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Philosophy as conceptual engineering : inductive logic in Rudolf Carnap's scientific philosophy French, Christopher Forbes 2015

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Philosophy as Conceptual EngineeringInductive Logic in Rudolf Carnap’s Scientific PhilosophybyChristopher Forbes FrenchB.A., Kansas State University, 2008A 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 2015c© Christopher Forbes French 2015AbstractMy dissertation explores the ways in which Rudolf Carnap sought to make philosophy scientificby further developing recent interpretive efforts to explain Carnap’s mature philosophical workas a form of engineering. It does this by looking in detail at his philosophical practice in hismost sustained mature project, his work on pure and applied inductive logic. I, first, specifythe sort of engineering Carnap is engaged in as involving an engineering design problem andthen draw out the complications of design problems from current work in history of engineeringand technology studies. I then model Carnap’s practice based on those lessons and uncoverways in which Carnap’s technical work in inductive logic takes some of these lessons on board.This shows ways in which Carnap’s philosophical project subtly changes right through his latework on induction, providing an important corrective to interpretations that ignore the work oninductive logic. Specifically, I show that paying attention to the historical details of Carnap’sattempt to apply his work in inductive logic to decision theory and theoretical statistics in the1950s and 1960s helps us understand how Carnap develops and rearticulates the philosophicalpoint of the practical/theoretical distinction in his late work, offering thus a new interpretationof Carnap’s technical work within the broader context of philosophy of science and analyticalphilosophy in general.iiPrefaceThis dissertation is an original and independent work by the author, C. F. French.Some of the ideas for section 4.5 were first explored and discussed in my forthcoming publi-cation (expected fall 2015): C. F. French, “Rudolf Carnap: Philosophy of Science as EngineeringExplications.” In Recent Developments in the Philosophy of Science: EPSA13 Helsinki. (Eds.)Uskali Mäki, Stephanie Ruphy, Gerhard Schurz and Ioannis Votsis. I am the sole author of thispublication.I originally intended there to be an additional chapter in this dissertation discussing Carnap’scorrespondence with Richard C. Jeffrey. Unfortunately, I was forced to cut this material. Seemy forthcoming publication: C. F. French, “Explicating Formal Epistemology: Carnap’s Legacyas Jeffrey’s Radical Probabilism.” In Studies in the History and Philosophy of Science. Guestedited by Sahotra Sarkar and Thomas Uebel. I am the sole author of this publication.This dissertation makes extensive use of archival material from the Carl Hempel, RudolfCarnap and Richard C. Jeffrey papers at the Archives for Scientific Philosophy at the Universityof Pittsburgh. Quoted by permission of the University of Pittsburgh. All rights reserved.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Carnapian Wissenschaftslogik as Conceptual Engineering . . . . . . . . . . . 72.1 Carnap’s Wissenschaftslogik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Wissenschaftslogik : Critiques and Reappraisals . . . . . . . . . . . . . . . . . . . 222.3 Carnapian wissenschaftslogiker as Conceptual Engineer . . . . . . . . . . . . . . 292.4 Carnap and the State of Inductive Logic at mid-Twentieth Century . . . . . . . 372.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Philosophical Method as Conceptual Engineering . . . . . . . . . . . . . . . . 533.1 Engineering as Means-End Reasoning . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Engineering Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3 Satisficing Wings and Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4 Changing Designs and Braking Barriers . . . . . . . . . . . . . . . . . . . . . . . 683.5 Herbert Simon and Satisficing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73ivTable of Contents3.6 Carnap as Conceptual Engineer . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 Designing Inductive Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Carnap’s Confirmation Function c . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3 From Confirmation to Estimation Functions . . . . . . . . . . . . . . . . . . . . 984.4 Carnap’s Continuum of Inductive Methods . . . . . . . . . . . . . . . . . . . . . 1024.5 Finding Optimal Values of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265 Constructing Rational Decision Theory . . . . . . . . . . . . . . . . . . . . . . . 1305.1 Carnap on Hume’s Problem of Induction . . . . . . . . . . . . . . . . . . . . . . 1325.2 Ramsey’s Decision Theory as Qualified Psychologism . . . . . . . . . . . . . . . 1355.3 Feigl, Reichenbach and Justifying Induction Pragmatically . . . . . . . . . . . . 1405.4 Inductive Logic, Expected Utility Theory and Decision Theory . . . . . . . . . . 1515.5 Rationalizing Decision Theory and Justifying Inductive Logic . . . . . . . . . . . 1585.6 The Aim of Inductive Logic and Robot Epistemology . . . . . . . . . . . . . . . 1725.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189vList of Figures3.1 Means-end Model of Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Hierarchical Model of Engineering Design and Knowledge . . . . . . . . . . . . . 594.1 A “Well-connected” System of Inductive Concepts . . . . . . . . . . . . . . . . . . 127viList of AbbreviationsThroughout the dissertation I use the following abbreviations to refer to various archives:ASP Archives for Scientific Philosophy at the University of Pittsburgh.CH Carl Hempel archives at ASP.HR Hans Reichenbach archives at ASP.RC Rudolf Carnap archives at ASP. For example, “RC 079-20-01” refers to thedocument numbered 01 in the folder numbered 20 in the box numbered079.RCJ Richard C. Jeffrey archives at ASP.I also make frequent use of the following acronyms to refer to Carnap’s published works:CIM The Continuum of Inductive Methods, 1952.ESO “Empiricism, Semantics and Ontology”, 1950 (reprinted and enlarged inthe second edition of Meaning and Necessity, 1956).LFP The Foundations of Logical Probability, 1950 (second edition, 1962).LSL The Logical Syntax of Language, 1937.viiAcknowledgementsAs an undergraduate at Kansas State University I made the transition from self-identifyingas an artist, a programmer and a wanna-be hacker (of the MIT/Richard Stallman, not thecriminal, variety) to being an academic philosopher. I’m appreciative to all of the faculty atKSU’s philosophy department who was there from 2003 to 2008. I especially want to thankBruce Glymour for his patience, advice and guidance in helping me not only gain expertise inthe philosophy of biology and causal modeling but also getting into grad school. I also want tothank Andrew Arana for allowing me to make a copy of a paper by Alberto Coffa discussingWittgenstein, Carnap and logical tolerance: I’ve been hooked ever since.For better or worse, I’ve always brought my programming sensibilities to traditional philo-sophical problems – for example, when I first read Carnap’s Aufbau as an undergrad, I somewhatnaively read him as painstakingly providing us with an algorithm for constructing the world onthe basis of pairs of elementary experiences. Perhaps as a consequence of this sensibility, I amnever easily impressed by appeals to philosophical authority, common sense or expertise andrarely do I put much stock, if any at all, in the justificatory value (as opposed to the rhetoricalor pedagogical value) of philosophical thought experiments and intuition-pumps. Argumentscome cheap: I want the dirty and messy technical, conceptual and empirical details – tell mehow to build up these epistemological, metaphysical or ethical world-views from scratch, brickby interlocking brick. After moving to Vancouver, I had tried to suppress this engineering read-ing of Carnap’s scientific philosophy as I’ve journeyed through the conceptual landscapes onoffer by Kant, Marburg neo-Kantians like Ernst Cassirer, the logical empiricists, the Americanpragmatists and contemporary philosophers of science. But as should be evident from the titleof this dissertation, I’ve come full circle to embrace a version of the engineering sensibility Ithought I had left behind; indeed, it turns out that it is exactly because Carnap as scientificphilosopher embraces this sensibility that I find his work so valuable and original.I would like to thank my intellectual peers and dearest friends and colleagues who have beenviiiAcknowledgementsthere from the beginning (more or less): S. Andrew Inkpen, Dani Hallet, Taylor Davis andRebecca Trainor. I would also like to thank my fellow graduate students and friends at UBC:Joel Burnett, Tyler DesRoches, Roger Stanev, Alirio Rosales, Jihee Han, A.J. Snelson, EmmaEsmaili, Gerardo Viera, Servaas van der Berg, Sina Fazelpour, Richard Sandlin, Jiwon Byun,Kousaku Yui and Laura Keith, Aleksey Balotskiy and Kaitlin Graves, Garson Leder and SerbanDragulin. A special thanks to Stefan Lukits for putting so much work into the UBC FormalEpistemology reading group and for providing me with valuable comments on chapter 4.While a resident at Green College at UBC from 2009 to 2011 I had the pleasure of meetingmany amazing people, including my friends Dan Randles, Wanying Zhao, Simon Viel, YuanJiang, Nathan Corbett, Maciek Chudek and Andrew MacDonald. I would also like to thank thefollowing people who I met as a visiting fellow at TiLPS in Tilburg, Netherlands: Jan Sprenger,Stephan Hartmann, Rogier De Langhe and Juan M. Duran. Thanks to Sahotra Sarkar andThomas Uebel for giving me so many comments on my contribution to a 2013 workshop atAustin, Texas on formal epistemology and the legacy of logical empiricism. I would also liketo thank the following friends and colleagues, past and present, I have met either at UBC orin Vancouver more generally: Flavia Padovani, Uri Burstyn, Jon Tsou, John Koolage, ScottEdgar, Samantha Matherne, Daniel Kuby, Dan Raber and Christina Marie Moth.I would like to thank the members of my committee: John Beatty, Christopher Stephensand especially Richard Creath. I would also like to thank several other faculty and staff at UBC(even if we haven’t always seen eye to eye on philosophical matters), both past and present:Margaret Schabas, Paul Bartha, Eric Margolis, Nissa Bell, Rhonda Janzen, David Silver, AdamMorton, Ori Simchen, Roberta Ballarin and John Woods. But most of all I would like tothank my supervisor, Alan Richardson. Although I’ve encountered my fair share of travails andtribulations while finishing the dissertation, Alan has helped me to become a more confident,independent, thinker who (I hope) doesn’t completely suck at writing.We can’t all write like Rudy or Alan, but we can keep on trying.C. F. French, July 15, Vancouver.ixDedication.To my parents, Janet and Donald,and my brother, Michael.xChapter 1IntroductionIn my view, the purpose of inductive logic is precisely to improve our guesses and,what is of even more fundamental importance, to improve our general methods formaking guesses, and especially for assigning numbers to our guesses according tocertain rules. And these rules are likewise regarded as tentative; that is to say, asliable to be replaced later by other rules which then appear preferable to us. We cannever claim that our method is perfect. I say all this only in order to make quiteclear that inductive logic is compatible with the basic attitude of scientists; namely,the attitude of looking for continuous improvement while rejecting any absolutism.— Rudolf Carnap, “Probability and Content Measure”, (1966)Rudolf Carnap was a twentieth century scientific philosopher who used logic to reformulateseemingly intractable philosophical questions about the nature of science into clearly definedtechnical questions formulated within a logical system. Influenced by philosophers and math-ematicians like Ernst Cassirer, Bertrand Russell, David Hilbert, Gottlob Frege and LudwigWittgenstein, he articulated an early version of this scientific philosophy in his 1928 book DerLogische Aufbau der Welt, a document which would quickly become a cynosure for members ofboth the Vienna Circle and analytical philosophy in North America and the United Kingdom.It was in the Aufbau that Carnap attempted to secure the objectivity of scientific knowledgeby showing how one could logically reconstruct the structure of scientific knowledge and thusdemonstrating how scientific knowledge is inter-subjectively communicable.1By the time Carnap published his Logische Syntax der Sprache in 1934, however, his earlierconceptions of logic and mathematics had undergone a radical transformation. He now embracedan attitude of logical tolerance according to which there is no “correct” logical system but insteadthere are infinitely many logical systems, each of which is more or less sufficient for reformulatingscientific language. Logic, for Carnap, was now understood as an instrument chosen for practicalreasons of expedience rather than correctness. This is the maturation of Carnap’s scientific1 See Friedman (1999) and Richardson (1998).1Chapter 1. Introductionphilosophy: traditional philosophy is to be replaced by the logic of science; the philosopher isnow envisaged as a wissenschaftslogiker – a member of a technocratic community tasked withsupplying new logical techniques, new logical technologies, to be used for the clarification andsystematization of scientific language and concepts.The fundamental question my dissertation seeks to answer – namely, the question: Howexactly did Carnap understand the way in which his practically minded logic of science couldpossibly be used to help clarify questions about the foundations of science, especially if we un-derstand such questions to be metaphysical or epistemological in nature – is not a new question.Indeed, there now exists an extensive Carnap reappraisal literature which, in part, attemptsto explain how exactly Carnap tried to marshal the conceptual and technical resources avail-able to him in order to reformulate traditional philosophical questions into either expressionsof one’s preference for one logical system over others or into questions about the logical syntaxor semantics of a logical system. And one of the ways in which philosophers working withinthis reappraisal literature have tried to explain Carnap’s debates with the twentieth-centuryscientific philosopher W.v.O. Quine regarding whether or not we should give up on the logic ofscience in favor of just looking to science itself (especially psychology) to answer questions aboutthe foundations of science is by interpreting Carnapian logic of science as a kind of linguistic orconceptual engineering activity (see chapter 2).My dissertation contributes to this Carnap reappraisal literature by examining Carnap’s logicof science not with regards to his work on pure logical syntax and semantics but rather from theperspective of his longest-running technical project, a project which has received surprisinglyscant attention in the reappraisal literature; namely, Carnap’s work on inductive logic fromthe mid-1940s until the late 1960s which he then uses to address foundational problems aboutinduction and probability in the sciences, especially the foundations of statistics and decisiontheory. In chapter 3, I draw on case studies from the history of engineering to articulate ahierarchical conception of engineering design and I then use this conception of engineering asan interpretative framework to reshape Carnap’s work on pure and applied inductive logic inchapters 4 and 5.In particular, I argue that Carnap was ultimately not in the business of searching for thecorrect account of inductive logic. But instead, he was engaged in a project which is similar2Chapter 1. Introductionto what Herbert Simon calls “satisficing”: one need only find a “good enough” solution to aproblem, especially when it is nearly impossible, for practical or theoretical reasons, to find theoptimal or correct solution (if it exists at all). There are many different ways Carnap couldformalize the “ill-structured” problems concerning probability and induction in the sciences andeach different inductive logic provides us with a different, more or less satisfactory way of formu-lating a “well-structured” problem using the instruments of logical syntax and semantics. Theresulting conception of Carnapian logic of science is that of conceptual engineering. Rather thanattempting to formalize the probabilistic structure of science in a one fell swoop, Carnap wouldrather have us design and construct inductive logics in a piece-meal fashion while remaining sen-sitive to the purposes and changing needs of the empirical sciences. Carnapian logic of science ishere understood not as a competitor to “naturalism” or the history of science but simply as anadditional method available to philosophers of science in their toolbox of conceptual resources.More specifically, in chapter 2, I introduce the reader to the main topics of this disserta-tion: Carnap’s Wissenschaftslogik, the current Carnap reappraisal literature and the historicalcontext for Carnap’s work on inductive logic. I explain Carnap’s distinction between pure andapplied logic by analogy with mathematical and physical geometry. I then discuss both Carnap’sattitude of logical tolerance and how he envisages the replacement of traditional metaphysicaland epistemological questions with the logic of science. Then I explain how a number of Carnapscholars – including Richard Creath, Michael Friedman, André Carus, Alan Richardson andSamuel Hillier – have articulated different conceptions of Carnap as engineer. Lastly, I discusshow those mathematicians and scientists working on probability and induction who most influ-enced Carnap themselves understood the task of providing an inductive logic, or a logical conceptof probability; specifically, I provide a quick exegesis of the work on probability and inductionby Harold Jeffreys, John Maynard Keynes and Frank P. Ramsey. I then discuss how Carnap’slater work on inductive logic marks an important transition from his earlier Wissenschaftslogikfor which induction was understood as a purely pragmatic matter which resists formalizationinto logical syntax.Next, in chapter 3, I examine several engineering case studies and isolate several generalprinciples of engineering design. Specifically, I elaborate on one view in the history of engineeringwhich notices that engineering design depends on a hierarchical distinction between the practical3Chapter 1. Introductionand theoretical. I argue that engineering is not a simple instantiation of instrumental reasoningfor which engineers simply supply a number of designs to an employer who in turns pickswhichever design best fits their needs. Instead, sometimes engineering designs are hierarchical:as the aim of an engineering project and the relevant technologies change, or problems areencountered with the actual construction of an engineering object, the different components ofa design will be altered and these changes may influence the engineer’s practical choices regardingthe other components of the overall design. After discussing Simon’s views on satisficing I thendiscuss how this hierarchical notion of engineering design bears on the analogy of philosophy asconceptual engineering: although the decision to adopt a linguistic framework may be a purelypractical matter, when we need to make logical modifications to that framework the decisionas to which modifications to make may still be practical but yet these decisions may need tobe informed by certain theoretical considerations (e.g., how this logic is to be used in someempirical investigation).In next chapter, chapter 4, I finally turn to Carnap’s work on inductive logic. I explainhow Carnap constructs inductive logics – or rather, pure inductive logics. I examine how, first,Carnap constructs various inductive logics based on a concept of degree of confirmation andthen, second, how he extends this project to construct his -system which parameterizes acontinuum of inductive methods. Indeed, it is exactly here that Carnap talks about the choiceof a value of  in engineering terms. I then discuss how Carnap used his concept of degreeof confirmation to define other inductive concepts, most notably, a concept of estimation foruse in theoretical statistics and semantic concepts of entropy and information. In the early1950s, Carnap suggests that his work on the concept of estimation may be used to restructurethe foundations of theoretical statistics. It is within this context that Carnap’s work on pureinductive logic seems to be answerable, at least to a certain degree, to how well a conceptof degree of confirmation can be used to define other inductive concepts which are central toparticular empirical sciences. Lastly, I suggest that Carnap’s attempt to find “optimal” valuesof  can be understood as a kind of engineering activity. This is yet another way in which thepractical decision to adopt an inductive logic may be sensitive to the empirical sciences.Finally, in chapter 5, I explain how Carnap applied his work on a pure inductive logic to thesciences – specifically, empirical and rational decision theory – by focusing on how Carnap tried4Chapter 1. Introductionto explain to his peers, like the philosopher Hans Reichenbach or the statistician Leonard J.Savage, how exactly the adequacy of an applied inductive logic need not depend on its empiricalsuccess. The focus of this chapter will be on how Carnap understands the connection betweeninductive logic, rational decision theory and empirical decision theory – indeed, we will see thatCarnap shows how one could design an inductive logic so that it is adequate for use in rationaldecision theory. This chapter is historical. We will discuss how Carnap is influenced by F. P.Ramsey’s work on a normative decision theory, how Carnap responds to the criticism from bothHerbert Feigl and Reichenbach that a logical meaning of probability cannot be a guide in life,how Carnap understands the application of inductive to decision theory as a methodologicalproblem and, finally, how Carnap later responds to criticism from John Lenz and Carl Hempeldirected at his unwillingness to talk about the justification of inductive logic. Crucially, Carnapexplicitly argues that he is not trying to provide a non-circular justification for inductive methodsbut instead is concerned with providing an application of inductive logic for those that alreadyaccept the validity of inductive reasoning. For Carnap, an interpreted inductive logic supplies uswith well defined, non-arbitrary, confirmation values – we can then use these values as a guidefor our scientific deliberations.In the final section of chapter 5 I explain how, taken as a whole, the historical episodesdiscussed earlier in the chapter lead up to Carnap’s 1962 paper, “The Aim of Inductive Logic”.For it is there that Carnap suggests that we can think of rational decision theory as supplyingto an idealized agent a credence or credibility function with which to make rational decisions,functions which are based on certain “requirements of rationality”. Crucially, however, Carnapargues that the adequacy of these credence and credibility functions need not be assessed interms of their actual empirical success but rather in terms of their “reasonableness,” i.e., theirinductive success for not just the actual state of affairs but for all conceivable state descriptions,for all possible “states of the world” according to a logical system. In this sense, Carnap en-visages a transition from empirical decision to rational decision theory, and then from rationaldecision theory to inductive logic. It is exactly here that we can think of Carnap’s conceptionof rational decision theory as a kind of “boundary space” between empirical decision theory,which is a fairly straightforward empirical investigation, and pure inductive logic. The interplaybetween the empirical and logical within this boundary space exemplifies, I suggest, a kind of5Chapter 1. Introductionconceptual engineering: Carnap shows us how to construct inductive logics in a hierarchical,piece-meal fashion which have been designed for rational decision theory via certain require-ments of rationality – requirements which, in turn, are sensitive to the empirical findings ofempirical decision theory.In the conclusion, chapter 6, I explain how we can use the notion of a hierarchical conceptionof engineering design from chapter 3 to frame Carnap’s construction of pure inductive logics –logics which have been designed to clarify the conceptual systems belonging to the sciences,especially theoretical statistics – that we saw in chapter 4 and to understand how rationaldecision theory provides Carnap with a kind of conceptual space to design inductive logic forempirical decision theory. I then discuss several weaknesses of my dissertation and how it willlead to future work.2 For example, I plan to compare the similar ways in which both Carnapand his student, Richard C. Jeffrey, treat the probability calculus as an instrument.3 I also thinkthere are important connections which have not yet been explored regarding Carnap’s place inthe history of twentieth-century philosophy of science: Herbert Simon, John von Neumann andCarnap all have projects which I would suggest trade in a common conceptual currency: thatscientific reasoning fundamentally works by finding “good enough” rather than the “correct”solutions to problems.2 For a slightly condensed version of my views, see French (2015b).3 This work is already underway, see French (2015a).6Chapter 2Carnapian Wissenschaftslogik as Conceptual EngineeringThe constant course of science is not diverted from its goal by the varying fortunesof metaphysics. It must be possible to gain clarity regarding the direction of this ad-vance, without presupposing the dualism of the metaphysically basic concepts. [. . . ]Are these [basic] concepts, the separation and reunification of which the whole his-tory of philosophy has labored, merely intellectual phantoms, or does a firm meaningand effect in the construction of knowledge remain for them?— Ernst Cassirer, Substanzbegriff und Funktionsbegriff, (1910)The fundamental question my dissertation seeks to answer is how the twentieth centuryphilosopher Rudolf Carnap attempted to clarify and even resolve foundational questions in thesciences by reformulating those questions in an artificial, logical language. What distinguisheshis technical projects from any number of other technically-minded projects whose purpose is tosomehow logically reconstruct the structure of science is the attitude Carnap takes toward logic.The attitude is this. One may freely choose from an endless stock of artificial languages thatartificial language which seems to them more fruitful or useful for clarifying and systematizingthe conceptual foundations of science. Logic, for Carnap, is an instrument chosen for reasons ofexpedience or fruitfulness rather than correctness.Nevertheless, Carnap’s reconception of philosophy as the logic of science, or Wissenschaft-slogik, may be viewed by some contemporary philosophers as an unnecessary, if not futile,endeavor. For contemporary analytical metaphysicians and epistemologists, for example, ques-tions about the robustly normative nature of knowledge and evidence, or what the structureof the world is actually like, cannot, as Carnap would have it, simply be dissolved by some-how translating these questions into a logical framework. I won’t attempt to defend Carnapianlogic of science against such contemporary philosophers. Instead, I am interested in illustratingthe merits and uses of Carnapian logic of science to those contemporary scientifically-mindedphilosophers who – even if they remain sympathetic to logical empiricism – would have us focus7Chapter 2. Carnapian Wissenschaftslogik as Conceptual Engineeringour attention away from the narrow confines of logical reconstruction and toward either thehistory of science or the science of science.4 In good Carnapian fashion, I do not claim thatthe logic of science is somehow superior to these contemporary approaches to the philosophyof science but only that, once we understand Carnap’s technical work as a kind of conceptualengineering, his logic of science has the potential to be a useful conceptual resource for formally-minded philosophers of science insofar as it provides a means to carry out technical projectswithout becoming side-tracked by traditional metaphysical and epistemological concerns.Thus the primary goal in this dissertation is one of philosophical and historical clarificationrather than then providing an argument for the claim that we should (or shouldn’t) adopt Car-napian logic of science ourselves. I draw on both Carnap’s work on inductive logic in the 1950sand 1960s and various archival materials, including personal correspondence and unpublishedmanuscripts, to paint a broad, historical, narrative explaining how Carnap tried to clarify andsystematize foundational questions about science – especially the foundations of both decisiontheory and theoretical statistics – by the practical construction and application of inductivelogics. To help me in this task I make use of an interpretive framework – that of philosophyas conceptual engineering – to help explain the radical nature of Carnap’s mature philosophicalmethod. Consequently, the dissertation has two audiences. The first are those philosophers andhistorians working on the Carnap reappraisal literature: my dissertation is the first extensivetreatment of Carnap’s work on inductive logic and it provides yet another refinement to ourunderstanding of Carnapian logic of science.5 The second are those contemporary philosophersof science for whom the idea that philosophy is conceptual engineering may prove to be a usefulframework to situate and motivate their own technical projects.For the rest of this chapter, I proceed as follows. I first explain Carnap’s Wissenschaftslogik,in part by explaining Carnap’s distinction between pure and applied logic by analogy withmathematical and physical geometry. Second, I provide several examples from the Carnap4 For the “received view” of how logical empiricism investigates the nature of science, see Suppe (1977) orvan Fraassen (1980). For more on the importance of the history of science for the philosophy of science, seeBurian (1977); Giere (1973), Kuhn (1962) and the articles in Domski and Dickson (2010). For more on a“naturalist” philosophy of science and criticisms of logical empiricism, see Giere (1985; 1988); Kitcher (1992;1993); Laudan (1996).5 Patrick Maher, however, has spent much effort attempting to explain Carnap’s inductive logic in termsof Carnap’s method of explication; see Maher (2010) and the references therein. Also see Uebel (2012a);Wagner (2011).82.1. Carnap’s Wissenschaftslogikreappraisal literature for how Carnap could possibly address criticisms which have been directedagainst his mature philosophical views by philosophers like W.v.O. Quine and Kurt Gödel.Third, I explain how various authors from the reappraisal literature have attempted to clarifyCarnap’s mature views by analogizing Carnap’s technical projects to a kind of engineering.Lastly, I quickly summarize the work by those mathematicians and scientists on induction andprobability who most influence Carnap’s own understanding of the problem space for a logicalmeaning of probability and induction, setting the historical context for chapters 4 and 5.2.1 Carnap’s WissenschaftslogikWhen Carnap scholars first compared Carnapian logic of science to a kind of conceptual engi-neering,6 it was done so in an effort to explain the philosophical differences between Carnap andQuine on the subject of analyticity. Neither the debate between Carnap and Quine nor the sub-ject of analyticity plays a prominent role in this dissertation. Nevertheless, a quick discussionof the disagreement between these two scientific philosophers may help illustrate to the readerwhy an engineering analogy is relevant at all.According to one conception of scientific inquiry, mathematics is the language of science inthe sense that scientific laws can be mathematically derived from a basic system of axioms;Newtonian mechanics, for example, can be derived from Newton’s three laws plus both the lawof universal gravitation and the resources of the infinitesimal calculus. Call such a system S.One ambition of logical empiricism, as the movement is commonly understood, is to give somegeneral theory of the language of science according to which one could clearly separate thosesentences of S which are true purely in virtue of the logical or mathematical consequences ofS from those sentences of S which are true in virtue of the empirical axioms of S – axiomswhich are in correspondence to certain facts of the world. To accomplish this task for not onlyS but for any scientific theory would be to clearly demarcate the analytic sentences of purelogic and mathematics from the synthetic, empirical, sentences making up the content of theempirical sciences (and if traditional metaphysical statements are shown to be neither analyticor synthetic they are revealed to be, literally, without meaning).6 For example, in Creath (1992, 154).92.1. Carnap’s WissenschaftslogikHowever, Quine famously argues that any such foundationalist epistemology is untenable.7Even if we did manage to agree that an adequate criterion for demarcating the analytic andsynthetic sentences of S could be found, no general distinction between the analytic and syntheticcould possibly be given for all languages. For in order to characterize the linguistic structureof any language L (called the “object language”) which is expressively more powerful than thelanguage of arithmetic with only addition, one must take advantage of the linguistic resources ofa separate language (called the “metalanguage”), e.g., a natural language like English, which isstronger than L to define a truth predicate in L over all the sentences in L. But herein lies theproblem: how do we know that this procedure for defining an analytic/synthetic distinction inthe object language could also be used for defining a similar distinction in the metalanguage (andif it holds for the metalanguage, that it also holds for the meta-metalanguage, etc.)? Quine’ssolution is that there can be no non-circular and principled distinction between the analytic andsynthetic. Instead, we must countenance a holistic and non-foundational conception of meaning:meaning, to use the famous metaphor from Quine, is best described as a web of beliefs withthe more “analytic” statements located at the center of the web and the “synthetic” statementslocated at the edges where they are constantly impinged by “experience”; here the distinctionbetween the analytic and synthetic is one of degrees.8Carnap, in his 1937 The Logical Syntax of Language (LSL),9 provides a general theory oflogical syntax and, later in the 1940s, a semantical definition for truth in a language, or L-truth(we will return to the notions of logical syntax and semantics below). Despite Carnap’s com-plicated logical constructions, as far as Quine is concerned, Carnap provides us with no reasonfor why his characterizations of logical truth, or analyticity, for certain classes of artificial lan-guages could possibly hold for natural languages like English. From Carnap’s point of view inthe 1940s, however, he is quite clear that he is only providing a clarification of analyticity asL-truth relative to a particular logical language; or, to use the language Carnap later adopts,7 See, for example, Quine (1951; 1969).8 Interestingly, as far as Quine’s criticism of Carnap’s empiricism is concerned, the historian Joel Isaac hasrecently pointed out that not only does Quine adopt the talk of “conceptual schemes” from the Harvardbiochemist Lawerence Henderson (who led the influential seminar “Pareto and Methods of Scientific Inves-tigation” at Harvard; see Isaac, 2012, 282) but that Quine first introduced his web of belief metaphor whilehe was still a graduate student in a student paper on Kant for a seminar taught by C. I. Lewis in 1931(Isaac, 2012, 140-142). So it seems like Quine had a nascent version of his criticisms of empiricism given inhis “Two Dogmas” paper already in 1931 – before Quine ventured to Prague to first meet Carnap.9 This is the expanded, English translation of his 1934 Logische Syntax der Sprache.102.1. Carnap’s WissenschaftslogikL-truth is only an explication of analyticity. Carnap stipulates beforehand which terms in thelanguage are logical and which are descriptive, or non-logical: the logical truths are those sen-tences formed just from logical terms which can shown to be true using the semantic resources ofthe language alone. Thus Carnap is clear that he is not attempting to provide a characterizationof analyticity for natural languages but only specially constructed logical languages.So here the disagreement between Carnap and Quine seems to rest on a misunderstanding.Quine is unhappy with the arbitrary nature in which Carnap demarcates the logical from thenon-logical terms of a language whereas Carnap would suggest to Quine that if he is unhappywith Carnap’s definition of L-truth relative to the language L then perhaps Quine should provideto Carnap what Quine would consider a more satisfactory explication of analyticity. But if thisis accurate, in what sense is Carnap still doing philosophy if he is no longer engaged, as Quinethought he was, in the project of providing a general analytic/synthetic distinction for anylanguage? This is where the engineering analogy comes in: at least by the 1950s, Carnap wasnot in the business of providing a general characterization of the analytic/synthetic distinctionwhich is “correct” for all languages; he just showed how this distinction could possibly be clarified.This is conceptual engineering: Carnap constructs languages as one would construct a ham-mer or cellular phone with some purpose in mind. However, just as we do not have to justify thechoice to use one kind of hammer over others to fulfill the task at hand, Carnap needn’t justifyhis choice of a language to Quine: some choices will lead to better or worse results than others,but no instrument is “correct.” But even as a conceptual engineer, Carnap is still working on thefoundations of science in the sense that he is engaged in the project of providing different, techni-cal clarifications for how one could possibly make sense of the meaning of the analytic/syntheticdistinction for specific classes of artificial languages. And perhaps if we search long enough, aclarification, or rather an explication, of analyticity will be found that appeases Quine.10 In thenext subsection we turn to Carnap’s views on logical syntax and semantics.Logical Syntax and Semantics.Distinctive to Carnap’s mature philosophical position is his adoption of a standpoint accordingto which questions about the foundations of science can be resolved by investigating the language10 For more on the Quine/Carnap debate and their correspondence, see Creath (1987; 1990a; 1991).112.1. Carnap’s Wissenschaftslogikof science. For Carnap, there is a certain degree of freedom available to us when choosing anartificial, logical language to translate the statements made by scientists when they are engagedin scientific activity. The occasion for this linguistic freedom is Carnap’s embrace of an attitudeof logical tolerance according to which there is no privileged or correct logical calculus thatmust be used to logically reconstruct scientific language. Consequently, from the perspectiveof Carnapian logic of science, traditional metaphysical questions are not questions about the“correctness” of scientific language but instead are revealed to be endorsements of one artificiallanguage over others and thus are without cognitive content.11 Likewise, traditional epistemo-logical questions about the “correct” or “rational” formation and justification of our beliefs arerevealed, again from the perspective of Carnapian logic of science, to tend to confuse logicalwith empirical questions.12 Thus, for Carnap, those traditional metaphysical and epistemolog-ical questions which have embroiled previous generations of philosophers are to be replaced bythe practical activities of the wissenschaftslogiker required to construct artificial languages asinstruments for the task of clarifying and systematizing the foundations of science.To see how Carnap thinks he can accomplish all of this it is important to keep in mind thatCarnap separates (if only as an abstraction) the study of language into three separate parts:(1) a theory of how the speakers of a language utter or write down sentences in particularcontexts, called pragmatics; (2) a theory of how certain expressions in the object languagedesignate the objects in some domain of discourse, like the class of red pandas in southwestAsia, called semantics; and (3) a theory for how the symbols or signs of a language can becombined to formulate syntactic expressions, like closed or open sentences, and rules for whencertain kinds of classes of expressions logically follow from other kinds of classes of expressions,Carnap calls this kind of investigation logical syntax.13 Besides this three-part separation of11 Simply speaking, a sentence a is without cognitive content if a is not only neither true nor false but thereis no way to evaluate it veridically at all; typically, it is instead said to express a preference or feeling.12 In short, Carnap rejects any conception of logic which entails any form of psychologism. Here, psychologismis simply the view that, however we conceive of logic, logic somehow affects either what rational personsought to believe or what they do, in fact, believe. For more on the history of psychologism in both nineteenthcentury philosophy and psychology, see Kusch (1995).13 The exposition of Carnap’s views in the next couple of sections follows closely Carnap’s work after thepublication of LSL – that is, when Carnap adopts something like Tarski’s method of defining (logical) truth;see Carnap (1939; 1942; 1943) for the details. My presentation of the technical material in this section,moreover, is not always historically accurate – I am much more concerned with Carnap’s views on syntaxand semantics starting in the late 1940s rather than explaining how he came to have these views duringthe 1930s and early 1940s. For more on how Carnap understands the difference between logical syntax andsemantics, see §39 of Carnap (1942); also see Creath (1990b); Ricketts (1996). See pages 146 and 153 of122.1. Carnap’s Wissenschaftslogiklanguage in pragmatics, semantics and logical syntax, Carnap also distinguishes between thelanguage under investigation, the object language (call it L) and the language with which westate the syntactical and semantical rules for L called the metalanguage, which is typically anatural language like English or German plus some mathematical resources.Linguistic rules stated in the metalanguage for L are formal in the sense that these rulesdo not refer to the semantic resources of the metalanguage; for Carnap, at least in the 1930s,what distinguishes the rules which belong to logical syntax as opposed to semantics is that theformer are formal while the latter are not. When it comes to logical syntax, there are twokinds of rules: rules of formation and rules of transformation. The rules of formation providerecursive definitions for how all the expressions of L, like sentences, can be formed using justtwo classes of signs: the logical signs like ‘(’, ‘⊥’, ‘⊤’, ‘∧’, ‘∃’ or ‘x1’ denoting logical notions likeparentheses, tautology, inconsistency, logical connectives, quantification and variables, and thedescriptive signs like ‘uj ’ or ‘Blue’ designating individual constants and predicates. The rulesof transformation state how kinds of expressions, like sentences, can be replaced or transformedinto other kinds of expressions: these rules characterize, for example, variable substitution andlogical implication. In addition to the inclusion of separate rules of formation and transformationfor the semantics of L, the semantical rules of L also include a recursive definition for truth inL. This is a deflationary, semantic notion of truth which uses the semantic resources of themetalanguage to state the exact conditions each sentence in L must satisfy to be true or falserelative to L. The semantic rules suffice to provide an interpretation of the logical calculus ofL if those rules are sufficient to determine truth criteria for all the well-formed sentences inthe calculus. An interpretation is true if, generally speaking, it is the case that both (i) thesyntactical and semantical notions of logical implication coincide and (ii) all sentences in thecalculus that are (not) provable are true (false) in the semantical system.14Once a logical syntax and semantics