concatenations of surds, on which weconfer referential powers by semantic interpretation.84 In the Tractatus, the vehi-cles of picturing become open formulas, wherein the natural proxies for objectsare free variables which, lacking quantificational governance, require an imposi-tion of value to contribute their share to the truth-bearing load. But what, inthe context of the Tractatus, is the source of this awaited imposition? At thispoint, it is all too tempting to fall back on the suggestion of Peter Hacker: ?amechanism of a psychological nature is generated to project lines of projectiononto the world? (quoted in Goldfarb 1981). In the context of Tractatus, it is pre-cisely backwards to follow the order of explanation established by Hilbert andTarski. Wittgenstein?s conception of logical structure begins with elementarypicturing facts, arrangings of objects which are ipso facto sayings that objects areso arranged in the world. Only then, once picturing has begun, does generalityarise. ?The truth or falsehood of a general proposition palpably depends on thatof elementary propositions?. Generality is inflicted on picturing facts, throughthe selective denaturing of their nominal constituents. Thus, we have in the Trac-tatus not naming achieved by assignment of value to anonymous variable, butvariation achieved through the anonymization of names.As we go along, the points just mentioned will mainly advance themselvesby rhetorical insinuation?the reader is hereby forewarned. But there is a third,more substantial claim, which is the eventual positive thesis of this chapter. Tostate the thesis, we need a little background. Wittgenstein famously insists thatevery elementary proposition has the possibilities both of truth and of false-hood.85 Moreover, the truth-value of every proposition is a function of thetruth-value of elementary propositions. But, given these claims, it is not at allclear how to accommodate statements of identity. On the one hand, a givenstatement of identity does not seem to have the two poles of truth and false-hood, and so does not seem to belong among the elementary propositions. Onthe other hand, since Wittgenstein rejects the identity of indiscernibles (5.5302),84The preceding few sentences are indebted to McCarty (1991).85Indeed, every genuine proposition has both possibilities, but I don?t need this subtler pointhere.91nor does identity seem to be explicable as a truth-function of elementary propo-sitions. Yet, it seems to be in virtue of the relation of identity that, togetherwith the quantifiers, we express such genuine facts as that there are two cupson the table. So, the story goes, Wittgenstein has a problem, which drives himinto a deviant reinterpretation of the objectual generality, one which had surelynever been palpably implemented in any rational discourse. This deviant read-ing, so it happens, enjoys the ideologically convenient feature that it allows allthe ?genuine facts? ordinarily expressible with the equality predicate to be ex-pressed without the equality. Thus, Wittgenstein is backed up into a distortionof the concept of generality, a concept whose analysis was a fundamental projectof his logicist predecessors?and indeed, a fundamental project of his own. Thisstory radically underestimates Wittgenstein?s power as a philosopher. As arguedin Chapter 1, Wittgenstein?s conception of analysis excludes all necessity but thelogical, and the resulting need to explain necessities of the world leads him to aconception of propositions as picturing facts. I?ll now argue that the conceptionof propositions as picturing facts supports a distinctive account of propositionalfunctions; in turn this account of propositional functions entails the redundancyof the equality predicate. Perhaps it is overstating the point to say that the redun-dancy of equality is merely a corollary to the idea that propositions are pictures.Rather, one may think of the role of relational structure and of identity in ordi-nary presentations of first-order logic as embodying one conception of the natureof mathematical multiplicity in representation. On this usual approach, althoughcertain rules of inference or conventions of model-theoretic treatment single outthe equality predicate as nominally logical, nonetheless identity manifests itselfgrammatically as a material relation. Wittgenstein reached another conception ofmathematical multiplicity on which identity becomes genuinely formal, pertain-ing not to what a sentence says about things, but rather to what a sentence musthave in common with things in order to say something about them. The concep-tion of propositions as pictures embodies a deep and, I think, mostly unplumbedlogical insight.86The chapter runs as follows. 2.1 and 2.2 build up some framework and termi-86Another appreciation of the unity of Wittgenstein?s ideas about equality and the variable ap-pears on Wehmeier (2012); there are also affinities with ideas of Kit Fine (2000).92nology for discussing first-order logic in the context of the Tractatus. 2.3 presentsa precise, perhaps even idealized, formulation of Fogelinite skepticism; I suggestthat this skepticism points to a philosophically important feature of Wittgen-stein?s discussion. 2.4 summarizes the Geach-Soames response, concluding withsome questions about the precise status of their interpretive claims. In 2.5, Ibegin by summarizing the Hintikka-Wehmeier reconstruction of Wittgenstein?snonstandard interpretation of the variable. I then pose the problem of recon-structing Wittgenstein?s analysis using purely Tractatus resources, and contrastWittgenstein?s account of propositional functions with the Fregean approach towhich Wittgenstein?s has been recently likened by Thomas Ricketts (2012). In2.6, I outline my own interpretation. The chapter concludes in 2.7 with an infor-mal statement of the underlying mathematical framework I propose.2.1 What is the issue?I take it that nobody wants to contest whether there is, roughly speaking, sucha thing as ?the system? of Frege?s Begriffsschrift, or whether it is ?adequate? tofirst-order logic. But it?s not quite obvious why this should be so. Frege jumpsfrom truth-functional logic to second-order logic, never pausing to isolate first-order logic as a system in its own right. Moreover, he does not explicitly define aclass of notations as the grammatical sentences of the system. Nor does he spellout completely the principles licensing passage from one judgment to another.And, of course, in 1879 Frege does not define any concept of semantic value of aformula, hence, also, no semantic concept of logical consequence. Still, it lookslike it is safe to attribute to Frege in 1879 some sure grasp of the ?essence? of first-order logic, because he demonstrates by example the structure of a sufficientlyintricate system of proof.In the Tractatus, however, no such system of proof appears. Indeed, this isno accident, for Wittgenstein holds that proof is merely a mechanical expedientfor recognizing tautologies in complicated cases. In the development of astron-omy the telescope becomes, at some point, practically necessary; and likewise,inasmuch as our judgments are, being judgments, suspended in logical interrela-tionships of implication and contradiction, proof may be practically necessary93to finding out just where we are. But these necessities are in each case no morethan practical; they are the needs of people with limited powers of vision or un-derstanding. When Frege advertises that the Begriffsschrift will illuminate thedependence of some truths on others lower down a tree at the base of which onefinds truths which belong there because their being truths is sufficiently evident,Wittgenstein complains that Frege injects into logic a kernel of psychology.87So it is not really clear how the question of adequacy to first-order logic getsa foothold in the Tractatus. Where, in that book, is logic at all? Here, I think itmight help to think not about proof, but instead about sense. First-order logic is,among other things, a recipe for language. Given some simple, primitive words?say, some names of individuals, and expressions of properties and relations be-tween individuals, then the recipe guarantees a whole system of possibilities fordistinguishing between ways in which the world might be. For example, givensome names of Alice and Bob, and expressions for being a person and for loving:the recipe guarantees the possibility of saying that Alice loves Bob, or that sheloves somebody who loves Bob, or that she loves only three people, and so on.On the other hand, for example, the system does not secure the possibility ofsaying that Alice loves only finitely many people: whatever it secures which youcan truly say of every situation in which Alice loves any finite number of peoplemust be true of some other situations as well. Nor, for that matter, does it give away to say only that she might?ve loved Bob.88At T4.5, Wittgenstein writes:It now seems possible to give the most general propositional form:that is, to give a description of the propositions of any sign-languagewhatsoever in such a way that every possible sense can be expressedby a symbol satisfying the description. . . .So our question begins to get a foothold. First-order logic is a recipe for a fam-ily of languages, which in turn guarantee the expressibility of many senses. Onthe other hand, in giving the general propositional form, Wittgenstein promises87Now, I?m not sure if this is actually Wittgenstein?s complaint. Wittgenstein?s complaint ?it isremarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as the criterionof a logical proposition? seems rather to object that the criterion makes logicality arbitrary.88Cf. 5.525.94to describe what is common to all the senseful expressions in all possible lan-guages. This general propositional form had therefore better somehow accom-modate first-order languages in particular. Now we do have some grounds forconcern. For Wittgenstein takes the existence of the general propositional formto have the following consequence.A proposition is a truth-function of elementary propositions (T5).But, this looks like an implausible, if not just completely garbled, description offirst-order formulas.For Wittgenstein, something is an X -function of some other stuff providedthe matter of its Xhood depends only on the matter of the Xhood of that otherstuff. So in particular, T5 says that the truth-or-falsehood of a proposition de-pends only on the truth-or-falsehood of elementary propositions. This impliesthat some propositions are such that the truth-and-falsehood of any propositionis entirely determined by their truth-and-falsehood. As Wittgenstein remarks,89every proposition is a truth-function of itself. So the teeth must be in the condi-tion that the propositions in the decisive class are elementary.By insisting that the truth-values of elementary propositions suffice to de-cide the truth-values of all propositions, Wittgenstein takes a crucial step awayfrom Frege and Russell, and toward a clear notion of first-order logical valid-ity. For under the assumption that every object has a name,90 a total verdict onthe truth-and-falsehood of elementary propositions becomes the elementary dia-gram of a structure as understood in standardmodel theory. Wittgenstein?s truth-functionality thesis is therefore simply the thesis that what determines the truth-value of a proposition is the elementary diagram of a structure. Logical validitythen becomes truth in all first-order structures over a given domain and signature.Although for Wittgenstein, it is possible to generalize with respect to every sim-ple meaningful constituent of an elementary proposition, such generalizationsyield no genuine higher-order resources.91 In particular, Wittgenstein?s approach89At T6.90If this boggles?as it does Hugh Miller III (1995), then grant him what we grant his logicteacher, who held that objects appear in propositions as names of themselves. Alternatively, seeShoenfield (1967), or even Quine himself (1998).91For details on my account of Wittgenstein?s treatment of higher-order generality, see Chapter95invalidates the second-order comprehension axioms which allow Frege to definethe ancestral (4.1252), and similarly invalidates Russell?s axiom of reducibility(6.1233).92 As Wittgenstein puts it elsewhere, the sense, or truth-functionality,of a proposition is its agreement and disagreement with truth-possibilities for el-ementary propositions. Far from obscuring the possibilities of first-order sense-making, the truth-functionality thesis opens a door to an account of them.The truth-functionality thesis therefore supports the interpretive conjecturethat in the Tractatus, Wittgenstein breaks from Frege and Russell by makingroom for an essentially first-order conception of logic. If the Tractatus has someproblemwith first-order logic, this must be a problemwith the details ofWittgen-stein?s account of the way in which propositions achieve their truth-functionality.For example, perhaps Wittgenstein holds that every proposition achieves truth-functionality because it is elementary. Since elementary propositions are logi-cally independent, Wittgenstein would hold that no proposition contradicts an-other. But then he could not accept even an uncontested fragment of first-orderlogic.The issue must therefore be as follows. According to Wittgenstein, somepropositions are elementary. But, logical interdependence between propositionsarises only with the introduction of propositions which are nonelementary. SoWittgenstein?s treatment of nonelementary propositions needs to account for thecomplexities of logical interdependence.Allegedly, this account is defective. For example, maybe Wittgenstein im-plies, on purpose or not, that not all apparent possibilities of first-order sense-making are genuine. Or maybe what he says is just so obscure that nothing coulddetermine whether it is incomplete or confused. In other words, the issue is notsettled without a cogent reconstruction that also fits the text.3.92For an early assertion of the invalidity of Russell?s axiom, see Letter 29, in McGuinness, ed.,(2008).962.2 Nonelementary propositionsWittgenstein holds that the phenomenon of logical interdependence betweenpropositions is already acknowledged, albeit falteringly, in the means by whichwe find them expression. As he puts it,The structures of propositions stand in internal relations to one an-other. (T5.2)In order to give prominence to these internal relations, we can adoptthe following mode of expression: we can represent a proposition asthe result of an operation that produces it out of other propositions(which are the bases of the operation). (T5.21)An operation is what has to be done to the one proposition in orderto make the other out of it. (T5.22)Consider, for example, two people who in some respect disagree, so that thingscannot at once be as the two think together. It may then arise that such disagree-ment can be traced to a single point. In particular, Dave may learn enough to findthat he and Carol disagree when he hears Carol claim that Alice loves Bob. InWittgenstein?s terminology, Dave finds, through this one claim by Carol, that hisposition lies outside of Carol?s. So, rather than expressing his contrary positionsui generis, Dave now follows an alternative custom: he expresses his position asthe result of denying what Carol said. Language already contains instrumentsfor representing a proposition as the result of, so to speak, doing something toanother proposition.This talk of ?doing something to a proposition? can mislead.93 What is actu-ally carried out by Dave is a verbal construction, whereby he represents his ownintellectual position as ?lying outside? of the position expressed in Carol?s claim.It is correct that his position, or better, his occupying the position, is in somesense the result of denying Carol?s. But, there is nothing in the nature of the po-sitions themselves such that one must be specified by means of the other and notvice versa. Maybe an analogy would be useful here. One might say that the Red93Compare Frege?s talk of theAnwendung and Ergebnis of aVerfahren in Begriffsschrift?s ChapterIII.97Sea is four hours? drive from the Dead Sea. But, the Red Sea is not itself the resultof a drive. Of course, one might happen to end up at the Red Sea, having drivenfrom the Dead Sea. But one could also have driven, walked, or ridden a camelthere from elsewhere, or perhaps even just be indigenous. Certainly, there is nointeresting sense in which the generic circumstance of being at the Dead Sea is,as such, prior to the circumstance of being at the Red Sea. Similarly for the posi-tions of Carol and Dave: you can get to each from the other. Note, furthermore,that this getting from each sea to the other by camel or by car does not make itthe case that the seas have such-and-such geographical relationship. Rather, thegeographical relationship between the seas makes it possible to travel betweenthem. Similarly for the intellectual commitments achieved by Carol and Dave: itis that the propositional content of the one commitment be the contradictory ofthe propositional content of the other, which makes it possible for Dave to reachhis commitment by expressing a denial of what Carol says.Wittgenstein thus takes propositions to stand to each other in logical or in-ternal relations. It is in virtue of such relations that, say, two propositions cannotboth be true, or, more generally, that a proposition excludes the truth of each ofseveral other propositions. To say that propositions bear such internal relationsto each other is just to say that the propositions depend on each other for theirtruth and falsehood. These internal relations are the wellspring of Wittgenstein?saccount of the truth-functionality of nonelementary propositions.Toward a sketch of this account, we need first to develop one slightly trickyidea. Wittgenstein inherits from Frege and Russell a heuristic for starting with arelation between items, and reconceiving the relation as a procedure for locatingan item by reference to the totality of items to which it bears that relation. Ina little more detail, the idea is this. Say that a relation is extensional when notwo items in its field agree exactly on the items to which they bear it. (Thus,being-mother-of is extensional, but being-parent-of is not.) Then an extensionalrelation has the feature that any item in its field can be seen as the unique itemwhich bears the relation to exactly such-and-such items. It is for such a relation,that Wittgenstein plans to find an associated operation, such that the result ofapplying the operation to those items yields that item which bears the relationexactly to them.98Let?s return to Carol and Dave. Carol made a claim which, as it were, leavesno room for things to be how Dave takes them to be. Of course, there is nosingle specific way in which Dave takes the world to be, but, rather, his beliefsleave no room for things to be as Carol says they are. Thus, Carol?s claim standsin an internal or logical relation to Dave?s overall view. But, this relation, con-trariness, is not extensional. For example, Dave?s position could be weakenedor strengthened in various ways while preserving its relation of contrariness toCarol?s claim. However, a consequence of Dave?s position, that we have alreadymentioned, is another, weaker position, namely that things are not as Carol says.This position is not only contrary, but rather, in the jargon, contradictory, towhat Carol says, because it is guaranteed to be correct whenever Carol?s claimis incorrect. Now, the relation of contradictoriness is extensional: there is onlyone position which allows the world to be in only those ways not allowed byany of some bunch of (other) positions. We now reach a means of characterizingany proposition in the field of the truth-exclusion relationship, namely, as thatweakest position which excludes the truth of such-and-such propositions. But ofcourse, Wittgenstein presumes that every proposition does have such a weakestcontrary, or contradictory. This is obvious, because to find the denial, we justneed to phone Dave. Thus, every proposition can be characterized as the denialof some other propositions.Here, finally, is Wittgenstein?s audacious account of the truth-functionalityof nonelementary propositions. Say that a conversation is maximal, providedthat absolutely everything that could ever be said under any circumstances will atsome point be said in that conversation. You are seated at a table with some otherpeople, and you begin to talk. Actually, people may have already been sitting atthis table, talking, perhaps for an unboundedly long time. You can say whateveryou want (at last!). But, each contribution must always assume one of two forms.First, you may assert an elementary proposition. Second, you can deny a bunchof stuff that some other people said. (More precisely, a move of the second kindwill find an expression ?that is all bunk?, where the reference of the demonstra-tive has been rigorously specified). The claim of proposition 6 of the Tractatusis that if people actually do this, as much as possible, subject to those two con-straints, then the resulting conversation is maximal. This claim of maximality99underpins Wittgenstein?s account of the truth-functionality of nonelementarypropositions.Let us now see how the idea just sketched is supposed to work. Consider acompletely arbitrary proposition; we aim to show that it is a truth-function ofelementary propositions. Since an elementary proposition is a truth-function ofitself, we can assume this proposition to be nonelementary. By the assumed max-imality of the table-talk, at some point this nonelementary proposition will havebeen asserted at the table. Thus, for the given proposition to be true is preciselyfor a truth to have been asserted at the table at that point. By the constraintson table talk, the nonelementary proposition must have been expressed in theform ?that is all bunk?, where the reference of the demonstrative has been rigor-ously specified. Of course, this reference will be to some bunch of propositionsthat had been asserted antecedently. We have then the following: that the givenproposition is true iff each of those other propositions is false. Each of thosepropositions must itself either be elementary, or be the joint denial of some otherstuff. Since truth-functionality of each elementary proposition is presupposed,it suffices to continue to unfold the reference of the demonstrative in each of thedenied nonelementary propositions. And so on.Will this, actually, work? Of course, it is clear that we do not have here aproposed practical guide to achievement of understanding. Rather the questionis: under the given conditions, must the truth-functionality of each propositionbe fixed? The answer, for all that I?ve so far said, is no. For nothing I?ve so farsaid rules out that the proposition whose sense we interrogate be the origin of aninfinite descending chain of denials.Fine. We reject by fiat the existence of any infinite descending chain. Thatis, every process of unfolding the demonstrative of nonelementary propositionsmust, for each of its branches terminate in an elementary proposition. Butnow, the unfolding process is guaranteed94 to yield a determinate condition ofagreement and disagreement with truth-possibilities for elementary propositions.That is, the truth-functionality thesis holds of everything expressed at the ta-ble. By the maximality assumption, we have therefore established the truth-94By the principle of recursion on a well-founded relation.100functionality of everything that can be said.In the account just sketched, we?ve slid over a big gap. Namely, when hasthe reference of a demonstrative been ?rigorously specified?? Note that these areplural demonstratives we are dealing with here. Maybe the speaker will not haveenough fingers to point to each of the referents. Suppose, more generally, onlyfinitely many propositions can ever be specified as a basis of denial. Then,95 everyproposition ever expressed in a maximal conversation will depend for its truth ononly finitely many elementary propositions. In that case, Wittgenstein could notacknowledge the expressive possibility afforded by first-order logic of depending,for the correctness of what you say, on the truth or falsehood of a priori perhapsinfinitely many elementary propositions. Then the Tractatus would indeed be,in the jargon, ?inadequate?. At another extreme, maybe these conversants at thetable can, as in a platonist dream world, for any collection of propositions what-soever, fix on it and deny precisely its elements.96 It is routine to show that undersuch a hypothesis, for every class of structures over our background signature,i.e., for every class of total possible ways for the world to be, there would thenbe a proposition expressing exactly that the world is in some one or other of theways in that class. The first-order definable classes of ways would be expressibletoo, piddlingly. But of course, we should remember who and what we are talkingabout here. This is Wittgenstein, student of Frege and Russell. Frege?s writingsthroughout his career are filled with amusing screeds about the purported con-cept of ?aggregates.? Similarly, Russell?s response to the inconsistency of na?vecomprehension was the opposite of the now-standard response?he felt that theunderlying problem was with the notion of a class-as-one-in-extension, an alleged?unity constituted by a many?. Russell thus retreats to classes-in-intension, find-ing logical safety in constraints on propositional structure.97 Wittgenstein inher-ited this suspicion from his teachers, and, like Russell, took the prior and clearernotion to be that of a class-in-intension, the class as characterized by its definingproperty. Consequently, we just cannot take the notion of set for granted here.95By Koenig?s lemma.96A basic problem with this suggestion is that it leaves completely obscure how a speaker com-municates to their audience which collection has been chosen.97Thanks to Ori Simchen here.101The control of infinite multiplicities must be exerted in some other way.It is at 5.501 that Wittgenstein attempts to fill the gap, by designating threeWays to specify the elements of the basis of an application of a truth-operation.The first Way is just to point out elements of the basis directly or list them your-self, but according to Wittgenstein this works only if the number of elements ofthe basis is finite. The second Way is Wittgenstein?s route to what ought to pass,in the wilds of the Tractatus, for what we should call quantificational generality.If the second Way leads into the mud, then that is where Wittgenstein?s analysisof quantificational expression leads.982.3 FogelinRobert Fogelin argued, first in his (1976) book on Wittgenstein, that Wittgen-stein?s analysis of quantificational expression does lead into the mud. Despitethe rejoinders by Geach (1981) and Soames (1983), he stuck to his guns in thesecond (1986) edition. Now, let me first be clear: his conclusion is not cred-ible. Nonetheless, I?m going to be somewhat patient, even charitable, towardFogelin?s discussion. The reason is that Fogelin?s skepticism gives him a motiva-tion to point out some peculiarities of the text which do not seem so interestingto readers with a more bullish attitude.Fogelin opens his objection with a summary of the T6 presentation of thegeneral propositional form. Then, however, he continues:It seems that every proposition that can be constructed using therecipe in proposition 6 will admit of a decision procedure, i.e., we willbe able to determine in finitely many steps whether it is a tautology,contingency, or contradiction. . . .According to Fogelin, on the one hand it is a basic epistemological commitmentof the Tractatus that it always be effectively decidable whether or not this or thatlogical relationship holds between propositions.99 Hence, according to Fogelin,it is Wittgenstein?s aim at T6 to construct an apparently quite general recipe for98There is a third Way too, which I?ll ignore here, but consider in chapters 3-5.99Frascolla (2007, 141ff) agrees with this assessment.102propositional expression, such that the obtaining of any logical connections be-tween propositions so expressible can be decided by an algorithm. Since, how-ever, first-order logical validity is not decidable, Wittgenstein?s aim at T6 is in-compatible with the acknowledgment of all first-order expressive possibilities.So, since Fogelin finds Wittgenstein?s account of propositional construction tobe somewhat vague and underarticulated in the first place, he ventures as a philo-sophical interpreter to refine Wittgenstein?s account in such a way that the classof acknowledged propositions yields a