has been given for L, Carnap then defines semanticalconcepts which either do or do not hold of all the sentences of the language in virtue of thesemantical system alone. Here I have in mind logical notions of analyticity and logical conse-quence whose meaning are fully specified by the semantics of L – Carnap calls such conceptsCarnap (1939) for what he means by “abstraction”; also see Sarkar (2013).14 See §§4-10, and especially pp. 164-65, of Carnap (1939). Famously, the Peano axioms have more than onetrue interpretation aside from the normal interpretation; see Carnap (1943).132.1. Carnap’s WissenschaftslogikL-concepts. For example, Carnap provides a definition of logical truth, or analyticity, in L asL-truth: a sentence S in L is L-true if the semantic rules alone suffice to show that S is true (inL). Importantly, this means that there may be sentences in the logical calculus which containdescriptive signs which, when interpreted, are neither L-true or L-false; rather, these sentences,according to Carnap, are the factual sentences of L. We may wish, for example, to state rulesfor defining a semantic truth predicate in L for those factual sentences that contain descriptivesigns: for example, we may wish to state rules which coordinate the individual constants in Lwith the balls in an urn and certain descriptive predicates in L with physical color properties,like having the property of being blue or red (as we will see below, this is one way in which Lcan be applied). Carnap, however, is not claiming that this notion of L-truth provides a generalclarification of analyticity and truth for all languages, both artificial and natural: it is only anotion of analyticity and truth for the language fragment L. According to Carnap, whetheror not a notion of L-truth in L is adequate depends on what interpretation we wish to give tothe logical syntax of L in the metalanguage.15 The semantical concepts in an object languagecan only be made precise relative to the antecedent semantic resources available to us in themetalanguage.Pure and Applied Logic, Mathematical and Physical Geometry.The idea that a semantical system can provide an interpretation for a logical calculus is central toCarnap’s mature views. Specifically, an interpreted pure logic can be applied in the sense that theprimitive descriptive terms in that logic are given an interpretation which coordinates them withempirical objects. In this section I discuss how Carnap explains this distinction between pureand applied logic by reference to the distinction between mathematical and physical geometry.In the late 1930s, Carnap distinguishes the application of an object language L from both theconstruction of a logical syntax and the provision of a semantical system for the syntax for L,or what Carnap later calls the formalization and an interpretation of a language, respectively.16An application of an interpreted logical calculus L is either a reinterpretation of the descriptive15 See, for example, Carnap’s remarks about L-concepts in §16 of Carnap (1942).16 See Carnap (1939). However, when Carnap has in mind axiom system specifically in his 1950 probabil-ity book, The Logical Foundations of Probability, the application of an axiom system is the same as itsinterpretation; see §6, especially pp. 16 and 59142.1. Carnap’s Wissenschaftslogiksigns of a semantical systems or, more typically, an interpretation of the primitive descriptivesigns through the use of semantical rules which make reference to the empirical measurements,observations or experimental results belonging to some scientific mode of inquiry.17 Next weexamine how Carnap explains the difference between pure and applied logic by comparing it tothe distinction between mathematical and physical geometry.18Generally speaking, geometry concerns the empirical measurement and comparison of spatialproperties and relations using mathematical concepts like “point”, “line” and “plane” definedrelative to some multi-dimensional mathematical space, like the three-dimensional Cartesiancoordinate system R3. Euclid was the first to formalize the mathematical part of geometry as aunique system of axioms and postulates which clearly stated how the concepts like “point” and“line” can be interpreted in terms of what can be drawn on a piece of paper using nothing but apencil and measuring instruments like a straight-edge and compass. It was only in the nineteenthcentury that it was discovered that the axioms of geometry can be studied independently of howthe concepts like “point” and “line” are interpreted. Specifically, it was discovered that thereexist geometrical axiom systems which are consistent but for which Euclid’s parallel postulatedoes not hold.19 Different choices of a set of geometrical axioms generate different geometries,each with their own class of mathematical theorems, conjectures and conventions.Mathematical geometry is concerned with studying what mathematical consequences holdfor different geometrical axiom systems. Specifically, different systems of axioms, e.g., thosesystems corresponding to Euclidean or non-Euclidean geometry, specify different extensionalrelationships for the primitive geometrical signs like “point” and “line”. Crucially, however, themathematical consequences of these different axiom systems do not on their own have anything17 This is sometimes done via coordinative definitions, or Zuordnungsdefinitionen; see Reichenbach (1920).Moreover, as we will see in chapter 4, Carnap later talks about the practical application of a semanticalsystem – especially a system with logical concepts of probability – via the imposition of certain requirementsrestricting the possible interpretation of that semantical system; for Carnap, such requirements concern themethodology of the semantical system and do not belong to the semantics of the logic itself. For example,see §44 of Carnap (1962b), especially p. 204.18 The subject of physical and mathematical geometry plays a prominent role in Carnap’s views beginning withhis 1922 dissertation, Carnap (1922); indeed, Carnap uses this distinction to explain the difference betweenpure and applied logic in one of his last (posthumously) published writings, see §4 of Carnap (1971a).19 For Euclidean geometry the shortest distance between any two points in R3 〈x1; y1; z1〉 and 〈x2; y2; z2〉 isequal to the magnitude√(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2. In non-Euclidean geometries, however, thisdistance formula does not provide the correct magnitude for the shortest distance between two points, orgeodesics. We would instead have to appeal to the more general mathematical conception of a metric spaceand then use this notion of a metric to calculate the distance between two points located on, say, a smoothmanifold with an affine structure. For more details, see Carnap (1995, Part III).152.1. Carnap’s Wissenschaftslogikto do with the physical world: they belong to the realm of pure mathematics. Physical geometry,by contrast, provides clear rules for how the primitive signs of an axiom system should correspondto physical locations, objects and relations; for example, to the points and lines in a generaltheory of space and time, or as longitudinal and latitudinal locations required for nauticalnavigation. In other words, physical geometry is the application of a mathematical geometricalsystem in the sense that coordination rules are given which specify how the primitive descriptivesigns of an axiom system correspond to specific classes of physical objects or properties.20In LSL, Carnap draws on this distinction between mathematical and physical geometry toexplain the difference between pure and descriptive logical syntax. Pure syntax, according toCarnap, is “nothing more than combinatorial analysis, or, in other words, the geometry of fi-nite, discrete, serial structures of a particular kind” (LSL, 7; emphasis in original).21 Just asdifferent mathematical geometrical systems can be constructed by investigating the mathemat-ical consequences of different axiom systems with primitive descriptive signs, pure syntax, forCarnap, studies the countlessly many different ways in which a logical calculus, with primitivedescriptive signs, can be constructed by choosing different rules of formation and transformation.Moreover, just as one may investigate different geometrical axiom systems from a purely math-ematical point of view (and so independently of the possible applications of those geometriesto specific scientific and engineering endeavors) Carnap invites us to treat the purely syntacticinvestigation of the possible kinds of logical form in a similar fashion to the investigation ofmathematical geometry, namely, as a mathematical activity independent of how logical calculicould possibly be interpreted and applied in order to express logical and empirical statements.Carnap’s envisaged plurality of logical forms – including especially those heterodox logicalcalculi which are either non-bivalent or intensional – is explained neither as the consequenceof any philosophical account of knowledge, reason or conception of the world but rather as theresult of the adoption of an attitude or standpoint which he takes toward both mathematics andlogic. Carnap expresses this attitude with a principle of logical tolerance: “It is not our businessto set up prohibitions, but to arrive at conventions” (p. 51).22 “In logic,” explains Carnap, “there20 For a more detailed explanation of how these constructions might go, see Carnap (1939).21 What Carnap seems to have in mind is this: pure logical syntax is the spatial investigation of how symbolslike ‘∃’, ‘)’ and ‘∧’ can be made to form finite, discrete and serial structures, like sentences and open formula.22 As Carnap points out in LSL, he was not the first to formulate such a view. Karl Menger, in 1930, was thefirst to express this attitude in writing; see Menger (1979, 11–16).162.1. Carnap’s Wissenschaftslogikare no morals”:Everyone is at liberty to build up his own logic, i.e., his own form of language, as he wishes.All that is required of him is that, if he wishes to discuss it, he must state his methodsclearly, and give syntactical rules instead of philosophical arguments. (LSL, 52)No longer “hampered by the striving after ‘correctness’ ” – that is, hampered by philosophicalarguments and projects concerned with the ontological commitments of the language we employ– we are free to investigate the syntactic properties of different logical calculi and then choosethat calculus which we have good reason to think is a fruitful candidate for structuring orframing scientific language. Simply put, Carnap rejects any foundational project which seeks tolocate the ontological or otherwise metaphysical consequences of adopting one logical calculusover others. Carnap instead pictures an endless oceanscape of possible language forms ripefor philosophical exploration: “before us,” says Carnap, “lies the boundless ocean of unlimitedpossibilities” (LSL, xv). We are free to explore whichever pure logical systems we wish.By contrast, descriptive syntax, like physical geometry, is concerned with providing an in-terpretation for a logical calculus; in particular, it is concerned with applying the primitivedescriptive terms in a language through a judicious process of selecting the right coordinativedefinitions which resemble the standard interpretation of some historical language already inuse.23 Thus, although we were initially free to construct whichever kind of logical calculi metour fancy, once we made the decision to interpret and apply that calculus in such a way thatit is intended by us to capture the structure of some part of the exact sciences, e.g., particulartheories in classical population genetics or particle physics, then we are no longer free to providejust any old semantical interpretation of the calculus. We would instead be in the businessof figuring out exactly how to construct a semantical system for our calculus which provides23 In Carnap (1939), Carnap distinguishes between two different ways of constructing both logical calculi andsemantical systems. The first way is what Carnap calls descriptive and it concerns theoretical investigationsof the linguistic properties of historically given languages. For example, we may wish to explicitly provide aninterpreted calculus for that snippet of English which corresponds to how scientists record their experimentalresults as declarative sentences. For Carnap, the question of how exactly an interpreted logical calculusshould be constructed so that it captures, loosely speaking, the logical structure of the language snippet isan empirical question best answered by an appeal to empirical linguistics and psychology rather than morelogic and philosophy. Alternatively, we could also construct an uninterpreted logical calculus or a semanticalsystem from scratch, so to speak, by freely choosing whichever rules of formulation and transformation(including rules of truth for semantical systems if required) one wishes without any pretense that thissystem is intended to resemble in any way any actual language-in-use. It is in this sense that logic, forCarnap, can be conventional: the question about how we should construct a logical syntax is not theoreticalbut it is rather a matter of preference and expedience relative to what we wish to accomplish with thiscalculus – after all, logical syntax is just the mathematics of how to combine together symbols which wecall logical and descriptive signs (see §11 of Carnap 1939).172.1. Carnap’s Wissenschaftslogikdefinitions for the primitive descriptive signs in our language that are sensitive to the modeling,experimental and theoretical activities of biologists or physicists.24According to Carnap, foundational questions about science which most contemporary ana-lytical philosophers would without much hesitation label as “metaphysical” or “epistemological”tend to confuse logical and empirical (most likely psychological) matters granted that we alreadyhave in our possession, so to speak, an adequate logical syntax and semantics for that language.The metaphysical question “Do neutrinos really exist,” for example, seems to ask a questionabout the world; namely, whether neutrinos really do exist or not. Carnap, however, does notassume that the notions of existence, logical truth or reference taken from natural languageare “correct.” Instead, when asked the question whether neutrinos actually exist Carnap woulddiagnose that question as either a question about the logical form of our language (e.g., arethe syntactic and semantic descriptive signs used to formalize and interpret the concept of aneutrino primitive or are they further reducible to expressions which contain other non-logicalsigns?) or as an empirical question which can be answered using the syntactical and semanticalresources of that language (e.g., whether certain classes of factual sentences which contain thosedescriptive signs designating neutrinos are true or false). In the first case, the existence questionbecomes, for Carnap, a practical question about which kind of logical form is most expedient oruseful for axiomatizing physical theories and, in the second case, the existence question becomesa theoretical question about what can be asserted using a language system which already haswell-defined syntactical and semantical rules.25 In other words, the metaphysical question itselfis transformed into either a question about which pure logic we prefer to use or, once we havechosen a logic, what can be expressed, as a theorem in that logic, once it has been applied tosome empirical situation.24 In other words, our syntactic conventions have to be put to empirical use: “In principle, certainly, a proposednew syntactical formulation of any particular point of the language of science is a convention, i.e. a matterof free choice. But such a convention can only be useful and productive in practice if it has regard to theavailable empirical findings of scientific investigation” (Carnap LSL, 332).25 See Carnap (1950), where he introduces the nomenclature of internal and external questions to help explainthis distinction; also see my chapter 4, pp. 112 ff.182.1. Carnap’s WissenschaftslogikCarnap’s Logic of Science.Now that we have a way to distinguish between pure and applied logic under our belt we canexplain Carnap’s logic of science in a bit more detail. In part V of LSL, Carnap remarks that thequestions of any theoretical field, like biology or sociology, can be expressed as either ‘object’ or‘logical’ questions, i.e., questions concerning the objects of the domain of a field, like a populationof Drosophilia in a biology lab, or questions about logical syntax of a scientific language, likethe logical syntax of the language used by evolutionary biologists when talking about fruit flies,respectively. Carnap is quite aware, of course, that from the perspective of logical syntax thisdistinction between ‘object’ and ‘logical’ questions is at best informal: ‘object’ questions are, forCarnap, really just ‘logical’ questions answerable by examining the logical syntax (and, later,the semantics) of that language.26 Nevertheless, according to Carnap, once we investigate thelogical syntax of traditional metaphysical or axiological philosophical questions formulated innatural language, like English, we discover that these questions belong neither to the ‘object’-questions of some scientific field nor are they ‘logical’-questions about the logical syntax andsemantics of a language. They are instead pseudo-sentences: “they have no logical content, butare only expressions of feeling which in their turn stimulate feeling with volitional tendencies onthe part of the hearer” (LSL, 278). “Apart from the questions of the individual sciences,” saysCarnap,only the questions of the logical analysis of science, of its sentences, terms, concepts, theories,etc., are left as genuine scientific questions. We shall call this complex of questions the logicof science. (LSL, 279)Next Carnap introduces the distinction between the material and formal mode of speech, adistinction with which he means to capture the difference between our customary ways of speak-ing when carrying out everyday, philosophical or scientific activities and the close examinationof language which can be garnered from the spelling out of the logical syntax of a language.For Carnap, it is only when traditional philosophical problems – problems which are typicallyframed informally in ordinary language, or what Carnap calls the material mode of speech – aretranslated into logical syntax, viz. what Carnap calls the formal mode of speech, is it possiblefor us to see why traditional philosophical questions are pseudo-questions.27 By translating the26 Presumably, the distinction between the ‘object’ and ‘metalanguage’ would, for Carnap, likewise be informal.27 As Carnap himself puts the point: “Translatability into the formal mode of speech constitutes the touchstonefor all philosophical sentences, or, more generally, for all sentences which do not belong to the language of192.1. Carnap’s Wissenschaftslogikinformal sentences scientists make when carrying out scientific activities into logical syntax, orthe formal mode of speech, a logician can pinpoint the exact logical relationships between scien-tific concepts and terms contained in these informal sentences. For example, by focusing on thelogic of science, philosophers can ask questions about the inter-definability and translatabilityof the terms in one language, say the language of evolutionary biology, into the language ofanother language, like the language of physics.Nevertheless, what some contemporary philosophers may not realize is that Carnap neverclaimed that the point of logical syntax was to formalize all scientific activities and processeswithin a single logical framework. The reasoning processes scientists go through in order tomake their judgments about the success of experiments, how experiments are performed orhow to evaluate the confirmability of theories given evidence may be left as pragmatic notions,i.e., they make use of concepts which refer to actual persons at a particular place in time.In Carnap (1936a; 1937a), for example, the notions of “testability” and “confirmability” aredefined pragmatically, i.e., in reference to what actual scientists do when they employ theseterms in scientific contexts. From the standpoint of logical syntax and semantics, “testability”and “confirmability” are then treated as primitive concepts which can then be used to definea plethora of scientific concepts.28 This fact will be of relevance throughout this dissertation:it was never the aim of Carnap, in LSL, to fully formalize inductive reasoning into a singlelogical framework and it is only in his later work on inductive logic that he begins to formalizefragments of the kind of inductive and probabilistic reasoning used by scientists.In LSL, for example, Carnap formalizes only the declarative sentences stated by scientistsusing the resources of logical syntax and he leaves any formalization of how scientific theorieschange over time, including the introduction of new theories in a logical syntax, to the method-ology and pragmatics of scientific activity. He outlines how a logician could go about providinga logical syntax for the language of physics in §82 of LSL: aside from providing a logical cal-culus with semantic L-rules, the logician would also introduce both primitive descriptive signs– including how to formulate protocol sentences in the language – and semantic P-rules, orprimitive physical rules, which characterize the basic physical laws of theoretical physics usinganyone of the empirical sciences” (LSL 313; emphasis in original).28 Carnap, for example, defines different notions of reducibility in terms of these pragmatic notions of ‘testa-bility’ and ‘confirmability’ in Carnap (1936a; 1937a).202.1. Carnap’s Wissenschaftslogiknewly introduced descriptive signs.29 Carnap is quite clear, however, that the P-rules do notformalize how new P-rules should be introduced into the physical system nor how the P-rules al-ready in the system can be altered as new protocol sentences corresponding to new observationalstatements are introduced into the system.Instead, it is the logician or scientist who must, so to speak, manually introduce, modify orremove the P-rules of the system – instead of being inferred, P-rules, according to Carnap, areto be treated as hypotheses relative to a body of protocol sentences in the language (318). Thesehypothetical P-rules are never in a strict sense either completely falsified nor fully confirmed :When an increasing number of L-consequences of the hypothesis agree with the alreadyacknowledged protocol-sentences, then the hypothesis is increasingly confirmed; there isaccordingly only a gradual increasing, but never a final, confirmation. Furthermore, it is, ingeneral, impossible to test even a single hypothetical sentences. [. . . ] Thus the test applies,at bottom, not to a single hypothesis but to the whole system of physics as a system ofhypotheses (Duhem, Poincaré). (LSL, 318; emphasis in original)In the 1930s, Carnap does not attempt to define a syntactic (or semantic) concept of “testability”or “confirmation” within the logical syntax of the physical language itself but instead treatsthese notions at the level of pragmatics: they concern how actual scientists or logicians cometo evaluate whether a hypotheses is testable or confirmable relative to some body of scientificevidence.30 But no L- or P-rules are sacred – any of these rules may at some latter point berevised or altered:No rule of the physical language is definitive; all rules are laid down with the reservationthat they may be altered as soon as it is expedient to do so. This applies not only to theP-rules but also the L-rules, including those of mathematics. In this respect, there are onlydifferences in degree; certain rules are more difficult to renounce than others. (LSL, 318)Indeed, according to Carnap, within the context of the logic of science our practical decisionsregarding the choice of a logical syntax are to be made on the basis of “practical methodologicalconsiderations”:The construction of the physical system is not effected in accordance with fixed rules, butby means of conventions. These conventions, namely, the rules of formation, the L-rules,and the P-rules (hypotheses), are, however, not arbitrary. The choice of them is influenced,in the first place, by certain practical methodological considerations (for instance, whetherthey make for simplicity, expedience, and fruitfulness in certain tasks). (LSL, 320)29 My labeling of the P-rules and L-rules as “semantic” is both anachronistic and slightly misleading: for Car-nap, the P-rules are the rules of the language which are not L-rules – there need not be a tight correspondencebetween “physical” and P-rules.30 See, for example, Carnap (1936a;b; 1937a).212.2. Wissenschaftslogik: Critiques and ReappraisalsThe L-rules and P-rules of a logical syntax for the language of physics are not provided to usas a consequence of either accepting some a priori realm of reasons or the existence of somenotion of transcendental agency or any docile deity but rather as a consequence of scientistsand logicians making “practical methodological considerations” on the basis of their scientificexpertise and knowledge. They will then modify these L-rules and P-rules to the extent whichthey find the current rules to be simple, expedient and fruitful.Thus Carnap provides us with no notion of a “meta-logic-of-science”: no rules for how scien-tists or logicians should modify the L-rules and P-rules of a language of physics. This too is apractical matter, but it is a practical matter informed by the projects and concerns of workingscientists and logicians. This, in a nutshell, is Carnap’s response to Quine: he, Carnap, is notin the business of providing the correct theory of analyticity but only a characterization of an-alyticity relative to some language which will suit our scientific purposes. Carnap sees himselfas offering to Quine different ways of applying a logical system, just as a mathematician couldoffer to a scientist different geometrical axioms systems. This is where conceptual engineeringas an interpretive framework starts to do work. Carnap, as conceptual engineer, shows howthe philosopher can contribute to foundational debates in the sciences: instead of appealingto subjects like psychology, e.g., under the rubric of “naturalized epistemology,”31 to examinehow scientists have used certain mathematical instruments produced throughout the history ofscience in their scientific reasoning Carnap instead shows us how we could construct these toolsfrom scratch.2.2 Wissenschaftslogik : Critiques and ReappraisalsThe image of Carnapian Wissenschaftslogik adumbrated in the last section may not be similarto the image of logical empiricism many analytical philosophers are familiar with. In “TwoDogmas of Empiricism” and his other writings, for example, Quine suggests that Carnap’s useof symbolic logic to investigate the foundations of science in the Aufbau should be understood asa continuation of British epistemology as Carnap, purportedly, tries to make good on Russell’sattempt to use symbolic logic to rationally reconstruct the empirical world from sense data alone.31 See Quine (1969).222.2. Wissenschaftslogik: Critiques and ReappraisalsFamously, of course, Quine argues that Carnap’s foundational epistemology fails in one of twoways. We have already encountered the first way, that Carnap cannot adequately characterizeanalyticity in terms of L-truth. The second failure is that Carnap provides us with no reason tothink that complicated, theoretical concepts, e.g, concepts from relativistic space-time theory,can be univocally logically reconstructed on the basis of observational concepts alone. In eithercase, Carnap, according to Quine, is engaged in an untenable foundationalist project. As analternative, Quine suggests that we instead adopt a non-foundational and holistic approach tothe foundations of science, an approach which does not countenance a clear separation betweenartificial and natural languages but instead draws on the conceptual resources from empiricalpsychology to inform our epistemological projects.Another worry about Carnap’s logic of science is that it is, quite literally, on the wrongside of history. In his 1962 The Structure of Scientific Theories (SSR), Thomas S. Kuhn had apermanent influence on the way historians and philosophers study science and its history. Ratherthan adopting a view about the history of science which tracks the logical structure of scientifictheories as they progressively get closer to the truth, Kuhn investigates the material history ofhow scientists are trained to do science using a specific set of assumptions, scientific concepts andtechniques, or a “paradigm”, and finds that, at least for cases of scientific revolutions, scientificcommunities do not smoothly transition from older to newer paradigms. The central insight isthat there is no straightforward way to isolate a single notion of progress defined over changesin scientific theories within scientific communities. For post-Kuhnian philosophers of science,Carnap’s logic of science is seen as embracing exactly that ahistorical and logical revisionistconception of scientific theories which Kuhn’s SSR rejects in favor of a philosophy of sciencewhich is invariably intertwined with the history of science.Fortunately, pioneered by scholars like Alberto Coffa, Michael Friedman, Warren Goldfarband Thomas Ricketts, there now exists a quite extensive Carnap reappraisal literature whichattempts to explain Carnap’s own philosophical views in his own terms rather through thehistorical narratives bolstered by Quine or Kuhn. Much of this literature has focused, in partic-ular, on Carnap’s philosophy of mathematics, including not only Carnap’s influences like Frege,Russell and David Hilbert, but also his later work on metalogic and his principle of logical232.2. Wissenschaftslogik: Critiques and Reappraisalstolerance.32 In contrast to Quine’s version of events, we now have much textual and histori-cal evidence that Carnap, in his Aufbau, was not concerned with the foundationalist problemof alleviating Cartesian doubts but rather with the problem discussed by nineteenth centuryGerman-speaking Marburg neo-Kantian epistemologists: this is the problem of showing howscientific knowledge, through the activity of rational reconstruction, is objective, viz. as in-tersubjectively communicable (see Richardson, 1998). Also, rather than Carnap and Quinebeing indefinitely at loggerheads, we find that not only do they both reject “intuition” or “com-mon sense” as an independent source of knowledge (see Creath, 1991) but that their separateapproaches to the philosophy of science are nearly identical aside from a few methodologicaldifferences (see Stein, 1992). When it comes to the philosophical differences between Carnapand Kuhn, not only do we find that Carnap was sympathetic to a manuscript of Kuhn’s SSR(see Reisch, 1991), there are plenty of similarities between Kuhn’s talk of revolutionary/normalscience in terms of “paradigm shifts” and Carnap’s own talk of making the practical decision toadopt a linguistic framework (see Earman, 1993; Friedman, 2001; Irzik and Grünberg, 1995).Finally, from a historical perspective, the supposed grip the logical empiricists had on NorthAmerican philosophy around 1950 doesn’t quite fit the facts: although it is true that logical em-piricism has left a lingering imprint on contemporary philosophy of science, logical empiricism,as a philosophical movement, was far from the dominant movement in post-World War TwoNorth American philosophy (see Creath, 1995; Reisch, 2005; Richardson, 1997a; 2002; 2007).33Consequently the Carnap reappraisal literature provides us with a subtle and complex ac-count of not only Carnap’sWissenschaftslogik but of logical empiricism in general. At the end ofthe previous section, for example, we found that in LSL Carnap does, loosely speaking, embracesome sort of holism for scientific concepts while simultaneously rejecting any foundationalistreading of his logic of science. And it is not as if Carnap leaves no room for sociological andhistorical investigations about the nature of science provided, of course, that we recognize thatsuch investigations belong to the methodology or pragmatics of science and not the logic of sci-32 See, for example, Awodey and Carus (2007); Carus (2007); Coffa (1991); Creath (1992; 1996; 2003); Friedman(1999; 2001); Friedman and Creath (2007); Frost-Arnold (2013); Giere and Richardson (1996); Goldfarband Ricketts (1992); Hardcastle and Richardson (2003); Reck (2013); Richardson (1994; 1996; 1997b; 2004);Ricketts (1994; 1996; 2003); Uebel (2007); Uebel and Richardson (2007); Wagner (2009; 2012).33 For more of the sociological and larger historical perspective of the Vienna circle, see Cartwright et al.(1996); Stadler (2001); Uebel (2007; 2012b).242.2. Wissenschaftslogik: Critiques and Reappraisalsence – in later chapters, we will even see that the history of probability theory and statistics doesin fact inform Carnap’s work on inductive logic. Nevertheless, despite these interpretive effortsto clarify Carnap’s mature philosophical project, we may still have lingering doubts about theadequacy of turning to logical machinery, like logical syntax, in order to answer philosophicalquestions. These worries are to be taken seriously. Far too often in contemporary philosophicaldiscourse genuine philosophical questions are seemingly hijacked by irrelevant technical detailsand problems.For example, in a paper originally intended for, but never published in, Carnap’s Schilppvolume, Kurt Gödel attempts to isolate a tension between Carnap’s attitude of logical toleranceand the application of logical systems to the empirical sciences.34 In what follows I outlineGödel’s argument as found in Goldfarb (1996) and Ricketts (1994). First, Gödel notes that,for Carnap, it seems that the logical relations or rules which fix the consequence relations of alanguage should satisfy the following constraint: that they don’t determine the truth or falsityof empirical propositions. For to do so would mean that those relations or rules improperlyclassify such propositions as “analytic.” Gödel calls those logical relations or rules which satisfythe above constraint “admissible.” However, if our logical rules really are admissible, by Gödel’ssecond incompleteness theorem, a stronger metalanguage is needed to show that our logicallanguage is consistent.35 However, now it seems like all the important philosophical work hasbeen relocated from the object language to the metalanguage. Carnap cannot now suggest thatthe decisions to adopt the rules of formation and transformation for the language are purelypractical as such decisions must now be informed by whether or not those rules are admissible.But now Carnap’s appeal to logical syntax does little to ameliorate Gödel’s concern aboutwhether the rules of transformation are admissible – isn’t this problem now best left to a logicalanalysis in the metalanguage, especially natural languages like English?As Ricketts (1994) points out, Gödel seems to presuppose that while the truth of analyticsentences is determined by the logical rules of the language, the truth of empirical sentences isdetermined, in some sense, by the world. In other words, “Gödel’s definition of admissibility,”says Ricketts, “employs a language-transcendent notion of empirical fact or empirical truth”34 See Gödel (1953); Goldfarb (1995).35 At least this is the case for sufficiently strong object languages.252.2. Wissenschaftslogik: Critiques and Reappraisals(180). Yet according to Ricketts, “Carnap, in adopting the principle of tolerance, rejects anysuch language-transcendent notions” (1994, 180). This is indicative of the philosophically radicalnature of Carnap’s views on the foundations of logic and mathematics and the application oflogic and mathematics to the foundations of science. Given an attitude of logical tolerance,we are free to investigate (and here I adopt a spatial metaphor) a space of alternative logicalforms or rules without presupposing that there are any antecedently given, well-defined, notionsof “fact”, “verifiable” or “confirmable” according to which a logical relation or rule could beevaluated as admissible.Of course, as Ricketts clarifies, Carnap can appeal to the standards and methodology ofscience in order to articulate what Gödel may have in mind by “admissibility”. But Carnap doesnot take such standards for granted; instead, Carnap understands his commitment to empiricismin a way similar to his commitment to tolerance. Neither is an assertion; rather, both areproposals. Thus Carnap’s commitment to empiricism is to be understood as the adoption of aparticular attitude; namely, that our current scientific language provides us with the standardsof rational inquiry and empirical significance. In adopting a principle of empiricism, Carnapcan appeal to empirical standards of our current scientific theories in order to better informour practical choices about which logical system will be satisfactory. Consequently, Carnap canonly understand Gödel’s concerns about whether our logical system is admissible after one hasmade the practical decision to embrace an empiricist attitude or stance – otherwise Carnap canat best only make informal sense of Gödel’s attempt to characterize a notion of admissibility, orsome other notion of “adequacy,” relative to the empirical world.Whatever we may think of Gödel’s argument and Ricketts’s rendition of how Carnap couldpossibly respond to it, we now have a better sense of what is so revolutionary about Carnap’smature philosophical views. In contradistinction to philosophical methods, like conceptual anal-ysis, which purportedly allow philosophers to “discover” the meaning of concepts or to obtainaccess to some realm of propositional facts in light of our intuition or a priori reason, Carnap’smature views emphasize the conventional, volitional and constructivist activities involved ininvestigating the foundations of science. Rather than answer philosophical questions about thenature of logic and mathematics by arguing that it is the case that X, Carnap, quite character-istically, instead constructs a language which contains the syntactical and semantical resources262.2. Wissenschaftslogik: Critiques and Reappraisalsto express a question like X – but he never claims that his own logical reconstruction of X islogically, empirically or conceptually identical to X. But what is particularly philosophical aboutthat?36 In a sense, the rest of the dissertation draws on the history of philosophy of science totry and provide some explanation using my own account of conceptual engineering (from chapter3) as an interpretive framework for explaining the philosophical upshot of Carnap’s work on apure inductive logic and his various attempts to explain how that inductive logic can be appliedto the empirical sciences, especially the foundations of statistics and decision theory (see mychapters 4 and 5).For the moment I want to discuss Carnap’s own attempt to explain his mature views when,in 1945, he adopts the vocabulary of explications instead ofWissenschaftslogik.37 The method ofexplication, according to Carnap, concerns the “replacement of a pre-scientific, inexact concept(which I call “explicandum”) by an exact concept (“explicatum”), which frequently belongs to thescientific language” (1963b, 933). More specifically, the method of explication is, for Carnap, atheory of scientific concept formation based on the historical observation that scientific concepts,after being initially introduced informally, later come to be replaced with more exact qualitative,comparative or quantitative concepts.38 The basic idea is that we first focus on an explicandumin natural language, call it W, the usage of which we agree is vague or inexact and then studythe ways in which W is inexact or vague by trying to clarify how it is used in ordinary speech. Inwhich contexts is the term used? In those contexts, if we all agree that it is being used correctly,why is it useful? When is it being misused? This is the clarification step in an explication. Afterthis step is finished, we next adopt some logical system, call it L, which already has well-definedsyntactic and semantical rules. We then define, in L, one or more semantical concepts, callthem ‘W’ and ‘Wy’, which are each possible explicata. Lastly, we can then give an interpretationfor ‘W’ and ‘Wy’ in L and then investigate the mathematical properties of these new concepts;36 As Peter Strawson puts the point in Carnap’s Schilpp volume: “For however much or little the constructionisttechnique is the right means of getting an idea into shape for use in the formal or empirical sciences, itseems prima facie evident that to offer formal explanations of key terms of scientific theories to one whoseeks philosophical illumination of essential concepts of non-scientific discourse, is to do something utterlyirrelevant – is a sheer misunderstanding, like offering a text-book on physiology to someone who says (witha sigh) that he wished he understood the workings of the human heart” (1963, 504-5).37 Carnap first introduces this method in Carnap (1945b): it is not a coincidence that this paper is also oneof his first published papers on the nature of probability and induction.38 In general, Carnap talks about this method in the following places (this list is not exhaustive): §§1-6 andchapter IV of Carnap (1962b), Carnap’s replies to Peter Strawson in Schilpp (1963) and Carnap (1956).272.2. Wissenschaftslogik: Critiques and Reappraisalsif we find these interpretations satisfactory, we can then apply the language L, which nowincludes the concepts ‘W’ and ‘Wy’, to some domain of objects. Thus we can then study howeach applied explicatum measures up, so to speak, to our expectations regarding the usefulnessand exactness of W in particular contexts. Carnap’s talk of explication is none other than theprocess of locating an adequate application of a pure logic.It is crucial to keep in mind, however, that what I call the measure of the “success” for anyprocess of explicating an explicandum with a particular explicatum is, for Carnap, not an all-or-nothing affair but is rather a matter of weighing the differing degrees to which the explicatumsatisfies a number of practical requirements; namely, the requirements of (i) similarity to theexplicandum, (ii) exactness, (iii) fruitfulness and (iv) simplicity (The Logical Foundations ofProbability ; hereafter LFP, 7). According to Carnap, the reason why the explicatum should beexact is so that it can be introduced “into a well-connected system of scientific concepts” and aconcept is as fruitful insofar as it can be used to formulate “universal statements,” like empiricallaws or logical theorems (LFP 7). Of all the requirements, simplicity is the least important.Lastly, for Carnap there is no limitation on how many explicata we can design and construct –this is a consequence, it seems, of his attitude of logical tolerance.We will return to the details of Carnap’s method of explication in chapter 4. Before wemove on, however, it is important to note that the explicit use of a logical system is not alwaysnecessary for the provision of an adequate explicatum. As Carnap clarifies his views in responseto criticism from Strawson’s contribution to Carnap’s Schilpp volume, Carnap says that he“[sees] no sharp boundary line but a continuous transition” between “everyday concepts andscientific concepts” (1963b, 934). In contrast to Carnap’s method of rational reconstruction inthe Aufbau, explications for concepts are not limited to artificial languages but can be carriedout in natural language too. But that doesn’t mean that artificial languages, like symbolic logic,have no use. “A natural language,” Carnap explains,39is like a crude, primitive pocketknife, very useful for a hundred different purposes. But forspecific purposes, special tools are more efficient, e.g., chisels, cutting-machines, and finallythe microtome. If we find the pocketknife is too crude for a given purpose and createsdefective products, we shall try to discover the cause of the failure, and then either usethe knife more skillfully, or replace it for this special purpose by a more suitable tool, or39 Strawson uses the tool metaphor himself to describe the difference between two philosophical methods,Carnap’s method of rational reconstruction and naturalism (here: ordinary language philosophy) (1963,503).282.3. Carnapian wissenschaftslogiker as Conceptual Engineereven invent a new one. The naturalist’s thesis is like saying that by using a special toolwe evade the problem of the correct use of the cruder tool. But would anyone criticize thebacteriologist for using a microtome, and assert that he is evading the problem of correctlyusing a pocketknife? (Carnap, 1963b, 938–9)The working analogy Carnap employs in this passage explores how using logic to study the foun-dations of science is similar to using a tool or instrument to accomplish some task. In the nextsection, after discussing how Carnap himself uses this analogy, I discuss a number of philosopherswho adopt this engineering analogy to help illuminate Carnap’s mature philosophical views.2.3 Carnapian wissenschaftslogiker as Conceptual Engineer“I admit that the choice of a language suitable for the purposes of physics and mathematics,”remarks Carnap in the second edition of his book Meaning and Necessity,40involves problems quite different from those involved in the choice of a suitable motor fora freight airplane; but, in a sense, both are engineering problems, and I fail to see whymetaphysics should enter into the first any more than into the second. (1956, 43)The context for his quotation is a discussion by Carnap regarding Quine’s views on ontologicalcommitment. For Quine, questions about ontological commitment boil down to the logicaldetails of how quantification over variables works in a language (42). For Carnap, of course,such questions about the logical form of a language amount to “a practical decision, like thechoice of an instrument” which “depends chiefly upon the purposes for which the instrument –here the language – is intended to be used and upon the properties of the instrument” (43). Thuswhereas Quine envisions ontological quandaries, Carnap discerns practical inquiries regardingwhich piece of linguistic machinery we could adopt. More succinctly put, Carnap states in hisautobiography thatWhether or not [the introduction of a linguistic framework – CFF] is advisable for certainpurposes is a practical question of language engineering, to be decided on the basis ofconvenience, fruitfulness, simplicity, and the like. (Carnap, 1963a, 66)Here, the term “language engineering” may be understood in the context of what Carnap latercalls “language planning” and is arguably indicative of his earlier interests in the developmentof artificially created natural languages, like Esperanto, after the First World War (Carnap,1963a, 68; see Friedman, 2007). In chapter 4, we will return to how Carnap himself employs40 For an earlier example of Carnap treating logic like a tool, see the last paragraph of LSL.292.3. Carnapian wissenschaftslogiker as Conceptual Engineerthis engineering analogy to help explain his work on inductive logic. But now I want to shiftthe reader’s attention to how this analogy has been used in the current Carnap reappraisalliterature.Richard Creath uses the engineering analogy to help explain how Carnap addresses worriesabout adopting a non-circular account for the justification of beliefs about the basic postulates ofa theory (Creath, 1992, 142-149). Typically, such a body of beliefs would be justified in terms of(metaphysical) intuitions but yet Carnap, according to Creath, rejects this presupposition. “Theaxioms or postulates,” Creath says of Carnap, “need no further epistemic justification becausea language is neither true nor false, and one is free to choose a language in any convenient way”(1992, 144). Instead, it is we who can lay down such axioms and postulates and it is we whoinvestigate where they lead us. For Carnap there is no further question about getting things“right” above and beyond the choice of these axioms or postulates: “the postulates (togetherwith the other conventions) create the truths that they, the postulates, express” (e.g. see Creath1992, 147). Any talk of epistemic justification is now to be replaced with a pragmatic inquiry:which postulates better or worse explicate some body of beliefs with respect to our theoreticalneeds?Creath calls this process of constructing and choosing satisfactory postulate systems —systems which are themselves revisable – “the engineering task of examining the practical con-sequences of adopting this or that system” (1992, 154). In what amounts to a crucial passagefor understanding this engineering analogy for the context of Carnap’s work on inductive logic,Creath applies this engineering conception to the example of the traditional problem of induc-tion. Because for Carnap the “pragmatic cost” of a language without “inductive rules” would betoo high, “the question is not whether to have inductive rules, but which”:Here again the matter is one of pragmatic comparison. If the rules are too weak, thenwe foreclose or complicate useful inferences. If the rules are too strong, then there is anincreased chance that one inference will conflict with another, thus requiring constant andcostly revision. The virtues of security as contrasted with those of educational adventurewill be weighted differently by different people, but we need not all agree so long as we makeour respective choices clear. There is no uniquely correct system, and the choice among thealternative is pragmatic. (1992, 154)Thus it seems the usefulness of alternative constructions, according to Creath, can be explainedin terms of an instrumental conception of rationality: relative to the adoption of some standard302.3. Carnapian wissenschaftslogiker as Conceptual Engineerof evaluation j as a measure of our intellectual ends, l is a better choice than an alternativem just in case l better satisfies j than m . Notice that nothing has been said about why wewould adopt j – all that is relevant is how the alternatives l and m measure up, so to speak,to the demands placed on them by j . The same seems true for engineering: our practical needsand wants provide the standards for what we want to happen in the world but engineering,by its very nature, cannot inform us what our needs and wants should be. When it comesto Wissenschaftslogik, all we can do, it seems, is to specify our logical languages in as muchdetail which seems necessary and then investigate and evaluate which of those languages willfit our theoretical needs. But that doesn’t mean that we must somehow produce a well-orderedpreference ranking of logical languages. “Inconsistent languages,” says Creath, “are pragmaticdisasters, and so are languages without inductive rules”:It is not necessary to establish that a language is maximally or even minimally convenientbefore using it, but philosophic discussion (where it is not wholly misguided) must bepragmatic. Qua pure logicians our job is merely to trace out the consequences of this orthat convention. This is an engineering conception of philosophy. (Creath, 1990b, 409)Exactly how Carnap can “trace out the consequences” of alternative inductive logics and thenweigh the extent to which those consequences satisfy the wants and needs of scientists is a topicwe will return to in chapters 3 and 5.An alternative way of understanding the engineering analogy, due to Samuel Hillier, ex-plains Carnapian Wissenschaftslogik as an engineering activity tasked with producing a lin-guistic model of some empirical phenomenon. Specifically, Hillier (2007) attempts to providean interpretive framework for understanding the Carnap reception literature by distinguish-ing between two independent interpretations of Carnap’s logic of science. The first project,which Hillier dubs “THERAPY,” focuses on the work of scholars like Thomas Ricketts and War-ren Goldfarb which, according to Hillier, is concerned with explaining why, for Carnap, mostepistemological and metaphysical problems are transformed into pseudo-problems, or problemswithout cognitive meaning (see Hillier 2007, 148 ff., especially 152-3). The second project,dubbed “EPISTEMOLOGY,” concerns the interpretive work by Michael Friedman and AlanRichardson. Here the emphasis is on Carnap’s Wissenschaftslogik in the 1930s as the study ofthe language of science, a study grounded in the clear separation of logical and psychologicalconcepts. Hiller explains this project with an analogy to physics: in lieu of questions about312.3. Carnapian wissenschaftslogiker as Conceptual Engineerthe justification of the use of ordinary language and concepts, both the scientific philosopherand physicist “[make] certain simplifying assumptions and construct a model of that language”(160).Hillier argues that, according to the second project, Wissenschaftslogik is made up of twoparts. First, guided by the logical principle of tolerance, the logician constructs any number ofmodels – here understood in the sense of logical languages – and, second, the logician locatessome notion of “fit” between these linguistic models and scientific activity; after all a model,Hillier explains, is “accepted only if it accords with the thing that is to be modeled sufficientlywell” (2007, 161). However this notion of “fit,” argues Hillier, is not characterized within alogical language but is rather defined relative to whether particular models offer “more accuraterepresentations, or are easier to work with, or whatever other advantages are usually associatedwith modeling in science” (162). In other words, when speaking of what it means, for Carnap,to prefer one language over others Hillier seems to assimilate together the syntactic preferenceswe may have for a language along with empirical measures of “fit” defined over pairs of linguisticmodels and the way the world happens to be.41 Consequently, Hillier’s Carnap no longer seemsto repudiate language-transcendent facts; instead, Carnap is now interpreted as appealing to anotion of “fact” precisely in the sense of what is being modeled or represented independently ofa linguistic framework (186).Hillier then argues that once we stitch together these two interpretive projects, THERAPYand EPISTEMOLOGY, we end up with a “linguistic engineering” interpretation of Wissenschaft-slogik (171). THERAPY is now understood as the conventional processes of designing models,and EPISTEMOLOGY is the empirical process of analyzing the language of science by “fitting”these models to the language scientists use (172). Specifically, Carnap’s principle of tolerance,argues Hillier, applies only to formal languages, languages which can then be freely constructed(169, 182-3, 186). Those freely constructed languages now not only function as tools but asmodels for the language of science. Thus, for Carnap, “there is a fact of the matter that needsto be respected, namely the actual, logical structure of the language of science” and there islikewise a fact of the matter “whether or not the chosen formal language is a good model for the41 This notion of fit, for Hillier, is a measure of how well an explicated concept is similar, really in terms oftruth-preservation, to the “target concept” (165).322.3. Carnapian wissenschaftslogiker as Conceptual Engineeractual language of science” (187).In light of the summary of Carnapian logic of science I provided at the beginning of thischapter, we should find Hillier’s discussion of Carnapian Wissenschaftslogik as depending onsome notion of “fit” between logical and scientific languages to describe Carnapian logic ofscience rather odd. For starters, this notion of “fit” is a notion neither Friedman nor Richardsonreadily adopt and, secondly, both Friedman and Richardson take seriously Ricketts’s suggestion(see above) that, for Carnap, there can be no appeal to language-transcendent facts. Indeed, thecentral presupposition of Hillier’s version of Carnap as linguistic engineer is that the logicianhas ready access to some notion of an “accurate representation” which can be used to gaugethe “fit” of any one of the language frameworks the logician may freely construct. But what isso revolutionary, philosophically speaking, about Carnap’s logic of science is its lack of any in-principle reliance on any robust notion of empirical or logical truth, representation or meaning.This is the difference between Hillier and Creath’s versions of the engineering analogy.Hillier’s notion of “fit,” however, is perfectly understandable to Carnap after both a proposalhas been made and accepted to adopt a principle of empiricism and a logical language hasbeen applied to some empirical science. Within this applied context, Hillier’s notion of “fit”can be defined pragmatically, viz. as denoting the sort of inter-theoretic considerations actualscientists employ to rank hypothesis given their evidence. Indeed, for Ricketts, Richardson andFriedman, Carnap’s commitment to empiricism is an expression of an attitude no different fromthe expression of an attitude of logical tolerance. “Carnap’s lessons are historical and formal,”says Richardson,the epistemic success of the exact sciences is revealed in their history and is due more toprecision and power of formal and mathematical techniques and how they are developed inempirical knowledge than to any other aspect of such science. Carnap sought to understandthat process through the introduction of the self-same techniques and the self-same toler-ance of formally precise linguistic forms in philosophy that one finds in the exact sciencesthemselves. This precision can then for the first time make tolerably clear what someone iscommitted to in being committed to, for example, empiricism. (Richardson, 2004, 74)The standards of scientific discourse provide us with an example of the use and power of formaland mathematical techniques and Carnap proposes that we adopt these standards as we in-vestigate the foundations of sciences using the artificial languages under active development bylogicians and mathematicians – logic is, for Carnap, an instrument but it is an instrument whichis not assessed as a part of Wissenschaftslogik on the basis of its representational properties.332.3. Carnapian wissenschaftslogiker as Conceptual EngineerExactly here, however, the reader may begin to worry that Carnapian logic of science rests onan untenable circularity: The proposal to adopt a principle of empiricism affords a wissenschaft-slogiker the conceptual resources required to apply their logical system to the empirical sciencesbut yet these resources are the very notions in need of philosophical clarification or explication.To adopt the language of explication, only through the creative, engineering, act of constructingmany different logical frameworks can we map out, so to speak, the possible ways constructingdifferent explicata. But because the explicandum is vague to begin with, there is no meaning-ful way to figure out whether any particular explicatum is “correct” or not without, it seems,appealing to extra-logical information about the applicability of each explicatum.One way of trying to make sense of this circularity is articulated in Carus (2007). There,Carus locates a “dialectical” relationship implicit in Carnap’s views which conceptually comesprior to Carnapian logic of science between, first, “the evolved systems of intuitively availableconcepts interwoven with ordinary language” and, second, “the constructional systems of scien-tific and mathematical knowledge” (x). Carus then periodizes Carnap’s intellectual thought intotwo stages. The first stage is similar to the project of the Aufbau; it is concerned with provid-ing an objective (or rather, intersubjective) rational reconstruction of our scientific knowledgewithin a constitutional system, a system which is understood to replace our “evolved” concep-tual system. The second stage is associated with Carnap’s adoption of a principle of logicaltolerance in LSL and amounts, according to Carus, to the implicit recognition of a dialecticalrelation between our evolved and constructed conceptual systems (x-xi).Importantly, it was this first stage of rational reconstruction which centers on the questionof how “to decide – from some overall viewpoint resting at any moment, of course, partly onintuitions – what intuitions we want; which ones to keep and which to supersede” that Carusdescribes as an “engineering task” (17).42 Here Carus turns to a distinction Carnap makes in1950 between internal and external questions – where internal questions are questions framedwithin a language system and external questions are practical questions about which languagesystem we are willing to adopt – to explain this dialectical relationship. In some places, forexample, Carus also adopts the vocabulary of “hard” and “soft” concepts to distinguish between42 Carus is here talking about our intuitions concerns which features of a logical languages we find preferableto others.342.3. Carnapian wissenschaftslogiker as Conceptual Engineerconstructed logical systems intended to replace “evolved” language and the decision to adoptsuch logical systems made from the standpoint of natural, “evolved” language, respectively.Indeed, for Carus, it is this standpoint of a “context of action, which overlaps to some degreewith the Lebenswelt in which the participants articulate the values and preferences that guidetheir choices” (279-80). Carus here points to work by Howard Stein (e.g. Stein, 1992; 2004) inorder to articulate a certain dialectical relationship between these “soft” and “hard”, or evolvedand constructed languages:[t]he explicative interaction between evolved and constructed systems takes the form not ofwholesale replacement or superimposition, for Carnap, but of piece-meal exchange withinthe context of a dynamic mutual feedback relation. (Carus 2007, 278)Unlike Carnap’s early method of rational reconstruction in hisAufbau, which replaces our evolvedlanguage with constructed language, explication, as Carus understands it, involves a feedbackrelation between the evolved and constructed concepts. This transition from rational recon-struction to explication signifies, according to Carus, the second stage. Here we are tasked withan engineering question concerned with whether the results of the above feedback relation aresatisfactory for our practical ends.It is important to Carus that when talking about explications that we distinguish betweenthe task of clarification, which amounts to a sort of initial analysis of a vague concept, and thepractical task of constructing and choosing between alternative explicata as (partial) replace-ments of the vague concept (2007, 20-1; 265-272; 278-9). Carus sees both tasks as crucial toan ideal of explication as a method suitable for reviving earlier Enlightenment projects: clar-ification concerns the tallying of preferences — as “desiderata” of different disagreeing parties(at least for the context of scientific communities) concerning which languages they prefer —and the choice of a language concerns the provision of a framework which the respective paritiesmay each modify until a satisfactory language is found (30-1). As Carus clarifies,[t]his is not simply a mechanical task of pasting together two incompatible languages; itobviously requires creative ingenuity — this is conceptual engineering. The outcome dependson the quality of this engineering. Occasionally a perfect synthesis can be found, but usuallythe solution in such cases is something of a compromise, which in practice fails to satisfyat least a few disputants on the fringes. These can then go on arguing, demanding thatthe compromise be reviewed, or they can walk out and start a new discipline. Such anengineering failure can always be attributed to the impossibility of the task, but it cannever be known for certain whether better engineering might not after all have done thetrick in the end. (Carus 2007, 31)352.3. Carnapian wissenschaftslogiker as Conceptual EngineerCarnapian conceptual engineering, according to Carus, is a piece-meal, dialectical process forwhich there is no guarantee of success. In the last two chapters of his book, Carus attempts toflesh out Carnapian explication as an ideal of explication – an ideal because Carnap, accordingto Carus, himself never saw these implications for his project clearly – continuous with Enlight-enment ideals which would function as conceptual resource helpful (say) for resolving disputesin political theory (like the debates between Rawls and Habermas) by allowing us to use theabove conventional framework to “engineer” concepts, for example, “to serve as tools for socialand political interaction” (305).Thus we have a picture of how the circularity of Wissenschaftslogik can be explained: thereis a dialectical relationship between (1) appealing to our “evolved” languages in order to clarifyconcepts and (2) replacing these “evolved” concepts with logically engineered concepts modeledloosely on the clarification of the “evolved” concepts.Another way of making sense of the circularity of Wissenschaftslogik is by drawing attentionto the fact that Carnap’s talk of treating languages as tools seems to coincide with Carnap’searly work on empirical concept formation (e.g. in Carnap, 1926) and Carnap’s early interestsin the study of measuring instruments, or Instrumentenkunde, as practiced in the 19th centuryGerman-speaking world (see Richardson, 2013, 61-5).43 Just as metrologists use instruments astools to define concepts of measurement, Carnap uses his metalogic, analogized as an instrument,as a tool to define, and so making explicit, certain scientific and philosophical concepts. AsRichardson points out, Carnap himself says as much in the preface to Carnap (1943); it is therethat Carnap tells us explicitly that he regards his semantics as “a tool, as one among the logicalinstruments needed for the task of getting and systematizing knowledge” (Carnap 1943, viii-ix).Logic for Carnap, then, is a tool which can be used to enhance our mental capacities. Theengineering analogy, for Richardson, is not an analogy (or metaphor) at all. Instead, “Carnap’sconsidered view,” says Richardson, “was that as a philosopher he engaged in the developmentof conceptual technologies for science and the science of science. This is Carnap the conceptualengineer” (2013, 65).4443 Carnap’s explications of prescientific concepts mirror, to a certain extent, the process scientific conceptsundergo over time of becoming more exact or precise (e.g. see Chang, 2004).44 Importantly, besides criticizing Carus’s reading of Carnapian explication as belonging to the tradition of theEnlightenment, Richardson also raises various worries about the received importance of Carnap’s “technical”conception of philosophy (2013, 71).362.4. Carnap and the State of Inductive Logic at mid-Twentieth Century2.4 Carnap and the State of Inductive Logic at mid-Twentieth CenturyStarting with this section, for the rest of this dissertation we will focus less on Carnap’s viewson logic and mathematics, including his conception of logical syntax and semantics, and moreon how he uses these conceptual resources as instruments for clarifying the foundations ofprobability and induction. By chapters 4 and 5, we will see how Carnap’s work on inductivelogic, which includes both his work on pure inductive logic and how inductive logic can be appliedto decision theory and theoretical statistics, can be seen as a conceptual engineering project:Carnap shows us how his work in inductive logic could possibly be of use in the empiricalsciences, especially when those sciences depend on a logical meaning of probability.45 Moreover,it is important to keep in mind that Carnap’s later work on inductive logic is an exampleof how his earlier Wissenschaftslogik can be extended : in the 1930s, after all, notions like“confirmability” and “testability” were understood to be pragmatic concepts which were treatedas primitive notions in the syntax and semantics of science – but by the mid-1940s, Carnapwas attempting to explicate qualitative, comparative and quantitative concepts of degree ofconfirmation as semantic concepts defined in terms of L-concepts.46By the time Carnap first published on logical probability in 1945, the year which marksthe end of the Second World War, other members of the scientific philosophy movement –including both members of the so-called Berlin Circle, like Hans Reichenbach and Richard vonMises, and the Vienna Circle, like Herbert Feigl – had been working on philosophical issueson probability for the better part of three decades.47 But even Reichenbach, von Mises andFeigl were investigating a mathematical theory which was by the 1920s already quite mature:Reichenbach’s own contributions to the axiomatization of the probability calculus aside, theearly twentieth century saw the rigorous axiomatization of the classical theory of probability45 Nowadays it is customary to talk about the philosophical problem of how to interpret probabilities and tospeak of different interpretations of probability. However, to avoid confusion with the interpretation of alogical calculus with the interpretation of probabilities I instead adopt the nomenclature of taking aboutthe meaning of probabilities.46 Although Carnap talks about all three concepts I focus excessively on classes of quantitative concepts.47 Influenced by scientists like Henri Poincaré and Johannes von Kries, Reichenbach writes (incidentally, duringthe middle of the First World War) his 1916 dissertation on the concept of probability, which is aptlynamed “The Concept of Probability in the Mathematical Representation of Reality” (published in Englishas Reichenbach, 2008). In the 1920s and 1930s, Reichenbach then writes a number of papers in which hearticulates a notion of probabilistic implication in a multi-valued logic; a summary and extension of his viewscan be found in Reichenbach (1935), which is in 1949 expanded and translated into English as Reichenbach(1949). See chapter 5 for more information regarding both Reichenbach and Feigl’s views on probability.372.4. Carnap and the State of Inductive Logic at mid-Twentieth Centuryoriginally formulated in the seventeenth and eighteenth centuries by scientists like Laplace,Leibniz and members of the Bernoulli family and a new measure-theoretic understanding ofprobability was deeply intertwined conceptually with the rise of probabilistic thinking in theempirical sciences, especially statistical mechanics and sociology.48 However, for members of theVienna Circle, the central philosophical issues concerning probability and induction didn’t somuch concern the mathematical details of the probability calculus but rather how to understandthe meaning of these probabilistic and inductive concepts – specifically, how to reconcile theirempiricism with scientific endeavors which purport to produce scientific knowledge which tradesin chances and uncertainties rather than certain knowledge and truth.49It will come in handy to first provide the reader with a simplified version of the probabilityaxioms. Without discussing too much mathematical detail, a continuous, finitely additive,probability function is characterized by the tuple (ΩOF O d ) satisfying the following axioms:1. d (Ω) = 1 (and, in virtue of d being a measure, d (∅) = 0).2. d (U) ≥ 0, for all U in F .3. (Finite Additivity) For any pairwise disjoint sequence of subsets of Ω, U1, ..., Un: d (⋃Ui) =d (U1) + · · ·+ d (Un) =∑d (Ui).Here Ω is called the outcome space and is a set of the most basic, exclusive and independent,events; F is the event space which is a set of subsets of Ω whose elements characterize all possibleevents; and, lastly, d is a measure function defined over the elements of F which satisfies axioms48 For systematic historical accounts of these developments see Gillies (2000); Hacking (2006); Keynes (1921);Porter (1986); Stigler (1986; 2002); Todhunter (1865); Von Plato (1998). Reichenbach and von Mises didattempt to provide their own axiomatization for probability theory; however, most modern textbooks onthe mathematical theory of probability follow the approach to characterizing probability theory, e.g., asfound in Billingsley (1995); Durrett (2005); Feller (1968), more or less follows the same approach as found inAndrey Kolmogorov’s 1933 axiomatization of probability theory using measure theory, published in Germanas Grundbegriffe der Wahrscheinlichkeitsrechnung ; other important early axiomatizations include Koopman(1940) and the work of the Polish, and Jewish, mathematician Janina Lindenbaum-Hosiasson, like Hosiasson-Lindenbaum (1940), before she and her husband, Adolf Lindenbaum, were murdered by the Nazis in 1942(see Pakszys, 1998).49 I say this only to begin to characterize the philosophical problems of probability that Carnap would havebeen familiar with in the 1940s (e.g., as discussed in Nagel, 1939). Of course, mathematical questions aboutthe foundations of probability theory have and continue to be of interest, especially questions concerningthe nature of how to handle infinite sequence or events which give rise to problems like the St. Petersburgparadox or how conditional probability functions should be defined; e.g., see Bartha et al. (2014); Easwaran(2014); Hájek (2003; 2012). However, because Carnap either did not concentrate or was not aware ofthese problems (and as important as they are to contemporary formal epistemologists and philosophers ofprobability) I will not discuss them in the dissertation.382.4. Carnap and the State of Inductive Logic at mid-Twentieth Century1-3.50 Conditional probabilities are then typically introduced by definition: the probability ofevent U given event V, or d (U|V), is defined as the following ratio,51d (U|V) =Def d (U ∩V)d (V)NFor self-proclaimed empiricists like von Mises, Reichenbach and Feigl, probabilities are de-fined over sets of hypothetically, but physically possible, sequences of events, viz., as the hypo-thetical limit of an observed relative frequency of some property which holds, or does not hold, ofeach event.52 For example, letting the possible results of flipping a coin infinitely many times becharacterized by countably many sequences of random variables, viz. a function Si : Ω→ {0O 1}where ‘1’ denotes heads and ‘0’ tails, if after a coin has been flipped n many times and m manyheads in this sequence have so far been observed, then the relative frequency of heads in theobserved sequence up to the n-th flip of the coin is the ratio mQn, which can also be expressedas the average Sn = 1n∑ni=1 Si.The trouble frequentists like von Mises and Reichenbach have, however, is that it is isn’t clearhow – if at all – probabilities should be assigned to singular events, i.e., events about which norelative frequencies have so far been observed. For example, consider one of the central resultsfrom classical probability theory: the (weak) law of large numbers (LLN). This law states that,for any infinite sequence of independent and identically distributed binomial random variablesl1O NNNO liO NNN with range {0O 1} and assuming that each trial li has the same mathematicalexpectation, or mean,  (i.e., E[li] =  for all i = 1O 2O NNN), thenfor all positive real numbers ", limn!1d(|ln − | R ") = 0NIn plainer English, the result says that for any error term with a value in the positive real numberline, as the number of trials approaches infinity, the probability that the absolute differencebetween the observed average and expectation of the trails is greater than the error term is equal50 More specifically, F is a sigma-algebra defined over Ω. This means not only that ∅; Ω ∈ F , but that if forall Xi; Xj ∈ F , where Xi; Xj are members of a countable set X ⊆ Ω, then: Xi ∈ F , Xi ∪ Xj ∈ F andXi ∩Xj ∈ F .51 Assuming, of course, that P (P) ̸= 0.52 Actually, von Mises defines probabilities relative to a kollektiv, which is an infinite sequence of independenttrials, like tosses of a coin. However, the technical differences between von Mises and, say, Reichenbach’swork on probability theory are not essential to the present point.392.4. Carnap and the State of Inductive Logic at mid-Twentieth Centuryto zero.53 The frequentist has basically two problems if they want to apply this mathematicalresult from probability theory to any actual empirical sequence of events; for example, as a wayto infer the value of the expectation that the same coin will land heads up when flipped basedon the current relative frequency of heads for a large number of trials. The first problem ismaking sense of the mathematical assumption that, if the random variables Smi , i = 1O NNNOm,record the observed results of the first m many flips of the coin, that for some value ,E[S1] = E[S2] = · · · = E[Sm] = NIt is not at all clear how a frequentist can justify the claim that E[S1] =  (after all, the relativefrequency of heads for only one flip which lands heads is 1Q1 – but surely that doesn’t license usto claim that  = 1). Obviously the frequentist can just assume that the coin does, in fact,exhibit a particular statistical distribution and then study different variations of LLN based onwhat kind of distributions Smi has for large m.54 The second problem is conceptually relatedto the first: how does the frequentist know, on the basis of their observed relative frequencies,that as m reaches infinity, the limit of Smi exists? Even if after a million flips of the samecoin in the same kind of way the observed relative frequency is, with an acceptable amountof error, very close to the value 1Q2, couldn’t it still be physically possible for “Nature” to allof a sudden “decide” to switch course and cause the coin to consistently land heads or tails,at least for the foreseeable future? The frequentist can provide no guarantees that this wouldbe implausible: the physical structure of the coin may cause it to exhibit radically differentstatistical distributions in the long run.55Alternatively we could instead explain what probabilities mean not in terms of observedfrequencies but instead we could represent the event space F in terms of the sentences (orpropositions) contained in a logical system L and then define a probability function over theselinguistic events. With a few modifications to the above probability axioms, the basic idea isthat we should be able to define a quantitative logical probability function, dr, over the basicsentences in L and then define a conditional logical probability function similarly as above, for53 Note that just because an event O has probability zero, that does not mean that O is impossible. If theprobability function  is a Lebesgue measure defined over the unit interval, for any finite set P ⊂ [0; 1],even if P ̸= ∅, it is the case that (P) = 0.54 This is essentially what Reichenbach does; see §§49-51 of Reichenbach (1949).55 For Reichenbach’s attempt at a solution, which we will return to again in chapter 5, see his chapter 9, called“The Problem of Application,” in Reichenbach (1949).402.4. Carnap and the State of Inductive Logic at mid-Twentieth Centuryany two sentences UOV in L (where V is not L-false)dr(U|V) =Def dr(U ∧V)dr(V)NConditional probabilities are of particular interest to proponents of the logical concept of proba-bility as dr(U|V) can be thought of a generalization of logical implication: if V logically impliesU, then dr(U|V) = 1 and, otherwise, dr(U|V) can be thought of as a relation of support orconfirmation for how much the sentence V supports or confirms the sentence U. The sugges-tion that a logical notion of conditional probability captures, in some sense, some liberalizednotion of logical implication is not my own invention; the idea can be found, for example, inthe writings of a member of the Vienna Circle, Friedrich Waismann.56 Following Johannes vonKries and Ludwig Wittgenstein, Waismann defines a logical probability function by assigningequal probability values to each sentential description of each basic event – in other words,equal prior probabilities are assigned to the most basic, exclusive and collectively exhaustive,events.57 Probability values for more complex sentences are consequently fixed by the choice ofthese prior probabilities and can be found by using the probability calculus.58Once provided with a probability function dr which is well-defined over the sentences of L,it is simple enough (if not cumbersome) to make sense of the weak LLN. For example, we couldcodify sequences of coin flips by interpreting individual constants, ‘u1O u2O u3O NNN’ as denotinginstances of a coin flip and then interpret the descriptive predicate ‘H(ui)’ to mean that thecoin flip denoted by the constant ‘ui’ landed heads up. Then we simply need to supply the priorprobabilities for the coin landing heads, e.g., as a single fixed value , dr(H(ui)) = , for allindividual contents indexed by i. Finally we could let the sentence Ym describe the observedresults of flipping a coin m many times and then define Ym as the relative frequency of howmany times the predicate H(x) holds for the first m many individual constants indexed with thenatural numbers 1O 2O NNNOm. There is the additional problem of how to handle infinite sequencesin L (L has to contain infinitely many individual constants) but stating a version of the weak56 See Waismann (1930).57 For the details, including Waismann’s connections with von Kries and Wittgenstein, see Heidelberger (2001).58 Two useful consequences of the probability calculus are Bayes’s rule and the principle of total probability.Bayes’s rule says that, for any tuple (Ω;F ; P ) as defined above, P (O|P) = P (M)×P (N|M)P (N)and the principleof total probability says that, if {Oi} is a partition of Ω, for any P ∈ F , P (P) = ∑P (Oi) × P (P|Oi).Combining these two rules, we can reformulate Bayes’s rule as: P (O|P) = P (M)×P (N|M)∑P (Mi)×P (N|Mi) , where {Oi} isa partition of Ω.412.4. Carnap and the State of Inductive Logic at mid-Twentieth CenturyLLN for logical probabilities is now just a mathematical exercise (which I won’t attempt todemonstrate here).What is important is that proponents of logical probabilities can use the expressive powerof a language L to characterize the entirety of the basic possible events as the atomic sentencesin L and then express the more complex events as those sentences in L which are formed bycarrying out logical operations, like logical conjunction or negation, on the atomic sentences.While the onus on the frequentist is to explain how the mathematical probability calculus canbe applied to observed regularities (e.g., whether the limit of a hypothetical relative frequencyexists in the limit, or how to assign probabilities to singular physical events), the onus of theproponent of a logical meaning of probability is providing some reason or justification for amethod or procedure which assigns probability values to all the sentences in L. This is thecentral concern with the logical meaning of probability: how do we assign prior probabilities toall the sentences in L and, if we decide to assign probabilities on the basis of some syntacticalor semantical properties of L, how could logical probabilities possibly be the basis for guidingour expectations about empirical events in the world; how could logical probability possibly bea guide in life?59The idea that the same probability values should be assigned to similar events is a persistenttheme of classical probability theory and is usually justified by reference to either the principleof insufficient reason or, which is arguably a consequence of the principle of insufficient reason,a principle of indifference, viz. equal probabilities should be assigned to those events which areequally possible – where “equally possible” can be explained in terms of the state of ignorance ofa reasoner or some physical symmetry, e.g., like assigning the probability that a coin will landheads to be equal to the probability that it will land tails because of the physical symmetries59 In the late 1950s and 1960s, there is resistance to the idea of induction being somehow dependent on amathematically-constructed language. First, Wesley C. Salmon, after corresponding with Carnap in thelate 1950s, argues that confirmation functions should satisfy the criterion of linguistic invariance (e.g. seeSalmon, 1963). Second, Nelson Goodman’s “new” riddle of induction, a problem first introduced (albeitin a different name) in Goodman (1946) and later clarified in Goodman (1955), was widely influential andthat problem, in a nutshell, suggests that there is a substantial epistemological problem concerning whichpredicates are, a priori, the “correct” predicates we should use to formulate inductive claims. As it turns out,however, in a handwritten note on the backside of a letter to Carnap, Goodman admits that he formulatedthe kernel of his worry only for Hempel, Helmer and Oppenheim’s purely syntactic concept of confirmationand before Carnap’s own work on inductive logic was published in 1945 (Goodman to Carnap, February 17,1947; RC 084-19-09).422.4. Carnap and the State of Inductive Logic at mid-Twentieth Centuryof the coin.60 Nevertheless, an important scientific event in the late nineteenth century forthe foundations of probability theory was the discovery, e.g., by the French mathematicianJoseph Bertrand, that multiple applications of a principle of indifference which assign probabilityvalues to the possible outcomes of a physical set-up based on different symmetrical or invariantproperties of that set-up may result in the assignment of different probability values to the sameoutcomes.61 Thus even grand appeals to metaphysical principles, like the principle of insufficientreason, do not afford the scientist a univocal method for assigning probabilities to basic events.But even if this weren’t the case, the very idea that probabilities – let alone a scientific method ofinduction – can be grounded or justified on the basis of metaphysical principles like the principleof insufficient reason or the uniformity of nature is anathema to empiricist strictures.Indeed, the sentiment that inductive methods are of little help to scientific reasoning isvoiced by influential scientists like Ernst Mach and Karl Pearson and these inductive suspicionswere shared by many members of the Vienna Circle (in contrast to the Berlin Circle, of whichReichenbach and von Mises where both members). In LSL, for example, although Carnap isquite amenable to the idea that the P-rules for a syntax for the language of physics may includeprobabilistic laws62 – presumably, Carnap has in mind here the kind of frequentist meaning ofprobability used in physics, especially statistical mechanics – and even though in the historyof science there are plenty of examples of scientists appealing to some notion of induction toexplain how new laws can be introduced in our scientific theories on the basis of the currentevidence, or protocol sentences, Carnap remarks that63this designation [viz., the method of induction – CFF] may be retained so long as it isclearly seen that it is not a matter of a regular method but only one of a practical procedurewhich can be investigated solely in relation to expedience and fruitfulness. That there canbe no rules of induction is shown by the fact that the L-content of a law, by reason of itsunrestricted universality, always goes beyond the L-content of every finite class of protocolsentences. (my emphasis; LSL, 317)Technicalities aside, like what Carnap means by “L-content”, the point is simple. Granted thatscientific laws hold universally and without restriction in the sense, presumably, that if a law isabout some kind of physical set-up then that law holds for all instances of that kind of physical60 For more on the classical theory of probability see Hacking (2006) and Todhunter (1865). Gillies (2000), inparticular, has a nice summary of the problems with the principle of indifference.61 Keynes (1921) is one of the first to clearly distinguish the principle of insufficient reason from the principleof indifference. Earlier critics of the principle include John Venn, Leslie Ellis and George Boole.62 See LSL, pp. 314.63 Similar sentiments can be found in section 1 of Carnap (1926).432.4. Carnap and the State of Inductive Logic at mid-Twentieth Centuryset-up, and, moreover, that no (non-trivial) inductive inference from a finite number of pro-tocol sentences to a scientific law implies that that law holds for all possible future protocolsentences, then no such inductive inference allows us infer the existence of a universally unre-stricted scientific law. Thus, if all scientific laws are universally unrestricted, there can be noinductive rules which govern the introduction of scientific laws into our physical language basedon a finite number of observational statements. Even if we could amass a collection of recordedobservations about whether the sun has risen every day since the invention of cuneiform writing,there is no logical implication from the sum of this solar evidence to the law that the sun willalways rise (even despite the fact that this sum of evidence would, given most frequentist andlogical meanings of probability, provide probabilistic support for the claim that the sun will risetomorrow). Induction, for Carnap in the 1930s, is an activity scientists engage in which resistsformalization into the logical syntax of the language.64What distinguishes Carnap’s earlier discussions of testability and confirmability from hislater work on inductive logic is that his later work treats of a semantic rather than a pragmaticconcept of degree of confirmation.65 It is such a semantic concept of confirmation which Carnapclaims provides a possible explication of the logical concept of probability and on basis of thissemantic concept Carnap then goes on in the 1950s to illustrate how it could be possible toconstruct an entire network of semantically