decidable notion of validity.Nonetheless, as Fogelin allows, Wittgenstein did not know that first-orderlogic was undecidable, and Wittgenstein fully intended at T6 to give an accountof all first-order logical structure. However, according to Fogelin, since T6 ul-timately acknowledges only a decidable fragment of first-order logic, this ac-count must be defective. Fogelin then proceeds to try to diagnose the confu-sion concretely. The diagnosis begins with a fairly natural and I think textuallywell-grounded reconstruction. Accordingly, Wittgenstein describes a system ofpropositional signs, and then claims that the propositions are what is expressedthose signs. The system of propositional signs is defined inductively in more orless the following way. An elementary proposition has its own primitively givensign; a propositional sign is also the result of prepending the fourteenth capitalletter of the English alphabet to the enclosure in parentheses of (or, for short, theEn of) a so-called propositional variable.100 In turn, the propositional variablesare an auxiliary class of signs defined by induction simultaneously with the classof propositional signs as follows. A list of one or more propositional signs, or ab-breviation thereof, is itself a propositional variable (a list); and what results from apropositional sign by replacing one of its constituent names with a variable nameis a propositional variable (a prototype). We now define by induction the sense(or truth-condition) of each such sign. The sense of the sign for an elementaryproposition is that proposition; and the sense of the En of a propositional vari-able is the joint denial of the values of that variable. The values of a list-variableare the propositions expressed by the signs listed; and the values of a prototype-variable are the propositions expressed by the results of replacing its constituent100The Enning of a formula is supposed to correspond to an attachment of the sign of joint denialto that formula.103variable name with a constant name.The Fogelinite now proceeds to identify a flaw in this framework. First, it isstraightforward to show by the construction of signs that(1) the En of the En of the result of replacing ?Anna? with ex in ?Anna lovesBob?is a propositional sign. Moreover, by the fixing of sense, it follows that (1) ex-presses the denial of the joint denial of the senses of the results of substituting aconstant name for ex in the result of substituting ex for ?Anna? in ?Anna lovesBob?.101 But on the other hand, consider(2) The En of the result of replacing ?Anna? with ex in the En of ?Anna lovesBob?.Clearly by the construction of signs this too is a propositional sign. Moreover, bythe fixing of sense, it follows that (2) expresses the joint denial of each denial of thesense of a result of substituting a constant name for ex in the result of substitutingex for ?Anna? in ?Anna loves Bob?.102 So, more casually speaking, it follows that(1) says that somebody loves Bob, but that (2) says that everybody loves Bob.Thus, (1) and (2) say different things. But a straightforward syntactical argumentshows that (1) and (2) are the same expression. So, in the account of logic juststated, to say that somebody loves Bob just is to say that everybody loves Bob. Inother words, this account does not respect some reasonably familiar distinctionsof first-order logic.Now, we have just seen that the account as stated here is actually incoherent.More specifically, according to Fogelin, it breaks under the strain of trying toaccommodate all first-order expressiveness while maintaining the possibility ofsome unspecified decision procedure for logical truth. Of course, at this pointthe interpreter might just stop here. But, Fogelin, acting in his rights as a philo-sophical interpreter, proposes to restore coherence by a minor adjustment. Theroot of the problem is that some signs have multiple formation histories, and the101|En(En([a/x]Rab ))| = j d (|En([a/x]Rab )|) = j d ( j d (|[a/x]Rab |)) = j d ( j d (|Rcb | : c ?U )).102|En([a/x](En(Rab )))| = j d (|[a/x]En(Rab )|) = j d (|En(Rcb )| : c ? U ) = j d ( j d (|Rcb |) :c ?U ).104interpretations generated by the different histories diverge. The fix is then just tocut down on the histories. Fogelin cites hereWittgenstein?s apparent explanationthat when the denial sign is attached to an open formula, the result is to expressthe denial of each of the instances of that formula.103 So, he infers, any resultof attaching a denial operator to an open formula should actually be regarded asa closed formula. Thus, the analysis (2) above violates Wittgenstein?s explana-tion, because it purports to find an subformula which is open, despite the factthat this subformula is the result of attaching the denial sign to a formula. Moregenerally, then, Fogelin?s resolution is that an open formula is always consideredto be closed by the innermost denial sign in whose scope it occurs. In this way,the Fogelinite settles on the conclusion that Wittgenstein?s account of logic is?expressively incomplete?. Although this looks unfortunate, we have have inci-dentally restored the metalogical coherence of Wittgenstein?s position, since theresulting expressible fragment of first-order logic is actually decidable.As I said, there is some appearance of textual ground for this reconstruc-tion in the text, especially with regard to what is most crucial for the accountof first-order concepts, namely the role of propositional prototypes.104 In par-ticular, note that in contrast to now-standard inductive presentations of first-order syntax, propositional prototypes are not introduced alongside elementarypropositions by the base clause of the induction. Rather, a prototype must beconstructed by a syntactical transformation on a proposition constructed an-tecedently. So, for example, on the now-standard approach, to find a way to saythat somebody loves Bob, we begin with the result of substituting ex for ?Anna?in ?Anna loves Bob?, which may in turn be explained as the combination under apredicate of two terms, one constant and one variable. To this expression we thenprepend further logical expressions, thereby forming a negated existential gener-alization with respect to the letter ex. In contrast, on the Fogelinite approach,one begins with the expression of some elementary proposition, say ?Anna lovesBob?, constructs a propositional variable by replacing the word ?Anna? with the103I?ll argue in 2.6 that this appearance trades on a use-mention confusion.104However, I?ll argue in 2.6 that Fogelin?s reconstruction trades on a use-mention confusion.Thus, Wittgenstein doesn?t describe attaching the En-operator to an open formula, but instead de-scribes applying joint denial to the range of values of a propositional function. A similar allegationof use-mention confusion appears in Miller III (1995).105letter ex, and finally forms the En of this variable.What the Tractatus itself says about the source of prototypes appears in anearly passage:Only propositions have a sense; only in the nexus of a propositiondoes a name have meaning. 3.3I call any part of a proposition that characterizes its sense an expres-sion. 3.31An expression presupposes the forms of all the propositions in whichit can occur. It is the common characteristic mark of a class of propo-sitions. 3.311It is therefore presented by means of the general form of the propo-sitions that it characterizes.In fact, in this form the expression will be constant and everythingelse variable. 3.312Thus, an expression is presented by means of a variable whose valuesare the propositions that contain the expression.(In the limiting case the variable becomes a constant, the expressionbecomes a proposition.)I call such a variable a ?propositional variable?. 3.313An expression has meaning only in a proposition. All variables canbe construed as propositional variables. (Even variable names.) 3.314If we turn a constituent of a proposition into a variable, there is aclass of propositions all of which are values of the resulting variableproposition. [. . . ] 3.315.This passage is subtle and rich, even by Tractatus standards. But the thrust isclear. Proposition 3.3 rejects an account of the sense of propositional signs whichwould assume nonpropositional signs to have their meaning fixed independently.The 3.31s elaborate on this rejection, by developing a replacement for the con-cept of independently meaningful names. This replacement is the concept of anexpression. Consider, first, the result of replacing ?Anna? with ex in ?Anna lovesBob?. Such a sign is a propositional variable, and it has no sense of its own.Rather, Wittgenstein holds, it serves to indicate several propositions by present-106ing their commonmark, which is the so-called expression. Thus, the whole pointof this stipulated concept of expression is to differ from the concept of a nameprecisely in manifestly explanatorily following, rather than apparently explana-torily preceding, the concept of proposition. Now, the general propositionalform surveys the totality of propositions as the totality of what can be expressedby propositional signs. The Fogelinite therefore finds it only natural that theinternal structure of the general propositional form should reflect the order ofexplanation which Wittgenstein advocates. Under the natural internal orderingexhibited through the general propositional form, a propositional variable doesnot arise in parallel with the propositional signs which express its values, but isconstructed by means of them.2.4 Geach-SoamesFogelin seems to suggest that one can tell a priori that something went wrongwith Wittgenstein?s analysis of first-order logic, because Wittgenstein thoughtthat logical truth is decidable. In particular, Fogelin concludes, Wittgensteinmust have just confused himself about the elementary character of variable-binding.Now, in response to Fogelin, many people have pointed out that such confusionwould be surprising in light of 4.0411, where Wittgenstein analyzes defective at-tempts to express quantificational generality, including an attempt with preciselythe defect that Fogelin purports to identify. But, Fogelin might say, Wittgensteintook his eye off the ball, because he was trying to ensure decidability. Still, youmight wonder: where does this kind of line stop? Should we also conclude that,say, Emil Post did not understand the basic formal patterns of first-order logicbecause he presumed that the Entscheidungsproblem was solvable? Of course Postunderstood the formal patterns of quantificational logic, and it is well-establishedthat Post spent many years seeking a decision procedure for truth in the system ofPrincipia (Davis 1965, 338). As for Wittgenstein, it seems fair to say that the sortof incompetence Fogelin alleges might have wearied Ramsey on his 1923 visit toWittgenstein in Austria, where the two of them discussed the logic of the Tracta-tus five hours a day for two weeks.105 Geach and Soames complain that the whole105Monk, 212ff; this point is emphasized in Rogers and Wehmeier 2012).107issue of decidability is a red herring, corrupting not Wittgenstein?s treatment ofquantification but Fogelin?s reconstruction of it.According to Geach and Soames, the Tractatus actually does contain sufficientresources for analysis of first-order generality. These resources lie very close tothe surface of the account of joint denial. For, joint denial has been explained byWittgenstein in such a way as to apply to arbitrary classes of propositions. How-ever, in the Tractatus framework, an ordinary quantified formula can be inter-preted as expressing the result of some truth-operation on a class of propositions.This is because, roughly speaking, the Tractatus presupposes a ?fixed-domain se-mantics?, according which the associated definition of logical consequence re-stricts consideration to structures with a common domain. For example, theclosed universal generalization of some formula with respect to some variableterm can be interpreted to express the joint affirmation of the propositions ex-pressed by each closed instance of that formula. Just as universal and existentialquantifiers are interdefinable via negation, so we might introduce a third quanti-fier, also interdefinable with those two, call it the En-quantifier, and attribute thisto Wittgenstein. Having introduced the En-quantifier, and continuing to allowthe secondary use of En to express finitarily-based joint denial, then it is obvi-ous that the resulting notational system is pretty much indistinguishable from astandard first-order system.So, according to Geach and Soames, it was Fogelin?s error to take Wittgen-stein?s informal talk of ?turning a constant name into a variable? to be a concretenotational proposal. The point of that talk was rather just to gesture toward onekind of multiplicity which can appear as basis of the denial operation. Nonethe-less, the talk does license the construction of a notational system which would beexpressively adequate. Such a system must, however, contain not just a sign forjoint denial, but also a sign to indicate just where in a formation tree a free vari-able becomes bound. Geach treats the variable-binding device as an inflection ofthe denial sign. Soames arranges his syntax under an analogous constraint, so thata variable-binding device appears in the construction of a formula only when it isapplied to a formula and this application is itself followed by another applicationof the denial sign.Geach and Soames both hold that Wittgenstein?s informal talk can actually108be cashed out in a concrete notation, but that insodoing, the talk should not betaken too literally. Here are the details. An atomic formula consists of a predicatefollowed by a sequence (with appropriate length) of either constants or variables.Now, every atomic formula is a formula, and the result of prefixing the denialsign to a set-representative is a formula. The result of enclosing in parenthesessome finite list of formulas is a set-representative, and the result of prefixing avariable to a formula is a set-representative, this prefix being understood to bindall unbound occurrences of the variable in that formula. A propositional sign isa closed formula.106Geach doesn?t explain systematically the determination of the sense of propo-sitional signs so constructed, presumably relying instead on the obvious possibil-ity of extending to this situation any ordinary approach to first-order semantics.Soames, however, does construct an explicit semantics. He defines, by simultane-ous induction on complexity of closed formulas and of closed set representatives,the sense expressed by the closed formula and the class of propositions corre-sponding to the set representative. This all goes as one would expect. The senseof the result of attaching a denial sign to a set representative is the joint denialof the propositions belonging to the denoted set. The propositions correspond-ing to a parenthesized list of propositional signs are the propositions expressedby the signs listed; and the propositions corresponding to the result of prefixinga variable to a formula are the propositions expressed by the formula?s closedinstances.107It seems to me that the aims of Geach and Soames are somewhat ambiguous.On the one hand, Wittgenstein himself tries to present a general propositionalform. This means, in particular, answering the question which classes of setsof elementary propositions correspond to truth-conditions of propositions. Thestrategy, at T6, is to answer this question by something like an inductive def-inition, according to which nonelementary propositions are constructed fromelementary propositions, by repeatedly applying joint denial to specifiable mul-tiplicities of propositions already constructed. Wittgenstein?s attempt to give aninductive definition of the totality of sense is important in its own right from106This version is due to Soames; I think it traces the text a little more tightly.107Soames presumes, as I do, that every object has a name.109the point of view of history of logic. No such attempt is even conceivable fromwithin the frameworks of his teachers Frege or Russell, for example.108 So, it isworth trying to understand how its details actually worked.Now, it may be that Geach and Soames propose, with their notational re-forms, to reconstruct Wittgenstein?s attempt to say which senses exist. On thisview, Wittgenstein?s attempt to state the general propositional form requires anotational amendment because Wittgenstein?s strategy implicates a certain nota-tion essentially. That is, Wittgenstein?s strategy would take two stages. First, hewould define a class of notations, or indices, which would be inductively gener-ated by constructive processes from the immediately given elementary proposi-tions. Second, he would define, by induction on syntactical complexity of theclass of indices, the class of sets of elementary propositions which correspondsto the truth-condition determined by an index. The totality of possible senseswould then be the totality of truth-conditions which are determined in this wayby indices.109On this interpretation of Wittgenstein?s strategy, a historically importantquestion arises: what actually is the class of notations which Wittgenstein putforward as the indices of all possible senses? How did Wittgenstein himself tryto define this class? Geach and Soames may be understood to propose that theclass of notations identified by Wittgenstein is loosely isomorphic to the class offormulas of a first-order language.110 On this interpretation of the Geach-Soamesprojects, Wittgenstein?s notational proposals are essential to his articulation ofthe general propositional form. However, although Wittgenstein?s strategy, sounderstood, does require constructing a definite class of notations for the possi-ble senses, his execution of the strategy is either incomplete or defective. In par-ticular, Wittgenstein?s inductive definition of the class of notations provides forturning a name into a name-variable in a notation already constructed. But, this108That is, Frege and Russell develop no analysis of the notion of truth-condition in terms of thenotion of truth-possibilities for elementary propositions, and so cannot form the general questionwhich classes of truth-possibilities correspond to truth-conditions of propositions.109Such a procedure is analogous to the definition of hyperarithmetic sets, as, for example inShoenfield (1967), 167ff.110Geach does acknowledge Way 3 variables, and sketches a way to deploy them along the linesof 4.1252.110provision does not yield the mathematical multiplicity required to express dis-tinctions of scope. If Wittgenstein did pursue such a notation-first strategy, thenthe oversight in his construction would seem rather peculiar. It seems, after all,fairly obvious that, in constructing a statement that all values of a certain propo-sitional function are false, one needs to say which function it is whose valuesare being considered. Such an oversight seems especially puzzling after Wittgen-stein?s lectures about the importance of finding a notation that is logically correct(3.325, 4.0411).On the other hand, another way of reading Geach or Soames is to take themto be reformulating interpreted first-order logic in such a way that it is then rea-sonably clear how its senses might be accommodated under the general proposi-tional form. That is, they would be observing that every closed first-order for-mula can be interpreted as a joint denial of the senses of some closed first-orderformulas of lower syntactical complexity. But then, it would be nice to knowhow the general propositional form itself is supposed to work. That is, con-sider an arbitrary propositional sign generated by the syntactical induction inSoames. We are told by Soames how the sense of such a sign is to be determined.But, how do we know that something actually answers to this predicted senseon Wittgenstein?s account? Must we just presuppose that Wittgenstein takes forgranted some kind of second-order quantification over elementary propositions?But then aren?t we just back to the platonist dream that every set of proposi-tions has a joint denial? In other words: how are we supposed to understand thisapparent use of second-order quantification in T6?2.5 Wehmeier and RickettsAlthough the papers just discussed are very nice, there is another interpretiveproblem that they pass over in silence. Wittgenstein does not just replace the or-dinary universal and existential quantifiers with a joint-denial quantifier. Rather,actually he develops a dramatic revision of the logic of generality. To bring outthe basic idea, notice that when Carol says that everybody loves Bob, then whatshe says entails that Bob loves Bob. That is, classically speaking, the variablename implicitly bound in Carol?s expression ranges over all people, over Bob in111particular. Wittgenstein, however, thinks that the proposition that everybodyloves Bob does not entail that Bob loves Bob. It entails only that, as we wouldput it, everybody but Bob loves Bob, and as for Bob himself, well, he may or maynot. More generally, the thought is that a variable ranges over all objects exceptfor those which are mentioned in its scope. Such variables are sometimes referredto as ?sharp?.111 Now, it turns out that when variables are read sharply, then theequality predicate becomes redundant: that is, whatever can be said with equalityand normal variables you can also say without identity but with sharp variables.Although this idea of Wittgenstein?s seems bizarre, it is not too hard to workout. Wittgenstein persuaded Ramsey that it worked in theory even if it is notespecially practical. The idea also showed up occasionally in the history of thesearch for solvable cases of the decision problem for validity of first-order for-mulas.112 Hintikka seems to have picked up the idea in the 1950s from Geachor perhaps von Wright, and produced the important (1956). More recently KaiWehmeier has published on this topic extensively (2012 with Rogers, also 2012,2008, 2004).Somewhat more precisely, the claim of expressive equivalence between first-order logic with equality and sharp first-order logic without equality is moreprecisely this. Consider a purely relational first-order signature,113 and now, forsome arbitrary nonempty set, consider the class of first-order structures over thissignature which have that set as domain. Call such a class a frame. Satisfactionconcepts for each of the two logics being defined as above, the sense of a closedformula with respect to a frame is the class of structures in the frame with respectto which the formula is true. Then, relative to any frame at all, every sense ofa closed formula of the language of first-order logic with equality over a purelyrelational signature is the sense of a closed formula in the language, over the samesignature, of sharp first-order logic without equality.114 This is so even thoughsome open formulas in the first system determine functions from assignments111Thanks to Goldfarb for noting this.112See in particular G?del (2002, 406 and 570; (thanks to Goldfarb for this reference)), and, speakof the devil, Geach and von Wright (1952).113I.e., containing no constant or function symbols.114See 5.2 for details. The first proof of this result appears in Hintikka (1956); a more elegantversion appears in Wehmeier (2004).112to senses which no formulas determine in the second system?for example, anequation between distinct variables.The reader may now be wondering why Wittgenstein thought it was a goodidea to depart in this way from the basic idea of Frege and Russell. For example,was he just not paying attention when his logic teacher was explaining the ba-sics? Many logic students often initially fail to appreciate that leading existentialquantifiers can be witnessed by the same constant.Actually, this question has a surprisingly good answer. The standard view ofcommentators is that Wittgenstein is driven to this reinterpretation of generalityfrom an unexpected place. Recall that a fundamental doctrine of the Tractatus isthat a proposition is a truth-function of elementary propositions. This means, atleast, that the truth-values of elementary propositions determine the truth-valueof every proposition.But there is really big snarl around the equality predicate. Perhaps, with asentence like ?Alice is Alice? or ?Alice is Bob? there is some feeling of languagegone on holiday (to say nothing of Hesperus and Phosphorus), and the Tractatusmight just dump such expressions as merely apparent. But equality appears inmany sentences where language is hard at work, for example as in ?Alice lovesat least two people?; the sense of this sentence cannot be expressed in first-orderlogic without equality?that is, in the system that Geach and Soames had gottenpretty close to helping Wittgenstein to accommodate.In response one might just wave a white flag and allow that the equality pred-icate may appear in propositions that are elementary. But this is out of the fryingpan and into the fire. For every elementary proposition has possibilities bothof truth and of falsehood. So the surrender leads to the even worse result that aproposition with respect to some object that it is the same as itself can be false.115There are various other stories in the literature about the way in which the equal-ity predicate presents a problem for basic Tractatus doctrines, but nobody seemsto disagree that it is a problem.115Michael Potter (2009, 290) says that a genuine relation of identity between objects would haveto hold contingently, because objects are simple. Presumably this is on the grounds that if objectsare simple and if identity is a genuine relation, then propositions to the effect that objects areidentical must be elementary.113Now our surprisingly good to the question why Wittgenstein adopts the ex-clusive reading of the variable is this. It so happens that, under some proviso,everything that can be said by means of standard first-order logic plus equalitycan be said by means of sharp first-order logic without equality. Thus, at least upto the proviso, and provided you are willing to swallow the reinterpretation, theequality predicate is redundant. For example, if you want to say that Alice lovesat least two people, you just follow the bad logic student: ?there is an ex and awhy such that Alice loves ex and Alice loves why.?To summarize, on an amended standard account, Wittgenstein revises the se-mantics of first-order logic with equality by restricting the space of assignmentsto those which are injective. This revision, while dramatic, is also well-motivated.For, the expressiveness apparently afforded by the equality predicate presents abig problem for the truth-functionality thesis. Upon restricting the assignmentspace, the equality predicate becomes contextually eliminable, under the provisothat the language contains no constants. In this way, a seemingly unrelated ide-ological fiasco over the equality predicate finds its resolution in Wittgenstein?stweak of the logic of generality.One cause for concern is just that Wittgenstein, as a historico-philosophicalpersonality, seems to have been highly averse to anything of an ad hoc or arbitraryflavor.116 Now, I would not wish to imply that this ingenious technical maneu-ver would be ad hoc by the standards of contemporary analytic metaphysics. Farfrom it. But, it might be ad hoc by the standards of Wittgenstein. And, here, weare doing Wittgenstein-interpretation, not contemporary analytic metaphysics.So there may also be some grounds to worry that, at least by Wittgenstein?s ownstandards, the sharp reading of the quantifiers is a flaw in the Tractatus. It ap-pears to complicate the program to develop a philosophical understanding of therelationship between a generalization and its instances.Ricketts (2012) acknowledges that Wittgenstein takes as a crucial philosoph-ical problem to understand the link between a generalization and its instances.Hence Ricketts quite rightly finds Wittgenstein?s account of quantificational gen-erality to be rooted in the 3.31s, which immediately follow the programmatic116In particular, this sensibility seems to animate some of his nastier mentions of Russell.114endorsement of the context principle. However, Ricketts remarks that Wittgen-stein?s account is ?in its way Fregean?, referring specifically to Frege?s explanationof the function-argument segmentation in 2.9 of Begriffsschrift. Later on, Rickettsraises the problem of understanding from a Tractatus point of view the quantifica-tion into results of truth-operations on elementary propositions, and maintainsthat Wittgenstein approaches the issue from a Fregean direction.Suppose we replace a name in an elementary sentence that occurswithin the representation of the construction