defined inductive concepts where, so to speak, thissemantic concept of degree of confirmation is the semantic knot which bundles the conceptualnetwork together. We will have a chance to untangle this remark in the following chapters and,in the next chapter, I provide a short narrative about the history and philosophy of engineeringwhich I will then use as an interpretive framework for characterizing Carnapian inductive logic.Before ending this chapter, however, I will, first, quickly point to the relevant scientists andmathematicians most influential on Carnap’s work on inductive logic and, second, highlight therelevant passages from Carnap’s own writings in which he himself provides synopses of the aimof inductive logic.There are four scientists and philosophers working in the 1920s and 1930s on logical meanings64 As Carnap poses the question to Reichenbach in 1929: “Could we, with the help of some inference process,infer from what we know to something “new,” something not already contained in what we know? Such aninference process would clearly be magic. I think we must reject it” (quoted in Coffa 1991, p. 329).65 In the 1962 preface to Carnap (1962b), Carnap states that in Carnap (1939) and earlier he had a pragmaticand not a semantic concept of confirmation in mind.442.4. Carnap and the State of Inductive Logic at mid-Twentieth Centuryof probability, including the closely related subjective meaning of probability, who are mostinfluential for Carnap when he starts working on probability and induction around 1941.66 Forthe remainder of this chapter, I discuss this earlier work on logical probabilities and explain howCarnap disambiguates his own work on inductive logic from these earlier views.When the topic of the University of Cambridge occurs in a conversation about Carnap’sphilosophical views, the intended context typically concerns the emergence of analytical phi-losophy by philosophical actors like G. E. Moore, Ludwig Wittgenstein and Bertrand Russell.After all, it was Moore and Russell who, in their own separate ways, demonstrated how tophilosophize using logical analysis and it was Russell (along with his co-author and teacher,Alfred N. Whitehead) who provided in the 1910s an axiomatization of logical type theory inPrincipia Mathematica. What is perhaps less well-known is the work on probability and induc-tion underway at Cambridge in the 1910s and 1920s, especially by the Cambridge logician W. E.Johnson, who articulated a logical meaning of probability as a logical relation between propo-sitions, and two of his more famous students: John Maynard Keynes and Frank P. Ramsey.67In his 1921 book, A Treatise on Probability, for example, Keynes not only provided a detailedphilosophical and historical summary of the foundations of probability but he also took theconcept of knowledge as primitive and defined probability as a relation between propositionsH and Y which corresponds to the quantitative degree of certainty, or partial knowledge, ofthe hypothesis H given the evidence Y. Keynes embeds his understanding of the probabilityrelation within a particular epistemological context which he borrows from both Russell andW. E. Johnson. Logical probability relations, for Keynes, are objective and real relations which66 Carnap first started to seriously think about problems of probability and induction at least as early as 1941,when Carnap was visiting at Harvard (Carnap, 1963a, 36). Feigl, however, dates Carnap’s involvementon problems of probability earlier, to 1938. It is was at an APA meeting in Urbana, Illinois that Feiglreports that he “urged Carnap to apply his enormous analytic powers to the problems of induction andprobability [. . . ]. Carnap immediately began sketching in many hours of intensive discussion of what laterbecame his great and influential work in Inductive Logic” (Hintikka, 1975, xvii). There is some evidencefor Feigl’s claim. The Western division of the APA did meet in Urbana from April 14-16 in 1938 and thereare documents at Carnap’s archive in Pittsburgh concerning a Gewichtslogik written in the summer of 1938(RC 079-20-02). As will become clear in chapter 5, I think there is reason to believe that Carnap was atthis time also thinking about Reichenbach’s notion of “weight” from Reichenbach (1938). In a documenttitled “Weight (degree of confirmation)” from 1941, Carnap defines an absolute notion of weight, ‘oet’,and a relative notion of weight, ‘ret’, and then defines, with a slight change to Carnap’s notion on mypart, ret(o; b) as mWt(m+n)=mWt(n), where o; b represent, arguably, state-descriptions but I can’t decipher theGerman short-hand (RC 079-20-01, p. 1, December 2, 1941).67 See Galavotti (2005; 2011b), Howie (2002) and, for more on Keynes in general, including G. E. Moore’sinfluence on Keynes, see Skidelsky (2003).452.4. Carnap and the State of Inductive Logic at mid-Twentieth Centurycan be located in the logic of scientific theories. An example from Keynes’ book illustrates thispoint quite nicely:When we argue that Darwin gives valid grounds for our accepting his theory of naturalselection, we do not simply mean that we are psychologically inclined to agree with him; it iscertain that we also intend to convey our belief that we are acting rationally in regarding histheory as probable. We believe that there is some real objective relation between Darwin’sevidence and his conclusions, which is independent of the mere fact of our belief, and whichis just as real and objective, though of a different degree, as that which would exist if theargument were as demonstrative as a syllogism. We are claiming, in fact, to cognize correctlya logical connection between one set of propositions which we call our evidence and whichwe suppose ourselves to know, and another set which we call our conclusions, and to whichwe attach more or less weight according to the grounds supplied by the first. (Keynes 1921,5-6)We will see in chapter 5 that Ramsey, in his 1926 article “Truth and Probability,” also conceivesof probability as a logical relation but is nevertheless critical of Keynes’s claim that probabilityrelations are real and objective. Rather than suggesting that there is the degree of belief orcertainty which is attached to a proposition attesting to the fact of the Darwinian theory ofevolution by natural selection as a consequence of the existence of a real and objective logicalrelation between that theory and a multifarious collection of empirical evidence (where boththeory and evidence are expressed as sets of propositions) Ramsey argues that the degree towhich a person is certain in Darwin’s theory given their current evidence can bemeasured as theirdegree of belief in that hypothesis given their evidence as a function of the betting quotients theywould be willing to take defined over the possible states of the biological world. Nevertheless,for both Keynes and Ramsey a logical conception of probability, broadly understood, is closelytied with human psychology and behavior.68In the 1920s and 1930s, Harold Jeffreys, a Cambridge-based astronomer, geophysicist andmathematician, co-authored with the mathematician Dorthy Wrinch a series of articles on thenature of the scientific method which provided a conceptual platform for his own conception ofprobability as an inductive logic, viz., a probabilistic logic based on a system of axioms in a waysimilar to how, according to Jeffreys, Russell and Whitehead based their own work on deductivelogic on a number of primitive postulates characterizing deductive reasoning (Jeffreys 1939, 7-8,16).69 Indeed, for Jeffreys, logical probability is central to the very idea of a scientific method:68 See the first two chapters of Keynes (1921), including Keynes’s notion of “secondary” propositions which heborrows from W. E. Johnson.69 Jeffreys is perhaps most well-known for his work on geophysics and as a vocal critic of the continental drift462.4. Carnap and the State of Inductive Logic at mid-Twentieth Centurythere is no way to reduce scientific method to merely deductive logic without, says Jeffreys,“rejecting its chief feature, induction” (1939, 2). Borrowing an idea from Karl Pearson’s TheGrammar of Science, Jeffreys argues that although the “materials” of scientific reasoning willchange across scientific disciplines and fields, the scientific method remains invariant: “[t]heremust be a uniform standard of validity for all hypotheses, irrespective of the subject” (7). Prob-ability theory is, for Jeffreys, such a uniform standard. Moreover, Jeffreys was not one to shyaway from appealing to the restricted use of invariance principles – like the principle of indiffer-ence – to assign similar probability values to physically similar or symmetrical events. However,although Jeffreys’s reasons for appealing to the symmetries of physical systems to assign logicalprior probabilities was couched in metaphysical language, his arguments for appealing to suchprinciples were based less out of metaphysical conviction than methodological necessity. This isthe strength of the logical meaning of probability: in the face of uncertainty, or a partial lack ofempirical evidence, there are well-defined procedures for assigning similar events the same priorprobability values.70The final actor I wish to mention is the Italian mathematician Bruno de Finetti. First,it was de Finetti who articulated a purely subjective meaning of probability closer in kind toRamsey’s conception of probability than Keynes’s or Jeffreys’s. Second, from the 1930s to the1980s, de Finetti made numerous contributions to the mathematical theory of probability theoryand subjective decision theory, including defining probability functions over a certain kind ofmathematical object called exchangeable sequences – a mathematical result which generalizesCarnap’s work on a continuum of inductive methods.71 Lastly, de Finetti’s work on a subjectiveconception of probability directly influenced the statistician L. J. Savage to attempt to laytheory of the formation of Earth’s surface as proposed by Alfred Wegener. It was with Wrinch that Jeffreys,partially as a reaction to Broad (1918), put together a general, probabilistic theory of scientific inferencebased on inverse probability which is in based on the idea that more complex laws are to be assignedhigher prior probabilities and more simple laws lower prior probabilities (Howie 2002, 106). These earlierpapers inform Jeffreys’s two later books, Jeffreys (1931) and Jeffreys (1939). Wrinch, who was a lecturerin mathematics at University College London when she collaborated with Jeffreys from 1919 to 1923, alsoattended lectures by both Bertrand Russell and W. E. Johnson, was an admirer of Wittgenstein and wassomething of a personal assistant for Russell until 1921 (Howie 2002, 109). Interestingly, it is may havebeen Wrinch who first introduced Jeffreys to the logical work of Russell and Whitehead (2002, 90).70 In this respect, Jeffreys also influenced the statistician and physicist E. T. Jaynes who, in the 1970s, publisheda series of papers in which he suggested that probability theory can be understood as an extension ofdeductive logic for which probability values can be assigned on the basis of empirically-informed invariancesof physical systems. He then makes use of this logical meaning of probability in his work on informationand entropy; see Jaynes (1957a;b; 2003).71 See Good (1965); Skyrms (2012); Zabell (2005).472.4. Carnap and the State of Inductive Logic at mid-Twentieth Centurythe foundations of theoretical statistics on the basis of decision making under uncertainty (seechapter 5). Indeed, de Finetti was instrumental in providing the mathematical and conceptualframework for Bayesian approaches to statistical and scientific reasoning. Nevertheless, becausedelving into the complexities of both de Finetti’s mathematical work and his philosophicalbackground (which would require us to discuss Henri Poincaré and the early twentieth centuryItalian pragmatists), save for a few sections in chapter 5, I say relatively little about de Finettior his various influences, latent or direct, on Carnap in this dissertation.72These four different ways of articulating a logical, and also a subjective, meaning of prob-ability provide the immediate historical context for how Carnap structured his own project ofproviding the foundations for a logical notion of probability, a project which is really taskedwith showing how to construct many possible inductive logics based on one of many semanticconcepts of degree of confirmation.73 For example, when Keynes and Jeffreys write full lengthtextbooks on probability and induction they do not try to re-invent the already extant field ofprobability theory but rather they attempt to show how their new conceptions of probabilitycan be used to reproduce already well-known mathematical results, like the weak LLN. Carnapfollows suit in his own textbook on probability and induction, his 1950 book LFP : although heleaves much of the mathematical details to an unpublished second volume to LFP, the maindesideratum for any pure inductive logic is that it can reproduce and recover most of the centralresults in classical probability theory, statistical inference and estimation theory.74Moreover, unlike Keynes and Ramsey, Carnap is at pains to separate the psychological andepistemological aspects of inductive reasoning from a purely logical meaning of probability. Itis for this reason that when Carnap begins to articulate how to construct an inductive logiche distinguishes between a pure and applied inductive logic, a distinction which is parallel tothe distinction between mathematical and physical geometry discussed at the beginning of this72 That isn’t to say that the topic of de Finetti’s influence on Carnap, and vice versa, isn’t of interest. Forexample, Richard C. Jeffrey points out that de Finetti cites Carnap’s Aufbau as an important influence inone the former’s earliest published works; see Jeffrey (1989).73 As we will see in chapter 4, Carnap in fact is a pluralist about the meaning of probability: the frequentistconcept is useful in the empirical sciences, especially physics, whereas the logical concept is more useful indecision making and statistical inference; see Carnap (1945b).74 I say “most” because Carnap, unlike Leibniz, for example, cannot appeal to any unrestricted use of a principleof indifference to assign logical probability values to sentences in a language (although see Hacking, 1971).Carnap, however, does acknowledge that he must appeal to a restricted version of a principle of indifferenceto assign logical probability values (Carnap, 1963a, 73).482.4. Carnap and the State of Inductive Logic at mid-Twentieth Centurychapter. It is for Carnap a pure inductive logic which can be applied, as we will see in chapter 5and 6, to either a normative or empirical decision theory just as a mathematical axiom systemcan be applied to physical geometry: specifically, a semantic concept of degree of confirmationcan be applied insofar it is coordinated, or interpreted, as a credence (or credibility) functionrepresenting the conditional (or absolute) degree of belief of an actual or ideal agent.75 Whenreading Carnap’s later work on decision theory it is easy to collapse this distinction betweenpure and applied logic; but Carnap is not one to reify logical concepts: pure inductive logic hasno direct implications for how rational agents should believe or act. Only by showing how hiswork in pure inductive logic could possibly be applied to the empirical sciences does Carnapexplain how a logical concept of probability could ever be used to help guide our expectationsabout empirical happenings.Because Carnap, as a matter of practical decision, commits himself to some principle ofempiricism he cannot follow Jeffreys in relying on any metaphysical principle (regardless ifits evocation is purely pragmatic) like a principle of indifference to assign probability valuesover the sentences, or propositions, of a logical system. Or rather, of all the possible logicalrules and procedures one could construct in the metalanguage for assigning logical probabilityvalues to each sentence contained in the object language, Carnap can at best state proposals orconventions which restrict the admissible rules or procedures which may be employed – but thereis no metaphysical or epistemological justification for adopting these proposals or conventions:it is a matter of practical decision. Exactly here the engineering analogy finds its niche: as aninterpretive framework, the engineering analogy helps us to explain why Carnap’s applicationof a pure inductive logic could possibly answer philosophical questions about the foundations ofprobability and induction without adopting any justificatory or more traditionally (normative)epistemological vocabulary.Lastly, one should keep in mind that just as certain members of the Cambridge milieu,like Keynes and Jeffreys, were concerned with providing a logic of probability, Carnap’s ownambitions for his work on inductive logic is that it will eventually be possible “to construct asystem of inductive logic that can take its rightful place beside the modern, exact systems ofdeductive logic” (1962b, iii). This language is not metaphorical: as Carnap later explains,75 See Carnap (1962a; 1971b).492.5. Conclusioninductive logic does not propose new ways of thinking, but merely to explicate old ways.It tries to make explicit certain forms of reasoning which implicitly or instinctively havealways been applied both in everyday life and in science. (1953, 189; italics in original)It is here in the early 1950s that Carnap describes deductive logic as a theory of deductivereasoning continuously developed, from Aristotle on through to Frege, which allows us to replacedeductive “common sense” with “exact rules” (189). Similarly, by “inductive reasoning,” Carnapmeans “all forms of reasoning of inference where the conclusion goes beyond the content of thepremises, and therefore cannot be stated with certainty” (1953, 189 ). Carnap, however, isclear that the point of inductive logic is not to eliminate any “non-rational factors” present ininductive reasoning resembling a “scientific instinct or hunch” (1953, 195). Rather the “function”of inductive logic, says Carnap, ismerely to give to the scientist a clearer picture of the situation by demonstrating to whatdegree the various hypotheses considered are confirmed by the evidence. This logical picturesupplied by inductive logic will (or should) influence the scientist, but it does not uniquelydetermine his decision of the choice of a hypothesis. He will be helped in this decision inthe same way a tourist is helped by a good map. If he uses inductive logic, the decision stillremains his; it will, however, be an enlightened decision rather than a more or less blindone. (1953, 195-6)The imagery Carnap employs in this quotation assimilates inductive logic to a kind of map orguide and as such highlights the instrumental nature of his work on inductive logic: just asCarnap in the 1930s, as a consequence of his attitude of logical tolerance, treated deductivelogic as an instrument, in the 1950s he likewise understands inductive logic as a kind of tool orinstrument which may be used, either effectively or poorly, by scientists to make “enlightened,”reasoned, decisions as opposed to “blind,” arbitrary, decisions.2.5 ConclusionCarnap tells us that “[t]he history of the theory of probability is the history of attempts to findan explication for the prescientific concept of probability” (1962b, 23). In 1949, Carnap writesto a young Kenneth Arrow saying7676 The phrase “theory of behavior under uncertainty” occurs only in one place in the published version ofArrow’s dissertation, which was funded by the Cowles commission, on page 88 (N.B. Carnap was readinga type-written draft of Arrow’s thesis); see Arrow (1951). Arrow uses this phrase in reference to the lackof a theory of behavior under certainty required for investigating optimal economic systems, viz., optimalsystems required for centralized planning.502.5. ConclusionYou speak of a lack of a well-developed theory of behavior under uncertainty (p. 111). Ithink that for such a theory not only psychology but also inductive logic would be necessary,and that the lack of such a theory at the present time is due to the lack of the development ofa satisfactory inductive logic. I hope to develop at least the foundations of such an inductivelogic in my book. (Carnap to Arrow, June 29, 1949; ASP RC 084-04-02)Unlike in LSL where Carnap was primarily worried about the foundations of mathematics andlogic, Carnap’s primary motivation for engaging in the technical project of constructing an in-ductive logic is to show how once a satisfactory pure inductive logic is found it can be applied foruse in the empirical sciences. This is an historical example for how a scientific philosopher mayattempt to use their logical machinery to help clarify the foundations of science. Inductive logicis, for Carnap, an explication of inductive reasoning based on a logical concept of probability;but unlike deductive logic, in the 1940s, the field of inductive logic is still in its infancy. Toinvoke Carnap’s own ocean metaphor, there is a vast ocean of inductive logics which have yetto be explored and only partial methodological guidance exists for Carnap in the 1940s fromthe statistical sciences regarding which seas are more likely barren than not. In a letter to HansReichenbach, Carnap says:As you will see from my book, my objections are not directed against your theory itself.However, I believe, that in order to be applicable to the procedures of science your theorymust be supplemented by genuinely inductive concepts. Some parts of your theory, forinstance, the rule of induction, inductive inferences, and the concept of posit, contain im-plicitely [sic] and in a hidden way inductive concepts. Genuinely inductive concepts which Iregard as necessary, cannot be reached from your basis, because you want to base everythingon the frequency conception. The hidden inductive concepts must be made explicit and besystematized. This, in my view, is the task of inductive logic. (Carnap to Reichenbach,November 18, 1949; ASP HR 032–17–15)According to Carnap, Reichenbach’s work on a frequentist notion of probability implicitly con-tains – if it is to be applicable to the procedures of science – “hidden” inductive concepts whichCarnapian inductive logic attempts to make both explicit and systematic.This chapter began with the claim that Carnap attempted to resolve foundational questionsin science by proposing that the adoption of a linguistic framework is, in part, practical. Morespecifically, insofar as foundational questions in science can be formalized and codified withina logical system, the decision to adopt that logical system in contrast to any number of othersystems is, in part, a practical decision: it is analogous to choosing an instrument (like a hammer)rather than any number of other instruments (like other carpentry tools) to achieve some task(like pulling a rusty nail from a solid piece of aged timber). The obvious philosophical objection,512.5. Conclusionhowever, is that surely there is some theoretical, or objective, sense in which this formalizationor codification is “correct” or “justified”. Carnap argues otherwise: there are numerous ways toconstruct purely deductive or inductive logics which can then be applied to the sciences in thesame kind of way that mathematical geometry can be applied in the empirical sciences. Theprocess of application, moreover, is a methodological process: it concerns how a scientist maychoose how to coordinate the logical and non-logical, or descriptive, terms in a logical systemwith their empirical observations, experiments and measurement devices. There is, for Carnap,no privileged and antecedent notion of the a priori or conceptual reason according to which wecan “get things right”: this, too, would be a matter of adopting of proposal: it is a practicalmatter.77As we will see in the later chapters, Carnap attempts in his work on inductive logic to doexplicitly, by a repetition of many purely volitional and creative acts, what over the course ofthe history of probability and induction several generations of scientists and mathematicianshave failed to produce implicitly through first-order scientific inquiry: an adequate inductivelogic applicable to the empirical sciences. Or at least this is how Carnap sees things; he isengaged in this task of making explicit and systematizing inductive concepts – for example, byconstructing a pure inductive logic and then showing how it could possibly be applied to theempirical sciences – that I suggest is best understood as a kind of conceptual engineering. Butthese engineered concepts are, as Cassirer’s epigraph at the beginning of this chapter may haveled us to believe, neither wholly intellectual fantoms nor are they the subliminal fundament ofscientific knowledge: for Carnap they are instead self-made concepts which have been designedby us and only imperfectly mirror the jumble of concepts already in use in the sciences anddaily life. For Carnap in the 1950s, I argue, there is no deeper philosophical task concerning thefoundations of science that we could be engaged in other than the task I would call conceptualengineering.7877 This historically accurate picture of Carnap stands in stark contrast, I would suggest, to the caricature ofCarnap as a foundationalist epistemologist recently popularized, for example, in Chalmers (2012).78 Of course, there exist deeper first-order mathematical and scientific tasks. The issue of how to characterizeCarnap’s earlier views (e.g., when he rejects empiricism in Carnap, 1923) and how best to historically narratethe winding path Carnap takes from his pre-Aufbau work to his later work on, say, normative decision theoryis a highly complex, historical task which I do not take up in this dissertation – excellent starting placesinclude Carus (2007), Richardson (1998), Uebel (2007) and Frost-Arnold (2013).52Chapter 3Philosophical Method as Conceptual EngineeringPhilosophically, Carnap was a social democrat; his ideals were those of the enlight-enment. His persistent, central idea was: “It’s high time we took charge of our ownmental lives” — time to engineer our own conceptual scheme (language, theories) asbest we can to serve our own purposes; time to take it back from tradition, time todismiss Descartes’s God as a distracting myth, time to accept the fact that there’snobody out there but us, to choose our purposes and concepts to serve those purposes,if indeed we are to choose those things and not simply suffer them.— Richard C. Jeffrey, “Carnap’s Voluntarism” (1994)The plaintive call for a new engineering morality expresses a yearning to returnto a time when engineers fancied themselves, in words which have already been quoted,“redeemers of mankind” and “priests of the new epoch.” With the religion of Progresslying in ruins about us, we engineers will have to relinquish, once and for all, thedream of priesthood, and seek to define our lives in other terms.— Samuel C. Florman, The Existential Pleasures of Engineering, 2nd ed., (1994)Section 2.3 of the last chapter was dedicated to explaining why certain Carnap scholars,like Michael Friedman, Richard Creath, Alan Richardson, André Carus and Samuel Hillier,have attempted to explain how Carnap understood the philosophical significance of his techni-cal projects in logical syntax and semantics by framing those projects as a sort of engineeringactivity. In this chapter I draw on contemporary work on the history of professional engineer-ing – specifically, on the activity of engineering design in contrast to engineering fabrication,production and maintenance – to help inform what I have in mind by the phrase “conceptualengineering,” a conception of engineering which I suggest is more complicated and subtle thana mere implementation of means-end reasoning.7979 Although software engineering or the history of computer languages is perhaps more closely related to whatI call “conceptual” engineering than the automotive and aeronautical case studies I discuss later in thischapter, a detailed examination of these very technical subjects would require more space than I have onoffer for this dissertation and I will return to this topic in future work. Moreover, the history of computingengineering is still in its infancy; but see Mahoney (2004). Another avenue of interest is how Wittgenstein’searlier work on aeronautics influenced his views in his Tractatus, see Sterrett (2002).533.1. Engineering as Means-End ReasoningIn the next section I motivate this more subtle conception of engineering in a roundaboutway. I first point out that if we choose to adopt both a more mainstream philosophical method-ology resembling something like logical or conceptual analysis and a conception of engineeringunderstood as the implementation of means-end reasoning then philosophical activity wouldseem to be completely orthogonal to engineering activity. Second, I provide an alternative,more subtle, conception of engineering design that I suggest, when used as an interpretiveframework, provides us with a more systematic account of how Carnapian logic of science is akind of conceptual engineering. The argument for this last claim, however, is an argument fromillustration which spans chapters 4 through 6 of this dissertation. The proof of the pudding, soto speak, is in the eating.3.1 Engineering as Means-End ReasoningIn a series of autobiography remarks from his 2004 posthumously published book, SubjectiveProbability: The Real Thing, the philosopher Richard C. Jeffrey says that instead of pursuinga doctoral degree in philosophy after receiving a masters in philosophy from the Universityof Chicago in 1952 under Carnap’s supervision he immediately left to work at MIT’s DigitalComputer and Lincoln Laboratories.80 The reason for his flight from philosophy, Jeffrey tells us,is that he had “observed that the rulers of the Chicago philosophy department regarded Carnapnot as a philosopher but as — well, an engineer” (2004, preface).81Unlike Jeffrey’s use of “engineering” in the epigraph at the beginning of this chapter, his80 In 1955, Jeffrey returns to philosophy as a PhD student at Princeton University co-supervised by CarlHempel and Hilary Putnam – it is in his dissertation, finished in 1957, that Jeffrey invents his “probabilitykinematics” (see Jeffrey, 1957). Before Jeffrey returns back to the Socratic fold, however, he works on aclassified project on the design of digital computer, codenamed “Whirlwind II.” The original Whirlwindcomputer was first designed in 1947 in what was then MIT’s Servomechanisms Laboratory but that lab,facilitated by funds from the Office of Naval Research, was soon merged with the Digital Computer Labora-tory (DCL) in 1951. Soon afterwards the DCL was incorporated into the much larger Lincoln Laboratories –which was then composed of five divisions – as a new “Digital Computer” division, or Division 6. Jeffrey waspart of Group 62 of Division 6, lead by one Norman Taylor, which was tasked with the “logical design” of anew prototype, “Whirlwind II” or, using the military designation, AN/FSQ-7. Numerous archival materialis now available online through MIT’s Dome archives testifying to this fact. For example, while at DCL, be-tween 1953 and 1955, Jeffrey wrote at least five internal memorandums on logical networks and their algebraand according to one internal report for Division 6, there is a now unclassified memorandum 6M-3268 titled“Crosstell Input Element Specifications” written by Jeffrey (and other authors) dated January 6th, 1955(MIT Dome, 6D-52-1, CASE 06-1104). For more on the history of Division 6 and their later contributionto the Semi-Automatic Ground Environment (SAGE) air defense system, see Redmond and Smith (2000).81 Apparently, in a 1938 letter to Richard McKeon – who was then the head of the philosophy department –Morris Cohen describes Carnap as a “technician” (personal communication with Alan Richardson).543.1. Engineering as Means-End Reasoninguse of the word “engineering” to describe the attitude of Carnap’s peers at Chicago towardCarnap’s technical work has a pejorative connotation. Jeffrey doesn’t spell out exactly whylabeling a scientific or technical philosopher an “engineer” is an intellectual slight but the basicidea seems simple enough to grasp. If the point of philosophical discourse is to arrive at, forexample, representations of what the world is really like, what we ought to value unconditionallyor how we ought to act in ethical dilemmas, then – seeing as how engineers are only concernedwith getting us from the state of affairs A to the state of affairs B – no amount of engineeringtechnique or know-how could possibly afford us with answers to either (i) metaphysical orepistemological questions like what states of affairs really are or how we can have knowledgeof them, or (ii) axiological questions like why we should value certain states of affairs overothers. Regarding the first set of questions, philosophers qua engineers must borrow theirmethodology and language from the empirical sciences and thus must already be committed toa metaphysics and epistemology consistent with a scientific worldview. Regarding the secondset, philosophers qua engineers cannot explain why we should value engineering solutions tophilosophical problems which promote human well-being and flourishing, reveal truths aboutthe world and ourselves, or are simple and elegant any more than professional engineers quaengineers (rather than, say, members of a democratic community) can explain why their projectsshould reduce harm, be economical or be aesthetically pleasing.82Another way of explaining the point is as follows. As I cannot pretend to know how to beginto accurately capture all the key features of contemporary, analytical philosophical methodol-ogy within a single framework, I ask for the reader’s patience as I outline the framework foran idealization of a philosophical method resembling Socrates’s elenchus or Moorean logicalanalysis. Simply put, these are methods which search for the “truth.”83 There are three stepsrequired to answer a philosophical question of the form “What is the nature of X.” The firststep is the provision of a well-defined space of possibilities for what X could possibly be; forexample, a space of reasons, possible worlds, concepts or sets of (true) propositions. The secondstep is the provision of empirical, conceptual, or logical plausibility constraints which are used82 Engineers, after all, can function quite well in totalitarian or communist societies. For general introductionsto the nature of engineering, especially the importance of engineering failure and the relationship betweenengineering and the humanities, see Florman (1996); Petroski (1992; 2012); Vincenti (1990).83 For a more sympathetic rendering of contemporary logical or conceptual analysis, see Glymour and Kelly(1992); Soames (2003); Williamson (2007).553.1. Engineering as Means-End Reasoningto evaluate, in some way, segments of this space of possibilities; here I have in mind not onlyconceptual notions like a priori or conceptual truths, notions of rational agency or accounts ofmental representation but also additional sources of information or knowledge, like common-sense, sense data or descriptions of phenomenological experience. Third there is some methodof search which, either literally or figuratively, searches through this space of possibilities in anattempt to locate, or at least get closer to, the correct, or target, possibility in the possibilityspace; most common, for example, is the method of putting forward arguments for the meaningof X until either a counter-example is found relative to the current plausibility constraints,upon which the argument is then modified, or an entirely new argument is put forward andanalyzed.84 According to this idealized model of logical or conceptual analysis, philosophicalactivity is likened to the formation of a kind of optimization problem.Figure 3.1: Means-end Model of Engineering.Engineering, as defined in an introductory book to engineering, is “in its most general sense,turning an idea into a reality — creating and using tools to accomplish a task or fulfil a purpose”(Blockley 2012, 1). Characterizations of professional engineering like this one tend to conveyit simply as the activity of physically implementing a plan or design which resulted from in-strumental, or means-end, reasoning. According to this view, what I call the “means-end modelof engineering,” the activities of engineering itself (represented by the lined box in Fig. 3.1)are sharply separated from the activities, products and decisions of non-engineers, includingwhomever hired the engineers and practicing scientists. In order to do their jobs, engineers onlyneed to know about the information from two “inputs” outside of their discipline (represented84 For example, an algorithm for this method of search may look something like this: (S1) Construct a newvalid argument Γ for the correct meaning of X which is located in the space of possibilities; (S2) Checkthe soundness of each premise in Γ against the plausibility constraints; (S3) If the meaning of X given byΓ does not pick out any concept in the space of possibilities given the current plausibility constraints thenstop the current search and repeat step (S1); (S4) If a counter-example is found for a premise in Γ thenmodify the faulty premise and replace it with a new class of premises, call the new argument Γ∗; Repeatstep (S2) for Γ∗; (S5) Output Γ as the correct characterization of X.563.2. Engineering Designby the directional arrows in Fig. 3.1): first, the “inputs” from the employer, including theengineering problem itself, design specifications and safety/economic/resource/time-sensitiveconstraints; and, second, the “inputs” from the mathematical and empirical sciences, includingempirical theories, predictions and models which can be adapted to specific engineering prob-lems and tasks. Thus, as technically complicated and interesting as the problems of engineeringmay be, the job of engineers is nevertheless essentially instrumental: they use their scientificexpertise to design and construct physical artifacts – the “output” – which they expect to satisfy,at least to the best of their ability, the values, needs and constraints specified by their employer(and professional codes of conduct, industry regulations and so on).It seems to be a consequence of both this means-end conception of engineering and theconception of philosophical method as a search for truth that engineering is only relevant tophilosophical activity after philosophers have formed some consensus as to how to formulatetheir philosophical investigation in the sense that they agree on which space of possibilities,plausibility constraints and a method of search should be used to solve the philosophical problem.All the engineer has to do is then use their expertise and technical know-how (e.g., by writingup a computer program or drawing up complicated flowchart diagrams) to search for the bestpossibility from the space of possibilities given the plausibility constraints. But then thereseems to be little overlap between engineering and philosophical activities: on the unlikelyoccasion that a group of philosophers actually does come to agreement as to how to formulate aphilosophical question then – paradoxically enough – philosophical inquiry itself seems to cometo an end. What is left is merely the figuring of a technical answer to a technical question. Allthat is left is engineering.3.2 Engineering DesignThere is a growing consensus amongst historians of engineering that a simple means-end concep-tion of professional engineering is misguided; specifically, there is active resistance to the ideathat engineering is best explained as an applied science.85 Rather than viewing engineers astechnicians who apply ready-made products from the empirical sciences to construct artifacts,85 For more nuanced discussions of the nature of engineering, including engineering design, see Dym and Brown(2012); Johnson (2009); Vincenti (1990).573.3. Satisficing Wings and Propellersengineering is instead viewed as an activity which must occasionally produce new and origi-nal scientific knowledge independently from working scientists in order to adequately designand construct artifacts. Engineering design, in particular, is one part technique and the otherpart science: it is the process of gradually transforming a vague and abstract design problem– like building a cheap personal computer – into a well-defined hierarchy of more manageablesub-problems, problems which may require engineers to produce new knowledge and technicalknow-how in order to solve.86 And as new technologies emerge, failures occur or aesthetic tastesor risks change, engineers will have to modify and even replace the components of this hierarchyin order to produce a satisfactory engineering product. The very notion that an engineeringdesign could be “correct” is illusionary – “correctness” is a moving target which requires a con-tinual and piece-meal process of finding better ways of solving a protean problem. At the endof this chapter I argue that it is this piece-meal and hierarchical conception of engineering thatprovides the appropriate interpretive framework for understanding the revolutionary featuresof Carnapian Wissenschaftslogik. But before I say any more about either this conception ofengineering or Carnap, I first discuss in each of the next two sections case studies from thehistory of engineering design.3.3 Satisficing Wings and PropellersHow should one design an aircraft? Engineers, of course, don’t typically start off with suchbasic questions: they get hired to design particular components of aircrafts which are intendedto fulfill very specific tasks while satisfying any number of economic, environmental, safety orlegal considerations. But employers themselves typically do not tell engineers how this imaginedaircraft can be turned into reality using the resources and technologies already available toengineers. A company like Boeing, for example, may decide to build a new line of jetliners thatwill have a large wing-span, fits X many passengers, has Y cubic meters of cargo space and thatis cheaper, more fuel-efficient and is easier to maintain compared to the kind of planes alreadyin operation at most major airlines. From an engineering point of view, the task of buildingthis new line of jetliners is an “ill-posed” or “ill-structured” problem: the problem itself offers86 The idea that engineering design problems are hierarchical comes from Vincenti (1990); see below.583.3. Satisficing Wings and Propellerslittle guidance regarding how one should make any number of important design and productiondecisions, like what overall design or archetype of the plane should be used, how many enginesit should have, or how to solve any number of more specific design decisions, like how to designthe wing airfoils and the fuselage for the body of the plane.87Figure 3.2: Hierarchical Model of Engineering Design and Knowledge. The figure is my own butthe engineering terminology is taken directly from Vincenti (1990); specifically the two lists on right-hand sideof the figure are cited verbatim directly from Vincenti. For the details about each level of the engineeringdesign hierarchy see p. 9, for the list of engineering kinds, or “categories”, see pp. 208 ff. and for the list ofknowledge-generation activity see pp. 229 ff.There are even further questions regarding how to design, construct and manufacture thecomponents of the aircraft, like what kind of materials should be used to construct the wingsor even more “mundane” questions regarding what kind of rivet should be used.88 The resultis a hierarchy of problems for which any given solution to one problem may have consequences,both practical and theoretical, for the other problems (e.g., adding a more powerful engine will87 Vincenti (1990) uses the terminology of an ill-structured versus a well-structured problems, which he inturns takes from Simon (1973). According to Herbert Simon this terminology was first used by W. R.Reitman in the 1960s.88 Vincenti (1990), for example, spends an entire chapter talking about the difficulties of finding an appropriatemethod to install rivets that are flush with the body of an aircraft.593.3. Satisficing Wings and Propellersnot only raise the production costs of the jetliner but engineers will also have to re-examinethe structural supports for the wings and fuselage). Vincenti calls this a design hierarchy, apictorial representation of which is given on the left-hand side of Fig. 3.2. Not only do changesto the “top” of the design hierarchy, like the project definition, reverberate down to the designquestions at the lower levels, but as engineers fail to find satisfactory solutions to the questionsat the “lower” levels, or new knowledge is produced or technical tools discovered, engineers mayfind it more convenient or efficient to alter the project definition itself.With a nod toward Thomas S. Kuhn’s distinction between normal and revolutionary sci-ence, Vincenti distinguishes between normal and radical design (Vincenti, 1990, 7-9; see Kuhn1962). While instances of radical design changes include decisions to switch the design ofan aircraft based on Boeing’s 747 “Jumbo Jet” to an airship like Deutsche Luftschiffahrts-Aktiengesellschaft’sGraf Zeppelin, normal design changes are more modest: the overall archetypeof the plane remains fixed but the components or sub-components of the aircraft design are al-tered. Most everyday engineering is concerned with such “normal” design problems and, becausewhen Carnap is working on inductive logic he is arguably worried more about “normal” designproblems than not, I focus exclusively on “normal” design.89But what does it mean to say that the archetype of a design remains fixed? Vincentiborrows Michael Polyani’s notion of an operational principle as a way to codify the requirements,purposes and goals that an engineering design, and ultimately the physical objects modeled onthat design, are intended to satisfy.90 Thus the operational principle for an airplane, or someother kind of object, provides an operational or functional definition of what kind of airplanethe object should be. According to Vincenti,it is the operational principle that provides the criterion by which success or failure is judgedin the purely technical sense. If a device works according to its operational principle, itis counted as being a success; if something breaks or otherwise goes wrong so that theoperational principle is not achieved, the device is a failure. (209)Airships and jetliners have different operational principles: they are designed to be successful orto fail in a number of different ways and their respective operational principles set the standard,89 Here I would suggest that examples of normal design change for inductive logic would be changing how onedefines semantic confirmation functions while a radical change would be to replace the semantic concept ofdegree of confirmation with another concept entirely, like providing a semantics for how scientists use theword “confirmation” in natural language.90 See Polanyi (1958, 176 and 208).603.3. Satisficing Wings and Propellersso to speak, against which engineers and their employers can measure the success and failureof the final engineered product. Consequently, normal design, for Vincenti, concerns engineersworking with both a similar operational principle and also a tacitly agreed upon “normal con-figuration” of the artifact — i.e. “the general shape and arrangement that are commonly agreedto best embody the operative principle” (209). In other words, the operational principle fixesthe task of designing the object while a tacit “normal configuration” operates in the backgroundto regulate how the actual physical objects are manufactured and produced. Below we willsee that practical and theoretical problems arise when an operational principle and a “normalconfiguration” of an artifact conflict; the way we design objects in the drawing room does notalways smoothly transfer over to the machine shop, and vice versa: there is always a certaindynamic or interplay between the design and construction of the engineering artifact.One example of how the design of different sub-components of an aircraft may have con-flicting practical and theoretical considerations is nicely illustrated in chapter three of Vincenti(1990). How should the flying qualities of an aircraft – i.e. “those qualities [. . . ] that govern theease and precision with which a pilot is able to perform the task of controlling the vehicle” –influence the design of the aircraft?91 These qualities, says Vincenti, “are thus a property of theaircraft, though their identification depends on the perceptions of the pilot” (53). In particular,Vincenti is interested in two kinds of flying qualities.92 The first quality is the physical controlthe pilot has over the aircraft using, typically, a stick and pedals to mechanically move boththe flaps on both the wings and the horizontal and vertical tails of the aircraft. The design ofthese controls determines both how well the pilot can control the aircraft in order to achievetheir plans and objectives and how much control the pilot perceives they have over the aircraft;in Vincenti’s words, “[t]he effort required by these tasks gives the pilot a feeling of confidenceor apprehension about the airplane” (53). The second quality concerns the inherent stability ofthe aircraft which93has to do with the ability of an airplane, by aerodynamic action alone and without anycorrective response by the pilot, to return to an equilibrium flight condition after a transitorydisturbance, as might arise, for example, from a gust. (54; emphasis in original)91 Vincenti 1990, 53.92 To make this discuss of flying qualities manageable, Vincenti only focuses on longitudinal flying qualities.93 Stability is defined in terms of an equilibrium of particular measurable properties of an airplane (Vincenti1990, 59).613.3. Satisficing Wings and PropellersThe more stable the aircraft, the less likely it is to deviate from a flight path due to externaldisturbances. But that means the pilot must put in more effort to perform aerial maneuverswhich deviate from the current flight path of the aircraft and thus the pilot may feel like theyhave little control over the behavior of the aircraft. According to conventional engineeringwisdom, “[i]nherent stability,” says Vincenti,is important to flying qualities because the stable airplane resists initiation of a change inflight condition to more or less the same degree as it does a transitory disturbance. Theunstable airplane, by contrast, responds readily, even perhaps excessively, to movement ofthe controls. Stability and control thus work at cross purposes, and the ease and precisionwith which a pilot can control an airplane depend as much on its stability characteristics ason the action of aerodynamic control surfaces. As they relate to flying qualities, stabilityand control are different sides of the same coin. (54)The notions of stability and control of an aircraft, then, are fairly straight-forward theoreticalnotions which can be studied using a variety of mathematical and physical methods. Moredifficult to quantify, however, is the pairing of the subjective experiences of pilots with specificcombinations of the flying control qualities: this is in part a practical problem which is sensitiveto the preferences and expectations of different kinds of pilots. What does it mean, for example,in the technical vocabulary of control and stability, when a pilot says that a plane feels “sluggish”when making tight turns? This is a problem for engineers: given that engineers are tasked withdesigning military aircraft which will allow pilots to effectively and efficiently perform combatoperations and maneuvers, how should they quantitatively measure the qualitative judgments ofpilots and then use these qualitative reports to coordinate specific stability/control qualities ofthe aircraft with the expectations of experienced pilots? Once these questions are addressed thedesign engineer can then better tackle questions about how much stability is too much stability.Crucially, this trade-off between control and stability did not arise because of some economic ortheoretical constraint but rather, says Vincenti,[...] it came into being because of the practical needs and limitations of the human pilot. Thebalance therefore could not have been achieved on purely intellectual grounds and withoutextensive flight experience. It summarized a practical design judgment (based in this caseon subjective opinion) of a sort that cannot be avoided in engineering. (107)Starting in 1918, a group of engineers working at a new laboratory at Langley Field inVirginia for the National Advisory Committee for Aeronautics first started to measure the sub-jective experiences of pilots.94 Using new measuring technologies (like an altimeter, tachometer94 For those interested in the historical details, see chapter three where Vincenti discusses how, before 1918,623.3. Satisficing Wings and Propellersand airspeed meter, p. 70), the engineers at Langley worked closely with test pilots to try toquantify the subjective experiences of pilots while simultaneously recording the quantitativeresults of these measuring instruments. The result was the creation of a quantifiable method formeasuring the control of an aircraft as a function of both stick-fixed stability (stability when thecontrol joystick is held fixed by a pilot) and stick-free stability (the stability when a pilot releasesthe stick) (68 ff.). In 1936 Edward Warner, who was then an engineer working for the DouglasAircraft Company, wrote a report of which Vincenti says, “embodied for the first time the notionthat desired subjective perceptions of pilots could be attained through objective specificationsfor designers” (81). Here is how Vincenti summarizes the results of these historical events:The road from the recognized but ill-defined problem of 1918 had been a long and compli-cated one. The idea that subjective pilot preferences could be embodied in objective designrequirements, itself the product of a decade and a half of learning, had been validated byproducing a set of requirements that accomplished that job. From here on, the problemof flying qualities was conceptually a different ball game. Research engineers could nowdevote themselves to refining and extending the requirements with confidence that the ideawas useful. Designers at the same time had a greatly improved understanding of what waswanted in flying qualities and explicit specifications at which to aim. They didn’t alwayssucceed, of course; knowledge of how to design a given requirement still left much to bedesired. [...] Their problem now, however, was mainly one of designing (i.e., proportioning)the airplane rather than deciding at the same time what to design for. (97)This quote, I suggest, is indicative of a distinction between the practical and theoretical that iscentral to the activity of engineering design. For any piece of machinery or technology there isof course a wide array of theoretical facts detailing what will, can and should happen to thatartifact under many different kinds of conditions: the stability of an aircraft with a particularfuselage, for example, is a property we can infer from our knowledge of aeronautics. Practicalconsiderations, nevertheless, play an important role in deciding which fuselage to use: here thesubjective preferences of pilots will play a nontrivial part, along with economic or strategicfactors, in deciding which fuselage to use in the design of the aircraft. There is, in a sense,a dynamic interplay, or feed-back loop, between the practical needs of pilots and military or-ganizations and the engineering knowledge generated by aeronautical engineers at places likeLangley airfield. Once theoretical results were found which could satisfy most of the practicaldemands of pilots, the original ill-structured problem of balancing control and stability becomescertain engineers on both sides of the Atlantic emphasized control or stability, or vice versa, until the early1920s, when engineers realized that both stability and control were required, especially for military aircraft;also see Bloor (2011); Gibbs-Smith (1960; 1966).633.3. Satisficing Wings and Propellersmore tractable. Of course, whether these practical demands were met, viz. whether the trade-off between control and stability used in various designs of military aircraft were ultimatelysuccessful, is a question that can only be answered when active military pilots actually use theaircraft:Though conformity with the quantitative flying-quality specifications can also be measuredin flight, the final test there remains the pilot’s subjective reactions. The flying-qualityspecifications retain their function as means – a design guide – and resist becoming anend. [...] Thus, for the designer, the quantities set down in performance specifications arethemselves objective ends; the quantities prescribed in specifications of flying qualities areobjective means to an associated subjective end. (100)Only after much engineering trial and error was it possible for design engineers to transformthe practical considerations of pilots into theoretical constructs which could be written downin blueprints required to manufacture and produce aircraft that balance control and stability.But this process was not a simple piece of means-end reasoning: engineers had to re-think, ondifferent occasions, how to define what it meant to quantify the subjective experiences of having“control” over an aircraft.Next I discuss another example from Vincenti (1990) about the work by two aeronauticalengineers, William F. Durand and Everett P. Lesley, who designed and empirically tested air-craft propellers in the 1910s and 1920s. From a certain perspective, the problem of designingpropellers is a simple optimization problem: after constructing models of different kinds of pro-pellers, one simply has to find some way to quantify the relevant properties of these propellersand then test them, e.g., in a wind tunnel, until an optimal propeller design is found. In otherwords, each propeller kind belongs to some point in a state space S and all we have to do isevaluate each point in S in terms of some utility function i and then use linear programming,or some other optimization method, to find that point s in S such that i(s) is the maximumvalue of i (restricted to S). In practice, however, this optimization process is not so clear cut:sometimes what matters most to an engineer is finding “good enough” states in S which comes“close enough” to a maximum value of i .As it turns out, it is both expensive and difficult, if not impossible, to build and test allpossible propellers characterized by a point in S. Instead, engineers must randomly test a finitepoint in S and they must make these tests using miniaturized and scaled propellers. Specifically,643.3. Satisficing Wings and Propellersengineers must appeal to what Vincenti calls the method of parameter variation, which is95the procedure of repeatedly determining the performance of some material, process, ordevice while systematically varying the parameters that define the object of interest or itsconditions of operation. (139)Engineers also need to depend on some theoretical law of similitude in order to extrapolate theperformance of a full-scale propeller from the performance of a smaller scale model propellerwhere the measure of performance for the scale-model is a special kind of quantitative magnitudecalled a dimensionless group.96 Using the method of variation engineers can classify togethersimilar propellers as a single propeller design the performance of which can then be measured, invirtue of the law of similitude, as a function of distinct quantities measured experimentally usinga wind tunnel. Some function of these quantities can then be shown to form a dimensionlessgroup, providing a measure of the performance of propeller designs. Then the engineer can tryto maximize the value of this measure over all possible propeller designs.However, the question of what exactly should be optimized is not trivial. Propellers work,basically, by transferring the rotative power of the engine into a propulsive, forward movementpower and so the “success” of a propeller can be understood in terms of how mechanically efficienta certain propeller is at transferring rotative into propulsive power (141). Thusly, as Vincenticlarifies, the question of whether a certain propeller design is successful or not depends on theprior choices which have been made concerning the engine and the aerodynamical features ofthe wings and fuselage of the aircraft as these are the sort of properties which would causallyeffect the forward movement of the aircraft (141). Moreover, even though engineers had knownhow to design propellers in terms of a finite number of parameters, like the mean pitch ratio of apropeller, in the 1910s there was no systematic collection of empirical data about the efficiencyof propeller designs, nor was there any systematic theory or mathematical model for how theefficiency of different propeller designs were related to each other.97 Thus there was no priortheoretical basis to which engineers could appeal in order to claim that certain kinds of propeller95 For a brief history of this method, see pages Vincenti, 1990, 138-141.96 Vincenti defines a dimensionless group as “a mathematical product of two or more quantities arranged suchthat their dimensions (length, mass, and time, or combinations thereof) cancel, leaving a “pure number,”that is, a number without a dimension” (140).97 The mean pitch ratio of a propeller is defined by Vincenti as “a measure of the angular orientation, relativeto the plane of propeller rotation, of the blade section at some standard representative radius” (148). Inother words, it is a measure of how much the blades of a propeller are “twisted” relative to the vertical axisparallel with the front of the aircraft.653.3. Satisficing Wings and Propellersdesigns would in fact be more or less efficient with different kinds of aircraft designs; indeed,the initial theorizing about air propellers was done by analogy with the work by naval engineerson marine propellers.98In 1916, the NACA funded a preliminary study to empirically test a limited number ofpropeller designs by building an air tunnel at Stanford University.99 The design of a propellerwas understood in terms of five shape parameters, whose values I denote by r1, r2, r3, r4, r5, oneof which is the mean pitch ratio; I here represent these values as a vector r (= 〈r1O r2O r3O r4O r5〉).For the 1916-17 measurements, only forty-eight propellers were tested (3 different values of themean pitch ratio and two other values for the other four parameters comes out to 48 distinctpossible propeller designs). Three other parameters were also included in the mathematicalmodel for the efficiency, or performance, of a propeller used by the engineers Durand and Lesley:j is the forward speed of the aircraft while X is the diameter and n the revolutions per unittime of the propeller (146). The result is a model of the performance of a propeller in terms ofa function F of j , n, X and r. Several different empirical measurements were then made foreach of the specially-constructed three-foot model propellers in the wind tunnel while the valuesof j and n were simultaneously varied. A law of similitude, based on the earlier work of theParisian structural engineer Gustave Eiffel,100 was then used to measure the efficiency of eachpropeller, , as a dimensionless group based on the ratio jQnX. The resulting equation for theperformance of scale-model is this (150): = F( jnXO r)NThe advantage of this simplified equation, according to Vincenti, is that, for any particular kindof propeller represented by some vector r we only need to plot the  values against the values ofthe dimensionless group jQnX, or what Vincenti calls efficiency curves, in order to figure whatwhen  is maximized.Simplifying the problem a bit, for any full-scale propeller with shape and diameter param-eters r and X, all we need to do in order to calculate the efficiency of the propeller with the98 See Vincenti, 1990, 141-2.99 For more details see Vincenti, 1990, 142-159.100 See Vincenti, 1990, 142; 151.663.3. Satisficing Wings and Propellersparameters r and X is to find, for any values n and j , the value of jQnX (for the scale-modelpropellers) that maximizes  — call it ∆jQnX. Then the most efficient values of j  and nfor the full-scale propeller design r and X, are all those empirically feasible values of j  andn (that is, feasible in terms of the specifications of the Stanford wind tunnel) such that thefollowing equation holds:j n= ∆jnX×XNNotice first that the resulting mathematical model is an example of the kind of empiricalknowledge engineers produce on their own, independently of collaboration with scientists; thisis an example of why engineering cannot be easily assimilated to mere means-end reasoning.101Second, notice that this example offers philosophers a glimpse at how difficult engineering canactually be: even for reasonable values of r where each ri only has l possible values, there areexactlyl5 many possible propeller designs which would need to be constructed (as scale models)and empirically tested (with co-varying values of j and n) in order to explore then entire “space”of propeller designs; if l = 8, for example, then engineers would have to perform over thirtythousand tests – one for each different kind of propeller – in the wind tunnel. Notwithstandingtheir resolve, few engineers would have the time, funds or resources available to them requiredto test every single possible propeller design for large values of l.102 For very large, multi-dimensional state-spaces, engineers have to find some theoretical crutch, like a law of similitude,which would allow them to reduce the number of possible solutions that need to be tested.Thus even though in later tests Durand and Lesley extended the number of shape parameters,including three values of the mean pitch ratio, to nearly eight thousand propeller designs theydid not build and test the required eight thousand scale models; instead, as “judicious samplingbecame necessary,” they only constructed an additional fifty scale propellers which they thenadded to their study (151-152).103101 Or rather, engineering is no more simply understood as means-end reasoning than most of the empiricalsciences; also see Vincenti (1990, 160-6).102 Moreover, even for analytic or computational approaches to the problem, depending on how large theproblem space is there is no guarantee that an optimal algorithm exists which will solve (if ever) theproblem in linear time (e.g. using linear programming algorithms to traverse the problem space). Insteadone would need to turn to certain “sub-optimal” algorithms which will differ in their respective benefits andcosts (e.g. see Simon, 1996).103 Of course, similar problems of simulation and optimization crop up in the empirical sciences, especiallyphysics and biology. Relevant here is the discovery of Monte-Carlo methods to “randomly” search throughstate spaces in physics; e.g., see Galison (1997).673.4. Changing Designs and Braking BarriersAfter amalgamating their various reports into a single, more comprehensive, report the resultis the discovery of the following empirical generalization:To optimize propeller performance at a fixed flight condition, one value of pitch ratio willsuffice. The designer need only calculate the value of VQnD for that condition and selectfrom the data the pitch ratio giving maximum efficiency at that value (interpolating betweencurves if necessary). (152-3; see figure 5-5 in Vincenti 1990, p. 153)The result is a certain trade-off between propeller designs: although an aircraft with a certainpropeller will be more efficient at higher speeds, i.e. for high values of j , the same aircraftmay be less efficient for lower speeds because the propeller is less efficient at lower values ofj . Thus because the values of n and j vary all the time during normal flight conditions themost efficient propeller is not a propeller fully specified by some value of r at all: instead itwould be a propeller designed so that the mean pitch ratio of the propeller could be modifiedin flight. Although, as Vincenti notes, Durand and Lesley did provide a model in 1918 forsuch a propeller, the technology required to construct so-called variable-pitch propellers onlybecame available in the 1930s (153). But even with the discovery of the variable-pitch propellersolution, the aeronautical engineer can still provide no guarantee that a more efficient propellerdesign doesn’t exist: with the emergence of new technologies and the growth of engineeringknowledge there are infinitely many possible ways in which the design of the propeller could bechanged, both radically and otherwise, to help maximize performance – especially as the designof airplanes themselves change.3.4 Changing Designs and Braking BarriersI next turn to a case study from the history of automotive engineering which offers a betterillustration of how engineering design problems can change over time; specifically, how designproblems can transition from vague operational principles to well-formed technical and mechan-ical problems. This section elaborates on this point from the vantage point of the history of thedevelopment of anti-lock braking systems for automobiles as found in Johnson (2009). Specif-ically, Johnson argues that engineering knowledge is developed co-extensively with the modi-fications of engineering communities, communities which are in turn formed around a volatile“attractor,” or a “communally defined problem” (5).104 In particular, she discusses how anti-lock104 Importantly, Johnson’s “attractor” framework is not a theory about how engineering communities form inthe first place, but how disparate engineers break off and cluster around a problem, an attractor, and how683.4. Changing Designs and Braking Barriersbraking systems (ABSs) were developed to reduce the problem of vehicular skidding from the1950s to 1980s.After the Second World War the initial problem was to figure out how to mitigate the risingnumber of accidents and deaths due to the increase in the number of automobiles on NorthAmerican roads (2009, 26). As Johnson points out, unlike possible socio-political interventionslike preventing drinking and driving through better driving education, preventing skidding was adistinctively mechanical solution to the problem of mitigating accidents which could be tackledhead-on by engineers (26). Thus the original problem of curtailing accidents quickly morphedinto the more tractable problem of increasing the safety of vehicles by designing (preferably,profitable) automobiles that are less likely to skid. As Johnson illustrates in her book, however,providing a solution for how to reduce skidding turns out to be, from both a technical andconceptual standpoint, a very complicated problem.First, there is the issue about how to define the problem of skidding: is it just a problemabout the change of the coefficient of friction between a tire and the road, or is it a more holistic“interaction problem” between the car, the driver, tires and road?105 Without a well-definedstatement of the problem, it is difficult to articulate a space of possible designs from whichengineers can entertain which designs best satisfy the constraints of the problem.106 In otherwords, the engineers lacked an operational principle: only once a clear statement of the problem,and definition, of vehicular skidding was given could any sort of technical solution be proposedand implemented. Second, there is the issue of knowing what kind of instruments and tools canor should be used to help solve the problem. Johnson stresses, for example, that automotiveengineers had the genuinely difficult problem of measuring, in real time, the deceleration of a tire(and, moreover, the simultaneous measurement or calculation of deceleration for all four tires)in order to provide a real time measurement of skidding. It wasn’t until the 1980s that digitalsensors were able to provide the reliable, precise and, most importantly, real-time measurementsrequired to estimate exactly when tires are skidding. Third, even if one could adequately measurethe deceleration of a tire in real-time, there is still the problem of designing a mechanical systemthese engineers form a new, sub-community engaged in finding solutions to the attractor (4-6). Immigrationfrom other engineering communities is especially important because “[n]ew ideas and tools move into thecommunity in part because participants move between communities, and ideas require human vectors” (6).105 See Johnson, 2009, 42–4.106 See Johnson, 2009, 105–7.693.4. Changing Designs and Braking Barrierswhich can, in real-time, modulate the brakes (which turns out to require many modulationsper second) in order to change the friction coefficient of a tire and thus, ultimately, preventskidding. Moreover, such mechanical systems have to modulate the brakes relative to real-time measurements of the other tires: any such system has to not only perform simultaneousmeasurements on each tire, but it also has to compare and perform calculations on measurementsin real-time in order to help determine how to modulate the brakes for each tire. The upshot ofthis for understanding engineering, according to Johnson, is that “[a]t its core, ABS is a systemfor measuring, comparing, and responding” (80).Moreover, the story of how, at the end of the Second World War, it came to be the casethat there were no engineers that specialized in automotive anti-skidding technology to thestate of affairs in the 1980s, when there were engineers who specialized entirely on ABSs, is acomplicated matter. Indeed, Johnson argues that in order to understand how the engineeringknowledge concerning ABSs was developed, we have to look at how the skidding problem —along with skidding measuring instruments and technology — changed within separate commu-nities of engineers and how various engineers from other communities become professionalizedinto a community of engineers with various kinds of expertise which focused on the problemof skidding. For example, Johnson emphasizes that, in several instances, aeronautical engi-neers had to initially provide their expertise concerning the braking systems of aircraft to thoseworking on automotive skidding; in fact, sometimes these engineers migrated entirely from theaeronautical to automotive engineering communities. More specifically, Johnston explains howanti-skidding devices were already developed for disc-brakes for aircraft after the Second WorldWar and it was in Great Britain that these devices were hastily modified to work for auto-mobiles. Unfortunately, not only were these early devices unreliable but they were also veryexpensive and thus were not mass-marketable.107The problem of how to measure skidding was first tackled in Britain during the 1950s. Thestory Johnson tells is complicated, but the basic point is that there was, in general, disagree-ment on how to design the instruments for measuring torque or angular velocity of tires, calleddynamometers, as a means to calculate changes in deceleration in tires. These disagreements107 Only expensive cars, like Rolls-Royces and Jaguars, used disk brakes in the 1950s, whereas cheaper carsused drum brakes; disc brakes only became less expressive in the 1960s (Johnson, 2009; 49-53, 70).703.4. Changing Designs and Braking Barrierswere generated because of differences in opinion concerning how automobiles work in the firstplace. Initially there were disagreement about whether laboratories or road tests should be usedto test different braking designs (79; also see chapter five in Johnson 2009). It was during thistime that some British engineers made, according to Johnson, the often “fatal” assumption that“all the wheels decelerated at the same rate” (66). It was not until the advent of electronicsthat skidding could be reliably measured but even this was at first done using analog electronicswhich depended on vacuum power (76-77).108Whereas British engineers tended to treat anti-lock braking systems as a modification to analready extant braking system, it was American engineers, working at companies like Ford andChrysler, that designed braking systems to include ABSs. Specifically, Johnson talks about thekinds of competing design decisions engineers had to make concerning the design of a particularABS, called “Sure-brake,” at Chrysler and Bendix.109 The sort of design questions about the“Sure-brake” system are the kinds of questions best understood as being made relative to thedesign hierarchy for an automobile and not just a design hierarchy for a braking system. Forexample, consider the question about whether there should be sensors on just two or all fourwheels. Two-wheel designs are cheaper, but are less effective. Four-wheel designs, on the otherhand, are more expensive but offer better performance. There is also the choice engineershave to make about whether sensors for measuring angular velocity should be mechanical orelectric (using analog controls).110 Thus, the original skidding problem — which was originallya problem about braking systems — is now a problem about how to design automobiles withan ABS. These ABS designs, however, could, and did, fail because of the unexpected physicallimitations of the technology involved. For example, the analog sensors used for the Sure-Brake ABS turned out to be unreliable: salt on the roads corroded wires and radio/TV towersinterfered with the analog sensors.111 Yet such failures are not always debilitating; in fact, failureis an essential part of engineering design – it weeds out those designs which are impracticable108 And so after laws were passed in the United States stating that all cars should have Catalytic converters,there was a reduction in the amount of vacuum power that could be allocated to analog electronics and thusengineers had to re-think how braking hydraulics should be powered (Johnson, 2009, 117).109 See Johnson, 2009, 111 ff.110 See Johnson, 2009, pp. 112-13 for a list of requirements engineers decided the Sure-Brake ABS should meet.111 The sensors, Johnson reports, were discovered to fail because of “nighttime test drivers [playing] the radiowhile driving” (2009, 114).713.4. Changing Designs and Braking Barriersor impossible.112 There are no a priori guarantees regarding which designs won’t fail: instead,engineers have to try out particular designs in order to acquire expertise and know-how in orderto come up with new designs will have better expectations of success.Despite the various attempts to better design ABSs like Sure-Brake ABS in the 1970s, thesesystems were never an economic success. The cars were just too expensive; in fact, the firstinexpensive and integrated ABS was only first introduced into North American mass marketswith the 1983 Ford Scorpio (133). The major breakthrough which allowed for the proliferation ofinexpensive ABSs happened on the other side of the Atlantic in places like Sweden, France andWest Germany: by the 1970s digital electronics had become much less expensive and many Eu-ropean engineers were gaining expertise in computer programming languages, like FORTRAN,which were necessary to implement the algorithms for performing the calculations required tocompare the measurements from electronic dynamometers (98). Moreover, new technologies(again, from aeronautical engineering) made their way into automotive engineering. In thiscase, high-speed valves used for airplane instruments were borrowed by automotive engineersto quickly modulate brakes; these new values could modulate brakes around 60 pulses per sec-ond – quite a dramatic increase over the 4-6 pulses per second used by earlier ABSs (122-3).The upshot is that each tire could have its own high-speed brake-modulating device, deviceswhich could then be controlled using digital circuits. Moreover, these new technologies openedup new design possibilities: expectations were adjusted concerning what was theoretically andpractically possible. For example, Johnson claims that these engineers argued their “system wasderived from theories of tire friction” and as such[t]heir design goals were aimed at realizing what was theoretically possible according totheories of vehicle and tire dynamics, rather than seeking simple improvements over existingbraking system technology (124).The problem of skidding had shifted: due to new technologies, it was possible to measure wheelslippage directly from changes in tire deceleration while simultaneously modulating brakes atvery high speeds to prevent and not just correct for skidding (2009, 125-135). This was atechnical achievement. The result was a piece of engineering knowledge.The relevant point of this history of ABSs is this: Johnson’s historical work and analysisallows us to see how a specific engineering problem, an “attractor,” has changed over time as112 See Petroski (1992).723.5. Herbert Simon and Satisficingengineers create new technologies and reformulate the current problem into a new problem madetractable by the new technologies. The original engineering problem of making cars safer bytrying to prevent them from skidding morphed into a more well-defined problem: namely, theproblem of figuring out how automobiles should be designed from the ground up so that theyhave a very specific and cheap kind of anti-lock braking system, i.e. an ABS built using a hostof new technologies like quick-pulsating values and electronic sensors. Moreover, these changesconstitute a continual redefinition of the central operational principle at the heart of brakingdesigns: the conditions of success and failure for braking systems changed as the technologiesand practical limitations (like economic success) changed.3.5 Herbert Simon and SatisficingEngineering problems, like those we just saw from the history of aeronautical and automotiveengineering, are rarely first articulated in the form of what the computer and social scientistHerbert Simon called “well-structured problems” (WSPs); namely, as problems sufficiently spec-ified so as to make the finding of their solution obvious using some general method. Instead,most problems begin their life as “ill-structured” problems (ISPs) – as the “residual concept” of“a problem whose structure lacks definition in some respect” (1973, 181). Simon explains thetransition (which he admits is a relation of degree rather than kind) from an “ill” to a “well”structured problem as relative to a procedure for solving problems, whether it be the cognitionof humans or an algorithm: a WSP is an ISP which has been reformulated, codified and alteredso that it is now a well-defined problem for a specific problem solver (186).113What is crucial to recognize is that there are at least as many ways of transforming an ISPinto a WSP as there are ways to solve problems and that each such way provides us with adifferent perspective for how to visualize, so to speak, the potential layouts for the internal logicof an ISP. Humans and computer programs can be trained (at least for the case of machinelearning algorithms) to use heuristic reasoning to play chess but they will rarely formulate and113 Simon was motivated in this article to explain how his General Problem Solver could possibly solve ISPs.Simon’s work on artificial intelligence, however, is conceptually linked with his notions of “satisficing” andbounded rationality from his work on decision theory: his earliest work with Allen Newel on a programthat finds logical proofs for Whitehead and Russell’s Principia Mathematica, for example, found proofs byusing “heuristic” rules to find “good enough” but not necessarily optimal proof solutions; e.g., see Newelland Simon (1956). For more on Simon’s notion of a “bounded rationality,” see Simon (1957; 1996).733.5. Herbert Simon and Satisficingimplement the same heuristics in the same kind of way. The method we use to solve a problemchanges how we conceptualize and reformulate that problem. Consequently the task of figuringout how best to optimize an ISP is not always obvious – but most of the time we just need tofind some way of transforming the ISP into a WSP whose solution is “good enough” for the taskat hand. Instead of finding the globally optimal solution, e.g., in terms of finding the maximumand minimum values of some expected utility function over a complete space of possible statesof the world, we instead “satisfice” by scaling back the requirements for what it would mean tofind an acceptable solution, like calculating one’s expected utilities over a limited set of plausiblestates of the world.114 “An earmark of all these situations,” later says Simon,where we satisfice for inability to optimize is that, although the set of alternative alternativesis given in a certain abstract sense (we can define a generator guaranteed to generate all ofthem eventually), it is not given in the only sense that is practically relevant. We cannotwithin practicable computational limits generate all the admissible alternatives and comparetheir respective merits. Nor can we recognize the best alternative, even if we are fortunateenough to generate it early, until we have seen all of them. We satisfice by looking foralternatives in such a way that we can generally find an acceptable one after only moderatesearch. (1996, 120)We have already seen several examples of how engineers can, firstly, transform a design probleminto a hierarchy of more tractable and technically feasible engineering problems – this is thetransition from an ISP to a WSP. Secondly, we have seen how engineers, as with the caseof propeller design, must settle with a “good enough” solution – they satisfice rather thanimpractically search for globally optimal solutions.Of course, I do not mean to suggest that satisficing or even the problem of design are distinctto engineering – scientists trade in these concepts and issues too.115 But whereas the computerscientist runs up against the mathematical limitations of computability and complexity theoryor the behavioral economist the limited reasoning capabilities of decision makers, the designengineer has to make due with the limitations imposed by the results of the empirical sciences,contemporaneous engineering knowledge and the practical needs of their firms and companies.114 Also see chapter 14 of Simon 1957.115 The distinction between scientist and engineer is frequently blurred with the advent of large-scale, coop-erative projects like the construction and operation of the Large Hadron Collider (also see Petroski 1992,esp. ch. 4). Vincenti (1990) also cites the example of one Irving Langmuir who received a Nobel Prize inchemistry for his work at General Electric’s Research Laboratory (227). Moreover, it seems reasonable tosuggest that engineers and scientists routinely coordinate and transfer between them the knowledge requiredto produce the technological discoveries and products associated with R&D labs and companies like BellLabs, RAND, Xerox, Hewlett-Packard, Microsoft and Google.743.6. Carnap as Conceptual EngineerNevertheless, while scientists are primarily concerned with getting at the truth, explanation,prediction or cultivating any number of scientific virtues, design engineers have an impressivedegree of creative freedom afforded to them by the initial vagueness of engineering problemsto find effective, but not necessarily optimal, ways to make something happen in the worldaccording to some plan or schema.116 Or as Vincenti puts the point:In general, all knowledge for engineering design (as well as for the engineering aspects ofproduction and operation) can be seen as contributing in one way or another to implemen-tation of how things ought to be. That, in fact, is the criterion for its usefulness and validity.