of a particular truth-function of that elementary sentence. The elementary sentence-functionin the formula collects together its values so that the entire formulacollects together expressions of that particular truth-function of val-ues of the elementary sentence-function. (Ricketts 2012).I?m sympathetic to Ricketts aim to see how Wittgenstein himself understoodthe notion of propositional function, and also to the thought that Wittgenstein?sunderstanding was somehow indebted to Frege (or Russell). However, it doesseem that Wittgenstein?s explanation differs from Frege?s in a way that Rickettsdoes not record. Frege?s explanation is this:If, in an expression [. . . ], a simple or a compound sign has one ormore occurrences and if we regard that sign as replaceable in all orsome of these occurrences by something else (but everywhere by thesame thing), then we call the part that remains invariant in the ex-pression a function, and the replaceable part the argument of thefunction. (Frege 1967, 22)On Frege?s account, then, a sign which occurs in a sentence may be regardedas replaceable by other signs. Frege?s informal examples make clear that by re-placement he means paradigmatically, replacement of a name by a name. It isthus the replaceability of a name with another name in a sentence that underpinsthe segmentation of the expression into function and argument. On this con-ception, the values of the function are naturally those sentences which result byreplacing that name with another name. Note, in particular, that Frege?s formu-lation explicitly allows for the possibility that, when an sentence contains many115occurrences of a single sign, then we may distinguish from among the totality ofoccurrences some subcollection to be regarded as variable in contrast to the otheroccurrences which remain considered fixed. Thus, Frege explicitly intends thata name may be regarded as replaceable in just one fixed one of its occurrencesin a sentence ranges two occurrences of that name. The resulting function thusindifferently encompasses values which contain two occurrences of that name,alongside values which contain one occurrence of that name and one occurrenceof another name. In other words, the range of a function may in general con-sist of sentences containing varying numbers of different names. As one wouldexpect, this conception is entirely consistent with normal mathematical practice.Let me now summarize Wittgenstein?s own account in a way that should beuncontroversial. As Ricketts says, Wittgenstein agrees with Frege that one canbegin with any whole proposition. Now, according to Wittgenstein, we maytake a name which occurs in the proposition, and turn this name into a variable.By so turning a name into a variable, one obtains a propositional function. Thepropositional function presents what Wittgenstein calls an Ausdr?ck. The Aus-dr?ck is the common mark of the sense of a multiplicity of propositions. Thevalues of the propositional function are the propositions whose sense is markedby the Ausdr?ck which the propositional function presents.There are two differences between the accounts of Wittgenstein and Frege.Frege?s explanation begins by introducing the notion of replacement of a namein a proposition with another name. In contrast, Wittgenstein does not talkabout replacing names with names. Rather, he talks about turning names intovariables. It is by turning names into variables that one obtains the presentationof a common mark of the sense of propositions, an Ausdr?ck. Second, Frege?saccount explicitly allows for the possibility of distinguishing among many occur-rences of a single name some particular occurrences to be regarded as replaceable.Wittgenstein does not explicitly make such allowance. These two differenceshave an important consequence. Frege allows that a function may take as its val-ues sentences containing variable numbers of names. However, Wittgenstein?saccount precludes such variability. Any propositional function constructible onWittgenstein?s account takes as its range a multiplicity of propositions which allcontain the same number of names.116After claiming thatWittgenstein?s account of functions basically derives fromFrege, Ricketts then introduces a scope-indicating device following Geach. Fi-nally, he concludes, ?there is a complication here that deserves mention??the in-terpretation of propositional variables ?in conformity with Wittgenstein?s viewson identity? (20012, 12). On Ricketts? Fregean explanation of the Tractatus ac-count of propositional functions, a function collects together all results of re-placing the considered-replaceable occurrences of a name with another name.Ricketts must then agree with the assessment of Rogers and Wehmeier that theTractatus account is defective. For, as he acknowledges, Wittgenstein?s views onidentity demand that the result of combining the sign of denial with a sign of apropositional function must be the denial of those values of the function whichresult from assigning the variables free in the function sign values which do notalready appear as objects mentioned in the function sign. So, Ricketts must findthat although Wittgenstein shares with Frege a project to explain the relationshipbetween a generalization and its instances, Wittgenstein?s own attempted expla-nation seems marred by an ad hoc maneuver required to eliminate the equalitypredicate contextually. It might then be doubted whether Wittgenstein?s own at-tempt to explain the relationship between a generalization and its instances couldbe seen as successful.117So, it seems to me that the question why Wittgenstein reconstrues the work-ings of name-variables does have an a priori surprisingly good answer. But, thisanswer, at least insofar as it is developed in the literature, retains some interpre-tive and philosophical deficiencies. In particular, as I?ve argued, on this reading,Wittgenstein?s commitment to the independence of elementary propositions ul-timately undermines his attempt to explain the relationship between a general-ization and its instances. For, the independence of elementary propositions rulesout an equality predicate. The ensuing adjustments to the concept of generality117Ricketts avoids attributing to Wittgenstein any distinction between occurrences of a name inan elementary proposition. To this extent, then, he does find a departure in Wittgenstein fromFrege. But Ricketts does allow Wittgenstein to distinguish between occurrences of a name in anonelementary proposition. Ricketts finds in the Tractatus a sharp distinction between ?the pictur-ing structure of elementary sentences and the iterative structure of portrayals of truth-functions.?In contrast, I hold that such a distinction cannot be reconciled with Wittgenstein?s treatment ofgenerality. See 2.6.117do not square with normal mathematical practice. In fact, ?normal? is an under-statement. It is probably safe to say that no mathematician in recorded historyhas seen the need for five different laws to express the associativity of multipli-cation, depending purely on the pattern of identity and distinctness among thearguments of the corresponding instances.118 The profundity of this aberrationmakes it hard to take Wittgenstein?s approach seriously. Even if one can techni-cally make sense of the variant system of logic, it can?t possibly be rooted in astable conception of representational structure. In particular, the concept of anAusdr?ck looks like a little greasepot of equivocation.2.6 PictorialityLet me begin by setting aside for a moment the questions of 2.5, and return to aquestion that lingers from 2.4. Did Wittgenstein really overlook the need distin-guishing the possibilities of scope of a quantifier?Recall that Fogelin?s puzzle addresses the relationship between(1) the En of the En of the result of replacing ?Anna? with ex in ?Anna lovesBob?(2) The En of the result of replacing ?Anna? with ex in the En of ?Anna lovesBob?.In (1) a formula is presented so as to seem towant to say that somebody loves Bob.In (2) a formula is presented as though it wants to say something else, namelythat everybody loves Bob. And yet (1) and (2) present the very same formula.So something has gone wrong. But what? Fogelin infers that the phrasing of(2) somehow tricks us into thinking that the formula says something other thanwhat it does say. Is that really the right diagnosis of the situation? That (1) and(2) cannot say different things, because they are the same formula? That formula(2) just tragically wants to say what it can?t say? Believe it or not, it is possible tobe even more pedantic. Consider(1a) ?Anna loves Bob?118Let alone, say, a physicist who requires several reformulations of Newton?s laws of motion.118(1b) The result of replacing ?Anna? with ex in (1a).(1c) The En of (1b).(1d) The En of (1c).(2a) ?Anna loves Bob?(2b) The En of (2a).(2c) The result of replacing ?Anna? with ex in (2b).(2d) The En of (2c).Clearly, so it seems, the last entries of the two sequences must say the same thing,because they too are the same formula. Is that really clear? I think that the answeris no. Our original na?ve and allegedly incoherent Fogelinite semantics, profferedonly as a diagnosis of confusion, was actually perfectly sound. To see this, recallthe picture from 2.2 of an allegedlymaximal conversation. The one sequencemaybe a single strand of this conversation, and the other sequence may be anotherstrand. Of course, you have to actually follow the dynamics of the conversationin order to make out these implicit connections between the utterances, whichare conveyed, in our metalinguistic report of the conversation, by nesting thelabel of one utterance in the description of the next utterance. In other words,the last entries of the two sequences are intrinsically indistinguishable, but whatmatters in logic is not intrinsic structure of any one thing that is said but theinternal relations between all that is said.We have now reached what so far as I know is a novel account of a system ofpropositional notations for the Tractatus. This system has exactly the formationrules predicted by Fogelin. However, the semantics would run not on individ-ual propositional signs, but rather on trees of signs which record conversationalorigins. Although intrinsically indistinguishable signs may be assigned differentmeanings, these differences will always derive from differences in the underlyingtrees with respect to which the different interpretations were determined. Suchan approach can readily be seen to recover the power of the Wehmeier-Soames119hybrid while avoiding adventitious postulation of a special scope-indicating de-vice.As many commentators have urged in response to Fogelin, Wittgenstein didnot himself single out any particular notational system for a special role in thebasic program. So, these commentators continued, there is no reason to supposeWittgenstein had any particular commitment to the system of signs that Fogelinascribes and indicts. I agree. We are essentially arguing about the form of wordswhich participants in amaximal conversationmust use in constructing a complexdemonstration of a basis of for joint denial. It is not clear, a priori, why these peo-ple should have to talk one way or another. What is the purpose of the conceptof the maximal conversation? I.e., of the general propositional form? That is adeep question I do not propose to settle here. However, one might notice that,in the relationalist sign-system just described, the only role for syntactical struc-ture of a noninitial notation is to indicate the means by which it immediatelyarises from its predecessors on a tree. Every part of the structure of a noninitialnotation which does not contribute to this role is logically inessential and reallyjust a distraction. There is no good reason at all why the last entries of the twosequences described above have the same internal structure.Nonetheless, as I argued in 2.3, it looks like there is a pretty good textual foun-dation for identifying a Fogelinite syntax in Tractatus. For, so it seems, the 3.31stalk about turning a name into a variable name to produce a propositional vari-able. And likewise, 5.501 explains that attaching an expression of joint denial to avariable name produces the joint denial of the values of that variable. I think thatFogelin has committed here a natural confusion of use and mention. The 3.31sdo not describe turning a name into a variable in a propositional sign. Rather,they describe turning a name into a variable in a proposition, which yields apropositional function. Likewise, 5.501 talks about forming the joint denial ofthe elements of the range of a propositional function. These are movements inthe realm of meaning?although, as I?ve emphasized, to speak of movement hereis just to trace metaphorically the internal relations between senses which arethemselves unchanging.Thus, the relevant point of the 3.31s is not that in the allegedly maximal con-versation, the result of replacing ?Anna? with a variable in a sentence is a form of120words that a speaker uses. Rather, it is that the result of replacing Anna with avariable is a form of sense that a speaker means. What words might we supposea speaker to utter in meaning such a thing? I give a formal syntax and semanticselsewhere, and my present concern is with interpretive justification. For presentpurposes, we may simply suppose that a speaker says, to express the joint de-nial of the values of the propositional function obtained by replacing Anna witha variable in a proposition already expressed by a speaker, something like this:?nothing is as you say Anna is.? In accordance with the program announced atT4.5, Wittgenstein?s analysis of this kind of expression involves resources not id-iosyncratic to this or that particular system of signs, but rather only resourcesinternal to the very possibility of a system of signs in general. The apparenteccentricities noted by Fogelin stem from the Wittgenstein?s insight that name-variables are not after all essential to the expression of quantificational generality.Let?s now return to the questions that arose in 2.5-2.6. There we seemed tofind that, setting aside the questions raised by Fogelin, Wittgenstein was driveninto an account of quantificational generality that could not be grounded in anystable account of representational structure. For example, suppose what is saidabout Bob is that Anna loves him. Then, Wittgenstein must find that ?nothingis as you say Anna is? is entirely consistent with Bob?s loving himself. That is,Wittgenstein needs the latter form of words to mean, as we would put it, thatnobody other than Bob loves Bob. Now, our stipulation is that the words usedby the speaker express the joint denial of the values of the propositional functionobtained by replacing Anna with a variable name in the proposition that Annaloves Bob. But isn?t Bob himself a value of this variable name? But then isn?t theproposition that Bob loves Bob a value of the resulting propositional function?Should we just insist that he isn?t? That is capitulation.Our question thus reduces to the question what are the values of a proposi-tional function which results by replacing Anna with a variable in the proposi-tion that Anna loves Bob. I suggest thatWittgenstein?s concept of a propositionalfunction is rooted in the concept of what he calls somewhat stiltedly an ?expres-sion?, a logically meaningful commonality amongst various propositions. Infor-mally, youmay suppose an expression to be what a proposition says of something(or of some things). The key idea is that if the proposition that Bob loves Bob is a121value of the result of replacing Anna with a variable in the proposition that Annaloves Bob, then the proposition that Bob loves Bob says of Bob what the proposi-tion that Anna loves Bob says of Anna. But consider the proposition which saysof Anna what the proposition that Bob loves Bob says of Bob. This propositionsays that Anna loves Anna. On the other hand, the proposition which says ofAnna what the proposition that Anna loves Bob says of Anna is just the propo-sition that Anna loves Bob. And to say that Anna loves Bob is not to say thatAnna loves Anna. Hence, the proposition that Bob loves Bob is not a value ofthe result of replacing Anna with a variable in the proposition that Anna lovesBob. Rather, the values of this expression are precisely those propositions whichresult from it by replacing the variable name with somebody other than Bob. Inother words, I contend that it is Wittgenstein?s concept of an expression whichunderpins the sharp reading of the variable.119But this last characterization is still a little misleading. To bring this out, letme suggest a somewhat vulgar heuristic. The expected way of finding the rangeof a propositional function is just to plug random objects into the place wherethere was a variable. But Wittgenstein?s way of finding the range of a function isto pull an object out of a proposition and see if the result is that function.Now, why can?t we just pull out the ?first? occurrence of Bob in the proposi-tion that Bob loves Bob? The problem lies, I think, in Wittgenstein?s departurefrom Russell?s conception of propositional structure. With respect to Russellianpropositions, one can talk about the position of an object. Consider, for exam-ple, the conjunction that Bob loves Bob and Bob loves Bob. Russell thinks thatthe conjunction sign represents a relation between two propositions: thus, thatthe connective vocabulary expresses relations between propositions in the waythat the amorous vocabulary expresses relations between people. Wittgenstein is119Strictly speaking, I?ll take expressions in the sense of the 3.31s to be notational items. Differentsigns may belong to the same proposition (i.e., may say the same thing); hence different signs maysay the same thing of a given object. The expression associated to a propositional sign by a name isthen a notation for what the sign says of the object. When time comes to get more careful, I?ll notspeak of what a proposition says of an object, but of what a propositional sign says with respect to aname. However, saying-of-an-object is the fundamental notion. The idea is that a proposition saysof something what the proposition that Bob loves Anna says of Bob if and only if it is expressed bysome propositional sign which says with respect to a name of that thing what ?Bob loves Anna?says with respect to ?Bob?. Thanks to Roberta Ballarin for raising this issue.122a philosophical rejectionist but he rejects this as much as anything. Connectivevocabulary does not contribute representational structure. In particular, conjoin-ing a proposition with itself does not take you beyond the original proposition.The duplication is idle, a samesaying (I would use the word ?tautology? here,but that is already taken). But there seems to be no single inevitable identifica-tion of occurrences of Bob in the conjunction with occurrences of Bob in theproposition conjoined. Worse yet, there are more choices of occurrences of Bobin the conjunction than there are in its conjunct. I think that such an apparentproliferation of choices of segmentation is not really grounded, for Wittgenstein,in what the proposition says about Bob. Rather, one might say, it is not onlyan elementary proposition which is a hanging-in-one-another of objects. Everyproposition is such a hanging-together. What we call logical composition doesnot, for Wittgenstein, multiply the ?occurrences? of an object, any more than, ifit turns out that you or me loves both of Bob and Anna, there will all of a suddenturn out to be two ?occurrences? of you or me. Generality, for Wittgenstein, isrooted in the pictorial character of propositions, in which names hang togetheras the objects do. This grounding of generalizability in pictoriality is not, as bothFogelin and Ricketts were in their ownways led to suppose, intelligible only withrespect to those propositions which are elementary. The apparent multiplicity ofoccurrences of an object, conjured by our means of expression, and the attendantappearance of possibilities of meaningful distinctions among those occurrences,is logically speaking an illusion. So, then, is any distinction between the internalstructures of elementary and nonelementary propositions.It is for this reason that by Wittgenstein?s lights, Russell?s way of determin-ing the range of a propositional function is confused. For there are really twodifferent concepts here. One concept is that of the class of propositions to whichan expression is common. I.e., for any object, and any proposition, there is theclass of propositions which say about some object what that proposition saysabout that object. For a proposition to belong to this class is for the expressionto result from it by removal of the object. But then there is another concept,the class of results of ?plugging? an object into the ?gap?, ?in? the propositionalfunction. Here, I suppose, one takes out the binoculars and scans the savannafor candidates?any one will do, that is, if it fits. It is from such perspective that123one must never forget the taboo?perhaps, a taboo on incest, or onanism??thatan object which already occurs in the expression, even if it, so to speak, fits inthe gap, does not belong there. In other words, it is only the image of the rangeof a propositional function as the results of plugging which demands an ad hocrestriction. As I contend, this image is specious. Rather, the range of a functionis to be explained as the class of propositions to which an expression is common.An expression is like a necklace from which one link has been lost. When a linkis torn out of a chain and left on the ground, it doesn?t just so happen to remainelsewhere on the chain.120Let me conclude this section by summarizing what I take to be the basic mo-tivation for an alternative reading of Wittgenstein?s treatment of quantificationalgenerality. It is standardly held that Wittgenstein is driven to the sharp treat-ment of name-variables by an ideological fiasco entangling the equality predicate.This gets the matter backward. As argued in Chapter 1, Wittgenstein?s basic con-victions about the nature of logic stimulate the development of a conception ofpropositions as pictures of reality. And as I?ve just argued now, the conception ofpropositions as pictures demands the exclusive reading of name-variables. Thus,the redundancy of the equality predicate is a corollary to a deeper insight.2.7 The pictureAfter all of this harangue, let me finally sketch in crudest outline my own under-standing of Wittgenstein?s conception of quantification.Given any proposition and any object we may ask what the proposition saysabout that object. What a proposition says about an object is, in Wittgenstein?sstipulative sense, an expression. Thus, the proposition that Bob loves Anna saysof Bob that he loves Anna, and the proposition that Bob loves Bob says of himthat he loves himself. This concept of expression extends in an obvious way toarbitrary finite sequences of names. In general, we expect that what a propositionsays about some objects depends on the order in which the objects are presented120I said Russell, rather than Frege, because Frege?s account does not suffer the same asymmetrybetween abstraction and instantiation. Wittgenstein?s departure from Frege needs independentjustification in terms of the pictorial conception of representational structure.124outside the proposition. For example, what the proposition that Bob loves Annasays of Bob and Anna isn?t what it says of Anna and Bob, though it may be whatthe proposition that Anna loves Bob says of Anna and Bob. Of course, we canalso ask of the proposition that Bob loves Anna what it says of Carol, and ofcourse the answer is that it says of Carol that Bob loves Anna.Wittgenstein?s concept of an expression leads straightforwardly to an accountof quantificational generality. Given an arbitrary proposition and name, therearises an expression, what the proposition says about the object named. An ex-pression is thus a logically distinct part of the meaning of the given proposition,which the given proposition may have in common with other propositions. Theexpression marks the sense of another proposition as well, provided that there issome, presumably distinct, object, such that what the second proposition says ofthe second object is what the first proposition says of the first object. Considernow an expression determined by an arbitrary proposition and name. This inturn determines the totality of propositions whose sense the expression marks.A quantificational generalization is the result of a truth-operation on such a to-tality.The reader may be excused for wondering what this talk of what a proposi-tion says of an object really comes to. Can we make mathematical sense of it? Ormust it wither in the never-never land of irreducibly metaphorical metaphysics?Let me conclude by sketching the mathematical underpinnings of my account.For expository simplicity, I will confine myself to that fragment of the Tracta-tus construction which has occupied the various commentaries discussed in thischapter.121Assume as primitively determined concepts the concepts of name and ele-mentary proposition. It is natural to assume that no names are elementary propo-sitions. Furthermore assume that there exists a relation of samesaying. This re-lation is assumed to answer every question whether what one elementary propo-sition says of some objects is the same as what another elementary propositionsays of some other objects. We do stipulate that samesaying is an equivalence re-lation, but make no further assumptions about it. We also make no assumptions121With the exception of one paragraph of Geach (1981, 170).125whatsoever about the structure of elementary propositions, other than that theextension of the samesaying relation be determinate.In this way, we take as primitive every answer to a question when an expres-sion marks the sense of a given elementary proposition. Given such primitives,we aim to construct all first-order definable truth-functions of elementary propo-sitions. The key technical problem will be to show how to extend the concept ofexpression from elementary propositions to all propositions. The guiding prin-ciples are straightforward. For example, the denial of a proposition says of somelist of objects what another proposition says of some other list of objects, pro-vided that the second proposition denies a proposition which says of the secondlist of objects what the first denied proposition says of the first list of objects.Now, recall the idea of a maximal conversation from 2.2. We introduce a newsystem of rules, permitting three kinds of move. An act of the first kind is theassertion of an elementary proposition. By an act of the second kind, somebodycan single out any finite bunch of acts that people have performed already, anddeny that any of propositions expressed by those other acts are true. In an act ofthe third kind, somebody can single out any one act that somebody already per-formed, emit a name, and then deny every proposition which says of somethingwhat the proposition expressed by the singled out act says of the thing named.Note that this second case allows for straw men, i.e., for the denial of proposi-tions which nobody has yet asserted. I do not think that this phenomenon isincomprehensible or even actually unprecedented. However, for technical rea-sons it is useful to note that straw men can always be fleshed out in principle, forexample by paying one?s friends to assert what one wishes to deny.122Let us momentarily presume an answer to the question when a propositionsays of something what another proposition says of something. We are then ina position to determine the sense of arbitrary propositions. Consider a conver-sation that really is maximal, so that every possible tree of moves descends fromsome act that has already been performed. We renew our insistence that a conver-122In this way, sufficiently wealthy speakers can overcome the obstacles raised by Fogelin (1987,80). Just kidding. Fogelin does not manage to identify an obstacle in the first place, because thisis a piece of pure mathematics. As Geach (1981) said, nobody actually has to list all the proposi-tions denied. Mathematical constructions do not require construction workers to skolemize thequantifiers. I am just using the concept of conversation as an expository metaphor.126sation be wellfounded, so that although some speech act may point to an earlierone, and those to earlier ones, and so on, every single chain of indicated acts musteventually terminate in the assertion of an elementary proposition. Suppose alsothat the conversation contains no straw men, so that nobody ever denies some-thing that was not yet asserted. We may assume the sense