(1990, 237)Engineers and their clients are not bound by how things are but only by how things couldpossibly be: they are free to change the measure – by modifying the operational principle, the“ought” – by which engineering designs as evaluated as better or worse. In this sense, designengineering is as much a practical as it is a theoretical activity.3.6 Carnap as Conceptual EngineerThe last section provides an interpretive framework, or a working analogy, which I use through-out the rest of the dissertation to describe Carnap’s work on inductive logic.117 The languageof explication, for example, parallels Simon’s language of “ill-structured” and “well-structured”problems: the philosophical problem of clarifying and systematizing an explicandum like thelogical concept of probability is analogous to an “ill-structured” problem whereas the use of bothsyntax and semantics as tools to construct an explicatum like a quantitative concept of degreeof confirmation is analogous to the use of problem-solving tools and methods to formulate a“well-structured” problem. When a pure inductive logic is applied for use in the empirical sci-ences, like theoretical statistics or information theory, an inductive logic qua logic may have tobe redesigned and extended by the logician to better meet the demands of these sciences: this116 Of course, the same could be said for many experimental scientists, especially those actively engaged in thedesigning of experiments and scientific instruments, see Baird (2004); Radder (2003).117 Of course, the engineering analogy is just an analogy: there will always be dissimilarities between explicationand engineering projects. Nevertheless, I talk of analogies instead of metaphors in part due to the work ofMary Hesse who argues, in Hesse (1966), that at least for analogical reasoning using models in scientificcontexts, an analogy between X and Y includes three kinds of components: how X is similar to Y, how X isdissimilar to Y and, most importantly, there are open (typically empirical) questions about whether specificparts of X is similar or dissimilar to different parts of Y. According to Hesse, it is this third component whichis most important for analogical reasoning in the sciences: as we learn more about the “open questions”about either X and Y we will come to learn, by analogy, more about the other. For a more detailedtreatment of analogical reasoning, see, e.g., Bartha (2010).753.6. Carnap as Conceptual Engineeris analogous to how the operational principle of a design hierarchy of problems can change overtime, especially as new technologies emerge or with the increase of our scientific or engineeringknowledge. Lastly, there is for Carnap no “correct” logic – there is no “correct” explication ofa concept of logical probability, or inductive reasoning more generally, but only better or worseexplications: this is analogous to engineers who satisfice rather than optimize. Explication is forCarnap the on-going, gradual, process of improvement of a system of concepts designed specifi-cally for clarifying the logical structure of scientific theories and concepts.118 This is a kind ofconceptual engineering. But conceptual engineering differs from most professional engineeringinsofar as concepts need not be “tested” empirically. Indeed, for Carnap pure inductive logic isnot tested directly but instead we must stipulate our own requirements and restrictions – ourown operational principle – for what to means for an applied inductive logic to be successful. Alogic can then be designed from the ground up to serve any number of scientific purposes justas an aircraft can be designed to serve any number of industrial or military purposes.The guiding idea for why this hierarchical account of engineering design is an appropriateinterpretive framework for explaining and clarifying the philosophical significance Carnap him-self assigned to his technical projects is that, as we saw in the previous chapter, he conceives oflogical syntax and semantics as tools or instruments chosen not because of their correctness ordue to some highly theoretical process of justification but rather because of their expected ca-pacity to satisfy our intellectual ends. In the previous chapter we discussed a number of Carnapscholars who have already explained Carnap’s mature philosophy method, with an emphasis onWissenschaftslogik as spelled out in LSL, through the lens of conceptual or linguistic engineer-ing. In this section I explain how this hierarchical account of engineering can accommodatemany of the insights made by these scholars while providing an original take on how Carnapcould be understood as a conceptual engineer.Before we move on, however, I want to make it perfectly clear that I am not claimingthat Carnap understood himself as a conceptual engineer in the above hierarchical sense ofengineering. I also do not endorse in this dissertation any historical account for how Carnaphimself understood the activity of engineering. Richardson (2013) has suggested, for example,that Carnap’s talk of treating logic as an instrument or tool can be traced back to his interest in118 Reck (2012) also emphasizes the point that, for Carnap, explication is a process.763.6. Carnap as Conceptual Engineernineteenth century German-speaking metrology, Instrumentenkunde. This may very well be thecase but the engineering case studies I draw on are from, for the most part, the mid-twentiethcentury and I make no claim whatsoever that the conception of engineering design as practicedby twentieth century professional engineers is at all similar (or dissimilar) to nineteenth centuryconceptions of design engineering or the making of scientific instruments. My treatment ofCarnap as conceptual engineer is purely an interpretive gloss on his technical work which ismeant to better explain his philosophical projects for the mental consumption of contemporaryphilosophers of science.As we saw in the last chapter, Richard Creath has employed an engineering analogy tohelp explain why Carnap does not appeal to traditional philosophical notions of justification orintuition in his technical projects. More recently, he puts the point like this:philosophers can devise, refine, and explore a variety of conceptual or linguistic frameworksand test their suitability for various practical purposes. These frameworks are tools, so wedo not have to prove that they are correct. Nor do we have to agree on which ones touse. We just have to be clear enough to see what follows from what. Then a new result,whether it is a newly clarified concept or a new theorem is a new and permanent and positiveaddition to our stock of tools. And Carnap can offer the preceding three decades and morein logic as an example of the sort of continuing progress that he is describing. Logiciansoften disagreed about which systems to use, but they almost never disagreed about whatwere the results of another’s systems. (Creath, 2009, 211)In place of philosophical arguments we instead can show how a system of concepts can be definedwithin a conceptual framework. The debate between Quine and Carnap, from Carnap’s pointof view, is a debate of logical preference and fruitfulness: after each of them has shown how todefine a notion of analyticity in their own separate frameworks they can then re-formulate theirdisagreement as a disagreement about whether they prefer differing logical consequences of theirlogical frameworks or how poorly one framework can be applied in the empirical sciences whencompared to the other.As admirable as Creath’s contribution is, however, it relies on a means-end notion of engi-neering: the job of the logician is merely to study an endless array of logical frameworks – toadd to the current stock of tools – and it is up to the scientist to choose that framework from thecurrent stock which seems to them most useful. However, as we will see in chapter 4, Carnapdesigns and constructs pure inductive logics with the aim of clarifying and systematizing theinductive concepts scientists already use. This means that there has to be some interplay, some773.6. Carnap as Conceptual Engineercommunication, between logicians and scientists. It is for this reason, however minor a point itseems to be, that the hierarchical view of engineering design provides us with, I argue, a moreapt analogy of Carnap’s mature method than a means-end conception of engineering.For André Carus, however, this point is not so minor. Indeed, Carus – who in turn draws onthe work of Howard Stein – emphasizes that explications exhibit a certain ‘dialectical’ propertyor feed-back relation between ordinary and artificial, logical language. There is a give and takebetween the theoretical and practical. Here is how Stein puts the point in the context of choosinga linguistic framework appropriate for theoretical physics:It may very well be – I am inclined to think it is – that the possibilities to be contemplatedin a framework for a theoretical physics as we know it today or as it is likely to develophave to be restricted by the general principles of the theory itself – principles that onewould be loth to call ‘analytic’. This is a serious modification of Carnap’s view. It locatesfundamental theory change in change of framework, and therefore outside the scope of thesort of inductive logic Carnap was trying to construct – which itself would, of course, beinternal to a framework. That, it seems to me, entails a development of Carnap’s views ina direction that I should characterize as ‘dialectical’; for it entails a certain blurring of thedistinction, dear to Carnap, between the purely cognitive, or theoretical, and the practical.(Stein, 1992, 291; emphasis mine)One way of reading Carnap’s distinction between the practical and theoretical in works like LSLis that the separation between the practical and theoretical must be sharp: there are practicaldecisions we need to make concerning how to set-up a logical system and then there are theoret-ical questions we can formulate within that system. Alternatively, to adopt Carnap’s languagefrom his 1950 paper “Empiricism, Semantics and Ontology,”119 there are external, pragmatic,questions about the choice of a framework and theoretical questions expressible using a singleframework: there is no mixture of the practical and theoretical relative to the same framework.Stein’s point seems to be this: if no inductive logic, itself a part of a language framework, canfully characterize when theory change should occur – here understood as revisions to that frame-work – then such changes can only be made within a broader, more compressive framework:namely, the “framework” used by practiced physicists as a highly specialized combination of or-dinary language and mathematics. But now practical questions about the choice of a languageframework for a language of physics may in fact be influenced by the answers to theoreticalquestions formulated in a different framework, the framework used by physicists. Granted thatthis physical language framework will eventually influence which concepts physicists adopt when119 See chapter 4, 112 ff., this dissertation.783.6. Carnap as Conceptual Engineertalking about physics, we then have a ‘dialectical’ relationship between ordinary and artificiallanguage.120In contrast to Carus and Stein I would argue that this mixture of the practical and theoreticaldoes not constitute a major change to Carnap’s mature position. In fact, as we saw in section2.2 from the last chapter, Carnap, in LSL, never attempted to formalize inductive processesas P-rules in a logical system and in later chapters we will encounter numerous examples forhow the practical and theoretical can “mix” in the transition from pure to applied inductivelogic. Indeed, even during the heyday of Carnap’s work on inductive logic he was never in thebusiness of defining confirmation functions over entire scientific theories, like Einstein’s generaltheory of relativity.121 My hierarchical conception of engineering design can help illustratethis aspect of Carnapian logic of science. I suggest that we can make sense of how theoreticalassertions, especially from sciences like theoretical statistics, can influence the decision to designan inductive logic in a particular way through the lens of hierarchical engineering.Lastly, I want to turn to my criticism of Hillier’s notion of linguistic engineering from thelast chapter. I argued against Hillier that Carnap does not, at least as a matter of principle,appeal to a notion of “fit” between the world and a linguistic model. But this raises an importantquestion. In cases of engineering design it seems fairly obvious when engineering projects fail:planes can turn out to be slow and clumsy, budgets go from the black to red and bridges fallapart. Engineering success, it seems, is ultimately tied to empirical success. So if we takeseriously the idea that Carnap’s work on inductive logic can be fruitfully understood as anengineering activity, surely we need to measure the success of an inductive logic by its empiricalsuccess – probability theory, after all, is only as good as it makes successful empirical predictions(just ask insurance adjusters and casino managers). Here we can reformulate Hillier’s claim assuch: the success of a linguistic framework, understood as an engineering design, must ultimatelybe measured by some measure of empirical success. In chapter 5, I argue that this is not thecase: for the case of how inductive logic can be applied to decision theory, Carnap uses rationaldecision theory as a kind of conceptual space by which inductive logics can be outfitted, so to120 “The practical realm kicks backs. Ordinary language is still to be overcome and improved, but is also [. . . ]the medium of practical reflection, the medium within which we choose among theoretical frameworks”(Carus, 2007, 21).121 See Carnap, 1962b, 243–244.793.7. Conclusionspeak, to conceptually test ideal agents in hypothetical empirical situations – no actual empiricaltests need ever be carried out. For Carnap, the success of a language framework need not betied to its empirical success. Instead, we have an example of how something like an operationalprinciple is characterized methodologically – as the requirements an adequate inductive logicmust satisfy – and this principle is then tested “conceptually,” i.e., tested in the possible statesof the universe according to a logical language. The result is a matter of finding an adequateenough inductive logic: it is a matter of satisficing.1223.7 ConclusionIn conclusion, the caricature of a philosophical method based on search – a method that seeksthe correct answer to a philosophical question – outlined at the beginning of this chapter dif-fers from philosophy as conceptual engineering in a way similar to how satisficing differs fromoptimization: the aim is to find a good enough solution relative to a practical set of criteriarather than guaranteeing that the “correct” solution will eventually be found. However the kindof conceptual questions Carnap is concerned with, like how to clarify analyticity or a logicalconcept of probability, are inherently vague: not only do such questions fail to distinguish be-tween different explicanda but they tend to unclearly mix psychology and logic. How exactlysuch concepts should be clarified is, for Carnap, more of an expression of one’s philosophical orscientific preferences rather than a search for some timeless truth. Carnap was not attemptingto do the history of science or to do psychology: he only provided us with a method, a methodwhich I suggest we can apply ourselves while incorporating norms and methods beyond that ofCarnapian logic of science, including, perhaps, from the history of science or feminist critiquesof science. It is a method one can use, for example, to detach oneself from foundational worriesabout the epistemology and metaphysics of a logical concept of probability and examine logicallyconstructed inductive concepts within some inductive logic: this is not a method to once andfor all settle foundational questions but instead it is a method to help systematize and clarifysome of the possible ways of thinking about probability and induction.Philosophers, for Carnap, are not priests: it is not our job to tell scientists or the layman122 This point is similar, I think, to the William C. Wimsatt’s work on articulating a conception of philosophyfit for epistemically limited beings such as ourselves; see Wimsatt (2007).803.7. Conclusionwhat they should or shouldn’t do, what they should or shouldn’t think or what they should orshouldn’t value. Rather, our job is simply to pinpoint misunderstandings and to help facilitateuseful and clear dialogue between interested parties. Conceptual engineering best captures, Isuggest, the final evolution of Carnap’s attitude of logical tolerance – an attitude which extendsto any technical machinery: for Carnap, there is no a priori restriction on what conceptualresources are not modifiable – the very resources central to contemporary philosophy, like con-cepts of mental representation, propositional facts, semantic content and modal reasoning, areno more sacred for the Carnapian conceptual engineer than concepts of analyticity and logicalprobability. And I suppose it is in this sense that Carnap’s mature philosophical attitude ismore pluralistic, democratic and tolerant than those philosophical attitudes characterized bymethods like conceptual analysis which would have us search until we found some truth aboutourselves or the world.123123 Compare, for example, Jeffrey’s epigraph at the beginning of this chapter. For more on whether explicationscould be an alternative to conceptual analysis, especially in collaboration with the X -phi literature, seeJustus (2012); Shepard and Justus (2014). Also see Kitcher (2010); Kuipers (2007) and, hot off the press,Dutilh Novaes and Reck (2015).81Chapter 4Designing Inductive LogicIt is conceivable that “you” could design a language so as to make Carnap’s theoryconsistent with the one presented in the present work. All probability judgmentwould be pushed back into the construction of the language. Something likeCarnap’s theory would be required if an electronic reasoning machine is ever built.— I. J. Good, Probability and the Weighing of Evidence (1950)Carnap suggested that in epistemology we must, in effect, design a robotwho will transform data into an accurate probabilified description of his environ-ment. That is a telling image for modern ambitions; it also carries precisely theold presuppositions which we have found wanting. We can already design prettygood robots, so the project looks initially plausible. But the project’s pretensions touniversality and self-justification send us out of the feasible back into the logicallyimpossible – back into the Illusions of Reason, to use the Kantian phrase.— Bas van Fraassen, “The False Hopes of Traditional Epistemology” (2000)When the mathematician and statistician I. J. Good says in 1950 that for a Carnapian induc-tive logic “all probability judgment would be pushed back into the construction of a language” heis attributing to Carnap what the contemporary philosopher of science Bas van Frasssen wouldlatter call a robot epistemology: The probabilistic judgments made by human reasoners are tobe logically reconstructed in terms of the logical probability values cranked out by an adequateconfirmation function – a function which is, as a matter of logical stipulation, completely well-defined for any possible experience an epistemic agent could ever encounter. This account ofCarnap’s inductive logic isn’t fanciful; Carnap did, after all, actively search for such a function.Nevertheless, I claim in this chapter that by the time Carnap wrote his monograph TheContinuum of Inductive Methods in 1952, he guaranteed neither that such a fully adequatefunction exists nor, assuming it does, that we would ever find it.124 This situation is analo-gous to an hierarchical engineering design problem – a problem best approached by satisficing.Indeed, in his work on inductive logic Carnap only specifies the scaffolding, so to speak, of a124 For further retrospective discussion of Carnap’s inductive logic, see French (2015a); Jeffrey (1970; 1973;1974; 1990; 1992a); Kuipers (1978).824.1. Historical Backgroundsemantic system L with a fixed, but not fully specified, interpretation within which measure andconfirmation functions can be defined using the semantic resources of L. Using this scaffolding,different inductive logics can be constructed and the requirements and restrictions placed onthe semantics of L altered to satisfy certain practical needs. The construction of an inductivelogic intended to explicate inductive reasoning is, for Carnap, as indelibly a theoretical as it isa practical matter.125The basic trajectory of this chapter is as follows. After a brief historical discussion of hiswork on inductive logic, I introduce the terminology and technical issues required to explainhow Carnap constructs a single quantitative confirmation function called c. I then go on torecap how Carnap distinguishes “pure” from an “applied” inductive logic in a way analogous tothe distinction between mathematical and physical geometry. Afterwards I concentrate on Car-nap’s construction of a pure inductive logic and explain how he sees the role of inductive logicin possibly clarifying the problem of estimation, a problem which concerns the very foundationsof theoretical statistics. In particular, I discuss how Carnap investigates a particular continuumof inductive methods by constructing a parameterization of confirmation (and estimation) func-tions called the -system. I then discuss how Carnap attempted to use this -system to locate“optimal” estimation functions which are nonetheless biased (in the statistical sense of the term).I then argue that Carnap’s attempt to find “optimal” estimation functions using the -systemcan be understood as a kind of engineering problem: it is an example of how an “ill-structured”problem can be transformed into a “well-structured” problem and this transformation process isbetter understood, I suggest, in terms of satisficing rather than searching for the truth.4.1 Historical BackgroundIt will be useful to split Carnap’s work on inductive logic into four periods. The first periodcoincides with the last half of his time at the University of Chicago from roughly 1941, when hefirst becomes interested in problems about probability and induction, until he leaves Chicagoin 1952 to take up a visiting fellowship at the Institute for Advanced Study (IAS) at Princeton125 That Carnap embraces a kind of pragmatism or voluntarism with his -system (explained below), however,is not lost on van Fraassen; see van Fraassen (1989, 176) and, for his own views, van Fraassen (1984).834.1. Historical BackgroundUniversity.126 It is during this time, from 1940 to 1941, that Carnap visited Harvard and, from1942 to 1944, used a Rockefeller grant to temporarily relocate to Santa Fe, New Mexico wherehe could work without distraction on pure semantics, modal logic and inductive logic.127 Thiswas a productive time for Carnap: in the 1940s he not only published three books on semantics(Carnap 1942; 1943; 1956, first published in 1947) and numerous papers on semantics, modallogic and inductive logic from 1945 to 1947, he also published what is arguably one of his mostwell-known and anthologized papers, “Empiricism, Semantics and Ontology.” Finally, he alsopublished both his probability book Logical Foundations of Probability (LFP) in 1950 and, in1952, his monograph The Continuum of Inductive Methods (CIM ). Although LFP was originallyplanned as a two-volume book called “Probability and Induction,”128 Carnap never managed tocomplete the second volume; nevertheless, a majority of the planned content for volume twoended up in Carnap’s 1952 monograph, CIM. By and large, the technical developments I discussin this chapter come from both LFP and CIM.129 What is also of note during this initial timeperiod is that Carnap, prompted by criticism of his two papers on inductive logic in 1945,130engaged in serious philosophical discussions about probability and inductive logic with his peers.For example, Carnap had extended discussions with Carl Hempel and Nelson Goodman in 1946sparked by their initial conversations at the annual meeting of the American Association for theAdvancement of Science (AAAS ) held in St. Louis.131 Also of relevance to the historical contextfor this chapter is that, while at Chicago in the late 1940s, Carnap influenced a new generationof philosophers of science in North America to work on probability and inductive logic. Therehe taught inductive logic from his apartment in 1948 to a group of students, including Abner126 Carnap moved to the University of Chicago from Prague in the fall of 1935; for more details see Creath(1990a).127 See Carnap, 1963a, 34–5, 41–2. For more on Carnap’s visit at Harvard, see Frost-Arnold (2013).128 LFP, vii. All references to LFP, including this one, are to the second, 1962, edition of the book.129 It is important to point out that despite the fact that LFP is published before CIM, one shouldn’t necessarilytreat CIM as a modification of Carnap’s views in LFP – this is because Carnap had a draft of CIM alreadyin the summer of 1949; Carnap to Quine, Nov. 26, 1950; reproduced in Creath (1990a, 420-422). Apparently,Carnap had also worked on the idea of the lambda parameter as early as 1947 (Carnap, 1980, 93).130 Carnap (1945a,b).131 Carnap (1945b) is part of a three-volume symposium on probability and induction in Philosophy of Phe-nomenological Research (Vol. 5 (4) Jun. 1945; Vol. 6 ( 4) Sept. 1945; and Vol. 6 (4) Jun. 1946). The othercontributors are Hans Reichenbach, Henry Margenau, Gustav Bergmann, Felix Kaufmann, Richard vonMises, Ernest Nagel, and Donald Williams. There is extensive correspondence resulting from the meetingbetween Carnap, Goodman and Hempel in St. Louis from 1946 to 1947 and the publication of Goodman(1946) at Carnap’s archives at Pittsburgh; box 084, folders 14 and 19.844.1. Historical BackgroundShimony, Howard Stein, John W. Lenz and Richard C. Jeffrey.132 And as we will see in thenext chapter, Carnap’s influence led to some important results in inductive logic and rationaldecision theory (most notably, results by John G. Kemeny, Shimony and Jeffrey). The secondtime period is from 1952 to 1954 when Carnap was a visiting scholar at Princeton’s IAS.133 Thethird time period is from 1954 until 1962 which spans from the moment he moved to UCLA untilboth his publication of “The Aim of Inductive Logic” in 1962 – the published version of the talkwhich he was personally invited by Patrick Suppes to give at the 1960 International Congressfor Logic, Methodology and Philosophy of Science134 – and his retirement from academia.135The fourth and last time period spans rest of Carnap’s life until his death in 1970 duringwhich he routinely works on and amends his manuscript “A Basic System of Inductive Logic”– this manuscript is subsequently posthumously published in two parts, each in one of the twovolumes of the periodical Carnap co-edited and planned with Jeffrey, Studies in Inductive Logicand Probability.136In this chapter we will focus on the first two time periods, between 1941 and 1954 (thenext chapter focuses on the third time period).137 In particular, we will be interested in howCarnap’s work on pure inductive logic is informed by how it could be applied to help clarify thefoundations of the empirical sciences. For example, Carnap spends part of his time at Prince-ton’s IAS working on a semantic concept of information based on concepts from an inductivelogic. However, much of this work was already completed from 1949 to 1951 and although heis invited to participate in a Cybernetics conference at Princeton in 1952, Carnap, for the mostpart, leaves it to Yehoshua Bar-Hillel (who previously held a research position at Chicago in1950) to disseminate this work to information scientists.138 Instead, Carnap spends the ma-132 See Shimony (1992).133 Apparently, Carnap’s time at IAS was not better than his time at Chicago; as Carnap would later tell Quinein 1955, these two years were “somewhat difficult years for me” (Carnap to Quine, Sept. 22, 1955; In Creath1990a, p. 440-1). Also of historical interest is a letter by Carnap’s second wife, Ina, to John Kemeny inwhich she discusses her and Carnap’s disillusionment with Chicago, their inability to stay at Princeton andthe uncertainty of moving to UCLA (Ina to Kemenys, Feb. 28 1954, RC 083-18-12).134 April 25, 1960, Ina to Kemeny, RC 083-15-03.135 Carnap occupied the same chair at UCLA previously held by Reichenbach who had died from a heart attackin 1953.136 See Jeffrey (1980); Jeffrey and Carnap (1971).137 Although his later work is interesting and is in need of examination, I say relatively little about Carnap’sfinal work on inductive logic in this dissertation; although see Skyrms (2012) and Zabell (2005).138 Carnap tells Kemeny in 1952 that he intended this definition “(in January 1949) as an analogue to thestatistical concept “amount of information” (see e.g. Wiener’s Cybernetics, pp. 75 ff), replacing statisticalprobability by inductive probability” (Carnap is referring to Wiener, 1948). However, in the same letter,854.1. Historical Backgroundjority of his time in 1952 working out the mathematical details of his work on inductive logicwith the mathematician John G. Kemeny.139 In particular they worked on (i) simplificationsof Carnap’s continuum of inductive methods, the -system, (ii) less idealized inductive logicsbased on Kemeny’s work on semantics and set-theoretic models and (iii) extensions of Carnap’swork on inductive logic to analogical reasoning, including what Carnap and Kemeny called the“two”- and “many”-family problems.140 Although there is no attempt in this dissertation to dis-cuss these results, it would be difficult to downplay the importance of Kemeny’s mathematicalcontributions to Carnap’s work on inductive logic. In 1959, for example, Carnap tells Kemenythat,Our meeting in Princeton was pretty much a miracle and revelation to me. In addition, itcame just at a time when I had need and use for miracles! (Carnap to Kemeny, May 5,1959, RC 083-15-12)Tellingly, in the same letter Carnap admits that of all his peers working on probability andinduction only Kemeny understood more mathematics than himself. Indeed, it is this collabo-rative work with Kemeny on inductive logic in 1952 that informs the vast majority of Carnap’slater work on inductive logic, including Carnap’s adoption of mathematical measure theory todefine measure and confirmation functions in the “Basic System” manuscript.Nevertheless, in the 1950s, unless you were lucky enough to be included within a tight-knitcommunity working on inductive logic – a community which included Jeffrey, Hempel, Kemenyand Feigl – you would have been unlikely to have had access to Carnap and Kemeny’s mostrecent technical results in inductive logic. A solution to their two-family problem, for example,Carnap says that he “found no time” to work out the theory so instead Carnap “dictated [his] notes on sixhalf-hour wire-spools and sent them to Bar-Hillel” (April 29, 1952; RC 083-18-20); the result is Carnap andBar-Hillel (1952). Bar-Hillel was actively engaged for a while in presenting Carnap’s work to a Cyberneticsgroup at MIT (Bar-Hillel to Carnap, March 15, 1952; RC 102-02-102). Apparently, the reason that Carnapcouldn’t attend the Cybernetics conference at Princeton was because of problems with his back (Carnap toL. J. Savage, April 11, 1953; RC 084-52-22). Presumably, Carnap is referring to the 10th, and last, Macy’sConference held at Princeton in 1953; Bar-Hillel, however, did give a presentation on a semantic concept ofinformation at this conference (Bar-Hillel, 1964, 11); for more on the Macy’s conferences and the history ofcybernetics, see Heims (1991).139 Interestingly, before Kemeny and Carnap first meet at Princeton, through correspondence Hempel introducesKemeny’s work to Carnap and Carnap realizes that Kemeny’s “index of caution” is actually the same as his parameter. (see Carnap to Kemeny, December 3, 1951, RC 083-18-30; Kemeny to Carnap, Dec. 10, 1951,RC 083-18-27).140 Supposing the -system helps us represent the “one-family” problem of specifying the probability that thenext ball pulled from an urn is blue, the two-family problem concerns how the -system should be modified tocalculate probabilities concerning two modalities, or “families”, like color and whether the ball is translucentor opaque. The “many-family” problem, then, concerns how the -system can be generalized to n-manyfamilies. Kemeny worked with Carnap to figure out how to quantify an inference by similarity or analogyin order to calculate these probabilities.864.1. Historical Backgroundwas only first published, in German, in section eight of the B appendix to Carnap and Stegmüller(1959) and, in English, in Carnap’s Schilpp volume.141 Moreover, the “Basic System” manuscriptwas nearly ten years in the making as Carnap first started sending out sections of the manuscriptas he wrote them to a few select peers in 1959. Indeed, on one of the few occasions that Carnapactually discussed his recent technical work in person – a two-day workshop organized by Hempelat Princeton in 1965 made to coincide with Carnap’s journey from Germany back to California– only a very limited number of scholars and graduate students were invited.142;143 But evenat this meeting, Carnap was only interested in technical improvements to his recent work ininductive logic: as Hempel reports in a letter from 1965 to the participants, “Carnap told methat he would not want to discuss broader philosophical questions concerning inductive logic,but certain technical problems related to his more recent axiomatic work in this field.”144But before Carnap became solely focused on technical questions about pure inductive logiche was also concerned with showing how his work on pure inductive logic could be appliedto the empirical sciences. For example, while Kemeny spent the academic year 1953-1954 inEngland (and before Kemeny was hired away by Dartmouth’s mathematics department in 1954),Carnap spent the majority of his time figuring out how his work on inductive logic could beused to construct an adequate explication of a semantic concept of entropy.145 Carnap finisheda two-part manuscript on entropy in January 1954.146 Part one – instead of using the semantic141 Carnap apparently had completed most of the “Replies” for that volume as early as 1958 but due to a varietyof reasons the volume was ultimately delayed until 1963, one year after Carnap’s retirement from UCLA.Actually Carnap is “officially” retired from UCLA in 1958 at the age of 67 (Carnap was born in 1891), butis reappointed after that on a year-by-year basis (Creath 1990a, 445).142 Sadly, aside from issues with Carnap’s health, there is another reason for the decrease in his academicoutput. Carnap’s wife Ina, who had been suffering from depression for some time, committed suicide onMay 26, 1964 (Creath, 1990a, 39). Afterwards, Carnap went to Germany to visit his daughter from aprevious marriage and his grandchildren. It is a testament to the friendship between Jeffrey and Carnapthat Carnap arranged for a telegraph to be sent to Jeffrey informing him of what happened the day afterIna’s death (RCJ box 9, folder 9 ); indeed, the correspondence between Carnap and Jeffrey (including theirwives) is quite regular from June 1957 until Carnap’s death in 1970.143 The full list of those invited is: Peter Achinstein, Paul Benacerraf, Herbert Bohnert, Herbert Feigl, RichardJeffrey, David Kaplan, John Kemeny, Henry E. Kyburg, Hughes Leblanc, Richard M. Martin, Sidney Mor-genbesser, Ernest Nagel, Robert Nozick, Hilary Putnam, Wesley C. Salmon, L. J. Savage, Abner Shimonyand Wolfgang Stegmüller.144 June 24, 1965; Hempel’s Archives at Pittsburgh ASP.145 According to Bar-Hillel (1964), Carnap first started worrying about entropy at while Princeton in 1952 afterCarnap and Bar-Hillel discussed conceptual problems with John von Neumann’s AAAS talk in St. Louis forwhich they were in attendance. In the talk von Neumann, according to Bar-Hillel, had apparently suggested“a triple identity between logic, information theory and thermodynamics” (Bar-Hillel, 1964, 11-12). For anextended discussion of these issues, including the differences between Carnap, Pauli and von Neumann’sviews on entropy and information, see Köhler (2001).146 The manuscript consisted of two parts: Part I as “A Critical Examination of the Concept of Entropy in874.1. Historical Backgroundnotions of state-description and range – started out with the notions of the description of amicro-physical state, X, and the number, z(X), of descriptions “similar” to X, which are thenused to define a concept of degree of order, a concept of disorder and then finally several differentversions of a concept of entropy, S. He then used these concepts to characterize the differentconcepts of entropy introduced by two physicists, Boltzmann and Gibbs, in order to articulatewhy he thought those concepts found in physics textbooks to be unsatisfactory. Part two ofthe entropy manuscript is more theoretical: basically, Carnap generalizes Boltzmann’s conceptof entropy to talk about how a density function can be defined for the “volumes” of abstract“environments” in order to define an abstract concept of entropy, S. However, what is mostinteresting for us, as we will discuss in the last section of this chapter, is that Carnap then goeson to use this concept to define concepts of degree of order and disorder, from which particularmeasure and confirmation functions, m and c, can then be defined (see Figure 4.6 on page127).We already know from Carnap’s autobiography that he discussed the first part of the entropymanuscript with physicists at Princeton – in particular, with Wolfgang Pauli, Leon van Hoveand John von Neumann. Apparently, however, this meeting did not go too well. Althoughall three disagreed with Carnap, Carnap was frustrated that together their criticisms were notconsistent. Nevertheless, in a letter to Kemeny, Carnap suggested that:my criticism is perhaps not valid with respect to what physicists actually do, in distinctionto what they write in the books. I still believe that many of the customary formulations arequite questionable; but this fact in itself would not make my lengthy discussions worthwhile.(Carnap to Kemeny, May 29, 1954, RC 083-18-14)Carnap, however, didn’t give up on the entropy manuscript; he sent copies of it to Abner Shimonyin 1955 and Howard Stein in 1957, asking both of them for their advice.147 Interestingly, Carnapnot only tells Stein that he “had to go back to studying statistical mechanics more closely thanI had done in the time of my studying physics way back,” but also that:Since the physicists did not understand my logical language, and since I was not completelysure of my physics, the ms. was laid to rest. (Carnap to Stein, August 29, 1957; RC090-13-24)Classical Physics” and Part II as “An Abstract Concept of Entropy & its Use in Inductive Logic” (Jan. 22,1954 Carnap to Kemeny; RC 083-18-13, pg. 2-3).147 Shimony to Carnap, July 9, 1955; RC 084-56-01 and Carnap to Howard Stein, August 29, 1957; RC 090-13-24.884.1. Historical BackgroundThe reception of his work on entropy was not as successful as he would have liked and so Carnap,when he wasn’t working on his autobiography and replies for his Schilpp volume, returned toworking on inductive logic after relocating to UCLA.148However, Carnap did not give up on the idea of applying his work on inductive logic tothe empirical sciences. But now instead of theoretical physics, Carnap became interested inapplying his work on inductive logic to rational and empirical decision theory. Indeed, 1955 wasa good year for the field of inductive logic as it saw the publication of results which showed howto define a notion of rationality in terms of the “consistency” or “coherency” of beliefs (relativeto a betting system) and when one’s degrees of belief must obey the probability axioms.149Basically, as Carnap understood the situation after 1955, it was Ramsey and de Finetti whohad shown, in Ramsey (1926) and de Finetti (1937), respectively, that a belief function iscoherent if it satisfies certain axioms of the probability calculus.150 More importantly, however,it was Kemeny who showed the reverse (although Kemeny and Carnap would later learn that deFinetti had shown this much earlier): if the belief function satisfies the probability axioms, thatfunction is coherent. Details aside, these results made up the theoretical backbone of subjectiveBayesianism and “probabilism,” or roughly the idea that rational degrees of beliefs should becashed out in terms of probabilities. And this was all going on simultaneously with the riseof Bayesian approaches to probability and statistical inference prompted, for example, by thepublication of Savage (1954). But this is a story best left for chapter 5.151The primary reason for providing this short history before discussing Carnap’s inductivelogic is to illustrate that Carnap’s work on inductive logic is not a single technical project;rather, it is a research program: based on the discovery of an adequate inductive logic, an148 While at UCLA, besides Haim Gaifman who was around in the late 1950s, Gordon Matthews and John L.Kuhns worked with Carnap as research assistants from 1955 to 1962 (Carnap to York, April 28, 1965; RC082-23-07). Most interesting is that Carnap worked with Matthews and Kuhns on a computer program tocalculate different confirmation functions, for different values of , and the printed outputs of this programcan be found at Carnap’s archive in Pittsburgh.149 See Kemeny (1955), Shimony (1955) and Lehman (1955).150 de Finetti’s paper is translated into English in Kyburg and Smolker (1964).151 Understanding the relationship between “necessitarians” like Carnap and “subjectivists” like Savage or deFinetti is a complicated question, not only from a contemporary point of view, but for Carnap and hispeers as well (e.g. Carnap tries to suggest that there really isn’t much difference between the views of deFinetti and himself in a letter to de Finetti, July 30, 1963, RC 084-16-01; also see de Finetti to Carnap,October 27, 1961; RC 084-16-02). For example, in 1952, Savage even points out to Carnap the similaritybetween Carnap’s -system and work by Bruno de Finetti on “equivalent” (i.e., what de Finetti later calls“exchangeable”) events (L. J. Savage to Carnap; Feb. 24, 1952; RC 084-52-25). For a more precise treatmentof these similarities, see Zabell (2005) and Good (1965).894.2. Carnap’s Confirmation Function centire conceptual edifice is to be constructed, an edifice conceptually tied to the foundations ofinformation theory, statistics and physics.152 I argue that such a research program should beproperly seen as a kind of conceptual engineering: it is the construction of an inter-connectedand hierarchical logical system which can be redesigned and interpreted to be of use to scientists,especially a new breed of social scientists studying decision-making under uncertainty. Beforewe can begin to understand the inner structure of this program – a journey that will take usthe entirety of the rest of the dissertation – we have to start, so to speak, from the ground up:with semantic measure and confirmation functions.4.2 Carnap’s Confirmation Function cAlthough Carnap delayed a more detailed discussion about c until the second, never completed,volume to LFP he explains the definition of this function in both the appendix to LFP andin Carnap (1945a). It is there that Carnap claims that c is an especially good candidate toexplicate the logical concept of probability. Nevertheless, Carnap is also quite clear that c, evenif it is an adequate explicatum, may not be the only such explicatum (LFP 563). The reasonwhy Carnap would think it an adequate confirmation function is that it has an interesting logicalproperty: the definition for this function characterizes a single function – for all sentences h, ein the logical system on which the inductive logic is based c(hO e) always has a unique and well-defined quantitative probability value.153 Conversely, most of the time when Carnap defines ameasure or confirmation function the definition picks out a class of functions, a class which canthen be made smaller by imposing more strict restrictions and requirements on the definition ofa measure or confirmation function. The task of an adequate inductive logic, then, is to figureout what kind of requirements and constraints to impose on measure and confirmation functionsso that we end up characterizing a single function.We next discuss how Carnap constructs c using the semantic resources of a logical system.First we have to discuss the logical system itself. When it comes to a quantitative inductive logic,Carnap defines confirmation and measure functions as semantic functions (or really, “functors”)152 For a very different vision of Carnap’s research program, but as a research program nonetheless, see Lakatos(1968).153 Provided that s is not logically false in order to avoid division by zero.904.2. Carnap’s Confirmation Function cdefined in the metalanguage (typically English plus a few Fraktur symbols) which gives numericalvalues to pairs of sentences in the object language (54).154 Carnap defines the object language,L, as including the following b + 1 many logical systems (see LFP 55-60):• The infinite system L1; viz. a first-order logic with identity and individual variableswhich contains both (i) an infinite sequence of individual constants, ‘u1’, ‘u2’, ..., and (ii)a finite number of primitive predicates of any degree (represented by capital letters, ‘d1’)designating properties.155• For all positive integers b , the finite systems LN ; viz. those logical systems with the samefinite predicates as L1 but only containing the first b individual constants from L1.Crucially, L by itself is just a logical calculus: the named individual constants and the finitenumber of predicates and relations are so far left uninterpreted. For Carnap, however, aninductive logic is built, so to speak, on the back of a semantic interpretation for this calculus.Although the technical details can be found in Carnap (1939; 1942), the basic idea is that, inthe metalanguage, a recursive definition of ‘true in L’ is defined over the primitive terms ofL (see LFP §17).156 However, Carnap does not demand that we provide a complete semanticinterpretation for L at the outset; instead, only what he calls a “skeleton” of an interpretationis to be given initially (59).A complete inductive logic can then be constructed by filling in the details of this inter-pretation. For example, Carnap first assumes that whichever interpretation of L we adopt, itmust satisfy what he calls the requirements of independence and completeness (which we willdiscuss below). Second, technically speaking, the definitions of the measure and confirmationfunctions, just like the semantic rules of truth, are to be explicitly defined in the semantics ofL.157 Lastly, the interpretation will specify what the individual constants and properties of L154 See chapter VII of LFP for Carnap’s work on a comparative inductive logic.155 For the quantifiers, I use the symbols, ‘∀‘, ‘∃’, and for the connectives, ‘¬’, ‘∧’, ‘∨’, ‘→‘. Note that I use thesymbols ‘¬’ and ‘∧’ instead of ‘∼’ and ‘’.156 As an example of an interpretation of primitive axiomatic terms, Carnap mentions, for example, Reichen-bach’s Zuordnungsdefinitionen (LFP 16). Moreover, there can of course be several interpretations for thesame axiomatic system: for Peano’s axioms, for example, Carnap remarks that “[t]here is an infinite numberof true interpretations for this system, that is, of sets of entities fulfilling the axioms, or, as one usually says,of models for the system” (LFP 17). Also see Nagel, 1939, 38-43.157 Interestingly, Carnap points out that we could follow Keynes and Jeffreys in (implicitly) representing prob-ability functions in terms of an operator in an intensional modal logic (LFP 280-1).914.2. Carnap’s Confirmation Function cwill represent, e.g, space-time points and physical properties of objects, or perhaps organisms ina population and the individual fitness values for these individuals. Whatever the case, however,it is important to clarify that Carnap distinguishes practical questions about the constructionof L and a semantics for L from methodological questions about which interpretation will bemost useful for empirical investigations (see LFP §44).158Before we can discuss these requirements of independence and completeness that the inter-pretation of L must satisfy, we need to introduce a bit of Carnap’s technical terminology; viz.the semantic concept of a state-description, a concept meant to explicate the notion of “possiblecases or states-of-affairs” (LFP 71). Roughly speaking, the atomic sentences of L belong tothe smallest set of sentences formed, for every predicate dn in L of degree n, by applying dnto any of the n many individual constants in L. A state description of L, then, is simply asentence formed by the conjunction of all atomic sentences such that each conjunct may or maynot be prefixed with a negation sign (71-2; see D18-1).159 The set of all state descriptions in L,according to Carnap, then describes all the possible cases the “universe” could be in; relative,of course, to those atomic sentences in L representing the “basic events” of that “universe.” Therequirements of independence and completeness ensure that this is the case. The requirementof independence concerns the interpretation of the non-logical signs of L: simply speaking, thisrequirement states that all atomic sentences are pair-wise logically independent (72).160 Thesecond requirement, the requirement of completeness, states that primitive predicates of L are“sufficient for expressing every qualitative attribute of the individuals in the universe of L, thatis, every respect in which two positions in this universe may be found by observation to differqualitatively” (74); or as Carnap alternatively expresses this requirement:if a system L is given and a universe, real or imaginary, is to be chosen as an illustrationor model for L for the purposes of inductive logic, then this universe must be neither richer158 For example, Carnap has an extended conversation about the construction of a deductive system L′ whoseconstants are interpreted as temporal series of events. However, although such a system is far less idealizedthan the logic Carnap actually constructs, the drawback of such a language is that it is too complex to beof much use (at least when we are forced to work out the computational details for such a logic by hand)(LFP 62-5).159 Although Carnap, at least in the 1950s, cashes out “possibility”-talk in terms of the sentences in a logicallanguage he reports that one could instead talk about propositions “provided it is done in a cautious way,that is to say, in a way which carefully abstains from any reification or hypostatization of propositions [...]”(LFP 71).160 More specifically, it includes two clauses: first, that the individual constants in L “designate different andseparate individuals” and, second, that the primitive predicate of L likewise represent pair-wise independentrelations and properties (LFP 73).924.2. Carnap’s Confirmation Function cnor poorer in qualitative attributes than L indicates. (LFP 75)These two requirements are not metaphysical assumptions. They are instead empirically-informed, methodological, restrictions the logician places on any adequate interpretation suitablefor L.161With these restrictions on the interpretation of L in place, we can now see how Carnapdefines the semantic concepts of measure and confirmation functions. However, we first needto introduce another important semantic concept: the concept of the range of a sentence. Forany sentence i in L, the range of i, call it R(i), is the class of state descriptions such thati “holds in” those state descriptions.162 It is with this semantic concept of the range of asentence that Carnap, for example, defines the semantic L-concepts central to deductive logic,viz. the semantic concepts of L-truth and L-entailment which are understood by Carnap to beexplications of analytical or logical truth and logical entailment, respectively (83).163 Likewise,the semantic concept of the range of a sentence in L is used to define the semantic concept ofa measure function. Specifically, Carnap defines m as a regular measure function over all thestate descriptions in LN as those functions that satisfy the following two conditions: (i) for anystate description ki in LN , the value of m(ki) is a positive real number and (ii) if  indexes allstate descriptions in LN , the sum of all m(k) is equal to one (D55-1; LFP 295).164 Carnapthen extends this definition to define regular measure functions defined over the sentences inLN using the following two definitions: (iii) for any logically false sentence h in LN , m(h) is bydefinition equal to zero and for any non-logically false sentence h in LN , the value m(h) is bydefinition equal to the quantity∑ m(k), where  indexes the state descriptions in the rangeof h, R(h) (D55-2; LFP 295). Finally, for any functions m and c defined over the sentences andpairs of sentences of LN , respectively, we say that c is based on m if the following holds: for any161 Carnap will relax these restrictions later in the early 1950s; see Carnap (1951; 1952) and Kemeny (1953;1956a;b). Carnap admits he always felt uneasy about the completeness requirement in Carnap (1963b).162 The basic idea is that the range of the sentence w are all those state descriptions consistent with the truth ofw; “holds in” is defined recursively but we should not read “hold in” as being synonymous with true in someset-theoretic model; see D18-4 in LFP, 78-9, for the details.163 See §19-20 of LFP for the details; roughly speaking, the sentence w in L is L-true if it holds in all state-descriptions (83). Likewise, for the sentences w, x in L, w L-implies x just in case R(w) ⊆ R(x) (83).164 Note that, for the sake of readability and continuity between the notations in LFP and CIM, I will notalways acknowledge explicitly the distinction between the metalanguage and L, e.g., instead of y Carnapuses the Fraktur symbol S to designate a definition in the metalanguage. The interested reader is invitedto consult LFP for the technical details.934.2. Carnap’s Confirmation Function csentences h and e in LN , if m(e) = 0 then c(hO e) is not defined and otherwise, (D55-3, 295)c(hO e) =m(e ∧ h)m(e)N (4.1)Finally, c is a regular confirmation function if it is based on a regular measure function (D55-4).In order to finish our construction of c we next need to introduce two more semantic notions:symmetrical measure functions and structure-descriptions. Carnap motivates the definition of asymmetrical measure function with an analogy to deductive logic: “we require that logic,” saysCarnap, “should not discriminate between the individuals but treat them all on a par; althoughwe know that individuals are not alike, they ought to be given equal rights before the tribunalof logic” (485). He captures this idea of the “non-discrimination” of individuals in terms of theconcept of isomorphic state descriptions.165 A symmetrical measure function m is defined as aregular measure function which assigns the same value to isomorphic state descriptions.166 Asymmetrical confirmation function is then simply defined as a function based on a symmetricalmeasure function. Next, paraphrasing Carnap’s technical definition, the structure-descriptioncorresponding to a state description ki in LN is the disjunction of all state descriptions isomor-phic to ki (116). Then a structure description K in LN is simply defined such that there is astate description ki in LN for which K is the structure-description corresponding to ki (116).The measure function m is then defined as that function fulfilling both the condition thatm is symmetrical and the condition that m gives the same numerical value to all structuredescriptions in L (LFP 563). The function c is then that confirmation function based on m.What is so nice about c is that the unique numerical values of c(hO e) for any sentences h,e (where e is not logically false) can be directly calculated using a number of logical theorems.Nevertheless, it took a lot of work to get here. Not only did we have to make choices aboutthe logical syntax of L, we also had to make assumptions about the interpretation of L andthen place further restrictions on our definition of measure functions, viz. that it is regular,symmetric and assigns the same value to structure descriptions, before we could define anadequate confirmation function. These are all practical choices: we could have chosen to use165 Simply put, two state descriptions yi and yj are isomorphic if there is a one-to-one relation R whose domainand image is the set of all state-descriptions in LN and yi equals yRj , where R is applied to all the individualconstants in the sentence yj ; for a more explicit definition see (LFP, §26) and (Carnap 1945a, 79-80).166 See §90, especially D90-1,2.944.2. Carnap’s Confirmation Function cdifferent definitions for measure and confirmation functions (i.e., by using different semanticresources of L) or by placing different restrictions on the interpretation of L. Moreover, thesechoices are not merely choices about “linguistic frameworks,” e.g., between whether to choosethe logical syntax and semantics of a system like L against the possibility of dealing with,perhaps, higher-order logical systems. Instead, the inductive logician has to make very specificdecisions about how measure and confirmation should be assigned their quantitative values, likewhether a restricted principle of indifference should be adopted to assign probability values tothe sentences of L.Moreover, the making of such decisions is in no way epistemological or metaphysical. Asfar as Carnap is concerned, he is just constructing a logical system L and suggesting possiblesemantic interpretations for this system. It is a purely logical activity. Now, according toCarnap, questions about which interpretations are useful or how to apply such a system totackle some empirical problem using inductive reasoning, are, indeed, methodological questions.It will be useful at this point to distinguish, to use Carnap’s terminology, between two “problems”for inductive logic, i.e., “pure” and “applied” inductive logic. The relationship between pureand applied inductive logic, Carnap points out, “is somewhat similar to that between pure(mathematical) and empirical (physical) geometry” (1971b, 69). For the case of mathematicalgeometry, according to Carnap, “we speak abstractly about certain numerical magnitudes ofgeometrical entities” and then prove theorems about those entities (69). However, accordingto Carnap, no “procedure of measuring these magnitudes” is provided; instead, such questionsbelong to physical geometry the task of which is “to lay down rules for various procedure ofmeasuring length, rules based partly on experience and partly on conventions” (69).Likewise for pure and applied inductive logic. In pure inductive logic, all we do is providea logical system without an interpretation of the non-logical constants with rules for definingmeasure and confirmation functions. Applied inductive logic, on the other hand, is concernedwith providing an interpretation of this logical system, i.e., we provide rules for interpreting theindividual constants and primitive predicates of the logical system as, for example, a systemof gas particles for which relations like density can be defined over collections of individual gasparticles in this system. Moreover, we may also wish to interpret the measure and confirmationfunctions themselves. Indeed, in the next chapter, we will discuss in detail how Carnap gave954.2. Carnap’s Confirmation Function cwhat he sometimes calls a “quasi-psychological” interpretation to measure and confirmationfunctions as credibility and credence functions (Carnap, 1962a, 303; 1971b). As Carnap laterputs the point,In applied IL, the theorems [from pure inductive logic, or IL - CFF] are used for practicalpurposes, e.g., for the determination of the credibility of a hypothesis under consideration ina given knowledge situation, or for the choice of a rational decision. Justifying an inductivemethod and, or specifically, offering reasons for the acceptance of a proposed axiom, is akind of reasoning that lies outside pure IL and takes into consideration the application ofC -functions. What is relevant in this context is not merely the consideration of actualsituations, but rather that of all possible situations. (1971b, 105).Notice here the distinction between providing reasons or justifying an applied but not a pureinductive logic. Indeed, Carnap suggests the case is similar to deductive logic.167 The analogyis this: on the one hand, there are the problems involving inductive logic and “methodologicalproblems and, more specifically, problems of the methodology of induction” and on the otherhand, there is a pair of problems involving deductive logic qua a field of pure mathematics andthe activity of carrying out the “procedures of deductive logic and mathematics” (LFP 202-3). More explicitly, methodological rules include not only useful rules of thumb or hints forusing an interpreted logic, including theories of approximation and the like (203), but also ruleslaying down requirements for an adequate interpretation. The requirements of completeness andindependence, for example, are such methodological rules (73). This also includes rules detailinghow inductive logic may be used; for example, Carnap’s principle of total evidence, i.e., that“the total evidence must be taken as a basis for determining the degree of confirmation,” is itselfa methodological rule (see §45B, especially p. 211).Moreover, in light of this distinction between the problems of a pure and applied inductivelogic, Carnap is well aware of the fact that what he calls the “application” of logic, includinginductive logic, to the activities of scientists “involves a certain simplification and schematizationof inductive procedures” (209). More specifically, Carnap says that the application of inductivelogic involves what he calls an “abstraction”; namely, that we abstract away from the actualvague or inexact concepts found in scientific practice and instead assume that “we deal only withclear-cut entities without vagueness” (209). Carnap’s language here is reminiscent of HerbertSimon’s distinction between “ill-structured” and “well-structured” problems we encountered in167 Carnap employs a similar analogy between logical syntax and geometry in section 25 of LSL.964.2. Carnap’s Confirmation Function cthe last chapter. Just as “well-structured” problems frequently must impose some kind of extrastructure on the original problem, Carnap is also cognizant that such abstraction comes at aprice:In any construction of a system of logic or, in other words, of a language system with exactrules, something is sacrificed, is not grasped, because of the abstraction or schematizationinvolved. (LFP 210 ).But Carnap is not arguing that there is some quantity of the physical world that cannot befaithfully captured by logical abstraction; “it is not true,” continues Carnap,that there is anything that cannot be grasped by a language system and hence escapes logic.For any single fact in the world, a language system can be constructed which is capable ofrepresenting that fact while others are not covered. (LFP 210)Instead, the main restriction on the method of logical abstraction, according to Carnap, is thatno single logical system can ever be expected to capture faithfully all facets of the world.168However, logical abstraction can only get us so far; after all, the point of logic, Carnap tells us,is not merely to clearly express facts but rather to help inform practical decisions:169The final aim of the whole enterprise of logic as of any other cognitive endeavor is to supplymethods for guiding our decisions in practical situations. (LFP 217)Nevertheless, as the theory of inductive logic itself is in its infancy, Carnap argues we muststart with an inductive logic based on simple languages, languages which can provide the basicscaffolding for later generations of logicians and philosophers to construct more complicatedand realistic inductive logics that can then be more fruitfully applied to actual scientific prob-lems (213-5). That inductive logic can be so schematized illustrates the conceptual impor-tance of inductive logic as an instrument for informing practical decisions: whether it be for afarmer, insurance agent, engineer or physicist “[t]he decisive point,” says Carnap, “is that justfor these practical applications the method which uses abstract schemata is the most efficientone” (218).170Next we will turn to an example of Carnap uses his work on inductive logic to try and makeprogress in science by using his work on confirmation functions to lay a single foundation for atheory of estimation in theoretical statistics.168 See, for example, Carnap discussion of using quadrangles to cover a circular area (LFP 210).169 Carnap, however, adds in parentheses that “This does, of course, not mean that this final aim is also themotive in every activity in logic or science” (LFP 217).170 Also see Carnap’s discussion of a trade-off between “extroverts”, or those that prefer the complexity ofnature, and “introverts”, or those that prefer the abstraction of schemata; in particular, Carnap says “it isclear that science can progress only by the cooperation of both types, by the combination of both directionsin the working method” (218-9).974.3. From Confirmation to Estimation Functions4.3 From Confirmation to Estimation FunctionsFor Carnap, the importance of constructing an inductive logic on the basis of an adequateconcept of degree of confirmation, e.g., a function like c, is not merely to explicate the logi-cal concept of probability. We may also be interested in explicating other inductive concepts,including concepts of relevance, estimation, information and even of entropy which are concep-tually tied to the logical concept of probability.171 Specifically, once an adequate explicatumfor the logical concept of probability is found, this explicatum can then be used to constructadequate explicata for a host of related inductive explicanda. As Carnap puts it, the conceptof degree of confirmation, understood as an explicatum for the logical concept of probability,is “the fundamental concept of inductive logic” (513). It is in this wider sense of explication –of explicating an entire system of concepts based on the explication of a single concept at theconceptual core of this system – that Carnap’s work on finding an adequate quantitative induc-tive logic is an explication of inductive reasoning. Yet finding an adequate explication of logicalprobability which could then be used to explicate an entire system of inductive concepts is nota trivial task; as Carnap puts it, we can only find such a concept by providing the right sort ofreasons for adopting it, e.g., reasons like “the fact that in many actual or imagined knowledgesituations the values of c are sufficiently in agreement with the inductive thinking of a carefulscientist” (540).Turning our attention to the problem of estimation in theoretical statistics, Carnap says thatthe state of the field of theories of estimation, at least from the point of view from “treatises onprobability and statistics” isa startling spectacle of unsolved controversies and mutual misunderstandings, all the moredisturbing when we compare it with the exactness, clarity and possibility of coming to ageneral agreement in other fields of mathematics. (LFP 513)The problem of estimation is basically the problem of finding an adequate estimation, basedon both an estimation function and past observations, of the value of some unknown physicalquantity, or rather, an estimate of some parameter representing a physical quantity.172 AsCarnap puts the point, one can think of an estimate given by an estimation function for aphysical quantity as a sort of guess – not an arbitrary guess but rather a reasonable guess (512).171 I don’t discuss Carnap’s work on relevance in any detail in this dissertation; see LFP, chapter VI.172 For example, see Fisher (1922).984.3. From Confirmation to Estimation FunctionsOnce found, such a concept will not only play an important role in everyday scientific activitybut also a foundational role in any theory of rational decision making (LFP §§49-51; also seechapter 5 of this dissertation). But the problem, at least according to Carnap in 1950, is thatthere is no general theory of statistical estimation. Rather, as Carnap notes, there are insteadseveral, competing, theoretical accounts of statistical inference and estimation, including R. A.Fisher’s work on maximal likelihoods, Abraham Wald’s work on statistical decision functions,Jerzy Neyman and Egon Pearson’s statistical hypothesis testing relative to type I and II errorsand Neyman’s confidence intervals (515-518). Moreover, these statistical accounts of estimationfunctions are all based on a frequentist or statistical concept of probability. But then “[w]hydid statisticians,” asks Carnap, “spend so much effort in developing methods of estimation,i.e., methods not based on a [logical concept of probability - CFF]?” (518). The short answer,according to Carnap, is that because of the historical association of a principle of indifference(or principle of insufficient reason) with the logical concept of probability – a principle found tolead to contradictions by scientists as early as Carl Gauss – only a theory of estimations basedon a frequentist concept of probability could possibly be adequate (518).In response, Carnap articulates two possible options. The first is to suppose that no adequatequantitative inductive logic will be found; thenthe methods developed by Fisher, Neyman, Pearson, and Wald or new methods of a similarnature are presumably the best instruments for estimating parameter values and testinghypotheses. They are ingenious devices for achieving these ends without making use of anygeneral explicatum for [the logical concept of probability - CFF], as far as the ends can beachieved under this restricting condition. (518)Alternatively, however, suppose that an adequate inductive logic is found, viz. an inductivelogic which does not depend on any unrestricted application of the principle of indifference.Then, according to Carnap,the main reason for developing independent methods of estimation and testing would vanish.Then it would seem more natural to take the degree of confirmation as the basic conceptfor all of inductive statistics. (518)This question of which of these two alternatives is more likely is connected to a problem Carnaphad raised a few pages earlier in LFP. The unsatisfactory state of the theory of estimation is dueto a problem that besets most theoretical fields in science: “any procedure of estimation dependsupon a choice, which is a matter of practical decision and not uniquely determined by purely994.3. From Confirmation to Estimation Functionstheoretical, logico-mathematical considerations” (514). As Carnap points out, many proceduresof science involve such a choice, like choosing a geometry for physical space. However, whatis advantageous about the question of whether we can find an adequate inductive logic thatcould be used as a basis for a theory of estimation is that, says Carnap, “only one fundamentaldecision is required” (514). As Carnap continues to say:As soon as anybody makes this decision, that is to say, chooses a concept of degree ofconfirmation which seems to him adequate, then he is in the possession of a general methodof which makes it possible to deal with all the various problems of inductive logic in acoherent and systematic way, including the problems of estimation. Thus this method helpsto overcome what seems to me the greatest weakness in the contemporary statistical theoryof estimation, namely, the lack of a general method. (514)This is an insightful passage into Carnap’s understanding of the theoretical issues at hand. Byreconstructing the results of theoretical statistics and probability as depending on the choice ofa single inductive concept of degree of confirmation, Carnap suggests that one could provide agrand foundation for all of statistics and probability – a general method capable of clarifyingand systematizing inductive reasoning, including reasoning about how to construct non-arbitrary“guesses” or estimations for physical, but unknown, quantitative properties. It is in this waythat Carnap hopes to contribute to the foundations of theoretical statistics.Now that we have a better sense of the potential import for Carnap’s work on estimationfunctions, I next turn to the details of his work on estimation functions. Suppose, firstly, thatR(u) is a discrete random variable representing the result of observing some physical magnitude,relative to the physical input u, which ranges over the possible values r1O r2O NNNO rn and,secondly, that one of the ri is really the actual value of this physical magnitude. Provided wehave evidence for previous instances of R(u), call it e, and that the sentences h1, ..., hn denotethe (logically exclusive) hypotheses that the actual value of the unknown quantity is r1, ...,rn, respectively, then Carnap suggests we can define the estimate of R(u) as a weighted mean(where the weights are confirmation values). More specifically, assuming ‘e’ logically implies‘h1 ∨ · · · ∨ hn’, the estimate e is defined as follows, (see D100-1)e(RO uO e) =n∑i=1[ri × c(hiO e)]N (4.2)Importantly, as Carnap will later show in Carnap (1952), this definition can be used to define1004.3. From Confirmation to Estimation Functionsunique estimation functions based on a particular class of confirmation functions. Specifically,supposing we had a continuum of different confirmation functions to choose from and that wecould define a unique estimation function based on each such confirmation function, we couldthen investigate how well particular estimation functions behave for different “states of theuniverse,” or to use a more formal mode of speech, for different state descriptions. Of course,then Carnap would need to have some notion of how “reliable” different estimation functionsare. Although Carnap considers several different ways of explicating such a notion, I will cutto the chase and quickly discuss the explicatum Carnap focuses on (see LFP §100B and §102).Assuming that rˆ is the actual but unknown value of the physical quantity measured by R(u),the error of the estimate e, or v, is defined asv(RO uO e) = e(RO uO e)− rˆN (4.3)As is standard (because the estimation of this error term is always zero), Carnap takes for theexplicatum of the reliability of estimation functions the estimate of the squared error, f2, i.e.,the weighted average of these error functions, squared, where the weights are given, like above,in terms of confirmation functions.173 Importantly, the estimation of squared error is useful ifthe actual value of R(u) is genuinely unknown. However, one can easily calculate a value ofrˆ relative to some fixed state description. For example, suppose we assume that a single statedescription in L is the actual one, then, if R(u) is a measure of the frequency of individuals inthat state description which hold of a , rˆ is simply the actual frequency of a ’s in this statedescription. Then instead of explicating the reliability of an estimation function in terms of f2,we can instead use the mean squared error, m2, defined relative to rˆ to investigate the relativereliability of estimation functions relative to a fixed, completely known, state description.174In section 4.5 of this chapter, we will see that Carnap uses this notion of the mean squarederror and his -system to try and find “optimal” estimation functions. This work constitutes, Iargue, one of the clearest examples of how Carnap uses his work on inductive logic to solve afoundational problem and that this process resembles a kind of conceptual engineering activity.173 Specifically, f2(R; u; s) =Df e(v2; R; u; s) =∑i[(e(R; u; s)− ri)2 × c(hi; s)].174 Relative to our current observed sample of s-many individuals, the mean squared error of e is defended asm2(e; rˆ) = v2 (Carnap, 1952, 56-59).1014.4. Carnap’s Continuum of Inductive MethodsHowever, before we can discuss that example we first need to examine the -system in detail.4.4 Carnap’s Continuum of Inductive MethodsCarnap tells us in the opening pages of CIM that he is concerned with two kinds of inductiveinference in the sciences. The first are inductive judgments whether to “accept or reject” ahypothesis based on prior and/or new evidence (CIM 3).175 More specifically, according toCarnap, an individual l “possesses” a method of confirmation if they can determine – even if“not necessarily by explicitly formulated rules” – some confirmation function c(hO e) such thatthe values of this function “represent” to l their degree of confirmation for the hypothesis hgiven the evidence e (4). The second kind of inductive inference is just what we have beendiscussing above: namely, the estimation of the unknown value of some physical quantity (3-4).More specifically, an agent l “possesses” a method of estimation if they have some procedurefor determining the values of the mathematical function e(rfOaOKO e) such that those values“[represent] to l the estimate of the rf of a in K on the basis of e” (4).176 Here rf denotesthe relative frequency of some magnitude defined relative to a , the property of interest, andthe class K the elements of which l has not observed and “is not described in e” (4).Ideally for Carnap, of all the possible c and e functions which practicing scientists couldpossibly “possess,” we would find wide-spread agreement concerning their preferred inductivemethods. Indeed, based on such a consensus, we would have more than enough reason to singleout a unique confirmation function to serve as an adequate explicatum to construct an adequateinductive logic fit to guide most, if not all, rational decision making. This truly would be a kind ofrobot epistemology. Moreover, supposing we could pinpoint the inductive disagreements betweenscientists, Carnap suggests that this situation would be similar to a controversy surrounding thenature of deductive inference between intuitionist and classical mathematicians, i.e., a debate“based on different interpretations of the basic logical terms rather than as genuine differencesin opinion, i.e., incompatible answers to the same question” (CIM 6).177 Carnap, however,175 The language of acceptance/rejection is one Carnap drops in his later work, especially as a response tophilosophers of science like Henry Kyburg Jr. who treat such notions as a part of epistemology and in needof explanation by articulating normative rules of detachment; for example, see Kyburg’s article in Swain(1970), Carnap (1968b) and, for a general overview, Hilpinen (1968).176 Generalizing from the example based on the random variable R(u) above.177 Of course, the latter controversy is at the center of LSL.1024.4. Carnap’s Continuum of Inductive Methodsis not so sanguine that the inductive differences between practicing scientists can be so easilyexplained away. “If we look at the contemporary situation in the field of logic, the theory ofinductive inferences,” says Carnap,178we notice the remarkable fact that a variety of mutually incompatible inductive methodsare proposed and discussed by authors of theoretical treatises and applied in practical workby scientists and statisticians. None of the authors is able to convince the others that theirmethods are invalid. I shall not try to decide the difficult question whether the situationin inductive logic is in this respect fundamentally different from that in deductive logic,including mathematics. [...] Whatever the solution of this philosophical problem may be, itseems to me that there can hardly be any doubt about the historical fact that, as mattersstand today, the differences of opinion concerning the validity of inductive methods go muchdeeper and are much more extensive in their scope than the differences in deductive logic.(CIM 5-6; my emphasis)That different scientists prefer different, incompatible inductive methods – methods sometimescentral to their understanding of scientific method and inference – is, for Carnap, a foundationalproblem in the sciences which is in need of philosophical attention. In deductive logic andmathematics, it seems only a minority of mathematicians reject the classical notion of logicalimplication (and consequence) in favor of intuitionist and other non-classical logics. Indeed, inLSL, Carnap constructs two different logical systems, one classical and the other intuitionist, inorder to evaluate and compare the logical consequences of each system; however, he takes forgranted the full power of classical mathematics to do so.179 That’s because the aim wasn’t toconvert those logicians who rejected the principle of the excluded middle; rather, the point wasto illustrate how a plurality of logical systems could be constructed.With inductive logic, it seems we have the opposite problem. There already is a plurality ofinductive methods, but it seems like there is little or no consensus regarding which particularmethods are more or less satisfactory than the others. Troubling for Carnap, however, is the ideathat this problem about the non-consensus of inductive methods goes deeper than the worry fordeductive reasoning for which once a single notion of, say, logical consequence is adopted thenalternative ways of spelling out the notion of logical consequence becomes “meaningless” (6).Instead, for the case of inductive reasoning, it seems that two scientists worried about the samehypothesis h and evidential basis e can both adopt their own inductive methods, methods which178 Note that “incompatible” inductive methods is a technical term for Carnap. The functions c1 and c2 areincompatible if, all relative to the same object language L, there exists one pair h-s such that c1(h; s) ̸=c2(h; s) (CIM, 5). Incompatible estimation functions are defined similarly.179 For more details about this latter claim see, for example, Friedman (2009).1034.4. Carnap’s Continuum of Inductive Methodsboth parties consider to be perfectly reasonable, but nevertheless end up recommending entirelydifferent confirmation values for h given e using their methods (6-7). The worry is that there isan inherent indeterminacy or subjectivity to the very nature of inductive decision making.But what is the source of this subjectivity for inductive reasoning or judgments? On the onehand, Carnap suggests that perhaps these inductive differences are “merely a matter of historicalcontingency due to the present lack of knowledge in the field of inductive logic” (7). Indeed, ifthis were the case “it would be conceivable,” says Carnap, “that at some further time, on thebasis of deeper insight, all will agree that a certain inductive method is the only valid one” (7).The initial stumbling block of there being scientists who find it reasonable to prefer competinginductive methods will eventually be overcome once we discover an inductive method which allscientists could simultaneously endorse (e.g. like the inductive method corresponding to c).Carnap presumably has in mind here scientists like Keynes and Jeffreys who argue that, just asdeductive logic is to be based on general epistemological principles or postulates, an objective,or rational, probabilistic relation p(eQh) relative only to the meaning of two propositions e,h is similarly based on general epistemological principles or postulates. However, unlike forclassical logic for which propositions or sentences are assumed to be truth-functional, Keynesargues – especially in cases from the use of probabilistic reasoning in law and gambling – thatnot all probabilities have sharp, quantitative values, while Jeffreys appeals to controversialsymmetry principles to guarantee that probabilities do, as a matter of epistemic stipulation,have sharp, well-defined values.180 Alternatively, one can embrace the subjective nature ofinductive reasoning, for example, by assuming that this rational probability relation p(eQh)is also a function of some mind or agent; e.g., as I. J. Good puts it, “you.”181 Specifically,for probabilists like Ramsey, de Finetti or Good, terms like “degrees of belief,” “belief,” or“judgment” are treated as primitive notions relative to a subject or agent; probabilities are thento be measured or elicited relative to a system of bets, i.e., a system explicitly defined relative to“beliefs” or “judgments” underlying the actions, preferences or expectations of a subject or agent.Ideally, as a product of the subjective nature of scientific judgment, objective, inter-subjective180 Nagel (1939) also argues against the idea that degrees of confirmation or belief need always be quantitative.181 According to Good, beliefs are a function of three variables: the propositions denoting what is “believed”and “assumed” and, thirdly, “the general state of mind [...] of the person who is doing the believing” (1950,2). This person, says Good, is who “you” describes.1044.4. Carnap’s Continuum of Inductive Methodsrelationships are then to be shown to hold for certain kinds of subjective probabilities (even ifthe subjective element of probabilistic judgments is never entirely eliminated).182On the other hand, even though Carnap also considers himself to be constructing an objectiveconcept of probability and, later, suggests confirmation values can be fruitfully interpreted withrespect to a system of bets (see my next chapter), Carnap cannot simply ground inductivelogic, as a piece of logic, with general epistemological principles or the empirical facts aboutthe subjective judgments of agents. For to do so, presumably, would be to violate one of thecentral strictures of Wissenschaftslogik, viz. that a sharp line must be drawn between logicaland empirical questions, a line which epistemological theories frequently blur. But this is whyit is important to clearly distinguish between the explicandum and an explicatum: whetheror not inductive methods are somehow inherently subjective or piece-meal is a thesis aboutinductive reasoning qua explicandum and not as an explicatum. Thus when Carnap considersthe possibility of whether “the multiplicity of mutually incompatible methods is an essentialcharacteristic of inductive logic” and says that, if so, “it would be meaningless to talk of “theone valid method”,” Carnap is talking not about inductive logic as a piece of logic but ratherabout the inductive practices of scientists (CIM 7).183Moreover, it is in virtue of this incommensurablity between inductive methods that Carnapthen suggests that the decision to adopt an inductive method over others is a practical and nota theoretical matter. More specifically, Carnap says this rejection of any talk about “the onevalid method”184[...] does not necessarily imply that the choice of an inductive method is merely a matter ofwhim. It may still be possible to judge inductive methods as being more or less adequate.However, questions of this kind would then not be purely theoretical but rather of a prag-matic nature. A method would here be judged as a more or less suitable instrument for a182 Deriving some kind of objective, or rational, results for subjective probabilities is the entire point of so-called“Dutch book” and representation theorems more generally. Whether such results are “normative” in anystrong sense, however, remains a controversial question (see Meacham and Weisberg, 2011).183 Indeed, even today prospects for a truly general theory of inductive inference are dim. Although Carnapwas aware of the similarities between finding effective or computable solutions to inductive and deductiveproblems, Putnam (1963) and, more recently, especially Kelly (1996) have done a great service by clarifyingthese similarities. For recent attempts that study the limitations of using probability theory to captureinductive problems, see Earman (1992); Norton (2003; 2010). Alternatively, statistical and machine learningtheory provides a slew of fruitful, technical frameworks for investigating the nature of inductive in a morepiece-meal fashion, e.g., see Bishop (2006); Hastie et al. (2010); Ortner and Leitgeb (2009); Vapnik (2000).184 Carnap continues: “ I shall not try to discuss this problem here, still less try to solve it; but I may indicatethat at the present time I am more inclined to think in the direction of the second answer” (CIM 7). The“second” answer is in reference to this idea that there is no one valid inductive method – I invite the readerto read this as an implicit nod toward satsificing.1054.4. Carnap’s Continuum of Inductive Methodscertain purpose. (CIM 7)Although Carnap leaves open the possibility of perhaps discovering this “one valid inductivemethod,” he never tells us how we would know it if we stumbled across it and throughout therest of the text of CIM he discusses the decision to adopt inductive methods, characterized bythe task to construct a continuum of inductive methods, in instrumental terms.Carnap’s -systemPart I of CIM is concerned with the provision of “a systematic survey of all possible inductivemethods” in the form of a parameterization of confirmation functions which a scientist can useto help them make more informed decisions – it is a means to help explicate their inductivereasoning practices. Really, Carnap distinguishes between two separate tasks. The first task,on the one hand, is to provide an ordering of inductive methods with respect to a linguisticparameter of a logical system. The second task, on the other hand, is just the inverse of thefirst: if everything turns out the way it is supposed to then from any given value of this linguisticparameter it should be possible to uniquely determine an inductive method (7-8). It will turnout that the  parameter from Carnap’s -system, a restricted continuum of inductive methods,satisfies both of these tasks.185 Moreover, aside from the fact such a parameterization wouldthen allow us to use the standard techniques from calculus and analysis to further investigatethese inductive methods, such a system, says Carnap,would enable us not only to compare any two of the historically given methods in a moreexact way than was possible so far but also to study new methods quantitatively. It wouldbe easy to discover one or several new methods which fulfil any given condition or whichare most useful for a specific purpose. (CIM 8)Carnap restricts his investigation to LN , which is the same as LN above except with theadded restriction that the only predicates are -many one-place predicates.186 Carnap thenintroduces the following technical notions required to construct his -system. The Q-properties,as Carnap calls them, represent a collection of  many exclusive and exhaustive predicatesrepresenting, so to speak, all the possible ways these -many unary predicates can hold of the185 The question of how to generalize Carnap’s -system is by no means trivial; see Good (1965); Kuipers(1978); Zabell (2005).186  is here a positive natural number assumed to be finite and larger than one.1064.4. Carnap’s Continuum of Inductive Methodsb individuals in a given state description.187 Then if a is any molecular property, i.e., aproperty formed using any of the  many predicates using the usual connectives, all occurrencesofa in a sentence in LN can be replaced by a disjunction of particular Q-properties or negationsof Q-properties. Lastly, the number of the Q-properties in this disjunction required to replacea is called the logical width of a , which is denoted by w.Carnap then restricts his investigation not to all possible inductive methods, but rather tothose methods represented by regular confirmation functions, i.e., those