of propositions ex-pressed by acts of the first kind to be fixed. Acts of the second and third kindsare denials. Since the conversation contains no straw men, the propositions theydeny have been expressed already, and by the wellfoundedness assumption, wemay assume their sense to have been fixed. But the sense of a denial of proposi-tions with fixed senses is clearly fixed.123Now let?s take up the task to extend the samesaying relation from elementarypropositions to arbitrary propositions. Consider two speech-acts, Act One andAct Two, and two corresponding lists of names, say Team One and Team Two.We want to determine the answer to a question, whether or not what the propo-sition expressed in Act One says of TeamOne is the same as what the propositionexpressed in Act Two says of Team Two. I will just use talk about what an actsays of some objects as short for talk about what the proposition the act expressessays of them.We first stipulate that Act One says of Team One what Act Two says of TeamTwo only if the two acts are the same kind of move. Now, note that if both ActOne and Act Two are assertions of elementary propositions, then our questionis already answered by the primitive part of the fixing of the behavior of thesamesaying relation. So assume the Acts to be nonelementary. By the assumptionof wellfoundedness, to determine the answer to the question, it suffices to reducethe question entirely to similar questions concerning acts singled out by ActsOne and Two. The answer to our question is no unless the Acts are of the samekind, so suppose they are. Suppose the Acts are of the second kind, so that eachsingles out a finite bunch of prior acts. Then, Acts One and Two are the sameexpression if and only if the acts in the bunches can be paired off in such a waythat, in each pair, the element drawn from the bunch indicated by Act One saysof Team One what the element drawn from the bunch indicated by Act Two123This paragraph is formalized in Definitions 5 and 6 of ?5.1.127says of Team Two. Suppose on the other hand that Act One is of the third kind.Then each Act involves emitting a name and singling out one earlier act. Appendto Teams One and Two the names emitted in Acts One and Two respectively.Now the answer to our question will be no unless the singled out acts say thesame thing of the expanded teams, in which case the answer is yes.124There is another way to visualize the process of checking whether two actssay the same thing of some objects corresponding to each. Consider each act as anupside down tree. The tree is built up out of inch-long segments of copper wireconnected by balls of clay. On the wire beneath a given ball of clay, there mayor may not sit a colored bead. And at the end of each branch, there hangs a littlemosaic consisting of colored beads glued to a piece of cardboard. To representwhat an act says of some team of names, simply suppose a corresponding seriesof colored beads to sit on the topmost wire of the associated tree. Now, supposethat two trees can be laid on top of each other in such a way as to align everybead, ball of clay, and piece of wiring on the one tree with a bead, ball of clayand piece of wiring on the other tree. Consider two complete, linear brancheson the two trees which correspond to each other under this alignment. Thesebranches each bear a series of beads which, as we move downward, begins withTeam One or Team Two, followed by further beads as they occur on the chain,followed finally by its terminating mosaic. With respect to any such chain, con-sider just its descending series of beads and the terminating mosaic. The mosaic isan elementary proposition, and the series of beads is an expanded team of names.We stipulate that two Acts say the same thing of their respective initial Teams ofnames if and only if each pair of corresponding decorated mosaics say the samething of the expanded teams of names denoted by the series of beads above them.The question then remains when two mosaics say the same thing with respect totheir corresponding series of beads.The question what a mosaic says with respect to the beads above it is to beanswered primitively. But for the reader?s amusement, let me fix an image. Whata mosaic expresses with respect to one bead is simply a matter of the shape ofthe region of the mosaic which that bead occupies, together with the rest of the124This paragraph is formalized in Definition 3 of ?5.1.128contents of that mosaic. If we ask what the mosaic expresses with respect to aseries of beads, then, roughly speaking, this is a matter of the series of shapes ofthe regions which are occupied by beads as we move up the series, together withthe remnants of the contents of the mosaic. In conclusion, note that although amosaic may appear to be made of many individual beads, what really stand forobjects in a mosaic, i.e., the names of which the mosaic is really a nexus, are notthe individual beads but rather their colors. As Wittgenstein put the point, ? ?A?is the same sign as ?A? ? (3.203). I have been urging that this point applies notjust to names in elementary propositions but to names in all propositions. Everyproposition is just a nexus of names. Propositions are pictures, and names are thecolors of which they are formed. Of course, the intricacies of logical picturingcan only really be grasped by examining the internal relations which makes pos-sible the totality of logical pictures. For example, the beads on the wires do notcontribute objects corresponding to the colors they bear, but denature the simi-larly colored beads in the mosaics beneath them, so that a mosaic contributes tothe proposition none of the objects which correspond to the colors of the beadsabove it.125125The metaphors of this concluding paragraph are cashed out in ?5.2. I apologize for the delay,which arises because I?ll cash out the metaphors in the context of a reconstructed system whichincludes variables constructed not just by Ways 1 and 2 but also Way 3.129Chapter 3Formal generalityIn what is perhaps not the single obscurity of the Tractatus, Wittgenstein writes:Only in this way is the progress from term to term of a form-series(from type to type in the hierarchies of Russell and Whitehead) pos-sible. (Russell and Whitehead did not admit the possibility of suchsteps, but repeatedly availed themselves of it.) [5.252]I aim to provide an interpretation of this remark.The obvious question to arise is the following: what does Wittgenstein meanby ?this way?? Now, it is an unfortunate fact of life in the Tractatus that onecannot always resolve the antecedent of a pronoun by straightforward appeal topreceding entries. For as is well known, Wittgenstein composed the Tractatusby assembling and rearranging individual sentences or paragraphs until he foundan order which suited him. And sometimes this process produced grammaticalpossibilities of interpretation which he did not intend. Nonetheless, the relevantearlier entries are as follows:The occurrence of an operation does not characterize the sense of aproposition.The operation says nothing, only its result, and this depends on thebases of the operation.126(Operations and functions must not be confused with one another.)[5.25]A function cannot be its own argument, whereas an operation cantake one of its own results as its base. [5.251]126Here I?ve slightly modified Pears-McGuinness. The original runs: Die Operation sagt ja nichtsaus, nur ihr Resultat. . . .130According to 5.2521, an operation can take its own result as its basis. So, Wittgen-stein?s point at 5.252 seems to be that this iterability of operations is what makesprogress from term to term of a form-series possible. Specifically, the basis ofthe operation would be one term of the form-series, and the corresponding re-sult would be the next term. Similarly, the result of the operation can itself betaken as a basis, which the operation takes to another result, the third term of theform-series. And so on. In this way, an operation takes us through an unendingsuccession of terms; thereby a form-series is generated.That much, I think, is relatively clear, and not in dispute. For, a page earlier,Wittgenstein has introduced the concept of operation. Our explanation so far fitswell with some those remarks:An operation is the expression of a relation between the structuresof its result and of its bases. (5.22)The operation is what has to be done to the one proposition in orderto make the other one out of it. (5.23)And that will, of course, depend on their formal properties, on theinternal similarity of their forms. (5.231)The internal relation by which a series is ordered is equivalent to theoperation that produces one term from another (5.232)Thus, I take the unparenthesized part of 5.252 to be at least verbally fairly clear.If we knew what an operation was supposed to be, then we would understandthis much of 5.252 pretty well.The very first example Wittgenstein gives of an operation is this:Truth-functions of elementary propositions are results of operationswith elementary propositions as bases. (These operations I call truth-operations.) (5.234)[. . . ] Negation, logical addition, logical multiplication, etc. etc. areoperations. [. . . ] (5.2341)Part of the point of the idea of an operation, then, is to be marked off fromthe idea of a function: operations and functions must not be confused with each131other (5.21). Here, Wittgenstein has Russell in mind. For Russell as for Wittgen-stein, a function is a distinguishable part of its values: if a proposition is a valueof a function, then one can discern the function in the proposition by regardingcertain objects in the proposition as variable. But Russell also thinks that nega-tion is an example of such a function, and thus a distinguishable part of its values.Now Wittgenstein parts ways with Russell. For Wittgenstein, negation is not afunction, but an operation. Russell thus confuses operations with functions. Anoperation, for Wittgenstein, is not a distinguishable part of its values. An oper-ation can vanish: thus, for example, ??p = p (5.254). Operations are ways ofgetting from one place to another. They are merely procedural: ?operations can-not make their appearance before the point at which one proposition is generatedout of another in a logically meaningful way? (5.233).So, Wittgenstein?s idea of the operation is shaped by the Grundgedanke, bythe fundamental thought that there are no logical objects, that ?the ?logical con-stants? are not representatives? (4.0312). The ?logical constants? are merely meansby which we represent a proposition as the result of an operation on other propo-sitions (5.2). They present a logical place by giving directions to that place fromanother one. So the idea of the operation is shaped by theGrundgedanke, becauseoperations are key to Wittgenstein?s understanding of truth-functionality. Re-sults of truth-operations are truth-functions of their bases: the result of negatingp is a truth-function of p. Paradigm instances of operations are truth-operations,the counterparts in Wittgenstein?s thinking to the truth-functions of Principia.Let?s now turn back to 5.252, to the idea that an operation, repeatedly appliedto its own results, thereby enumerates a whole series of propositions. As it hap-pens, we have skipped over the hard part. Parenthetically, Wittgenstein comparesthis repeated application to a passage from type to type in a Russell-Whiteheadhierarchy. Thus, the parenthetical remark seems to imply that each term of aform-series is followed by a term of higher logical type.It seems safe to say that we are now in the thick of a mystery. There is cer-tainly no explicit endorsement in the Tractatus of the ramified theory of types.There is very little talk of higher-order quantification in general, andmuch of thatis pejorative. Despite this near-silence, does Wittgenstein buy into some kind oftheory of types after all?132Worse yet: wasn?t the whole point of the idea of operation that it doesn?tcharacterize the sense of its results? Doesn?t Wittgenstein illustrate this point at5.254 by remarking that ??p = p? But isn?t negation a paradigmatic operation,which should take us from term to term of a form-series p,?p,??p, . . .? Howin the world can this form-series be seen as climbing through a system of Russell-Whitehead types? As 5.254 tells us, it goes around in a circle!The questions raised by 5.252 and related passages are perhaps the hardestquestions about the workings of logic in the Tractatus. But as we?ll see, the ques-tions raised by 5.252 are immediately implicated by the question of the signifi-cance of the general propositional form. So one cannot really have a good gripon the book until these questions are properly answered. Among the few in-terpreters who have worked on these questions concertedly, some conclude thatWittgenstein must accept some sort of classical system of type theory.127 Yetothers sharply deny this.128 Such a chasm between leading interpreters of thebook indicates that there are serious problems here. One aim of this chapter is todo some of the textual spadework that?s essential to resolving the questions abouthigher-order logic in the Tractatus. Nonetheless, I do myself have a clear position,and in ?1 I begin by staking it out: there is no interesting genuine higher-orderquantification in the Tractatus. The Tractatus contemplates, and, in principle,can contemplate, only first-order quantification, perhaps of a multi-sorted vari-ety. But, my position is, though clear, also only tentative. So I want to backoff a little on the systematic considerations here, and try to look at the evidencehonestly.Let me conclude this introduction with an executive summary of the sectionsto follow.129It was essential to the programs of Frege and Russell that logic involve somekind of higher-order generality. For they considered logic to be a body of sub-stantive truths which were applicable to all of science. This conception of logicalready strongly supports quantification into predicate position. If quantifica-127For example, Michael Potter (2008, 193ff) and Peter Sullivan (2004, 50).128Ricketts (2012), Goldfarb (unpublished).129The topic of this chapter involves a combination of conceptual and exegetical challenges whichmay tax the patience of all but the truly fanatical. Moderates might just read the executive sum-mary.133tion reaches into object-positions alone, then the logical theorems which legiti-mate inferences in the special sciences will mention properties special to those sci-ences, and so not be really general. Another conceivable way to topic-neutralitywould be to abstract out this reference to properties, by regarding the predicatesas merely schematic. But Frege and Russell would resist this approach, since thenthe theorems can no longer be regarded as substantive truths in their own right.If, on the other hand, we allow theorems of logic to generalize over all objectsand properties whatsoever, then the theorems of logic can be seen as broadly ap-plicable, substantive truths. Such truths would characterize the common portionof all science: purely logical objects and functions like truth, falsehood, negation,entailment, identity, and quantificational generality itself.130So the philosophical understanding of logic as a body of broadly applica-ble yet substantive truths is intertwined with the use of higher-order generality.Of course, higher-order generality had another rationale too, because Frege andRussell were logicists, aiming to show that arithmetic is just a highly developedbranch of logic. But, the logicist program is only remotely conceivable if logicamounts to more than just the syllogism. It was conceivable because, to Fregeand Russell, logic looked very powerful.Now, Wittgenstein learned logic from Frege and Russell. Thus, he inheritedtheir intuitions about logical expressiveness. In particular, the founding docu-ment of the new logic, Frege?s Begriffsschrift, culminates in a logical analysis ofthe ancestral of a relation. So Wittgenstein simply expected that the ancestralwas a logical notion. For what else could logic be?131 On the other hand, Iargue in ?1, Wittgenstein has strong philosophical motivations of his own fortaking genuine quantificational generality to be generalization over objects, andin particular over simple objects. First, his understanding of quantificational gen-erality requires that the instances of a generalization receive their meaning di-rectly by arbitrary convention, rather than meaning what they do as a matterof what must be the case once the conventions are in place (6.124). So, the in-stances of a generalization must be simple names. There can be quantification130This paragraph is heavily indebted to Goldfarb and Ricketts?see for example Goldfarb (2001)and Ricketts (2012).131For a discussion of the alternatives, see Wittgenstein (1913).134into so-called ?predicate-position??or, more accurate, quantification into everysimple, content-contributing articulated propositional segment; but such a gen-eralization can only be instantiated by another such minimal segment. Second,as argued in Chapter 1, analysis as Wittgenstein conceives it aims to show thatpropositions are truth-functions of elementary propositions, by exhibiting themas resulting from the elementary propositions by successive applications of truth-operations. Quantificational generality is then to be understood as a particularmeans of specifying the basis of truth-operation; but then instances of a general-ization must in some sense have lower complexity than the generalization itself;this at least rules out all higher-order quantification that is impredicative.Nonetheless, the student of Frege and Russell did inherit their expectationthat the ancestral was a logical notion. But, he rejected the resources Frege andRussell used to construct it. So, I argue in ?2, Wittgenstein takes another ap-proach: he defines the ancestral by means of a countably infinite disjunction.This raises a question of principle: just when can an infinite class of formulas beso taken as basis of a truth-operation? I argue that for Wittgenstein, the crucialconstraint stems from the picture theory. On the one hand, an infinite multiplic-ity of propositions can?t itself be surveyed, so the elements of the multiplicitymust be presented by a common mark. But, the picture theory requires a formalunity between the proposition and the situation it depicts. Hence, the commonmark of propositions in the multiplicity must be a formal one, so that it is alsocommon to the situations which the propositions in the multiplicity present.Now, Wittgenstein thinks that the propositions in a form-series do have such acommon formal feature: this is what legitimates the analysis of the ancestral interms of the concept of form-series.In ??3 and 4, I argue that Wittgenstein envisaged two quite distinct kinds ofapplication of the concept of the ancestral. In ?3, I show that Wittgenstein in-voked the notion of ancestral as early as 1914, toward the program of analysis ofstatements about complex spatial objects such as tables and chairs. In the develop-ment of this program, we find a striking illustration of a main thesis of ChapterI: a proposition borrows its internal structure from its formal unity with a situa-tion. In ?4, I summarize Wittgenstein?s early use of the ancestral in sketching thegeneral propositional form, and observe that he imagined this use to have quite a135different logical status.?5 gives my intepretation of 5.252. I discuss a notebook entry of 7.1.17 whichis the first appearance of the general propositional form in something like the ver-sion we find on the first page of Prototractatus. This entry shows that Wittgen-stein, as he completes the analysis of the general propositional form, sees the iter-ated application of truth-operations as the ultimate fate of the Russell-Whiteheadstratification of the totality of propositions. In the view of the Tractatus, propo-sitions do stratify into hierarchies, but this stratification is merely aspectival. Noproposition has a single position in a hierarchy essentially, but only relative tothis or that completed take on the totality of propositions. On this view, theN -operation does take us from one level of a hierarchy to another, and carries usup through the layers of the totality of propositions, just as Russell and White-head hoped to achieve off-the-books with their device of typical ambiguity. AfterWittgenstein?s rethinking, a hierarchy is no longer a system of logical constraints,but a survey of notational freedom.In ?6, I return to the tension that began to arise in ??3 and 4 between thetwo distinct applications of the form-series device. Wittgenstein does presentthe general propositional form itself by means of a form-series. Does this meanthat he thinks that the totality of propositions can appear as the basis of truth-operations? I present two possible answers. According to one answer, Wittgen-stein thinks that stipulable bases of truth-operations stand to the general propo-sitional form as set-constituting multiplicities stand to the class of all sets. Ac-cording to another answer, Wittgenstein meant what he said at 4.5, that the gen-eral propositional form is a variable, and thus available as stipulable basis fortruth-operations. I tentatively favor the second conclusion, but find the textualevidence to be inconclusive, even, perhaps, disconcertingly muddied.In ?7, I present an alternative interpretation of 5.252, suggested by ThomasRicketts in his valuable (2012). On Ricketts? view, 5.252 alludes to the possibilityof using the form-series device to simulate higher-order predicative quantifica-tion. Ricketts also sketches a speculative reconstruction of this development ofpredicative quantification. Since as Ricketts agrees, any detailed account of theapplication of form-series must be somewhat speculative, differences here are tosome extent a matter of taste. In any case, I hope that our readings that not sep-136arated by the chasm that almost surely separates the readings of two randomlychosen interpreters.The critique of Ricketts is a kind of negative advertisement for a specula-tive reconstruction of my own, which I offer in the following chapter. I iden-tify a very simple class of operations?essentially, substitution of a formula for apropositional variable in a formula?and show that this device suffices to expressthe ancestral and other?11 logical notions relevant to the program of logical anal-ysis. The resulting system forms a recursive sublanguage of the finite-variablefragment of L?1?.3.1 Two problems with higher-order generalityAs I?ve argued in Chapter 2, Wittgenstein?s account of the construction of propo-sitional functions differs importantly from that of Frege. Frege?s account beginswith the idea of replacing a name with a name in a sentence, thus leading us toconsider the name as variable while the remainder of the sentence is fixed. Byintroducing the antecedently intelligible notion of replaceability, Frege purportsto elucidate, though not to define, the new and questionable idea of somethinginvariant under the replacements, namely the Fregean function. Wittgenstein?saccount makes no primitive appeal to the idea of replacing a name with a name,for such an appeal leads to the standard fixing of the range of propositional func-tions, which Wittgenstein must reject. Does this mean that Wittgenstein just hasto take the idea of propositional function as primitive? This would be unfortu-nate, since Wittgenstein?s concept of propositional function, unlike Frege?s, isnot at all a natural generalization of the mathematical concept of function. Soit stands even more in need of explanation. Rather than talking about replacinga name with a name in a proposition, and thereby coming to regard a proposi-tional constituent as variable, Wittgenstein instead talks directly about turning aconstituent into a variable. But what does this talk come to?If we turn a constituent of a proposition into a variable, there is aclass of propositions all of which are values of the resulting variableproposition. In general, this class too will be dependent on the mean-ing that our arbitrary conventions have given to parts of the original137proposition. But if all the signs in it that have arbitrarily determinedmeanings are turned into variables, we shall still get a class of thiskind. This one, however, is not dependent on any convention, butsolely on the nature of the proposition. [. . . ] (3.315)Wittgenstein?s talk of arbitrary convention here is essential to his distinctive un-derstanding of the variable, and thus of his construction of propositional func-tions. Suppose that we construct a ?logical prototype? by beginning with aproposition turning all of its constituents into variables. Then, Wittgensteinsays, the resulting prototype differs from the initially given proposition in thatthe prototype does not depend on any of the arbitrary conventions by which theconstituents were assigned their meanings. Consider now another case, wherewe turn just a single constituent into a variable: the result then differs from theoriginal proposition just in that it no longer depends on the convention whichassigned a meaning to that one constituent.The talk of arbitrary convention gestures toward the following idea. It isobviously possible to effect local changes to linguistic convention by drawing achalk circle on the ground and stipulating that the name ?Kant? when uttered inthe circle means what ?Spinoza? actually means. Such a stipulation really has twocomponents: first, cancelling the actual meaning of the name ?Kant? and second,assigning the then-meaningless word ?Kant? a new meaning, that of ?Spinoza?.But now suppose we enact just the first half of the stipulation, step into the re-sulting circle, and then utter ?Kant was a great philosopher?. I take it that thisis basically Wittgenstein?s understanding of how from ?Kant was a great philoso-pher? one constructs the propositional function ?x was a great philosopher?.This interpretation explains why Wittgenstein puts so much emphasis, in his ac-count of generality, on the idea of ?arbitrary convention?. Such rhetoric is, afterall, rather peculiar: for example, so far as I know, it never appears in Frege orRussell?s expositions of corresponding ideas.But this account also sets up a striking departure from Frege and Russell withrespect to what parts of a proposition can be regarded as generalizable: namely,only those parts whose meaningfulness is expressly a matter of some arbitraryconvention. Now, some aspects of an expression of a proposition do not con-138stitute parts in the relevant sense at all. Clearly signs which occur as names doconstitute such parts. In contrast, for example, the ?cat? in ?cattle? could doso only under extraordinary circumstances. Wittgenstein also holds that sincethe sign of a truth-operation do not mark the sense of a proposition by means ofwhich it?s expressed, also no part of a proposition corresponds to truth-operationsigns either.132 But, suppose we begin with a proposition, turn a name in it intoa variable, and consider this resulting ?leftover part?. Returning to the originalproposition, can we turn the leftover part itself into a variable? To the extentthat this can be done, it must be done by the selective local repeal of meaning-securing conventions. Thus for example we can surely turn into variables eachof the names which occurs in the leftover part. But as far as I can see, that ispretty much all that we can do. And it is a poor substitute for genuine higher-order quantification. The essential problem is that the Tractatus faithful mustsharply distinguish between that which is fixed arbitrarily and what must be thecase once things have been arbitrarily fixed (6.124). Inasmuch as the leftover parthas a meaning at all, this is not thanks directly to something arbitrary but rathersomething that must be the case once we have begun to talk under the given con-ventions. One cannot simply single out for repeal some joint effect of a variety ofinterlocking conventions, but must repeal the conventions underlying the effectthemselves. Wittgenstein?s occasional use of notation for quantifying into predi-cate position is shorthand for turning all but one of the names in the propositioninto variables, thus a mere ersatz requiring nothing beyond ordinary polyadicfirst-order generalization.133132I would be pleased to put it like this: since ??p is the same proposition as p, therefore ?doesn?t mark the sense of ??p.133This suggestion is due to Ricketts (2012); I?m absolutely committed to something like this bythe argument of ?3.1.The 3.31s may seem to suggest (as apparently to Potter (2009, 270)) that Wittgenstein counte-nances abstraction of expressions other than names. However, in my view, that is a misreadingof the passage. Perhaps the crucial point is the first sentence of 3.315: ?Verwandeln wir einen Be-standteil eines Satzes in eine Variable, so gibt es eine Klasse von S?tzen, welche S?mtlich Werte desso entstanden variablen Satzes sind.?In particular, we need to determine the intended meaning of the term ?Bestandteil.? Onmy view, ?Bestandteil? in this context evokes the Russellian terminology of ?propositional con-stituent?; and propositional functions are not, according to Russell, constituents of their values.The word ?Bestandteil? occurs previously in the Tractatus at 2011, 20201, and 324. 2011 saysthat it is essential to a thing to occur as ?constituent? of a state of affairs. 20201 and 324 use the139Of course a there is a second andmuch better-recognized problem thatWittgen-stein finds with the use of higher-order generality amongst the logicists. Thisproblem specifically affects that higher-order generality which is impredicative.On the logicist treatment of arithmetic, cardinal numbers are identified withclasses of equinumerous classes (or classes of equinumerous concepts); having de-fined the cardinal analog of the successor relation, one then marks out the finitenumbers as those objects which bear the ancestral of the successor relation to theclass with which zero has been identified.134 Now, it is second-order generalitywhich lets us define the ancestral of a relation. On Frege?s version, a is said tobear the ancestral of R to b in case b has every R-hereditary property a has, i.e.,R?xy ??X (HerR(X )?Xa ?X b ),term to refer to the items into which analysis resolves a complex, thus presumably to simples.These antecedents suggest that Bestandteil denotes something simple, be it an object in a state ofaffairs, or a name in a sentence.At 4.024 and 4.025 there are slightly more apposite occurrences. For example, 4.024 says thatone understands a sentence upon understanding its Bestandteile. 4.025 says that translation ofone language into another proceeds by translating only the Satzbestandteile. The use at 4.024may be somewhat equivocal (depending on whether the point is that one needs to understandall its Bestandteile), but 4.025 is unequivocal: Satzbestandteile must refer to names rather than toarbitrary expressions.Wittgenstein also uses ?Bestandteil? in another sense, where what is under consideration isa proposition presented as the result of a truth-operation on other propositions. In this case,Wittgenstein occasionally refers to those other propositions, or rather to their signs in their oc-currence in the presentation of the result of the truth-operation, as ?Bestandteile? of the result.However, this usage only makes sense on the level of signs, since the bases of a truth-operationneed not characterize the sense of its result. In contrast, the answer to the question whether aname characterizes the sense of a proposition must be invariant under truth-operational rephras-ings and more generally independent of any particular means by which the proposition gets madesensibly perceptible.Now, it?s true that the remainder of the paragraph 3.315 employs ?Teil? anaphorically on ?Be-standteil?. Moreover, 3.31 has ?Jeden Teil des Satzes, der Seinen Sinn Charakterisiert, nenne icheinen Ausdruck (ein Symbol).? I think that the 3.315 occurrence of ?Bestandteil? in contrastto ?Teil? cancels any possible reference back to the 3.31 use of ?Teil?. The hypothesis of suchback-reference would be questionable anyway: according to McGuinness-and-Schulte, 3.315 de-rives from NL, whereas 3.31-3.313 derive from the PT3.20s (and a bit from NB2).I think, then, thatWittgenstein intends no verbal link from the ?Bestandteil? of 3.315 back to the?Ausdruck? of 3.31-3.313. ?Bestandteil? means a propositional constituent, whereas ?Ausdruck?means a part or aspect of a proposition. In the 3.31-3.313 use of ?Ausdruck?, Wittgenstein refersnot to what is removed, but to the leftover. There seems to me to be no clear implication in the3.31s that the leftover can itself be removed.This footnote is transcribed almost verbatim from an email to Ricketts.134I borrow this formulation from Goldfarb (naturally, his version is more elegant).140where a property is R-hereditary in case no thing with the property bears R tosomething which lacks the property, i.e.,HerR(X )??x?y(X x ?Rxy ?Xy).Supposing that only a few properties exist, say for example that there only existproperties with finite or cofinite extension, say, then a and b may satisfy Frege?sdefinition of the ancestral of R even while a does not bear the ancestral of R tob . And, although of course Frege or Wittgenstein couldn?t say this, the same istrue even if all first-order definable properties exist.A natural response to this challenge is: ?but all properties exist, includingthat of bearing the ancestral of R to a?. Frege?s axiomatic implementation ofhis definition embodies this response, since it licenses instantiating the boundsecond-order variable in the definition to the property of bearing the ancestral ofR to a. Wittgenstein, perhaps evoking Poincar?,135 charges that such a responseis viciously circular. As Ramsey and G?del urged, this circularity seems reallyvicious only if one regards a property as being constructed by the definition,rather than as being merely singled out by it.136As with some other of Wittgenstein?s objections to Frege, however,137 theimpredicativity complaint can also be seen to tell us something significant aboutWittgenstein?s own thinking. As I?ve argued in Chapter 1, Wittgenstein sees logi-cal structure as articulating the truth-functionality of propositions. In particular,a connective serves to present a proposition as the result of a truth-operation onsome other propositions, with those propositions obtained by truth-operationson others and so on; this branching process terminates ultimately in the proposi-tions that are elementary. After truth-operation has been separated from general-ity proper, first-order quantificational notation is then seen to articulate genuinepropositional structure: since analysandum is represented as giving way to propo-135This suggestion is due to Goldfarb (2012).136Peter Sullivan pointed out in a discussion that Wittgenstein?s accusation of vicious circular-ity is directed not just against Frege but also against Russell, whose higher-order quantification ispredicative. Wittgenstein may have been inured to this point by Russell?s adoption of the axiomof reducibility.137For example, 4.063.141sitions of a lower notational complexity, we can be sure of not going around ina circle. But impredicative quantification cannot be so redeemed, because theproposition to be analyzed will recur essentially in its analysis. So even if Fregewouldn?t be much flustered by Wittgenstein?s borrowing from Poincar?, the pas-sage does illustrate some central ideas of the Tractatus.3.2 Wittgenstein?s alternativeFrege?s analysis of the ancestral thus conflicts with Wittgenstein?s truth-function-ality thesis. At this point, Wittgenstein might simply have concluded that theancestral is not a logical notion. However, for someone who learned logic fromFrege and Russell, the definability of transitive closure of a relation might haveseemed as natively logical as the definability of, say, its symmetric closure. More-over, it seems hasty to abandon the logicality of transitive closure because ofproblems with one particular analysis. For example, a similar reaction mightinspire us to reject the logicality of symmetric closure because of the impredica-tivity of the definitionRs xy ??X (?u?v(Ruv ?Rvu ?X uv)?X xy).And this would be silly, since one has the unworrying alternativeRs xy ? Rxy ?Ryx.The alternative analysis of symmetric closure is unworrying, of course, be-cause as we?d now say, it is purely first-order. Unlike symmetric closure, tran-sitive closure is not first-order definable, and so there is somewhat better evi-dence that the concept of transitive closure intrinsically presupposes the notionof arbitrary subset of the domain. But, this is still too hasty. From an extra-Wittgensteinian point of view at least, the concept of transitive closure can becoherently taken to be independent of such higher-order notions. For example,the concept of transitive closure is closely tied with the concept of the order typeof the natural numbers, i.e., with the concept of finite ordinal. In particular,once the finite ordinals are fixed, then concepts definable by induction on finite142ordinals would seem to be fixed as well. But the transitive closure of a relationis definable by induction on the finite ordinals.138 And, the concept of finiteordinal is acceptable from a point of view which would take for granted the to-tality of natural numbers but reject full second-order quantification. Now, it?snot clear that Wittgenstein himself could reason in this way, from the determi-nateness of the natural numbers to the determinateness of transitive closure. But,the existence of this line of reasoning indicates that one might coherently acceptthe determinateness, or perhaps even the logicality, of the concept of transitiveclosure, while rejecting as indeterminate or as nonlogical the use of impredicativequantification.Since transitive closure is not first-order definable, we need some essentiallynot first-order logical device if transitive closure is to be a logical notion. Fromthe Tractatus point of view, the first question to ask about putatively logical de-vices is whether they can be reconciled with the truth-functionality thesis. In-tuitively, it should count as evidence that some device can be so reconciled ifwe can understand it simply to embody a new way of expressing agreement anddisagreement with truth-possibilities for elementary propositions. But it is nowworth recalling what every good student wants to say when asked to define ?an-cestor? in terms of ?parent?, namely just that there is at least one truth amongthe following bunch of formulas:Rab ,?x(Rax ?Rxb ),?x?y(Rax ?Rxy ?Ryb ), . . . .This characterization certainly does seem to show that whether or not a standsto b in the ancestral of R is merely a truth-function of elementary propositions.For once the list is fixed, then each formula in the list is a truth-function of ele-mentary propositions; but then it would seem that whether or not at least oneof them is true is also just function of which elementary propositions are true.Truth-functionality, on Wittgenstein?s scheme, demands only that the result ofintroducing a new representational device preserves the possibility of tracing,from each constructed position, through chains of agreement and disagreement138As in R?(0, x, y) ? Rxy and R?(s(k), x, y) ? ?z(R?(k , x, z) ? Rzy), and then R??(x, y) ??k ??R?(k , x, y).143with other positions, back down all the way to a pattern of agreement and dis-agreement with truth-possibilities of elementary propositions. It seems entirelypossible that some resources sufficient for rendering the na?ve analysis could meetthe truth-functionality constraint.There is an analogy between the na?ve approach to the ancestral and theWittgenstein?s treatment of existential quantification. For example, Wittgensteinshows how ?a bears R to something? is a truth-function of elementary proposi-tions,139 by reducing its truth to the circumstance that there be at least one truthamong the following bunch of formulas:Rab ,Rac ,Rad , . . . .Thus, the na?ve analysis of the ancestral and the Tractatus treatment of existentialgeneralization both represent a proposition as a result of a truth-operation onsome presumably infinite bunch of formulas. Moreover, this bunch is not a ran-dom set of formulas but the class of formulas instantiating some common feature.This analogy raises the question why existential quantification is a legitimate an-alytical resource. Perhaps the underlying principles license further resources thatwere overlooked by Frege and Russell, and which suffice to recover the ancestral.Wittgenstein?s conception of propositions as pictures entails that propositionand reality are alike, or one and the same, with respect to their total logical con-tent. It is then in virtue of this likeness that a proposition makes evident how theworld must be if it is true. Thus:We can see this from the fact that we understand the sense of a propo-sitional sign without having it explained to us. (4.02)A proposition shows its sense.The proposition shows, how things stand, if it is true. And it says,that they do so stand.140 (4.022)So, a proposition is in this way like a picture, that there simply is no space be-tween seeing a picture as a picture, and knowing how things are if they are as139NB: ??x f x means ?a true proposition f x? ? (9.7.16).140The last comma does not appear in Pears-McGuinness.144depicted.141 This overtness of pictoriality is what decides just when, for Wittgen-stein, an analysis may represent a proposition as the result of an operation onother propositions to be given not directly as constituents of the analysans butas values of a mere schematic representation. It is the very appearance manifestin the proposition which must itself be the way things are if the proposition istrue, even though this appearance finds its position in logical space only thanksto certain contingent facts.142 So, as Wittgenstein puts it in the Prototractatus:One must be able to see in the variable itself what it stands for. ?There must be a wholly determinate resemblance between it and itsvalues. (PT5.0054)This I suggest, is the principle which underlies the possibility of representing aproposition as the result of a truth-operation on some possibly infinite multi-plicity of propositions: that some common mark characterize exactly the propo-sitions belonging to that multiplicity. The analysis of a proposition may thenpoint not to the infinitude of other propositions directly, but instead to the char-acteristic mark itself. Such a mark is a formal or internal feature of propositions:The expression for a formal property is a feature of certain symbols.So the sign for the characteristics of a formal concept is a distinctivefeature of all symbols whose meanings fall under the concept.The expression of a formal concept is a propositional variable, inwhich this characteristic feature [Zug] is constant. (4.126f-h)141Wittgenstein?s puzzling critique of Frege at 4.063 elaborates on his commitment to this po-sition. There he complains that Frege finds that truth and falsehood are accidental properties ofpositions which are specifiable independently of appeal to the nature of those properties. Imagine,for example, a three-player game, in which two players are assigned to reconstruct a painting thatis only visible to the third player. The first player calls out a coordinate on the painting; the thirdplayer replies with the color on the painting at that coordinate on the painting, and the secondplayer paints in the corresponding region of the duplicating canvas. In principle, the first playermight be deaf and blind, and have no idea what properties the second player records.142I think that this respect for appearances sharply separates Wittgenstein?s approach to analysisfrom that of Russell, and in particular leads him to retract his early pronouncement that ?yourtheory of descriptions is quite certainly correct? (Letter 30 in McGuinness 2008). It is a misreading(cf. e.g., Hart (1973)) to suppose that the ?indefiniteness? in 3.24 alludes to the scope ambiguityunder negation which is predicted by Russell?s theory of definite descriptions.145It is only because the commonmark of the values of a propositional variable is aninternal or formal property that we can expect the analytical appeal to suchmarksto preserve the formal unity between proposition and situation. That is, becausethe mark is formal, its commonality amongst propositions just is its commonal-ity amongst the situations those propositions depict. Amark that is not formal inthis sense?for example, being expressed in English by a palindrome?justifies noexpectation of any commonality between the corresponding multiplicity of pre-sented situations, hence fails to explain how propositions constructed by appealto the mark could be pictures of reality.143Consider, for example, a propositional variable which results by denaturing aname in a given proposition. This variable presents a mark of those propositionsfrom which that same mark arises by the denaturing of a name; the propositionsso marked form its range. Propositions in the range thus palpably resemble thevariable. It?s easy to see which are the propositions over which it ranges. Theyconsist of the same number of names, arranged in the same way. The only differ-ence between the propositions in its range arises through one-one recolorings ofcertain constituent names which are flagged in the variable?s notation.144Now, Wittgenstein similarly holds that such a characteristic mark can befound for the multiplicity of formulas whose disjunction is the ancestral of agiven relation. For, he suggests that there is a formal operation which takespropositions to propositions, such that the results of repeatedly applying theoperation to some initially given proposition are precisely the elements of themultiplicity. It is stipulated that an operation is formal only if it produces a re-sult from a given basis purely in virtue of an internal relation between the basis143The first way of stipulating value-ranges at 5.501 is by enumerating a finite list of values. Thepropositions in the list can be chosen arbitrarily. Wittgenstein hesitates about whether the firstmethod does yield a variable at all, for he writes: ?instead of a variable we can simply write itsvalues?. Presumably, the reason he hesitates is because he doesn?t think that a variable so stipu-lated presents a formal commonality which distinguishes its values, yet he explains the variable aspresenting a commonality. He continues to refer to explicitly listed value-ranges as variables justbecause it is terminologically convenient to describe all presentations of multiplicities uniformly.Alternatively, since the list is finite, one could perhaps stipulate that the list itself is a kind ofdisjunctive formal concept.144So, propositional functions as characterized in Chapter II meet Wittgenstein?s conception ofthe variable better than Russellian propositional functions do, since the values of the latter kind offunction contain variable numbers of different names.146and result. A multiplicity of propositions so generated Wittgenstein calls a form-series. More formally, suppose O is the sign of an operation on propositions.Then, [A,? ,O(? )] is a propositional variable, whose values are the propositionsA,O ?(A),O ?(O ?(A)), . . ..The form-series notation appears to allow construction of a propositionalvariable with the range needed to express the ancestral of a relation. That is,writing R0ab for the formula Rab , and Rk+1ab for the formula ?xk(Rkaxk ?Rxk b ), then it suffices to find a formal procedure O ? such that O : Rkab 7?Rk+1ab . It seems obvious that under some reasonable interpretation of ?formalprocedure?, such an O ? exists.Now, Wittgenstein presupposes that the propositions in a form-series can becharacterized by a distinguishing formal feature and so can also be presented bya variable. He seems to take the very notation [A,? ,??(? )] of a form-series vari-able itself to show the common mark of the propositions in its range. I think it?sfair to say that however the concept of operation is explicated, the similarity ofthe propositions in the range of a form-series variable will not be nearly so palpa-ble as the similarity between the values of a propositional function. Nonetheless,the concept of operation should somehow respect the constraint that it be ?easyto see? whether a proposition falls in the range of a form-series variable whichthe operation serves to construct.3.3 Form-series in analysis in the NBSo, Wittgenstein accepted the position of Frege and Russell that the notion ofancestral is part of logic. Of course, belonging to logic means something quitedifferent to Frege and Russell than it does to Wittgenstein. In accepting that theancestral is logical, Wittgenstein concluded that it was written into the very na-ture of picturing. The notion of ancestral, as analyzed by the device of the form-series, realizes certain unities of proposition and situation. I?ll now investigatethe development through the Notebooks of this envisaged role of the ancestral,which, as we?ll see, appears already in 1914.Consider a proposition to the effect that the watch is on the table: this is amere appearance of a certain watch, on a certain table, and it must be the fact that147that watch is on that table which makes true the affirmation of that appearance.Yet there cannot be the possibility of such a fact unless some wheels and pins andsockets and housing and glass are assembled into that watch; and there cannot bethe possibility of such an assembly unless incomprehensibly many intrinsicallyindiscernible material points cohere into the wheels, pins, sockets, etc. But howcan all this look like, and, needless to say, be, a watch on the table? It is this funda-mental explanatory constraint which guides the invocation of covert generalityin analysis.As Wittgenstein acknowledges, the innumerable indiscernibles constitutingthe constituents of the watch are not, in the bald fact of their coherence, theappearance of the watch. The appearance of the watch somehow conceals thisfine structure. Moreover, there is another problem. Although these constituentsso configured altogether are the watch, the equation does not reverse: the watchdoes not reduce to them uniquely. They are, one might say, sufficient but notnecessary for the watch, or for its being on the table. As Anscombe puts it,There are hundreds of different, more minutely statable, and in-compatible states of affairs which would make that proposition true(Anscombe 1959, 34-35).Despite these problems,Wittgenstein insists that a proposition like ?the watchis on the table? must somehow genuinely depict a dyadic relation between twothings. The ground of this conviction is not phenomenological but logical. Thatis, we can soundly reason with such propositions as though they were dyadic; butsuch reasoning could be sound only if the dyadicity were pretty much genuine.But logic as it stands, e.g., in Principia Mathematica, can quite well beapplied to our ordinary propositions, e.g., from ?All men aremortal?and ?Socrates is a man? there follows according to this logic ?Socratesis a mortal? even though I equally obviously do not knowwhat struc-ture is possessed by the thing Socrates or the property of mortality.Hence they function just as simple objects. (22.6.15h)In this way, the picture theory demands a logical unity between the propositionand the situation signified. Let?s call this need to account for unity of ordinary148propositions with the ultimate form of what they depict, the pictoriality con-straint. The difficulty raised by this constraint is that logical structure mustexplain all necessary connections, which requires that in the last analysis, theapparent multiplicity of an unanalyzed proposition gives way to an incompre-hensibly larger and more complicated logical structure. How can the apparentlysimple representation be a logically adequate picture of the teeming monster un-derneath? As Wittgenstein insists, ?ordinary language is in perfect logical or-der as it is.? Given this aim, an approach via Russell?s theory of descriptionswould be inadequate, because Russell?s theory purports to expose the apparentsubject-predicate structure as specious. A dramatic rendering of the pictorialityconstraint appears in the Notes to Moore:When we say of a proposition of the form ?aRb ? that what symbol-izes is that ?R? is between ?a? and ?b ?, it must be remembered thatin fact the proposition is capable of further analysis because a, R,and b , are not simples. But what seems certain is that when we haveanalysed it we shall in the end come to propositions of the same formin respect of the fact that they do consist of one thing between twoothers. (NB111)Anscombe and von Wright remark that this passage was ?lightly deleted?; per-haps this was because of its apriostic stringency. Nonetheless, Wittgenstein con-tinues to pursue some account of how analysis could meet the pictoriality con-straint. At 5.9.14, he proposes that a proposition ?[aRb] about a complex con-sisting of a in the relation R to b might be analyzed into propositions whichretain the form ?, namely as: ?(a) ??