c functions which satisfyconditions C1-5 in CIM.188 Nothing of importance will be lost if, from here on out, we discussCarnap’s -system in terms of a specific example instead of adopting Carnap’s own, sometimesobscure, technical vocabulary. Let us interpret the language system LN as describing an urnwith b many balls and  = 3 many independent color properties: ‘eV’, ‘eG’ and ‘eR’ for blue,green and red balls. If some sentence e is an evidential statement describing any sample of smany balls from the urn (where s P b), eQ is a conjunction of s many Q-properties applied tothe s many balls in our sample.189 For example, suppose that from a sample of six balls – callthem ‘b1’, ‘b2’, ..., ‘b6’ – that we have the following evidential statement,eQ = ‘eR(b1) ∧eG(b2) ∧eV(b3) ∧eV(b4) ∧eV(b5) ∧eV(b6)’NIn other words, in our sample – in sequential order – a red ball, a green and four blue ballswere pulled from the urn. So relative to this sample of size s = 6, we can calculate the relativefrequency for each Q-property: if si (i = 1O NNNO ) represents the number of balls in the samplethat are ei, then siQs is the relative frequency of ei-successes in the sample of size s. In ourexample, with a slight abuse of notation, we have the si terms sR = 1, sG = 1 and sV = 4, and187 More specifically, if there are -many Q-properties, ‘Q1’, ‘Q2’, ..., ‘Q’, where  = 2, are just thoseproperties formed by the conjunction of all  primitive predicates closed under negation. For example, ifthere are only two primitive predicates, P1 and P2 there will be  = 2=2 = 4 Q-properties: Q1 =‘P1 ∧P2’;Q2 =‘¬P1 ∧ P2’;Q3 =‘P1 ∧ ¬P2’; and Q4 = ‘¬P1 ∧ ¬P2’.188 See CIM, p. 42. For the definitions themselves, see page 12; a slightly abbreviated list is the following. C1: Ifh and h′ are logically equivalent, c(h; s) = c(h′; s). C2: If s and s′ are logically equivalent, c(h; s) = c(h; s′)HC3: c(h∧h′; s) = c(h; s)× c(h′; s∧h). C4: If s∧h∧h′ is logically false, then c(h∨h′; s) = c(h; s)+ c(h′; s).C5: 0 ≤ c(h; s) ≤ 1.189 It is merely an artifact of our toy example that we don’t employ molecular properties to describe our sampleso that s is the same as sQ.1074.4. Carnap’s Continuum of Inductive Methodsso the relative frequencies of the red, green and blue balls in our sample aresRs=16OsGs=16OsVs=46NAll this notation will come in handy in just a moment. However, we first need to introduce justa bit more technical terminology before we can talk about Carnap’s -system.Besides requiring that the confirmation functions in our system be regular, Carnap imposesfive more conditions on these functions, C6-10. Let hi (i = 1O N N N O ) designate the hypothesisthat the next ball we see from the urn is the ith color. For example, again with a slight abuseof notation, hV says the next ball sampled from the urn will be blue. Finally, let ei be atransformation of the sentence eQ such that all Q-property conjuncts in eQ not equal to ei arereplaced with ¬ei. For example, using our sample eQ from above,eV = ‘¬eV(b1) ∧ ¬eV(b2) ∧eV(b3) ∧eV(b4) ∧eV(b5) ∧eV(b6)’NIt would be nice to ignore the order in which we see both the balls and color properties sothat the numbers 〈sV = 4O s:V = 2〉 capture, so to speak, all the information contained in oursample insofar as it is expressed by eV.190 In essence, this is what conditions C6-9 accomplish.Simplifying a bit, condition C6 states that, for all sentences hi and ei, the value of c(hiO ei)is the same for all the systems LN , independently of b (given that i P b) (13). ConditionsC7, C8 and C9 then make several symmetry assumptions about the individual constants andQ-predicates for our inductive system. If c is in the -system, then C7 is just the assumptionthat c is symmetrical and C8 states that c is symmetrical with respect to permutations of theQ-properties (14). Lastly, C9 states that no information is lost with respect to c, relative toany molecular property a and the hypothesis hM , when we transform eQ into eM (14).191Together, conditions C1-9 characterize all the confirmation functions in the -system.Specifically, Carnap argues in §4 of CIM that for any such function c in the -system, there ex-ists (relative to LN and some Q-property ei) a characteristic function G such that G(O sO si) =190 Of course, s = sN + s¬N .191 In symbols, c(hM ; sM ) = c(hM ; sQ).1084.4. Carnap’s Continuum of Inductive Methodsc(hiO ei), for i = 1O NNNO .192 Thus if different inductive methods are represented by differentc-functions, then the values of these functions, with respect to the values s, si and , are givenby the values of some characteristic function G(O sO si) (15-6).193 In §5, Carnap then providesa proof that for any given characteristic function G, that function uniquely determines theconfirmation values for all sentence pairs h, e in the language system LN (granted that e isnot logically false).194 A similar result then also holds for estimation functions: all the esti-mation functions based on confirmation functions in the -system are also determined by thecorresponding characteristic function (see CIM §6).This result marks the completion of the first task I mentioned above. By comparison, thesecond task is a bit more complicated. To recap what Carnap has done so far, it has been shownhow different characteristic functions, G, G′, G′′, ..., each uniquely characterize a differentinductive method in the sense that each such function uniquely determines the confirmationvalues for some confirmation function c in the -system for any hypotheses we can form about oururn of colored balls. Now the problem is to somehow parameterize this collection of characteristicfunctions with a single, logical parameter . Then, for some fixed logical system LN , it wouldbe possible to catalog different inductive methods based on this collection of G-functions whichwe could use, for example, to calculate the probability that the next ball in the urn, given thesample eV, will be blue. In creating such a parameterization, Carnap says he will be “liberalin the admission of inductive methods to the projected -system” while also “exclud[ing] thosemethods which practically everybody would reject” (24). He does this by distinguishing betweenan empirical and a logical “factor” and then defines a confirmation function relative to , where reflects the different “weighting” given to these two factors.More generally, Carnap points out that if we could catalog G-functions relative to somevariable x as a function of two quantitative parameters of G, say u1 and u2, where u1 P u2 and192 See pages 14-15; the relevant results are (4-5) through (4-8). I am well aware of the fact that both Kemenyand Savage pointed out to Carnap that  is independent of this function and U need only be defined interms of s and si; see Kemeny (1963), the manuscript of which Carnap originally received in April 1954, fora simplification of Carnap’s result using recursive functions, especially pp. 724-731.193 However, as any such characteristic function U is defined from R3 to (0; 1), there is no initial restrictionon these functions; indeed, there will be U-functions which do not correspond to c-functions in Carnap’s-system (CIM, 18).194 For the technical details, see pages 17-18 and results (5-2) through (5-4).1094.4. Carnap’s Continuum of Inductive Methodsx ∈ (u1O u2), then we can always re-express x as a weighted average,x =k1 · u1 +k2 · u2k1 +k2O (4.4)where k1 and k2 are real-valued “weights” for the parameters u1 and u2.195Returning to our example eQ above, suppose that we are interested in determining a valuefor the hypothesis hV that the next ball will be blue by first defining a continuum of inductivemethods and then choosing one of these methods to determine a value for c(hVO eV). Carnapreasons basically as follows. Supposing that inductive methods can be characterized by theweight they give to a logical and an empirical factor, then if no weight is given to the logicalfactor then only the empirical factor matters. Let the empirical factor be the relative frequency;for our example it is the relative frequency of blue balls in our current sample, sVQs = 2Q3.However, if all the weight is given to the logical factor, our empirical observations should haveno influence on the probability value. Let the logical factor be the relative width, wQ. In ourexample, because the property blue is a Q-property, the width of eV equals one, so the logicalwidth is just 1Q = 1Q3.196 Substituting in both these logical and empirical factors for ourend-points, i.e., u1 = sVQs and u2 = 1Q, and the parameter x with the value of G(O sO sV),all that is left to do now is determine the weights k1 and k2. Carnap chooses the samplesize s as the weight of the empirical factor. This choice, Carnap says, “requires no theoreticaljustification, since it does not involve any assertion” (27-8). Rather, all it requires is a practicaljustification; namely, that this choice “leads to an especially simple form of the parametersystem” (28). The logical weight is then simply assumed to be the inductive parameter, .197Plugging in our new values of u1 = sVQs, u2 = 1Q, k1 = s and k2 =  in equation 4.4, wehave G(O sO sV) = (sV +  · 1Q)Q(s+ ) = (4+Q3)Q(6+). As we change the value of  we get adifferent value of G(O sO sV), i.e., a different value of c(hVO eV).Generalizing, the result is the following equation which characterizes a continuum of induc-tive methods with respect to the parameter , for 0 P  P ∞ ( = 0 and  = ∞ are special,195 Assuming both (i) e1 +e2 = 1 and (ii) for the distance terms r1 = |x − u1| and r2 = |x − u2|, it is thecase that d1=d2 = W2=W1.196 Condition C10 is just the assumption that w is equal to one; see pp. 26-7.197 Actually, Carnap treats  as a function of , s and si.1104.4. Carnap’s Continuum of Inductive Methodslimiting cases):198G(O sO si) =si + Qs+ N (4.5)The smaller the logical weight, the more important the empirical factor, here the relative fre-quency. Thus as  approaches 0, the value of G will approach the value siQs. The larger thevalue of , the empirical factor becomes less important and all G values approach a fixed limit,regardless of whether new observations are made.199 Provided, along with a new condition C11,that conditions C1-10 hold, Carnap then goes on to show that the G-values determined by theabove equation relative to the parameter  also determine the values of particular confirmationfunctions for the hypotheses hi (or hM ) given the evidence ei (or eM ) and fixed values of s andsi (30).200 Finally, Carnap provides a general method to define, for any state description k inthe object language, the value of m(k) as a function of products of G-values.201 A confirmationfunction c, relative to , is then defined as that function based on this measure function. Inthis way, each value of , in the interval (0O∞), characterizes a specific confirmation function,a function which represents a unique inductive method.The equation we end up with is the familiar characterization of the -system:c(siO s) =si + Qs+ (4.6)So now that we have a smorgasbord of inductive methods to choose from and investigate,how could we know which values of  provide us with an adequate confirmation function?For Carnap the finding of an adequate c is not an isolated affair. However, when we havefound such a function, we can then construct an inductive logic and along with it a theory forinductive reasoning in general, including reasoning about estimates of physical quantities usingthe function e, viz. that estimation function based on c. However, for Carnap, the decision198 More generally, U(; s; si) = (si +  · w=)=(s+ ). The -values 0 and ∞ are not strictly speaking in the-system because they violate C1-9, e.g., c=0 isn’t actually a regular confirmation function, but ratherboth c0 and c∞ are defined by limiting conventions; see CIM §§13-14.199 Indeed, this is the problem with Wittgenstein’s inductive method in the Tractatus which basically says allstate descriptions have the same m-values; see CIM, pp. 39-40.200 Condition C11 states that, if c is in LN , the quantity [s · c(hi; si) − si]=[1= − c(hi; si)] remains invariantunder changes to s, si and the sentences hi and si (29-30).201 More specifically, the measure of any sentence h in our logical system can be expressed as a function of themeasures of all those state descriptions which hold of h and so m(h) equals a product of the U-values formeasure of these state descriptions, for a fixed , see CIM §10.1114.4. Carnap’s Continuum of Inductive Methodsto adopt a particular value of  is not to be justified on the basis of some epistemological ormetaphysical principle or argument; indeed, Carnap tells us that it “is fundamentally not atheoretical question” because theoretical questions are answered in the form of assertions, i.e.,as true or false statements which, if true, “demands the assent of all” (53). Instead,the answer consists in a practical question to be made by l. A decision cannot be judgedas true or false but only as more or less adequate, that is, suitable for given purposes. How-ever, the adequacy of the choice depends, of course, on many theoretical results concerningthe properties of the various inductive methods; and therefore the theoretical results mayinfluence the decision. Nevertheless, the decision itself still remains a practical matter, amatter of l making up his mind, like choosing an instrument for a certain kind of work.(CIM 53)The decision to adopt a value of , Carnap tells, is practical – it is analogous to choosing aninstrument to accomplish some task. Consequently, the choice of a value of  is adequate, itseems, insofar as the resulting confirmation (or estimation) function satisfies our given purposes,like whether it provides us with an easy to use inductive logic or whether the resulting estimationfunctions satisfy any number of methodological considerations. Moreover, the mathematicalconsequences of adopting c and e for a particular value of , for example, may also influenceour decision if we can’t use it to derive, e.g., the statistical theorems (or if those theorems turnout to be trivial, e.g., when  =∞). Nevertheless, the decision to adopt a particular value of itself is not a matter of right or wrong but only of better or worse. For example, Carnap tellsus that the agent l will have to decide whether or not  is assumed to be a function of  andthe answer to this decision – either ‘yes’ or ‘no’ – represents what Carnap calls methods of the“first” and “second” kind, respectively (see CIM §§15-16). Either choice will come with its owntheoretical and practical consequences, e.g., one method may be more mathematical tractablethan the other.Disambiguating Practical MotivationsBefore I discuss Carnap’s work on finding “optimal” values of , we should take a moment topause and discuss another place where Carnap explicitly draws a distinction between the prac-tical and theoretical. While Carnap was working on CIM between 1949 and 1951 he published“Empiricism, Semantics and Ontology,” or simply ESO.202 It is in ESO that Carnap attempts202 References are to the slightly altered re-print of ESO in the second, 1956, edition of Meaning and Necessity.1124.4. Carnap’s Continuum of Inductive Methodsto intervene in a debate between professional philosophers, whom we may want to call “realists”and “anti-realists” today, concerning whether abstract entities, like natural numbers or fictionalnames, “really” exist or not. According to Carnap, such ontological questions arise due to afailure to distinguish external questions about the ontological status of the terms in a linguis-tic framework and internal questions about the meaning of terms within that framework, i.e.,questions answerable in terms of the semantical resources of that linguistic framework.For example, suppose that the linguistic framework in question is the familiar “thing”-language most of us implicitly adopt on a daily basis, i.e., that the world is composed of thingslike chairs, neutrinos and corporations. Then, at least according to Carnap, any external ques-tion about what it really means for a thing in this thing-world to exist “cannot be solved becauseit is framed in the wrong way” (207). Indeed, questions about whether the entity x is “real” ornot should not be recast as questions about whether one believes in the reality of x but rather aswhether x is an “element” of the thing language or not. Of course, once a framework is adopted,at least in the sense that we make the practical decision to start using that language, this frame-work can be used to frame our experiences, including our own reports about the propositionalattitudes we experience. Nevertheless, “the thesis of reality of the thing world,” says Carnap,“cannot be among these statements, because it cannot be formulated in the thing language or,it seems, in any other theoretical language” (ESO 208). More specifically, linguistic frameworksare composed of a number of rules specifying the formation and interpretation of statements,including what it means to “accept” or “believe” such statements and, moreover, one can changebetween linguistic frameworks simply by choosing to adopt a new system of rules to frame, fromthat moment onwards, all of one’s evidential and/or theoretical statements. It is in this sensethat the decision to “adopt” a linguistic framework is a practical rather than a theoretical matter(e.g. see ESO 217-8).In other words, Carnap’s distinction between external and internal questions (relative to aparticular linguistic framework) is a particular example of Carnap’s more general distinctionbetween the practical and theoretical: external questions are examples of questions better di-agnosed as practical questions about the decision to adopt a linguistic framework and internalquestions are examples of theoretical questions answerable within a framework. So the external,or practical, choice of a linguistic framework seems to be an all or nothing affair, or what I1134.4. Carnap’s Continuum of Inductive Methodswill call coarse-grained change: from a host of alternative linguistic forms, an agent will chooseone, and only one, system of rules; specifically, rules which will then be used to “frame” theirstatements and expressions. Nevertheless, Carnap tells us that203[t]he decision of accepting the thing language, although itself not of a cognitive nature, willnevertheless usually be influenced by theoretical knowledge, just like any other deliberatedecision concerning the acceptance of linguistic or other rules. The purposes for which thelanguage is intended to be used, for instance, the purpose of communicating factual knowl-edge, will determine which factors are relevant for the decision. The efficiency, fruitfulness,and simplicity of the thing language may be among the decisive factors. And the questionsconcerning these qualities are indeed of a theoretical nature. But these questions cannotbe identified with the question of realism. They are not yes-no questions but questions ofdegree. (ESO 208)So even though the choice to adopt a framework is binary, the process by which this choice ismade is one of degrees. It is a delicate matter of weighing the consequences of formulating alogical system this or that way, including figuring out what kind of logical systems would makeit easiest to provide it with an interpretation in line with our methodological considerations.Different assertions have to be weighed and compared, for example, assertions about whatexactly one’s aims and preferences are and the inclusion of certain theoretical assertions takenfrom outside the object language (e.g., results from computability theory) about how one couldpossibly accomplish subsets of those aims relative to these preferences. However, how thisweighing is done is ultimately up to the agent; to adopt the language from chapter 2: thereis no “meta”-framework we can appeal to inform our practical decisions regarding how best toformalize the thing-language from a “neutral” point of view.As an example, suppose an agent is deciding whether to keep using the thing-language orto instead adopt a process-language: in place of events defined in terms of both the qualities ofthings at a certain point in time over some duration of time, they would instead talk about eventsas the organic interactions between physical processes during some period of time.204 Notions ofthe “efficiency,” “fruitfulness,” or “simplicity” for both the process- and thing-languages, it seems,would be assessments made of these languages from the perspective of some metalanguage (whichvery may well be the thing-language itself plus certain mathematical resources, like set theory)about the relative merits of the logical and empirical consequences that can be formulated ineither of these languages. Perhaps, just to provide an example, someone attentive to Carnap’s203 Also see ibid., 221.204 For example, along the lines of Dupré (2008).1144.4. Carnap’s Continuum of Inductive Methodspractical/theoretical distinction could reason as follows. Whereas the process-language lendsitself naturally to the formulation of mathematical dynamical systems required by ecology, thething-language instead lends itself to the creation of systems of partial differential equationsrequired by Newtonian physics. Then, in a way analogous to how one scientist may prefernon-Euclidean to Euclidean geometry, depending on whether one is worried about biological orphysical phenomena, a community of scientists may find it more useful or convenient to adoptthe process-language instead of the thing-language. Carnap is fairly explicit that something likethis is, in fact, possible:The acceptance or rejection of abstract linguistic forms, just as the acceptance or rejection ofany other linguistic forms in any branch of science, will finally be decided by their efficiencyas instruments, the ratio of the results achieved to the amount and complexity of the effortsrequired. (ESO 221)The process- and the thing-languages, so to speak, provide us with different frames or lenses forconceptualizing the activities of scientists; however, for Carnap whether one of these frames is“correct” is an external question: it is to be answered in instrumental terms, e.g., by measuringthe efficiency of each language as a ratio of the number of useful results to a measure of thecomplexity of the language itself.205 However, whatever the case, it would be, for Carnap,ultimately a practical decision whether to adopt a system of rules corresponding either to thething- or process-language. It is also a practical question, for Carnap, whether we would wantto use a measure like the ratio of desired results to their complexity required as a measure oflinguistic adequacy. However, once these decisions have been made by the agent they committhemselves to a new system of rules and these rules then “frame,” so to speak, any and alltheoretical assertions made by an agent. For Carnap, the sooner we recognize this lesson thesooner we can divert valuable cognitive labor away from worrying about traditional metaphysicalquestions to instead worrying about how to measure the efficiency, simplicity or fruitfulnessof different linguistic frameworks, or even how to more efficiently draw out the theoreticalconsequences of different frameworks. Only then can we combat premature prohibitions againstcertain linguistic frameworks rather than just straightforwardly “testing them by their successor failure in practical use” (ESO 221). Indeed, to make such prohibitions about linguistic form,205 Presumably, however, we would have to carry out such an investigation from either the thing or processlanguage, for example, by measuring and comparing the efficiency of those scientists who opt to adopt theprocess-language in contrast to the thing-language.1154.4. Carnap’s Continuum of Inductive Methodssays Carnap, “is worse than futile;”it is positively harmful because it may obstruct scientific progress. The history of scienceshows examples of such prohibitions based on prejudices deriving from religious, mythologi-cal, metaphysical, or other irrational sources, which slowed up the developments for shorteror longer periods of time. Let us learn from the lessons of history. Let us grant to thosewho work in any special field of investigation the freedom to use any form of expressionwhich seems useful to them; the work in the field will sooner or later lead to the eliminationof those forms which have no useful function. Let us be cautious in making assertions andcritical in examining them, but tolerant in permitting linguistic forms. (ESO 221; emphasisin original).If Carnap’s motivation for introducing the practical and theoretical distinction in discussionsabout ontological commitment is to grant us freedom and tolerance in investigating linguisticforms, is the choice of  practical in the same way we are free to adopt a language framework?Ostensibly, questions about how to choose a value for  resemble an external question, i.e., aquestion about how to make coarse-grained decisions about one’s entire inductive framework.206In CIM, for example, Carnap clearly distinguishes decisions about whether to change values of between empirical investigations in contrast to adopting a new value of  in order to fix asingle inductive method which will then be used for all of one’s empirical investigations duringa period of time (CIM 54). It is the latter kind of decision which concerns Carnap.207 Whenan agent adopts a particular inductive method, understood as a logical concept of probability,theywill apply it to all inductive problems, problems of confirmation for all kinds of hypotheses;of estimation for all kinds of situations [...]; of choosing a practical decision; etc. Oneinductive method is here envisaged as covering all inductive problems. (54; my emphasis)However, Carnap also acknowledges that when it comes to the inductive concepts used in scien-tific reasoning, it may be difficult for scientists to make such wholesale changes to their inductiveintuitions.208 That is, in orderto change a belief at will; good theoretical reasons are required. It is psychologically difficultto change a faith supported by strong emotional factors (e.g., a religious or political creed).(54-55)206 In his contribution to Carnap’s Schilpp volume Arthur W. Burks explicitly draws a parallel between Carnap’sexternal/internal distinction from ESO and the choice, outside a system, of choosing a confirmation functionversus the finding of confirmation values within a system; for Carnap’s reply, see Carnap, 1963b, 979–982.207 Rosenkrantz (1981), for example, suggests that once  is chosen, it is chosen for life (Ch.1, §3, p. 4).208 Although Carnap doesn’t put it in the following terms, perhaps the situation is similar to those who havemore Bayesian or Likelihoodist inductive intuitions versus those whose intuitions reside with more classicalstatistical hypothesis testing.1164.4. Carnap’s Continuum of Inductive MethodsNevertheless, as with the decision to adopt a linguistic framework, Carnap tells us that thedecision to adopt a value of  “is neither an expression of belief nor an act of faith, thougheither or both may come in as motivating factors” (55). Instead, “[a]n inductive method,”Carnap tells us,is rather an instrument for the task of constructing a picture of the world on the basis ofobservational data and especially of forming expectations of future events as a guidance forpractical conduct. l may change this instrument just as he changes a saw or an automo-bile, and for similar reasons. If l, after using his car for some time, is no longer satisfiedwith it, he will consider taking another one, provided that he finds one that seems to himpreferable. Relevant points of view for his preference might be: performance, economy, aes-thetic satisfaction, and others. Similarly, after working with a particular inductive methodfor a time, he may not be quite satisfied and therefore look around for another method.He will take into consideration the performance of a method, that is, the values it suppliesand their relation to later empirical results, e.g., the truth-frequency of predictions and theerror estimates; further, the economy in use, measured by the simplicity of the calculationsrequired; maybe also aesthetic features, like the logical elegance of the definitions and rulesinvolved. (55)It is worth some space, I think, to try and unpack this passage. First, to return to our previousdiscussion, Carnap’s appeal to the practical/theoretical distinction in ESO seems by in largeto secure freedom and toleration against the elimination of linguistic forms due to dogmaticontological restrictions. In the above passage, however, by suggesting that the choice of aninductive method, characterized by a value of , is practical, Carnap is instead showing how anon-dogmatic investigation of inductive methods can take place. Different values of , accordingto Carnap, end up representing different inductive methods, or instruments, useful for framingthe inductive deliberations of an agent. So here the practical choice of a value of  signifiesa positive, or constructive project: for all the different values of , an agent can decide whichvalue provides the best inductive instrument. Specifically, as a matter of practical choice, onewill choose that instrument which best satisfies any number of pragmatic features, like whetherthe resulting instrument itself is parsimonious or easy to use (however, as Carnap intimatedabove, this choice may also be informed by any number of theoretical assertions).209Carnap started off with a rather philosophical problem of figuring out how to study a con-tinuum of inductive methods. He designed and constructed the skeleton of a logical system, likeLN , and then defined a class of confirmation functions by stipulating a number of requirements,209 There is a certain similarity here between Carnap’s talk of pragmatic and theoretical features influencingpractical decisions with Kuhn’s talk of scientific values influencing, but not uniquely determining scientifictheory choice; for more on scientific values see Kuhn (1977; 1983) and, more generally, Douglas (2009).1174.4. Carnap’s Continuum of Inductive Methodsnamely C1-11, that these functions must satisfy. The result is the -system; it is, as Carnapsays, “an inexhaustible stock of ready-made methods systematically ordered on a scale” (55).Moreover, if an agent “feels,” says Carnap,that the method he has used so far does not give sufficient weight to the empirical factor incomparison to the logical factor, he will choose a method with a smaller  – a little smalleror much smaller, according to his wishes. On the other hand, if he wishes to give moreinfluence to the logical factor and less to the empirical factor, he will move up his mark onthe -scale. Here, as anywhere else, life is a process of never ending adjustment; there areno absolutes, neither absolutely certain knowledge about the world nor absolutely perfectmethods of working in the world. (55)In this passage the choice of  seems like an external question: it is a pragmatic matter how ascientist decides to adjust the value of  as she uses the -system as an instrument for workingwith the world, as a guide for making non-arbitrary decisions. Yet we will see in the next sectionthat Carnap adopts certain statistical notions, like that of a “biased” estimator, to show how fora fixed state-description certain values of  are “optimal” in the sense that e provides us withthe closest estimate of the “actual” values of some parameter. But surely, at least for Carnap,the question of whether  is optimal must be an internal question: the notion of “optimality”is a technical notion definable with the semantics of the object language.So perhaps this is the kind of “blending” of the practical and theoretical which Carus andStein suggest marks a break with Carnap’s mature thought? I agree that it is a kind of blendingof the practical and theoretical, but Carnap is cognizant of this blending and it is in no wayfatal to his project. Firstly, as the discussion above about ESO makes clear, as long as weare not trying to reify  itself there is no trouble to slip back and forth between external andinternal questions when talking about choosing an adequate value of  – or, likewise, betweenthe formal and material mode of speech – if it helps the scientist to use inductive logic as aninstrument or as a heuristic for the logician to design better inductive frameworks. Not evenCarnap, in all his published writings, always clearly indicates when he is speaking informallyat the level of pragmatics and methodology as opposed to stating a claim within a well-definedlanguage framework (but, my oh my, does he try). Secondly, it may be helpful to distinguish (theterminology is my own), even if only as a matter of degree, the choice of a language framework,like semantic system like LN , as coarse-grained practical decision as opposed to the piece-mealmodifications and extensions the logician makes to an already extant logical systems – these1184.4. Carnap’s Continuum of Inductive Methodsare fine-grained practical decisions. Thus the decision to modify the value of  need not beconsidered a whole-scale, coarse-grained, change of semantical system but rather a fine-grainedchange to the “same” semantical system – both are practical changes but they differ in degreeand severity of the changes being made.I suggest that we can think of these fine-grained changes to LN – especially when defining the-system, estimation functions, or semantic concepts of information and entropy – as the sort ofdesign changes made to a hierarchical engineering design. The conceptual engineering frameworkbecomes especially apt when we start to worry about how to apply a pure inductive logic: whatwe want to do with the logic is specified by something like an operational principle and as wemake fine-grained changes to some pure inductive logic, these choices have repercussions for theother semantic concepts in the system: this is especially the case if we define all other inductiveconcepts on a single choice of a function c: as we change the value of , we likewise alter themeaning of all the other inductive concepts based on c. The question of what value of  isadequate is now, in a certain sense, a design problem.Moreover, it is passages like the one quoted above that bring to mind Herbert Simon’s notionof “satisficing”; it is this notion, rather than global optimization, which I suggest captures thesense in which a practical choice can, for Carnap, be satisfactory, efficient, fruitful or whatnot.For example, notice that we have spent the majority of this chapter merely cataloging all thedifferent ways in which Carnap’s construction of an inductive logic, from the c-function tothe -system, can be construed as a series of practical decisions. First, (1) there is the choiceof L, understood as an axiomatic system of logical calculus. Of course, this choice will beinfluenced both by practical considerations of computational complexity. Second, (2) there isthe choice of a semantic interpretation for L, a choice which can again be split into logicaland methodological, or empirical, considerations. More specifically, we now have a series ofpractical choices about how to specify this interpretation. For example: (2a) we can placerestrictions on the interpretation of L, like the requirements of completeness and independence;(2b) there remain various decisions which have to be made about how to design and constructany number of inductive concepts based on an adequate concept of degree of confirmation, e.g.,like estimation functions; and finally, (2c) we have to made decisions about how to define a classof adequate confirmation functions, e.g. a single function like c or a system of functions, like1194.5. Finding Optimal Values of conditions C1-11. Lastly, (3) we have methodological decisions to make concerning how bestto apply our interpreted inductive system. For example, if we want to use our inductive logicin decision theory or statistics, (3a) we will have to somehow coordinate adequate confirmationfunctions with the credence or credibility functions of ideal or actual agents. Moreover, (3b) wewill have to supply methodological rules for applying our inductive logic, like the requirementof total evidence.This discussion is fairly schematic and abstract, but the point is this. It seems as though,according to Carnap, we can keep, for the most part, a series of choices about (1-3) fixed, savefor a decision about how to define the most basic semantic concept for the entire inductivelogic; namely, a choice for how to define a concept of degree of confirmation, c, in terms of thevalue of a parameter like . Exactly here (1-3) resembles a kind of hierarchy of design decisionswhich need to be made in order to construct a pure inductive logic that can be applied to aparticular scientific purpose. All of this talk of practical and theoretical decisions, however,is far too abstract to do much philosophical work. Fortunately, we will not have to deal inabstractions for too much longer. Within the context of developing a more general method fortheorizing about statistical estimation functions, Carnap himself shows how, through a logicalinvestigation, “optimal” values of  can be found. It is to this example which we now turn.4.5 Finding Optimal Values of Besides studying the “internal logical character” of an inductive method, Carnap says in additionthat wemay confront it with a given series of events or a whole world, either the actual universe oran assumed one described in a given state-description, and examine how well it performsif it is applied to various parts of the world in order to obtain degrees of confirmation orestimates concerning other parts. (59)More specifically, Carnap uses his -system to investigate the performance of a continuum ofestimation functions.210 Now it would take an empirical investigation, employing perhaps theestimate squared error of estimation functions, to study the performance of these functionsrelative to actual empirical predications made in scientific practice. Carnap does not engagein such an empirical investigation. Instead, as I discussed at the end of section 4.3, Carnapassesses the performance of estimation functions within a given state description; it is such an210 A version of this section can be found in French (2015b).1204.5. Finding Optimal Values of investigation to which Carnap says is “of a purely logical nature” (59). Indeed, to engage inthe empirical investigation above would require us to use, for example, estimations of squarederror, to measure the performance of estimation functions but that means we would already bepresupposing a particular inductive method to carry out these estimations. It is only through thispurely logical investigation of assuming that a state description is true that we can investigateinductive methods “on a neutral basis without presupposing the acceptance of one of them”(60).211Indeed, the purpose of Carnap’s investigation of estimation functions on a “neutral basis” isto show that the preference among statisticians for “unbiased” over “biased” estimation functionsis unwarranted. Specifically, Carnap shows that from a purely logical point of view, i.e., on theassumption that the state description k is the case, there exist, for large b , “biased” estimationfunctions with a smaller mean squared error relative to k than the mean squared error of aparticular “unbiased” estimation function relative to k. Assuming that e is an estimate for therelative frequency rf and that rˆ is the actual value of this frequency, then e is unbiased if, forany sample of size s, the mean value of e equals rˆ and the bias of such a function is the differencebetween e and rˆ (59). However, it turns out that there is only one unbiased estimation functiongiven by Carnap’s -system, viz., the function characterized when  = 0, e0.212 This inductivemethod is none other than that method which tells us probability values should be equal toobserved relative frequencies, i.e., what Reichenbach calls the “straight rule” (44).213 Indeed,Carnap notices that it is a consequence of Fisher’s method of maximal likelihood (e.g. in Fisher1922) that, if R(u) is the parameter for some relative frequency rf and we use Fisher’s methodto find the maximal likelihood of R(u) given our current evidence, the probability of R(u) mustbe given by the straight rule.214 It is in part for this reason, says Carnap, that statisticians,211 On the downside, however, “by framing the problem as a logical question, our investigation must necessarilyabstain from making any judgment concerning the success of an inductive method in the total actual world.A judgment of the latter kind is obviously impossible from an inductively neutral standpoint” (60).212 Technically speaking, e0 is not actually in the -system at all as it is not a regular confirmation function(42). Instead, Carnap calls c0 and e0 “quasi-regular” functions meaning they can be characterized as thelimits of regular functions as → 0 (42).213 We will return to Reichenbach’s work in the next chapter; for now all that matters is that Carnap noticesthat Reichenbach’s so-called “rule of induction” (e.g. in Reichenbach, 1949, 446) “is essentially the same asthe straight rule of estimation” (44).214 For example, if a random variable x is expressed in terms of the parameter  and xi is a sequence of samplesfrom x, then where the posterior probability is given by P (|xi) the likelihood Z(;S) is just P (xi|); forthe differences between the principle of likelihood and the law of likelihoods, including how likelihoods maybe used in scientific practice, see Edwards (1972); Hacking (1965); Sober (2008).1214.5. Finding Optimal Values of like Kendall (1948), prefer unbiased estimation functions (CIM 44).215 In other words, Fisher’sgeneral method is exactly the sort of empirical investigation of estimation functions whichpresupposes an inductive method, viz. the straight rule.The advantage that Carnap sees for his own “neutral” investigation of biased and unbiasedestimation functions is that it does not presuppose any particular inductive method. It isthis investigation which we will turn next and although the following couple of paragraphs aretechnical this discussion will be of use when we later reconsider Carnap’s practical/theoreticaldistinction. The crucial, logical, assumption is that some state description k in LN is assumed tobe the case. Relative to k, we can then investigate a continuum of estimation functions relativeto the parameter  of the relative frequency of predicates a for all b individuals in k, or whatCarnap calls rf. Specifically, for any observed sample of s many individuals, where the class Kcontains those bs many individuals not yet observed, such that sM is the number of individualsfrom the sample that are a , Carnap shows that estimation functions can be characterized bythe equation (where w is the logical width of a and  is 2), (1952, 62)e(rfOaOKO eM ) =sM + (wQ)s+ N (4.7)Because we are assuming that we already know k, the actual value of rf, i.e., rˆ, is fixed, Carnapexplains that we can then explicate the notion of the “measure of success” of e in terms of itsmean squared error, which Carnap shows us can be expressed with the following equation:m2(eOaO kO s) =s · rˆ(1− rˆ) + (wQ− rˆ)2 · 2(s+ )2N (4.8)Relative to the state description k and our sample of size s with sM many individuals which holdofa in LN , while varying the value of  we can try to find the smallest value of m2(eOaO kO s).In other words, we will find that value of  which minimizes the mean squared error of e inthe state description k. Importantly for Carnap, this procedure generalizes not only for any215 Indeed, Carnap later opines that “Many contemporary statisticians seem to regard unbiased estimates aspreferable, [...]. As far as I am aware, no rational reasons for this preference have been offered” (73). Indeed,similar remarks can be found in Howson and Urbach, 2006, 164-6; also Hastie et al., 2010, 52 and chapter7; and within the theory of statistical inference – in part due to a theoretical trade-off between the bias andvariance of an estimator where multiple parameters are being modeled – unbiased estimators are not alwaysthe most satisfactory estimators, e.g. see Efron (1975).1224.5. Finding Optimal Values of predicate a , but also for the most basic predicates in L, i.e., the Q-properties ‘e1’, ..., ‘e’.The basic idea is that, for a fixed state description k, one can count the number of times eachQ-property uniquely holds of the b individuals in LN ; following Carnap, let us denote thesenumbers as bi. Thus for each ei, the actual frequency of ei’s in k is rˆi = biQb (note that∑rˆi equals one). After showing that the values of e(rfO eiOKO eQ) are given by the quantity(si+Q)Q(s+), Carnap calculates the mean squared error of these Q-based estimation functionsrelative to  as,m2Q(eO kO s) =s− 2Q+ (2 − s)∑ rˆ2i(s+ )2N (4.9)At this point in Carnap’s investigation, the term∑rˆ2i plays a very important role; specifically, itis the only term in equation (4.9) which tells us anything about the logical properties of the statedescription k. More specifically, C