(b ) ? aRb . This proposal yields somepeculiar results: for example, from ?my watch weights n grams? it hardly fol-lows that the dial of my watch weighs n grams. Nonetheless, that Wittgensteinretained this peculiar proposal can be seen from obvious kinship similarity tothe remark of 2.0201: ?a statement about complexes resolves into a statementabout their constituents and into the propositions that describe the complexescompletely.?145145Wittgenstein?s use of ?describe? (beschreiben) here is not Russell?s. Objects, in this sense, can-not be described, for a description, in the relevant sense, is a reflection in language of the complex-149Now, the 5.9.14 approach certainly could not be Wittgenstein?s conclusivetreatment of complexes. For he maintains that the number of parts of a complexmight be infinite. But, following Russell, he holds that analysis could not intro-duce infinitely complex propositions explicitly. So, we have a kind of trilemma.A situation might be infinitely complex. But analysis cannot present a proposi-tion as infinitely complex. Yet the proposition is formally one with the situationit presents.It is under these theoretical pressures that he introduces the notion of logicalprototype.146 A prototype is an essentially general or schematic representationof a multiplicity of propositions, which nonetheless sufficiently resembles thosepropositions as to respect the pictoriality constraint on analysis. Propositionalfunctions are one kind of example of a prototype, form-series variables are an-other kind, but Wittgenstein?s understanding is deliberately open-ended.The appearance of prototypes in the underpinnings of truth-functionality ofpropositions is mainly tacit. But, Wittgenstein says:When a propositional element signifies a complex, this can be seenfrom an indeterminateness in the propositions in which it occurs. Insuch cases we know that the proposition leaves something undeter-mined. (In fact the generality-sign contains a prototype.) (3.24)What Wittgenstein means here by indeterminateness is precisely the appearanceof necessary connections of the given propositions to other propositions whichare unexplained by overt logical structure. It is precisely the contribution of theprototype to effect these connections. Let?s return to the case of the watch onthe table. Although Wittgenstein himself never spells out what sort of prototypethis case would involve, I speculate that the analysis might come to somethinglike the following: ?there are some objects, such that some of them are tablewise-arranged, the others watchwise-arranged, and each of the watchwise-arranged ob-jects is above at least one of the tablewise-arranged objects, and some watchwise-ity of what is described.146I take there to be two related senses of the term ?prototype? (?rbild) in the Tractatus. Oneis the sense of 3.315, which appears e.g. at 12.11.14 and descends from from Russell?s Theoryof Knowledge manuscript. The other is the sense of 3.24, which pervades the struggles of NB2(11.5.15, 19.6.15), and this is what?s in question here.150arranged-object is touching some tablewise-arranged object?. The terms ?tablewise-arranged? and ?watchwise-arranged? would presumably themselves be existentialgeneralizations over various concepts of spatial configuration: for example, exis-tential generalizations over possible positions of the hand on the watch?s dial.147The analysis as a whole clearly requires prototypical specification of the basesof truth-operations. It is moreover plausible that such specification will requirenon-first-order resources, in particular the concept of ancestral.148 For example,saying that some objects to be arranged watchwise requires saying that they areconnected.Marie McGinn (2006, 124ff) helpfully suggests that prototypes may be un-derstood as linguistic meanings149 of words, which determine a proposition onlyrelative to the context of use. Thus, a prototype would contribute, alongside themeanings of other words in a sentence, to determining a function from contextsto propositions which would be the fixed linguistic meaning of the sentence. Thefine structure of the prototypes themselves would be slurred over by the signs.150This fits well with the following remarks:The fact that there is no sign for a particular proto-picture does notshow that that proto-picture is not present. Portrayal by means ofsign language does not take place in such a way that the sign of a147As Ori Simchem pointed out, this analysis is compatible with a watch?s being embedded in atable. More generally, such a priori formalization seems pretty implausible. In the amusing entry22.6.15e, Wittgenstein acknowledged that he could not give conditions for a watch?s being on thetable. In the next couple of paragraphs, I?ll suggest that onWittgenstein?s view, the logical structureof incompletely analyzed representation is parasitic on the structure of what is represented.148Throughout this passage I?m indebted to unpublished work of Thomas Ricketts.149Or ?characters? in the sense of Kaplan (1977).150As McGinn puts it,Rather, we should think of our mastery of the ordinary language sentence in whicha sign for a complex occurs as grasp of the form of the proposition that wouldreplace it on analysis. It is in this sense that the proposition ?contains a prototype??something that is not yet an expression with sense?which specifies the form of theproposition that the sentence can be used to express, but not its sense. The form orprototype can be described by a general proposition: (Ex,Ey)xRy. However, ona particular occasion of using the ordinary language sentence to express a thought,the variable signs of the prototype are replaced by constants and the speaker usesthe resulting proposition to assert that a determinate possible state of affairs ex-ists. . . (2006, 128).151proto-picture goes proxy for an object of that proto-picture.The second sentence looks mysterious, but it fits fairly well with McGinn?s read-ing. In place of sign of proposition, Wittgenstein here considers the sign of aprototype. An object of a proposition would be an object denoted by one of itsconstituent names. But, a prototype doesn?t consist of names which go proxyfor objects. Rather, it contains variables, which do not go proxy for objects butrange over them.OnMcGinn?s view, prototypes are aspects of linguistic meaning, so that theirstructures are fixed by the context-independent nature of the words in the lan-guage. The context serves only to fill in the identities of the constituents of theeventually expressed proposition. The form of the proposition expressed by thesentence is fixed by linguistic competence with the sentence-type itself, and con-text contributes only identities of propositional constituents. For example, theidentities of the constituents of the watch would fill in the gaps in its prototyp-ical representation, thereby determining a definite representation of the watch.It is not clear that this is quite Wittgenstein?s idea. In particular, the passage justquoted continues as follows:The sign and the internal relation to what is signified determinethe prototype of the latter; as the fundamental co-ordinates togetherwith the ordinates determine the points of a figure. (8.5.15)Applying this passage to the example we?re considering, what are signified are thewatch, the table, and the possibility that the one is on the other. Thus, accordingto the passage, the signification determines the prototype of what is signified.It is the watch and the table that determine the structure of their prototypicalappearances.So, I agree with McGinn that the prototype articulates151 the internal struc-ture of what it presents, so that one and the same prototype cannot present statesof affairs with different internal structures. Moreover, McGinn is also right thataccording to Wittgenstein, the proposition a sentence expresses varies with con-text of use. But, McGinn furthermore holds that the sentence-type itself, inde-pendently of context, determines a prototype in such a way that, to obtain the151Or ?competely characterizes?, as at 16.6.15h of the previous footnote.152proposition expressed by some token in context, it suffices to resolve the valuesof the objectual variables in the prototype determined by the type. Thus, oneand the same sentence-type, under its fixed linguistic meaning, could be used toassert the existence of different states of affairs in different contexts; but thesestates of affairs will differ only with respect to their objectual constituents andnot with respect to their structures.However, in the notebook passages under discussion, Wittgenstein seems tohold that not just the constituents but also the form of the proposition expressedby a sentence may vary from context to context. That is, context contributes notjust content but also (?the real?) form.When I say this watch is shiny, and what I mean by this watch altersits composition in the smallest particular, then this means not merelythat the sense of the sentence alters in its content, but also what I amsaying about this watch straightway alters its sense. The whole formof the proposition alters. [16.6.15g]152As Wittgenstein repeated obsessively, the structure of a watch, let alone ofa mote in the air, is complicated and fluctuating. Moreover, a prototype artic-ulates153 the internal structure of what it represents, so that one and the sameprototype cannot present states of affairs with different internal structures. Butthen, prototypes are too complicated and variable to be determined by linguisticmeaning of sentence-types alone.Somehow, Wittgenstein?s idea must be that a prototype of the propositionexpressed by ?the watch is on the table? depends partly on the context of use ofthat sentence?in particular, on which watch is mentioned, and on what is theinternal state of that watch. Oddly, Wittgenstein reformulates this dependenceas follows:That is to say, the syntactical employment of the names completelycharacterizes the form of the complex objects which they denote.Now, if ?x completely characterizes y? is taken to mean means (at least) ?y super-venes on x?; at least this much of 16.6.15h does follow from 16.6.15g. The two152Another statement of this point appears at 20.6.15r.153Or ?completely characterizes?, as at 16.6.15h of the previous footnote.153paragraphs seem to present different directions of explanation; perhaps the rightconclusion to draw is that since this relationship between representation and rep-resented is logical, neither direction of explanation should be favored over theother. In any case, the essential point for my purposes is that the structure ofthe unanalyzed picture is somehow essentially tied to the particular situation itdepicts. The unity of proposition and situation is prior to the identity of theproposition: it is that the watch has the structure it has which shows the proposi-tion to have the structure it has, which shows the proposition to be what it is. Itis to the extent that a proposition depends for its nature on the articulable naturesof things that the proposition suffers indeterminateness. Analysis replaces suchcovert dependence with overt articulation.Let me just conclude this section by remarking that it was already in 1914that Wittgenstein recognized both the need for non-first-order resources in theanalysis of propositions and moreover that already by this point he had deviseda way to supply them. The second edition of the Notebooks includes an appendixwith facsimiles of notebook extracts which the editors of that volume deemedindecipherable. These extracts include the following:aRb ? bRc ? cRd ? dRe = ?(a, e)(?Rn).aRnetogether with Wittgenstein?s dating of 19.9.14. Another extract is assigned thesame date by the editors of the Notebooks, and transcribed in the body of theirtext:A proposition like ?this chair is brown? seems to say something enor-mously complicated, for if we wanted to express this proposition insuch a way that nobody could raise objections to it on grounds ofambiguity, it would have to be infinitely long. (19.9.14)These two extracts are all and only the material the editors date to 19.9.14. So,it seems that they belong to the same entry. The reference to an infinitely longsentence in the second passage must then refer to the formulation of the ancestralin the first passage. Moreover, the first sentence of the next day?s entry is this:154That a sentence is a logical portrayal of its meaning is obvious to theuncaptive eye. (20.9.14)So, I take the point of the discussion to be simple. In revealing the way in whichan ordinary proposition secures the determinacy of its sense, analysis may resortto representing the proposition as the result of a truth-operation on the terms ofa form-series. This conjecture receives some support from a later remark:The mathematical notation for infinite series like1+x1!+x22!+ ? ? ?together with the dots is an example of that extended generality. Alaw is given and the terms that are written down serve as an illustra-tion. In this way instead of ?x. f (x) one might write ? f x. f y . . .?.[22.5.15]There is no obvious grammatical antecedent of the phrase ?that extended gen-erality?. But, this passage appears in the context of a difficult discussion of thequestion how complex spatial objects could ?seem tome to be essentially things?I as it were see them as things.?And the designation of them by names seems tobe more than a mere trick of language? (13.5.15). Wittgenstein answers the ques-tion at T3.24, with the remark quoted above. But, already 15.5.15 anticipatesthis answer: ?So much is clear, that a complex can only be given by means of itsdescription; and this description will hold or not hold.? I conjecture that a weeklater, with the phrase ?that extended generality? (22.5.15), Wittgenstein refersto the generality which a form-series variable may contribute to a proposition.Thus, the 22.5.15 entry points to a diversity of logical resources which wouldunderpin the pictoriality of ordinary unanalyzed propositions. The concept ofancestral was held by Wittgenstein to be implicated in the very logic of pictur-ing. Needless to say, this possibility would have been overlooked by Russell inhis reliance on quantificational generality in the theory of descriptions.154154Further support for the claim that ?extended generality? at 22.5.15 means ?formal generality?appears at 22.6.15, where Wittgenstein mentions Whitehead?s ?Systematic statement of symbolicconventions? to explain how ?the sense of the proposition ?the watch is lying on the table? is morecomplicated than the proposition itself.?1553.4 Another role for form-series?So by April 1916, when Wittgenstein takes up the third wartime notebook, thedisjunctive analysis of the concept of ancestral was already firmly in hand. Theancestral is just one particularly natural special case of the extended conceptionof generality which analysis of the pictoriality of thought demands. It was al-ready plausible that generality could be somehow so extended, since the ancestralwould have appeared to the student of Frege and Russell as a paradigmatically log-ical notion.The third notebook begins with an attempt to state the general form of theproposition, as the result from elementary propositions by applying a formaloperation repeatedly. The ancestral then naturally reappears:(p) : p = aRx.xRy . . . zRb(p) : p = aRx [16.4.16]Here, the relation R is that relation which the result of an operation wouldbear to its basis, a relation which Wittgenstein would later identify as ?inter-nal? (T5.232). This contrasts sharply with the earlier uses of the ancestral, whoseunderlying relation would be an ordinary external relation such as that of spa-tial contiguity. At 16.4.16, then, an ordinary analytical tool is pressed into anextraordinary application.The form-series device is therefore not just another source of ordinary logicalcomplexity, but also distinctively required to explain how ?after the primitivesigns have been given we can develop one sign after another ?so on?? (21.11.16).Wittgenstein follows his sketch of this explanation at 16.4.16, by anticipatingT5.252: ?In this way, and in this way alone, is it possible to proceed from one typeto another?. Thus, to judge from the context of this thought in theNotebooks, thereason that Wittgenstein says that only a form-series allows us to pass from typeto type is that one must use a form-series to articulate the general propositionalform. If there are hierarchies of types of propositions, then somehow or otherthe general propositional form must traverse them. It is therefore because ofthe involvement of a form-series in articulating the general propositional formthat the form-series makes it possible to pass from type to type in a hierarchy of156propositions.The form-series which motivates the precursor to T5.252 is therefore not aform-series of the sort which would subserve the articulation of propositions in-volving the ancestral of ordinary external relations like contiguity or parenthood.Actually, the form-series in question isn?t supposed to belong to the articulationof a proposition at all:The above definition can in its general form only be a rule for a writ-ten notation which has nothing to do with the sense of signs. Butcan there be such a rule?The definition is only possible if it is itself not a proposition.In that case a proposition cannot treat of all propositions, while adefinition can. [17.4.16]Here, Wittgenstein insists that the articulation of a general propositional formcannot be regarded as a proposition, presumably because otherwise a proposi-tion might somehow talk about all propositions. He speculates that this use ofa form-series might rather be understood to express a definition, provided that itmakes reference only to signs, and not to their senses. This resembles some laterremarks on propositional variables, especially in that segment of the 3.31s whichderives from the PT5.00s.155 On my understanding, in the Tractatus, the pointof assigning values to a propositional variable is to represent one proposition asthe result of an operation as applied to the values of that variable; one therebyrepresents the proposition by means of the logical relations which it bears toothers. Thus, the specification of values of a variable becomes a primary tech-nique of analysis. Wittgenstein speaks of such specification of values as a matterof ?definition?. In this vein, he says that a name, being an unanalyzable symbol,cannot be ?pulled apart? by means of a definition (3.26; cf. also 3.24d). Its in-vocation of the concept of variable aside, I take this understanding of definitionto be essentially secure by the end of the second notebook.156 At 17.4.16, then,Wittgenstein seems to envisage a realm of formulations of analysis of proposi-tions, and to allow that formulations in this realm do not express propositions155Namely, 3.316-3.318.156See in particular 21.6.15i; and compare with 3.24.157but must instead be understood as codifying rules of translation between systemsof signs (3.343).157So, the form-series device plays two roles for Wittgenstein. On the one hand,it underpins the logical structure of ordinary thinking about genealogy, geog-raphy, epidemiology, and so on. In this way, it underpins what one might callthe structure of appearance, in particular the appearance of ordinary spatiallycomplex things as, indeed, really things. For, such appearances are essentially ap-pearances in signs, which partake of all the logical structure that there is. On theother hand, the very system of signs itself also invokes the concept of the ?andso on?. It is in the context of laying out the unity of language in analysis that theform-series device finds another use. This second use forms the context in whichWittgenstein initially poses the remark that the form-series device allows us topass from type to type in a hierarchy of propositions.3.5 T5.252 and the evolving conception of hierarchyNow, it seems to me that T5.252 is an example of a remark which means some-thing different in the Tractatus than its precursor does in the Notebooks. In par-ticular, I agree with Ricketts that in the Tractatus, the point of the remark israther that the form-series device lets us pass from type to type in a hierarchy,not just in formulating the definitional assemblages like that of T6, but in thisor that ordinarily senseful proposition of genealogy or geology or whatever. Yetin the Tractatus context the remark points to a deeper departure from Russellin Wittgenstein?s reception of the notion of hierarchy. This departure emergesin the gradual 1916 refinement of the account of how arbitrary propositions aregenerated from elementary ones by repeatedly applying an operation.As Jinho Kang has argued (2005, 15), in the early part of NB3 Wittgensteinhas not settled on the Tractatus idea that it is by means of just a single operationthat all propositions are generated.158 Indeed, at 11.5.16 he writes:157I don?t, at this point, propose to make sense of this notion of ?codifying? which would some-how be contrasted with the ?expressing? which ordinary signs do for propositions. I sometimespretend to understand the ?codification? of a rule as a kind of directive. But this pretense quicklygives way under pressure.158I think the internal evidence overwhelmingly supports Kang?s view, as against that ofMcGuin-158There are also operations with two bases. And the ?|?-operation is ofthis kind. |(? ,?) . . . is an arbitrary term of the series of results of anoperation.(?x).?xIs (?x) really an operation? But what would be its base?Here, the |-operation would be the Sheffer stroke, which already appears in theNotes on Logic. This bears obvious resemblance to the N -operation of the Trac-tatus but also two differences. First, a result of | is true when at least one of itsness (2002, 266), Potter (2012), that the Prototractatus was written after the third notebook. Potter(2012, 22) presents Kang?s evidence as ?Wittgenstein?s asking questions to which he already knewthe answers?. However, Kang?s evidence showsWittgenstein not only to be asking questions whichhe?d settled in the Prototractatus, but moreover couching those questions fromwithin an older con-ceptual framework. As Kang argues, Wittgenstein?s analyses of the general propositional form inNB3 show an incomplete progression toward the formulation which appears on the first page ofPrototractatus. In addition to Kang?s arguments, it?s also worth noting that the characteristic (Proto-)Tractatus usage of the term ?variable? to mean ?(structured) propositional variable? doesn?t appeareven in NB3; nor certainly, does the overbarring notation, the ternary square brackets, or the ideaof a formal concept. At the beginning of NB3 Wittgenstein is still working with the Sheffer strokeof Notes on Logic (i.e., with dyadic nand); mention of an infinitary truth-operation doesn?t appearuntil midway through NB3, and NB3 contains no mention of the shift from nand to nor. Kangslightly overreaches in claiming that ?nothing recognizable as the idea of formal series occurs inany manuscripts before NB3?, since the analysis of the ancestral appears already in 1914. But thisearlier provenance of the analysis of the ancestral actually supports Kang?s conclusion, for none ofthe mentions of the ancestral throughout Notebook 3 contain the ternary bracket notation whichappears on the first page of Prototractatus. (Actually, on that page they are not brackets but justvertical lines?but essentially the same thing.)Potter says (23) that the decisive consideration against Kang is that the NB3 remark ?my workhas opened out from the foundations of logic to the nature of the world? could only have been writ-ten after the first page of the Prototractatus. The first page of Prototractatus lists (approximately)the first six main propositions of the Tractatus. So, Potter infers, the first page of Prototractatusmust reflect an intention that Tractatus end with T6?that is, an intention that Wittgenstein endhis book with a statement of the general form of the truth-function. Thus, Wittgenstein, accordingto Potter, must have written out that list of six main entries before his sense of the significance ofhis work broadened out. In support of Potter?s view, it?s worth nothing that although PT6 itselfoccurs on the first page of the PTmanuscript, the rest of the PT6s don?t appear until page 64 (muchlater than the appearance of commentaries on the other single-digit entries).Potter?s argument seems somewhat tenuous. First, the fact that T7 does not appear with theother main sentences on the first page of PT does show that Wittgenstein didn?t intend to includeT7; unlike the other main sentences, T7 does not head its own expository section. Second, evengranted that, in writing the first page of PT, Wittgenstein intended T6 to be the last main sentence,it doesn?t follow that he intended T6 to be the last sentence. The PT6s like the T6s form a unifiedexposition, structurally quite different from the earlier sections, and it?s not surprising that heshould have recorded them all at once. If, furthermore, they summarize later developments in histhinking, it?s also plausible that they might appear later on in the PT manuscript.159bases is false, whereas a result of N is true only when all of its bases are false.Second, and much more importantly, Wittgenstein only considers applying thisoperation to one or two bases. Indeed, the tone of the remark suggests that hehad just hit on the idea that an operation might take more than one basis. At thisstage, then, there cannot yet arise any question of replacing ordinary quantifica-tion with applications of |. It is for this reason that Wittgenstein asks whether(?x). is ?really? an operation, as though this is a conclusion he?s forced to accom-modate but can?t yet see how. Part of the difficulty seems to be that its basis is nota priori finite. But Wittgenstein is also struck here by the fact that unlike p|q ,the notation (?x). f x does not express a proposition as the result of operating onpropositions which are themselves expressed by parts of that notation. This iswhy he asks what the bases of (?x). f x are supposed to be?he cannot yet imag-ine that they are the instances of f x, to all of which at once a single operationmight be applied.The Tractatus account of quantification under the general propositional formrequires Wittgenstein to distinguish truth-function from generality (5.521), andhence requires the full-fledged concept of propositional variable of 5.501 which,so far as I know, doesn?t appear until PT5.00s. At 9.7.16, Wittgenstein seems toapproach the basic idea:Don?t forget that (?x) f x does not mean: There is an x such that f x,but: There is a true proposition ? f x?. [9.7.16]I suspect, though, that at this point what Wittgenstein intends is that the quanti-fier is an operation which takes as its base not a proposition but a prototype, orcommon form of a propositional multiplicity. For, he goes on a few days later toseem to despair of the possibility of finding a single general form of all operations:If two operations are given which cannot be reduced to one, it mustat least be possible to set up a general form of their combination.?x,?x|? x, (?x)., (x). [13.7.16]Here, the ?x indicates the general form of the elementary proposition.159 Atthis pointWittgenstein seems to doubt even the definitional interchangeability of159For it alludes to the 16.4.16 proposed specification of the elementary propositions by meansof their function-argument structure.160the existential and universal quantifiers in the presence of negation, although thereason for this is not clear to me.160 The last sentence of the quoted 13.7.16 entryanticipates the remark of 5.503, which alleges instead the easiness of spelling outhow all propositions can be built up from a single operation.So it must be somewhere between December 1916 and January 1917 thatWittgenstein concludes that after all only one operation need be taken as funda-mental. For toward the end of the third notebook he writes:In the sense in which there is a hierarchy of propositions there is, ofcourse, also a hierarchy of truths and of negations [Verneinungen],etc.But in the sense in which there are, in the most general sense, suchthings as propositions, there is only one truth and one negation.The latter sense is obtained from the former by conceiving the propo-sition in general as the result of the single operation which producesall propositions from the first level. Etc.The lowest level and the operation stand for the whole hierarchy.[7.1.17]This passage presents the closest approximation in the Notebooks to the generalpropositional form as it appears in the Tractatus. As we?ll see, it illuminates oureventual quarry, the meaning of the talk of levels in 5.252. Evidently, given itsterseness and the sparseness of its context, its meaning must to some extent be amatter of speculation. But, it seems natural to suppose that when Wittgensteinrefers to a hierarchy of truths and negations, he may refer to the distinction oflevels of truth and falsehold which appears in Principia (Whitehead 1910, 42).161Thus, propositions at different levels of a hierarchy may in turn have differentlevels of truth and falsehood proper to them. But for Russell, truth and false-hood, branching as he thinks they do into a variety of levels, thereby branch intoa variety of levels of propositional function. This is because Russell takes truthand falsehood to be truth-functions, and takes truth-functions to be propositional160In the same vein he speaks at 29.8.16 of the ?usual small number of fundamental operations?;and similarly 26.11.16 has: ?All operations are composed of the fundamental operations.?161Thanks to Sanford Shieh for thinking so too.161functions. Now, I think that roughly speaking, Wittgenstein shares Russell?s as-similation of the apparent linguistic predications of truth and falsehood to whatwe would call the truth-functional connectives. So, transposed to Tractatus vo-cabulary, the passage might then be understood to be envisaging a stratificationof types of affirmation and denial. Perhaps, by ?the sense in which there are, inthe most general sense, such things as propositions? he alludes to the existence ofthe general propositional form. Thus, the suggestion would be that one graspstheoretically the general propositional form by reconceiving Russell?s stratifiedtruth-predicates (or, stratified connectives) as signalling truth-operations whichdo not mark the sense of their results. So understood, there would no longerbe any need for many affirmations and denials, but a need only for one of each.The general form of the proposition still represents propositions as standing ina hierarchy, but the hierarchy is only the hierarchy of iterated applications oftruth-operations, thus merely a hierarchy of notations and not of modificationsof sense.In the 7.1.17 statement of the general propositional form, then, Wittgensteinacknowledges a hierarchy of propositions. My contention is that this acknowl-edgement remains in the Tractatus and in particular reappears at T6 and at 5.2522.The grounds for this contention are that the entry at 7.1.17 simply is the attain-ment of the general propositional form as it appears in the Tractatus.162 Thus,the Tractatus acknowledges a hierarchy of propositions, namely, that hierarchyof ordered applications of the operation of denial by means of which all proposi-tions are eventually generated. This hierarchy, however, is independent of reality(26.4.16; 5.5561). For, it assigns a rank to a proposition simply by measuring thenumber of times a truth-operation is applied to construct it. One and the sameproposition will thus reappear at arbitrarily high levels, since it is expressed bythe double negation of what expresses it.163 Indeed, I?ve already claimed that el-ementariness does not characterize the sense of propositions. For, all there is tothe sense of a proposition is the way in which it depicts the world to be. And162Since the very first page of the manuscript of the Prototractatus contains a variant of T6, thefirst page of the Prototractatus manuscript must have been produced sometime after 7.1.17.163At least relative to a completion of analysis, one can talk about the smallest number of appli-cations of a truth-operation a proposition?s expression requires.162there are no truth-operators in the world, on whose application the geometry ofnotation forces well-foundedness. There is no hierarchy of propositions in theworld, not just as a matter of what is, but as a matter of what could be: there areonly internal relations between situations. But it is only in virtue of the struc-ture of notation, of iterated application of truth-operators, that propositions takepositions in a hierarchy.We?re now in a position to see the point of the claim at 5.252. Let me repeatthe passage:Only in this way is progress from term to term of a form-series (fromtype to type in the hierarchies of Whitehead and Russell) possible.(5.252)In the Tractatus, hierarchies characterize systems of propositional notation. Ahierarchy is simply what we would call a measure of the truth-functional com-plexity of formulas, i.e., the supremum of the lengths of the branches of theformation trees. Now, at 5.501 Wittgenstein lists three kinds of propositionalvariable. The first kind of variable has its values fixed by a finite list; the secondkind has its values fixed as the range of a propositional function. For a variable ofeither of these two kinds, some proposition in its range is such that the rank of theresult of applying a truth-operation to all propositions in its range is the successorof the rank of that proposition. Thus, arbitrarily deep nestings of the first twokinds of variable only yield propositional notations of finite truth-operationalrank. Contrast with this the use of the form-series variable in expressing the an-cestral of a relation. The range of this variable is a series of propositions, suchthat for all n, the nth proposition in the series is expressed by a notation contain-ing > n nested occurrences of the N -operator. Since it is the number of nestedoccurrences of N in its notation which determines the rank of a proposition ina hierarchy, the form-series device therefore allows the construction of a seriesof terms each of which gives way to a successor of higher rank. Thus, throughthe use of the form-series device, the ranks of propositions enter the transfinite.Or, to say at last what I take Wittgenstein to mean at 5.252: through the use ofa form-series device, one constructs a serial presentation of the bases for a truth-operation, a series which passes from level to level in the hierarchy of results of163truth-operational iteration.Now, at this point it?s natural to ask: doesn?t Wittgenstein?s remark suggestthat the form-series lets us move from one type to another in the hierarchies ofRussell and Whitehead? Those hierarchies are not, or not all, themselves imme-diately intelligible as hierarchies of results of iterated truth-operations, orderedby number of iterations. Rather, most canonically, they are hierarchies of propo-sitional functions, with a function assuming this or that position by means ofthe quantifier ranges it intensionally implicates. And, on my account, there?s lit-tle trace of this kind of structure left in the Tractatus, so it?s not clear how myexplanation of 5.252 could be right.Wittgenstein says that Russell and Whitehead repeatedly make use of for-mally generalized steps from one level of a hierarchy to the next, but that they donot admit the possibility of such steps. Perhaps another way to put this is: fromtime to time, Russell and Whitehead help themselves to formal generality, buttheir own principles preclude its official recognition. Now, if for ?formal gen-erality? we read ?typical ambiguity?, then it becomes something Wittgensteinmight have seen in Principia. Perhaps there is some significance to Wittgenstein?smention of Whitehead: although it would already be natural for Wittgensteinto mention him as a co-author of PM, Wittgenstein shared the understandingthat the theories of types are due to Russell. At 22.6.15 of Notebook 2, and inprecisely the context which I?ve argued is a crucial occasion of Wittgenstein?suse of formal generality, he mentions Whitehead?s ?Prefatory statement of sym-bolic conventions?, whose basic point is that ?whatever can be proved for lowertypes. . . can also be proved for higher types. Hence . . . it is unnecessary to knowthe types of our variables, though they must always be combined within onedefinite type? (Whitehead and Russell 1910-1913 volume 2, IX).It is then also clear to the UNPREJUDICED mind that the sense ofthe proposition ?The watch is lying on the table? is more compli-cated than the proposition itself.The conventions of our language are extraordinarily complicated.There is enormously much added in thought to each propositionand not said. (These conventions are exactly like Whitehead?s ?Con-164ventions?. They are definitions with a certain generality of form.)(22.6.15)The Tractatus concept of ?formal generality? codifies this ?generality of form?which Wittgenstein traces to Whitehead?s discussion. So, my understanding ofWittgenstein?s mention of Russell and Whitehead at 5.252 is this. The Principiapresents a hierarchical arrangement of the totality of propositions and proposi-tional functions. This hierarchy officially precludes generalizing over all proposi-tions. Still, Russell andWhitehead take implicit appeal to particular such general-izations to be licensed by the device of typical ambiguity. Now, the Tractatus alsopresents a hierarchical arrangement of propositions, which threatens to interferewith some possibilities of logical construction. Formal generality lets us con-struct a proposition which is a truth-function of a whole series of propositions,each succeeding term resulting from its predecessor by a formal operation. Inthis way, formal generalizations are what lets a single proposition climb throughWittgenstein?s series of logical types.Wittgenstein?s allusion to Russell and Whitehead elides what I take to be aphilosophically significant technical departure for the Tractatus. For in Wittgen-stein?s hands, the system of logical types is no longer a system of constraintserected for our (or God?s?) logical safety. So it would be wrong to think thatformal generality is a kind of ?system override? which can be safely exercisedunder certain conditions. Rather, for Wittgenstein, Russell and Whitehead sim-ply do not acknowledge all possible ways of stipulating values of a propositionalvariable. A proposition can be a truth-function of elementary propositions inany way that it can possibly be?that is, in any way that can be articulated. Wedon?t suffer a system of type constraints, but enjoy the full expanse of notationalpossibilities.3.6 The riddle of T6By the end of Notebook 3, then, Wittgenstein has reached the following picture.Ordinary sentences stand in logical relations of entailment and contradiction be-cause they express results of repeated truth-operations on elementary proposi-tions and are therefore truth-functions of elementary propositions. In particular,165the totality of propositions can be exhibited notationally in a series of stages, eachstage containing a joint denial of every specifiable multiplicity of propositions al-ready constructed. Thus, the ?theory of types? no longer describes a systemof constraints on what can be said, but rather clarifies how things are said, bymaking notationally manifest the logical relationships which are constitutive ofsaying one thing rather than another.Wittgenstein announces, in the Prototractatus: ?the theory of types becomesclear.? This is one of the five or six percent of the Prototractatus entries whichdoes not reappear in the Tractatus. The editors of the published version (Wittgen-stein 1971) observe that its numbering as PT5.00 must be incomplete. However,they suggest that it probably belongs between or immediately after the followingentries, which anticipate 5.251 and 5.252a:A function cannot be its own argument, whereas an operation cantake one of its own results as its base. [PT5.00161]It is in this way and only in this way that the step from one type toanother in the hierarchy is possible. [PT5.00162]However, it?s also possible that the whole of the PT5.00s can be read as an elab-oration of PT5.00. For, when he talks about the theory of types becoming clear,he refers to its transformation from a system of type constraints into an explana-tion of truth-functionality. And it is the PT5.00s which expound the results ofthis change.To put my interpretive conjecture as a concrete counterfactual: had Wittgen-stein preserved the PT5.00s in the Tractatus, then PT5.00 would have been num-bered T5.001, and PT5.00x might have become T5.001x. It should be noted thatPT5.00 is the only PT entry whose numbering is claimed to be incomplete by theeditors.164 Moreover, Wittgenstein?s draft of PT contains many corrections ofnumbering. It therefore seems implausible that Wittgenstein simply overlookedthe minor decision whether to append 1611 or 1621 to the string 5.00. A nat-ural hypothesis is that he didn?t complete the numbering of 5.00 because of the164One entry onmanuscript page 101, which becomes T5.154, lacks a number altogether, but thisonly gives the somewhat trivial mention of an urn containing variously colored balls; moreover,the absence of a number there is flagged by Wittgenstein, whereas the numbering of PT5.00 isunflagged.166extent of induced renumberings of other entries. Now, no entries with indicesof the form 5.0016. . . precede PT5.00 in the manuscript of PT, and PT containsonly three such entries 5.00161, 5.00162, and 5.00163. So, numbering PT5.00as PT5.0016. . .would not have induced much inconvenience of renumbering ofother entries. On the other hand, the required renumbering predicted my hy-pothesis is quite extensive, since most of the PT5.00s (twenty-two entries) doprecede PT5.00 in the manuscript.165The Prototractatus is evidently a close ancestor of the Tractatus. Much of thedifference between the texts consists in shifts of terminology or slight incremen-tations of numbering. So far as I can tell, there are only two really gross revisions.Michael Kremer (1997) has observed that Wittgentein elevates his statement ofthe ?context principle? from PT3.202 to T3.3, and argued that this indicatesthat the Tractatus is a ?transitional work?, occupying a position of philosophicalmovement from Wittgenstein?s pre-Tractatus views to the views of the so-called?middle? or ?later? Wittgenstein. But the grossest revision to the Prototractatusis not much discussed in the secondary literature.The PT5.00s form a sustained exposition of Wittgenstein?s distinctive logi-cal theory. This includes the explanation of the formula at T6, and the theoryof propositional variables. Save for a familiar point on truth-operations whichdate to the Moore Notes, the known antecedents of the PT5.00s mainly appear inthe attempts on the general propositional form from Notebook 3, which havebeen discussed already in this paper. The PT5.00s must culminate developmentswhich followed the end of Notebook 3 in January 1917. But, what must beenintended as a summary of these perfected developments disintegrates in the Trac-tatus, into an anonymous but distinctive diaspora which populates the 5.2s, the5.50s, the 3.31s, and the 4.127s. My hypothesis is that the PT5.00s expound whatWittgenstein in the Prototractatus to be his ?theory of types?. He broke up this165In support of McGuinness-Schulte, it?s worth noting that the relationship of PT5.00to PT5.00161 resembles the structure of various other continuous passages, including 3.332,PT3.00161, NB107, and NB96. This observation doesn?t refute my textual conjecture since it issurely PT3.00161 rather than PT5.00-PT5.00162 which is the antecedent of 3.332. But in any case,my interpretive claim doesn?t stand or fall with the textual conjecture. The point is just that thePT5.00s can be read as though they are a commentary on what is numbered as PT5.00. Readingthem in this way illuminates 7.1.17, 5.252, and, I hope, ultimately T6167exposition for the same reason that he excised its lead remark, PT5.00. But whatwas this reason? Doesn?t the theory of types become clear in the Tractatus?The theory of types does not become clear in the Tractatus, for the same rea-son that the formula at T6 is never properly explained. Many commentators(even quite good commentators such as McGinn 2006, 233) have taken Wittgen-stein to explain the notation of T6 at T5.2522, and assumption has led them toconcoct artificially ?linear? orderings of the totality of propositions, under theconceit that the totality of propositions itself constitutes a form-series. How-ever, this confuses two quite distinct roles into which Wittgenstein pressed theconcept of iteration. For, 5.2522 explains only a form-series generated by anoperation which takes just one proposition as base. In contrast, the purportedform-series described at T6 would be generated by an operation taking a propo-sitional multiplicity as its base. Thus, commentators confuse the articulationof the general form of the proposition, with a means of specifying a base for atruth-operation. Wittgenstein?s implementation of the truth-functionality thesisrequires that these two functions of the iteration concept be distinguished. For,the determination of truth-value of the result of applying a truth-operation to thetotality of propositions would depend on the result of that determination. EntryT5.2522 elaborates on the third way of stipulating the range of a variable, andtherefore ought to follow the T5.51s and T5.52s, as elaborations of 5.501f.The frustration of commentators? attempts to concoct a linear ordering of thetotality of propositions can be seen as a consequence of Wittgenstein?s attempt toenforce a separation between the two roles of the concept of iteration in his earlyphilosophy of logic. In its use to specify bases of truth-operations, Wittgensteinonly explains the iteration of unary operations. In contrast, Wittgenstein simplydoes not explicitly license the use of multigrade operations in specifying opera-tional bases. However, a proposition can be seen as the result of a truth-operationon any multiplicity which is formally specifiable, since the range of a variableis purely a matter of stipulation. And there seems to be no sense of ?formalspecifiability? which would include form-series generated by unary but not bymultigrade operations. Here it is striking to compare 5.2522 with its antecedentPT5.005351:168Let us write the general term of a series of forms like this:|x?0, x?,O?(x?)|The x?0?s are the initial terms of the series, the x??s are terms arbitrarilyselected from it, and O?(x?) is the term produced from the x??s bymeans of the operation O?(??) as the series proceeds.In the Prototractatus, the explanation of multigrade-generated form-series appearsin the place proper to explanation of a means of specifying the basis of a truth-operation, that is, as an elaboration of the antecedent of 5.501. But it appears thata multigrade-generated form-series is precisely the sort of multiplicity that thegeneral propositional formwould take as its range, were the general propositionalform to be seen as a variable. We thus seem to have in the Prototractatus anunflinching embrace of impredicativity.The change from PT5.005351 to T5.2522 is unquestionably deliberate, sinceit requires both a deletion of the bar superscripts and a change in number of theaccompanying explanation. I think this change evinces an attempt on Wittgen-stein?s part to distinguish between the presentation of the general propositionalform and the form-series presentation of bases of truth-operations. AlthoughI?m not exactly clear on how Wittgenstein wished to effect the distinction, let mesuggest two hypotheses.First, it is natural to suppose if [ p?,? ,N (? )] were really to be a variable, thenit would involve an implicit generalization over all formal concepts. For, the mid-dle term ranges over all specifiable multiplicities; together with the third term itsignifies the operation which takes any multiplicity to its joint denial. However,since we?re supposing the bracket expression itself to be a variable, therefore itwould itself have to be a value of the bound variable it contains. So as Sundholm(1992, 70) suggested, the T6 variable seems to involve some kind of circularity.166One tempting response, suggested by Peter Sullivan (2004), is to reject the state-ment of 4.53 that the general propositional form is a variable. Now, Sullivandoesn?t develop this proposal in detail.167 But here?s one way it might go. An ex-166I can?t resist quoting his remark: ?given Wittgenstein?s harsh words against Frege and Russellconcerning impredicativity (4.1273) some care on his part would not have been out of place here,if only to show that the author of the Tractatus was aware of the lurking impredicativity? (70).167Well, Sullivan?s strategy is to argue that Wittgenstein tries to forestall the possibility of con-169pression of generality is a variable which signifies a formal concept, and the valuesof the variable are the items which fall under it.168 So, the general propositionalform can be understood as a variable only if there is some common distinguishingmark of all formal concepts. Perhaps Wittgenstein implicitly rejects the requiredhypothesis that there is such a formal concept as that of formal concept itself.169This seems to be an insinuation of the phrasing of 5.501, with its emphasis on theentirely optional or even arbitrary character of catalogues of methods of stipulat-ing the bases of truth-operations: ?We can distinguish three methods of descrip-tion? (5.501, emphasis in original). Moreover, in the Prototractatus antecedent ofthis sentence, the word ?can? is unemphasized, which (mildly!) suggests that inthe revisions accompanying the dissolution of the 5.00s he decided to amplify thetheme of openendedness of formality.However, in the Tractatus, Wittgenstein says that it should be possible to findan exact expression for all the ways in which propositions may arise by repeatedlyapplying the operation N (5.503). The possibility of such an exact expressionseems to me to sit ill with the thought that formality might not itself be formal.Frankly, the maneuver feels like a bit of a dodge. And if he really thought itwas viciously circular say that the general propositional form is a variable, thenwhy does he still say it at 4.53? I don?t know of any better way of developingSullivan?s suggestion, but in the absence of something better, I don?t find thesuggestion satisfactory.170There is a simpler story. For Wittgenstein, analysis can represent a proposi-tion as the result of a truth-operation on any multiplicity of propositions distin-guished by some internal mark or feature. Any genuinely internal mark or fea-ture is a legitimate means of specification, so long as it respects the logical unitybetween proposition and presented situation. The unitymay be theminimal one:just, that this is how things are. Of course, there is no reason to consider suchstructing a variable which ranges over the elementary propositions. He doesn?t give any textualevidence for this claim, and doesn?t mention that Wittgenstein tried twice to construct a variableover elementary propositions in NB3. See Chapter 1, ?3.2 for more discussion.168This is 4.127, give or take a use-mention issue.169On this view, available bases of truth-operations might stand to the totality of propositions assets stand to the class of all sets.170Sullivan, strangely, doesn?t mention the revisions of the PT5.00s.170a representation in itself to be in any respect revelatory. One needs somehowto fit every proposition into the system of all propositions, and this ultimatelyrequires that analysis be wellfounded. Only if analysis constructs a wellfoundednotational arrangement of propositions can it serve as an elucidation of thought.Thus, it is entirely possible to apply truth-operations to the totality of proposi-tions, at least in the sense that, for example, it is possible to jointly deny a propo-sition and its negation. However, this possibility is irrelevant to the project ofanalysis, because it does not further the articulation of truth-functionality.171So, despite the revision from PT5.005351 to 5.2522, Wittgenstein maintainsno in principle objection to specifying the bases of a truth-operation by means ofa form-series variable which itself involves a variadic generating operation. How-ever, he may have recognized no pressing practical need for such an extension ofthe form-series device in analysis. The change to 5.2522 merely serves to clarifythe distinction between the two appearances of form-series variable in Wittgen-stein?s exposition. Wittgenstein may then have meant what he said, when he said?the general form of the proposition is a variable?.1723.7 An alternative readingThomas Ricketts (2012) presents a systematic reconstruction of the role of form-series in the Tractatus framework. Ricketts applies this reconstruction to someimportant interpretive questions, including the proper reading of 5.252. I?m sym-pathetic to much in Ricketts? program. In this section, I want to point out somedifferences between our readings.173Ricketts? paper opens by reviewing the importance of higher-order general-ity to the logicist projects of Frege and Russell, and then raises the question:?what becomes of higher-order generality in the Tractatus?? Ricketts maintains,171There are some affinities to Wittgenstein?s response to the paradoxes in Gaifman (2000).172Naturally, I?m not completely confident of this conclusion. But the two hypotheses justsketched are the only ones I know of, and the former strikes me as tantamount to alleging a kindof intellectual dishonesty. My own limited experience with the Tractatus, unlike that with certainother books, has been that more and more of it becomes intelligible with time and patience. Thismakes me want to give its author the benefit of the doubt.173However, I?m thoroughly indebted to Ricketts? work. The criticisms to follow should be takenas praise.171as do I, that Wittgenstein rejects the possibility of impredicative higher-orderquantification altogether. Nonetheless, Ricketts also argues that the Tractatuscontains resources to simulate Russellian predicative higher-order generalization.This ?simulation? is achieved by means of formal generality, i.e., by means of theform-series device.174Let me now summarize Ricketts? envisaged construction. Ricketts doesn?tconcern himself with the details of specifying form-series themselves. Rather, heassumes,wherever there is a procedure for recognizing members of a class ofRussellian first-order sentences in terms of logically significant fea-tures of their construction, there is a formal law that generates aform-series whose members are those sentences. (Ricketts 2012, 18)In particular, there will be a form-series C0(x),C1(x), . . . whose terms are exactlythe first-order formulas in the free variable x.175 Now, B be an atomic formula,174 Ricketts does allow that generality proper in the Tractatus might reach slightly beyond whatcan be achieved by purely first-order means (as I?ve identified them in Chapter II). Say that a (unary)elementary sentence function is the result of removing a name a from an elementary propositionA. Ricketts says that such an elementary sentence function can itself be removed from A; then,according to him, the values of the resulting variable include all elementary propositions contain-ing the name a. I hold that talk about ?removal? or ?replacement? must be explained in termsof abrogating some arbitary formula conventions directly in virtue of which signs belong to thebedeutungsvoll symbols to which they do belong. But it is only genuinely simple signs, i.e., names,which receive their meaning directly from arbitrary conventions. So, I?m committed to the viewthat only genuine names can be ?removed? or ?turned into variables?. One can remove a succes-sion of names from an elementary proposition, but the range of the resulting variable will be muchmore limited thanwhat Ricketts envisages: it will include only those elementary propositions fromwhich that variable can be constructed by removal of names. Thus, I individuate expressions (inthe sense of 3.313) much more finely than does Ricketts. That is, Ricketts appears to hold thatany name a determines (or is?) a single expression ? (a) which marks the sense of all elementarypropositions containing the name a, and then tries to explain how this expression could mark thesense of all propositions whatsoever that contain a. According to me, however, it is not correct totalk about a name as an expression or as determing an expression uniquely. Rather, an expressionis only determined by a name together with a proposition. There will be (presumably) infinitelymany different expressions ? (a,A1),?2(a,A2), . . . depending on the form of the proposition Ai .Each ? (a,Ai ) will mark the sense of some propositions containing a, of course including Ai itself.The classes of propositions correspondingly marked out by ? (a,Ai ) then disjointly exhaust thepropositions containing the name a. All propositions in the same ? (a,Ai ) are instances of thesame ?logical prototype? in the sense of 3.315; but moreover, they will result from prototype bycontaining a ?in the same place?.175Ricketts? construction therefore requires the existence of ?free variables?. I think that theTractatus conception of objectual generality excludes them, but do not press the point here.172presumably also in the free variable x, and letAbe an arbitrary containing at leastone occurrence of B . Then, Ricketts assumes, for any selection from amongstthe occurrences of B in A, there is a form-series D0,D1, . . . such that each Di theresult of uniformly replacing the selected occurrences of B in A with Ci .176 Thedisjunction of the Di will then express a generalization over all the first-orderdefinable properties. For example, we could in this way generalize with respectto the first occurrence of F in the formula ?x(F x ? F x), obtaining a formula?X?x(X x ? F x).Ricketts? plan extends well beyond generalizations over first-order definableproperties. A few refinements would open a path to generalization over first-order definable polyadic relations as well. But more importantly, we can nowconsider the totality of formulas in one free individual variable which involvesimulated generalization over first-order definable relations. These formulas,Ricketts assumes, may themselves also be collected into a form-series, and so thatgeneralization over them would be simulated as well. In this way, we might suc-cessively construct the hierarchy of Principian ?orders? of properties of individu-als.177 Introducing variables of higher type, ranging over properties of properties,relations between properties and relations, etc. opens the prospect of simulatedhigher-type generalizations as well, with higher-type pseudoquantifiers similarlystratified into Principian ?orders?. In this way, Ricketts speculates, the Tractatusmay recover the expressiveness of the entire ramified hierarchy. Finally, Rickettsconcludes, ?I should like to think that this is what Wittgenstein has in mind,when he speaks in 5.252 of advancing from type to type in the hierarchies ofRussell and Whitehead? (2012, 21).Now, before presentingmy response to all this, I should say that the foregoingsummary has been tailored to highlight points of disagreement. On the whole, Ifind myself sympathetic with much of Ricketts? philosophical-interpretive aims.But, let me turn to the disagreements. First, of course, is we read 5.252 quite dif-176This formulation is a bit more general than that of Ricketts. But I suspect it should be liber-alized somewhat further still. For example, we need to be able to accommodate generalizing withrespect to both occurrences of F in Fa ? F b .177The Principia meaning of ?order? is different from what is now standard: properties at a fixedlevel, say properties of an individual, recursively stratify into orders, according to the level andorder of the quantifiers required to express the properties.173ferently. Second, while I think that Ricketts has the right applications in mind forthe logical resources of the Tractatus, it seems to me that the underlying accountof those resources leads to some technical problems, and also to some interpretivepositions which I can?t quite follow all the way.Ricketts says that 5.252 alludes to the possibility of constructing, or sim-ulating, the ramified hierarchy in Tractatus framework. On Ricketts? account,form-series serve to generate the bases of the infinitary disjunctions and conjunc-tions which simulate predicative higher-type quantification. Thus, for example, aform-series may contribute to simulating predicative generalization over Princip-ian ?first-order? properties of an individual. Such form-series may appear nestedinside one another, thus simulating nested higher-type quantifiers as well. How-ever, Ricketts? account never invokes a form-series such that each of its terms isfollowed by a term of higher type. And yet, this seems to be just what 5.252 saysthat the form-series are supposed to make possible. So Ricketts? reconstructiondoesn?t seem to fit the letter of the text.178On my reading, in contrast, by ?operation? at 5.252, Wittgenstein?s meaningfits something as simple as the N -operation itself. Thus, an example of progressfrom ?type to type? would be the seriesA,NA,NNA, . . . which the variable [A,? ,N? ]would generate. Of course, such a series adds no expressive power, but other se-ries with the same property do, for example a series of formulas whose disjunc-tion expresses the ancestral. The problem forme, as Ricketts would quickly pointout, is that the step from A to NA (or to ?A) is certainly not a step to a higherRussell-Whitehead type. However, as alreadymentioned, the 7.1.17 notebook en-try explicitly mentions a hierarchy of ?truths and negations?, and he says that asingle operation generates all propositions in this hierarchy. So the only questionis why Wittgenstein says ?Russell and Whitehead?. I suggest that Wittgensteinsees the Russell-Whitehead stratification as dissolving into the notational strat-ification induced by iteration of the N -operator. In the Tractatus context, theform-series handles easily the transitions from level to level which aren?t possiblein the context of ramified type theory.178I also don?t know of any place where Wittgenstein mentions simulating higher-type quantifi-cation, or envisages applying it. In contrast, there?s good evidence he envisaged direct applicationof the form-series device in analysis, as I argued in ?3.174Let me now turn to the question of reconstructing Wittgenstein?s basic non-first-order resources. Here, Ricketts doesn?t bother with the details of specifyingform-series. Rather, he assumes that formulas can be assigned a logically arbi-trary lexicographic ordering, with respect to which there is a reasonable notionof effectiveness for infinite classes of formulas. Then, he stipulates that every?effective? class of signs may be generated as the terms of a form-series. It seemsthat Ricketts may envisage the possibility of arithmetizing the propositional signsof the Tractatus, because names, and eventually the signs of elementary proposi-tions, are constructed in the progress of analysis. Thus, it is just up to us, in theprogress of analysis, to stipulate a notation which makes sense of talk of effective-ness, perhaps simply by insisting on arithmetizability.179Ricketts? construction also requires some auxiliary syntactical machinery.For example, it presupposes the existence of formulas containing variables ofarbitrarily high type. Moreover, some principle must guarantee180 that, for anyform-series S of formulas in a free variable of some (ramified) type ?, and any for-mula A containing a free variable of type ?/?, there exists a form-series whichgenerates every result of instantiating the free variable in A to a formula in S.Moreover, the idea of instantiation here is not straightforward?even in the ear-liest stages, questions arise like how to spell out the instantiation of the X inXa ? X b to formulas like F x or even Rxy. The complications here seem tolead far from the characteristic forms and concerns of the Tractatus, and it is notclear what might be gained interpretively from spelling them out.But let me return to the stipulation that every ?effective? class of formulas isgenerated by a form-series. I am not sure that unqualified talk of effectiveness isnaturally suited to the problem of stipulating bases of a truth-operation. For onething, arbitrary effective classes of notations do not in general have any logicallysignificant commonality. Being a palindrome, or having an even G?del number,are natural examples of decidable properties of notations, yet they are not logi-cally significant. On the other hand, the property of being a formula is not such a179Here I?m indebted to conversation with Ricketts.180Actually, Ricketts doesn?t need the full strength of the following statement. But as far as I cansee, weakenings require further complication, and complication is the main worry here. Insofar asI have worries about excessive strength, these address the ?base clause? of Ricketts? characterizationrather than this ?closure clause?.175natural or obvious property, but requires specially tailored construction. With ageneral notion of effectiveness in hand (perhaps underpinned by a prior comandof a theory of arithmetic?), one then needs to erect on top of this general no-tion a theoretical superstructure of ?logical significance?. We might begin witha theory of arithmetized syntax, and then extend this to some (hopefully still ef-fective!) proto-Tarskian criterion of permutation-invariance. But, it is worryingthat it should take work to turn the basics of this analysis into something evenseems like logic, let alone like the Tractatus.181Another symptom of the decidedly nonlogical feeling of the blanket appealto effectiveness is that it seems to obviate the independent concept of objectualgenerality. By generating the class of first-order formulas, we a fortiori generatethe class of first-order formulas which differ from each other with respect to thename which appears at one particular position; and presumably this class can besifted out effectively. Any notion of effectiveness adequate to Ricketts? purposeswill be satisfied by the classes of instances of objectual generalizations. It seemsto me that a notion of complexity according to which the class of formulas witheven G?del number is simpler than the class of instances of a universal general-ization is not a fundamentally logical notion.In fairness to Ricketts, at 22.5.15 Wittgenstein does actually envisages thepossibility of generating the instances of a universal generalization by means of aform-series.182 But, I find this and related passages in the Notebooks to be philo-sophically problematic. There are two reasons for this.First, such passages as 22.5.15 belong to a worrisome strand of Wittgenstein?sthinking where he seems to assume that any class of propositions sharing a nat-ural common feature can be generated by a formal rule. This is explicitly sug-181Perhaps the clearest way for me to bring out this point is to refer the reader to AppendixA, which develops a connection between a simplified version of the Ricketts proposal and thehyperarithmetic hierarchy.182?The mathematical notation for infinite series like1+x1!+x22!+ ? ? ?together with the dots is an example of that extended generality. A law is given and the terms thatare written down serve as an illustration. In this way instead of (x) f x one might write ? f x. f y . . .? ?(22.5.15).176gested as late as 23.11.16, toward motivating a hypothesis that the class of elemen-tary propositions might be generated by means of a form-series. The worrisomestrand is most prominent in NB3. But it also seems to motivate the somewhatdispiriting 5.512, which says that a common rule governs the construction of allpropositions which are equivalent to ?p.183Given the subtle persistence of the worrisome strand, it seems indisputablethat Wittgenstein did not think his way through to any systematic account ofthe workings of the form-series device. But, there is a second problem with suchpassages as 22.5.15, which disappears from all mentions of form-series in the Trac-tatus. In the Notebooks we find repeated mention of form-series generated by op-erations which systematically introduce new names, or new elementary proposi-tions, into their results. However, in the Tractatus, none of the three form-series-like examples I know of?appearing at 4.1252, 5.512, and 6?have this feature.That is, none of the form-series-generating-operations in the Tractatus introducenew objects or elementary propositions into the constructed series. Instead, theymanipulate truth-operations and bound objectual variables. I take this shrinkingapplication of the device to reflect a considered judgment on Wittgenstein?s partabout the relationship between what we construct and foresee, and what we donot.184In particular, as Sullivan has pointed out, a proposition (or propositional sign,or formula) is supposed to be the result of an operation on another proposition invirtue of an internal relation between the two (2004, 52). But operations that suc-cessively introduce more and more new objects must distinguish between objectsby introducing them in one order rather than another, and the forms of objectswill not in general underwrite such distinctions. Now, perhaps in response tothis observation, Ricketts stipulates that an operation can exploit ?logically in-significant? features of the names of the objects?for example, perhaps the namesare indexed by integers. Such features would then yield a way of successively sin-183 Ramsey (1923, 472) noted that this would ?presuppose the whole of symbolic logic?. Thanksto Peter Sullivan for this observation.184The suggestion to simulate all predicative quantification, and a fortiori all objectual quantifi-cation, seems to me in particular to overstep this boundary. There ought to be not to be a presup-position that objects can be numbered and typed at the very outset of analysis. Yet the very firstinstance of simulated higher-type generalization will require this.177gling out each object of a given logical form. However, Wittgenstein holds thatan object can be singled out from others only if it has properties the others don?thave (2.02331). Hence, the Ricketts account, together with 2.02331, requires thatobjects of the same form respect the identity of indiscernibles. But Wittgensteinrejects the identity of indiscernibles (5.5302).Perhaps there?s a general interpretive moral I want to draw from this critiqueof Ricketts? account of the form-series device. On Ricketts? account, it appearsthat the names of objects are something that is stipulated.185 Insofar as the num-ber and forms of objects are reflected in the numbers and forms of their names,it then follows that the numbers and forms of objects are themselves open tostipulation?or, more broadly, to some kind of anticipation. Although the Note-books toy with anticipating the forms of elementary propositions, Wittgensteinturns this aside in the 5.55s of the Tractatus, a passage which almost entirely post-dates the Notebooks.186The general drift of the 5.55s seems to be this. Analysis lays out internalrelations between propositions. But, propositions are pictures of reality, andtherefore participate together with it in a common form. But then analysis isresponsible to reality, which we cannot foresee.187185With this last remark I have in mind especially the presupposition in Ricketts of the countabil-ity of names. For Ricketts, analysis must in its very first steps survey the totality of names?notwith the treatment of objectual quantification, but with the treatment of simulated predicativequantification. For me, the analysis of objectual quantification eventually requires the use of par-ticular names, because I take objectual quantification to be represented by removal of a name froma proposition. However, in my framework, it may be possible to give an account of the idea of re-moving a complex from a proposition, in such a way that analysis can put off the question whetherit is a name or a complex that has been removed. I speculate that it is this sort of problem whichmotivates the seemingly bizarre proposal at 5.9.14, although the treatment of quantification?inparticular, the separation of truth-function and from generality?was not at that point fully inplace.186In particular, I take it that in the Notebooks, Wittgenstein did contemplate the possibility thatthe forms of elementary propositions are constructed?in particular at the outset of the third note-book. The 5.55s ensue from the settling of his perspective on the role of elementary propositionsin the statement of the general propositional form, which takes place between November 1916 andJanuary 1917.187I don?t think that this talk of reality, or of ?things out there? subverts ?the notion that analysisproceeds, starting fromwhole judgments, being responsible to inferential patterns only? (Goldfarb2001a, 192). Goldfarb?s remark is in reference to Frege rather than to Wittgenstein, so I don?tdisagree with that remark. Wittgenstein?s setting is special, because of the formal unity betweenproposition and situation.178In the next chapter, I?ll sketch a lightweight, partial account of the concept ofform-series, and present some applications of the resulting device. I disavow anyclaim that Wittgenstein himself considered this account, let alone that he carriedout the constructions I exhibit. On the other hand, the account has the follow-ing virtues. First, it is simple and definite. Second, it requires only resources thatare incontestably available in the framework of the Tractatus. Third, it respects,perhaps as strongly as possible, the requirement that it be ?easy to recognize?whether something is a value of the resulting propositional variables. Fourth,it suffices for natural applications that Wittgenstein did envisage. For these rea-sons, the account might be considered an ?existence proof? of the feasibility ofWittgenstein?s form-series idea.I do think, again perhaps not in disagreement with Goldfarb, that Wittgenstein does not, in theTractatus, repudiate entirely the idea of the NB2, that ?when a proposition is at least as complexas its reference, then it is completely analyzed? (9.5.15). His understanding of complexity seems,in 1915-1916, to rotate up, around this fixed point, to meet the idea that ordinary language is inlogically perfect order as it is.179Chapter 4IllustrationsIn the Tractatus Wittgenstein says that all propositions can be expressed in a sys-tem of signs with a surprisingly simple underlying scheme of generation. Accord-ing to this scheme, every logically complex sentence consists of a verdict N and apointer ? . The pointer indicates some propositions, and the verdict is that all ofthem are false.Clearly the power of the resulting system depends on just which multiplic-ities of propositions can be indicated by a pointer. If only finite multiplicitiescan be indicated, then the resulting system coincides expressively with finitarytruth-functional logic. If every set of propositions can be indicated, then everyclass of truth-possibilities for elementary propositions corresponds to the truth-condition of some constructible sentence. Neither of these options is interesting,and neither is Wittgenstein?s intention.Rather, Wittgenstein says that the aim of the pointer must be somehow ?stip-ulated (festgesetzt)? (5.501). At 5.501 he gives three examples of such conditions.The first is to indicate a finite bunch of propositions by listing them all. Thesecond is to give a propositional function whose values are the propositions indi-cated. The third is to write down a proposition and a procedure for generatingpropositions from propositions, thereby indicating the totality of propositionswhich can be constructed from the given one by repeatedly applying the proce-dure.The first kind of pointer should be clear. On the other hand, the second andthird kinds require further explanation. I?ve developed an analysis of the secondkind of pointer in related work, and the aim of what follows is to present someresults concerning pointers of the third kind. Since my account ofWay 2 pointersis nonstandard, I will sketch it here quickly, and then proceed to Way 3. By theway, I?ll henceforth refer to pointers as ?(propositional) variables?.1804.1 Ways One and Two, a sketchOn my view, Wittgenstein?s analysis of quantification is rooted in his conceptionof propositions as pictures, and in particular in the question how a picture por-trays an object or some objects. I take Wittgenstein to think that for any objectand any proposition, there is a way in which the object is thereby said to be.Wittgenstein?s analysis of quantification on my view takes as fundamental notthe Tarskian concept of satisfaction sequences and so on but rather the identityand distinctness of the ways in which elementary propositions say that objectsare. Elsewhere I show how to reduce the concept of way in which an arbitraryproposition says some objects are to the concept of how elementary propositionssay some objects are.The concept of the way in which a proposition says an object is should not beconfused with a Russellian concept of propositional function. There is a familyresemblance between the notions but there are fundamental differences as well,and these differences havemathematically significant manifestations in extension.For example, an elementary proposition Rab with respect to a that it bears R tob . The totality of propositions which say this of something or other is the rangeof a propositional variable constructed from Rab and a. For example, Rcb sayswith respect to c what Rab says with respect to a and so it belongs to the rangeof the variable. Same for Rd b ,Reb , and so on. However, suppose Rb b said of bwhat Rab said of a. Well, Raa says with respect to a what Rb b says with respectto b . Hence Raa says with respect to a what Rab says with respect to a. Andthat is absurd. Thus, in particular Rb b does not fall in the range of the functiondetermined by Rab .This conception of the relationship between names, elementary propositions,and the resulting Ausdr?cken determined by arbitrary propositions yields an ac-count of quantification which is in a sense to be made precise expressively equiv-alent to first-order logic with equality. Making that precise is not my concernhere. Instead here is a rough and ready guide to notation.To construct variables ofWay 1, simply enclose a list of propositional signs inparentheses. Thus (A,B ,C ) indicates A, B and C . As for Way 2, we will simplywrite ?aA to mean what A says with respect to a. Thus, ?aA is a variable which181indicates all propositions which say that of something.The construction of variables can be iterated. Thus, ((A,B), (C ,D)) ulti-mately indicates A,B ,C ,D because it immediately indicates (A,B) and (C ,D),which respectively immediately indicate A,B and C ,D . Similarly, (?aA, ?bB) im-mediately indicates pointers ?aA and ?bB which in turn indicate propositions whichsay of something respectively what ?aA says with respect to a and what ?bB sayswith respect to b .Let?s now return to the interaction between pointers and verdicts. If A is apointer which ultimately indicates B ,C ,D , then the verdict NA says that all ofB ,C ,D are false. We will take sentences to derivatively indicate themselves, sothat if on the other hand A is a sentence, then NA expresses the falsehood of theproposition indicated by A, to wit, the falsehood of A.188 Note, crucially, thatanything of the form NAmust be a verdict, no matter whether A is a pointer ora sentence.Pointers and verdicts combine to yield almost familiar idioms. In the specialcase where A,B , . . . are sentences, then NN (A,B , . . .) and N (NA,NB , . . .) expressthe ordinary disjunction and conjunction of what A,B , . . . say, so in such a casewe will write instead A?B ? . . . and A?B ? . . .. But now suppose A is a pointer.Then NN (A) expresses the denial of the proposition that every proposition ul-timately indicated by A is false. Thus we abbreviate NN as?. In particular, ifB is a proposition, then??bB asserts the disjunction of propositions which sayof something what B says with respect to b . On the other hand, since one canask of an arbitrary proposition what it says with respect to b , it follows thatfor a given proposition C one can ask what NC says with respect to b . Havingthereby constructed the variable bNC we can form the denial N ?bNC of everyproposition which bNC indicates, i.e., the simultaneous denial of the proposi-tions which say of something that that thing is not how C says that b is. Thus,for every proposition C and every object b , there is an expression of the simulta-neous truth of all propositions which say of something what C says with respect188I think that probably Wittgenstein took expressions of the kind (A,B , ...) not to be variablesbut rather to be themselves the multiplicity of bases of application of N . That is, I take his atti-tude to have been that of Russell in PoM that only infinite multiplicities require the recourse tointensions.182to b . That is to say, there exists a uniform analogue to universal as well as existen-tial quantification. We will not introduce an abbreviation here because we havenot yet sufficiently surveyed the principles of pointer construction to prove thatfor each pointer A to propositions B ,C , . . ., there is a proposition to the effectthat their denials NB ,NC , . . . are all false, though this result is evident for point-ers constructed by iterations of Ways 1 and 2 in tandem with denial verdicts. Iregard Fogelinite skepticism about universal quantification itself in this contextto be completely out of the question.Since the notation just presented is slightly peculiar, let me hint at some waysof projecting into it soundly some familiar principles of interpretation. In anexpression ?aA, the governing ?a should be regarded to bind all occurrences of ain A which are not already bound. Once namelike occurrences of a become sobound, then their a-ness completely disappears. One might thus think of ?a an?anonymizer?. It turns the occurrences of the ?constant name? a into occur-rences of a variable name which is indeterminately any of the names a, b , c , . . .which are appropriate to the context.Upon falling under control of an abstract, a name a loses its individual name-like aspect, and occurrences of name a itself become merely apparent. Termsenjoying such an anonymous second life might well be supposed to have entereda new logical category. This change in category might emphasized syntactically.In particular, suppose we simply introduce a new syntactic kind of expressionsx, y, z , . . ., and replace the ?merely apparent? occurrences of names a, b , c , . . . withoccurrences of x, y, z, . . .. Then, an indefinite propositional constituent, a vari-able name, gives way to a definite constituent, a name-variable.The shift from variable names to name-variables underpins Wittgenstein?smore or less familiar usage of quantifier notation. Thus, for WittgensteinN ?aFa becomes ??xF x??aRab becomes ?xRxb , and??b (Rba ?N ?c(Rcb )) becomes ?x(Rxa ???y(Ryx)).However, the fact remains that for Wittgenstein, there is some a such that N ?aFa,183viz., ??xF x, is the denial of all propositions that say with respect to some nameFa says with respect to a. This, of course, is just the denial of the propositionsFa,F b ,F c , . . .. But then likewise, there is an a such that??aRab asserts thetruth of at least one proposition among those which say with respect to somename what Rab says with respect to a. And this is to assert the truth of at leastone of Rab ,Rcb ,Rd b , . . ., passing over the question of Rb b entirely. Thus, notfor Wittgenstein but for us,N ?aFa becomes ??xF x??aRab becomes ?x(x 6= b ?Rxb ), and??b (Rba ?N ?c(Rcb )) becomes ?x(x 6= a ? (Rxa ???y(y 6= x ?Ryx))).In general, the idea is for us to gloss NN ?aA as ?x(B ?A??), where B is the con-junction x 6= a ? x 6= b ? . . . with a, b , . . . the constants occurring in A??, and A??is the result of everywhere replacing a with a new variable x in the gloss on A.Thus, although Wittgenstein does introduce name-variables with apparentlyfamiliar syntax, their logical role is rooted in a distinctive conception of the re-lationship between names and propositions. The pictorial character of a propo-sition involves its exhibition of some definite mathematical multiplicity. Suchmultiplicity is, at least, a multiplicity of names, and the multiplicity of namesin a proposition requires, at least, an answer to the question how many namesit contains. A different answer is a different multiplicity. Wittgenstein?s name-variables receive a nonstandard reading because they signal results of denaturingof names in logical pictures. Familiar quantifier notation obscures the underly-ing patterns of logical construction as Wittgenstein conceives them. Since thesepatterns come under strain with the introduction of Way 3 variables, I?ll stick toa notation which keeps track of those patterns.4.2 Way ThreeLet?s now return to the original observation that Wittgenstein describes not twobut three ways of fixing the aim of the pointer. We have just seen that the firsttwo ways yield a variant of first-order logic. It?s therefore natural to suppose that184the point of Way 3 is to lead further to the expression of concepts which are notfirst-order definable. At 4.1252 and 4.1273 Wittgenstein suggests that the Way 3method allows the construction of a pointer which indicates the multiplicity ofsentences in the seriesRab (4.1)?x(Rax ?Rxb )?x?y(Rax ?Rxy ?Ryb ).Obviously, were A a pointer whose range were the sentences (1), then?Awouldhold of a and b just in case a bears to b the ancestral of R.Wittgenstein?s explanation of the construction of Way 3 pointers is incom-plete. He gives an official notation, roughly [A,? ,O(? )]. In this notation, thefirst entry is simply a formula; the second and third entries together signify an op-eration O : ? 7? O(? ) on formulas. The notation as a whole is a variable whichindicates the formulas A,O(A),O(O(A)), . . . which result by repeatedly applyingO to A. Relative to some determinate concept of operation, then, the working ofthe variable should be transparent. Supposing O : ? 7? O(? ) to be an operationwhich takes the nth term of (4.1) to the n + 1th, then?[Rab ,? ,O(? )] wouldexpress the ancestral of R in the way that Wittgenstein intends.However,Wittgenstein says little to determine the concept of operation itself.He says that there is associated to an operation O some distinguished ?formal?relation O? such that B is the result of applying O to A in virtue of the fact thatB bears O? to A. It?s not clear what is involved in characterizing a relation as?formal?. One possibility is that whether or not the relation holds depends onlythe forms (and perhaps also number) of objects, and does not distinguish betweenone object and another of the same form. In other words, one might speculate?although I wouldn?t insist?that formality implies some sort of invariance undertype-respecting permutations of names.189 On the other hand, he also says that?it should be easy to see? whether B bears O? to A, and more generally whethersomething is indicated by a variable. Thus it would appear that O should be in189Cf. in particular 2.0233.185some sense effective. However, the analysis of the concept of effectiveness in thiscontext raises difficult interpretive questions, indeed verging into realms whereWittgenstein?s thought may not be entirely unconfused.Rather than trying to determine the indeterminate concept of operation Isuggest considering a class of procedures that ought to count as operations evenunder a very miserly construal. Suppose we enrich the generation of formulas sothat it proceeds not just from elementary propositions but also from a denumer-ably infinite collection p, q , r, . . . of schematic propositional letters. Let A be aformula so generated which contains the single letter p. And now consider theoperation which takes an arbitrary propositional sign B to the result of every-where substituting B for p in A. I propose we consider the procedureO : ? 7?A[p/? ]as an operation. Let?s call the formula A associated to O the ?carried formula? ofO. I will sometimes refer to an operation O as a ?root substitution? because itgrafts the carried formula onto the root of the formation tree of the operand.190Slightly abusing notation, it will be convenient instead to write simply [B , p,A]for the series B ,A[p/B],A[p/A[p/B]] . . ..Let?s now consider a simple example of the working of root-substitution. LetA be the formula N p ?C for some C . Then [B , p,A] is a variable whose valuesare the formulasBNB ?CN (NB ?C )?Cand so on. (This method of typesetting the values of a Way 3 variable shows thatit is indeed ?easy to see? whether a successor term of the determined form-seriesbears the requisite internal relation to its predecessor.)Way 3 variables interact with the N -operator just like the variables of Ways 1190One could also consider ?leaf substitution? which grafts a carried formula onto the leaves ofthe operand, but I wi
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Logic in the tractatus Weiss, Max 2013
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Title | Logic in the tractatus |
Creator |
Weiss, Max |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | What is logic, in the Tractatus? There is a pretty good understanding what is logic in Grundgesetze, or Principia. With respect to the Tractatus no comparably definite answer has been received. One might think that this is because, whether through incompetence or obscurantism,Wittgenstein simply does not propound a definite conception. To the contrary, I argue that the text of the Tractatus supports, up to a high degree of confidence, the attribution of a philosophically well-motivated and mathematically definite answer to the question what is logic. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-10-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0165595 |
URI | http://hdl.handle.net/2429/45296 |
Degree |
Doctor of Philosophy - PhD |
Program |
Philosophy |
Affiliation |
Arts, Faculty of Philosophy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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