AN ESSAY IN NATURAL MODAL LOGIC by PETER APOSTOLI B. A., The University of British Columbia, 1984 M.A., The University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Philosophy W accept this thesis as conforming to the requ^tandard THE UNIVERSITY OF BRITISH COLUMBIA April 1991 © Peter John Apostoli, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) 7- cg \cO7 Abstract A generalized inclusion (g.i.) frame consists of a set of points (or "worlds") W and an assignment of a binary relation R w on W to each point w in W. generalized inclusion frames whose R w are partial orders are called comparison frames. Conditional logics of various comparative notions, for example, Lewis's V-logic of comparative possibility and utilitarian accounts of conditional obligation, model the dyadic modal operator > on comparison frames according to (what amounts to) the following truth condition: oc>13 "holds at w" iff every point in the truth set of a bears R w to some point where holds. In this essay I provide a relational frame theory which embraces both accessibility semantics and g.i. semantics as special cases. This goal is achieved via a philosophically significant generalization of universal strict implication which does not assume accessibility as a primitive. Within this very general setting, I provide the first axiomatization of the dyadic modal logic corresponding to the class of all g.i. frames. Various correspondences between dyadic logics and first order definable subclasses of the class of g.i. frames are established. Finally, some general model constructions are developed which allow uniform completeness proofs for important sublogics of Lewis' V. Table of Contents Abstract^ ii Acknowledgements^ iv Introduction^ 1 Chapter 0: Logical Preliminaries^ 10 Chapter 1: Presenting JS5^ 17 Chapter 2: Presenting JK and Some First Degree Extensions ^28 Chapter 3: Some Semantic Correspondences ^ 36 Chapter 4: Prestudy For Model Constructions^ 52 Chapter 5: Model Construction for Normal Dyadic Logic^74 Chapter 6: Model Construction for Normal Logics Containing STR^87 Chapter 7: Model Construction for Natural Dyadic Logic^102 109 Bibliography^ iii Acknowledgements This essay is the development of an approach to modal logic started fifteen fears ago by Ray Jennings in his manuscript Leibnizian Semantics: An Essay in Monotone Modal Logic. I want to thank Ray for introducing me to this area of modal logic and thus providing me with an ideal theses topic and a very stimulating and congenial research environment. I owe much to Ray for providing crucial financial and "modal" support. My supervisor Dick Robinson introduced me to modal logic in my undergraduate studies; I thank Dick for thus helping me start what has proved to be a very happy scientific journey. This thesis (and my logical style) has greatly benefitted from Dick's careful reading. Many thanks to due to Dick and the Department of Philosophy at the University of British Columbia for supporting my early research. Finally I want to acknowledge the support of the Killam Trust, Canada Council and the Simon Fraser Advanced Systems Institute. Also, many thanks to the Department of Computer Science at the University of British Columbia for generously providing computing facilities. This essay was produced there on a Macintosh IIcx running Microsoft Word 4 and printed on an Apple Laser printer. iv Introduction^ 1 Introduction. Kripkean accessibility semantics is a generalization of the so-called leibnizian truth condition according to which necessary truths are those true in all possible worlds. On the Kripkean account, the range of possibilities relevant to the evaluation of a necessity statement is relativized to the point of evaluation, according to the truth condition that a sentence is necessarily true just in case it is true in every accessible (or, as we sometimes say, alternative) world. The leibnizian account is restored when the relation of accessibility is taken to be universal, so following Scott [19], the leibnizian truth condition may be said to model universal necessity. It's worth noting that the Kripkean account of necessity is only incidentally leibnizian. In [12], the universality of necessity is held to be consequence of Leibniz's more basic intuition concerning the relation of entailment, viz., that necessary truths are those whose negations analytically entail a contradiction. (And hence "would also obtain if God had ceated a world with a different plan" ([20], p.438).) In accord with the conceptual priority Leibniz places on the analytical account necessity, this essay presents normal modal logic as the definitional extension of a theory of generalized strict implication obtained by defining necessary truths to be those whose negations imply a contradiction. In its strict implication form, universal necessity captures the central notion of classical entailment, according to which a entails 13 iff the set of possible worlds (read: valuations) where a is true is a subset of those where is true. This inclusion, or categorical, representation of strict implication is maintained in Kripkean relational semantics for strict implication, since a strictly implies 13 at a world w iff the set of w-alternatives where a is true is a subset of those where 13 is true. As Ray Jennings has argued (in [7], [8]), accessibility semantics is thus unsuited to the general study of dyadic modal logic, since the categorical representation forces a whole range of deductively independent classical Introduction^ 2 principles upon us (as shown in Chapter 3). Indeed, in terms of mathematical generality, we shall see that dyadic logic is in many respects a more suitable vehicle for the study of unary modal logic than the latter is for the study of the former. The present-day possible worlds literature contains an approach to modelling dyadic modal operators on relational frames that is not based on truth-set inclusion. Dyadic logics of various comparative notions, for example, Lewis's notion of comparative possibility (see [14], [15], [16]) , and utilitarian accounts of conditional obligation (as in [0], [3], [4], [6]) interpret dyadic modal operators on frames under partial orders determined by the point of evaluation. This is done according to (what amounts to) the following truth condition: [T C] a> (3 holds at w iff every point in the truth set of a bears R w to some point in the truth set of 13. wherew is the point of evaluation and 12,4, is w's partial order. Although these "comparison semantics" are not motivated as modifications of strict implication, they nevertheless fall under Jennings' notion of a generalized inclusion frame. A generalized inclusion (g.i.) frame consists of a set of points (or "worlds") W and a function which assigns a binary relation R w on W to each point w in W. Generalized inclusion models are defined on these frames according to CT C] above. Within the class of such structures, g.i. frames which place order-theoretic restrictions on the R w are called comparison frames. With the exception of Jennings [6], [7] and [8] the study of generalized inclusion semantics has been restricted to the theory of comparison frames. This seems odd, since even if one concedes the philosophical primacy of the comparison-theoretic interpretation of generalized inclusion, mathematical generality would still seem to require a characterization of the logic validated by the full class of g.i. frames. But perhaps a more pointed complaint Introduction^ 3 concerns the surprising absence of provably complete axiomatizations of proper subclasses of the class of comparison frames. The only successful axiomatic characterization of any class of comparison frames that I am aware of is Lewis' presentation of the V-logics in his [15]. But even here, consideration is restricted to a very small subclass of comparison frames, those featuring strongly connected, transitive and reflexive comparison relations. Since nothing in Lewis' philosophical account of comparative possibility requires such a strong brand of comparison, the technical problem of closing this gap in the literature should have philosophical merit. One of the goals of this essay is to axiomatize the logic of various classes of g.i. frames that properly extend the class of comparison frames, as well as axiomatize proper subclasses of Lewis' comparison frames. Another goal is to present a unified relational frame theory which embraces both accessibility semantics and g.i. semantics as special cases. This goal is achieved via a generalization of universal strict implication which does not assume accessibility as a primitive. This unified framework is an extension of Jennings' generalization of universal strict implication presented in [6], [7] and [8]. The motivation for Jennings' generalization of the categorical perspective given in [6], [7] and [8] is largely independent of order-theoretic considerations. It has its conceptual and historical roots in a temporal view of hypotheticals nominally attributed by Jennings to De Morgan. This view allows for (but does not require) an element of temporal lag to separate the situation in which an antecedent of a conditional holds from those in which the consequent holds. One metaphor operative in Jennings' analysis of conditionals is a generic notion of the causal or temporal "production" of situations or states: On this view, at least some natural language conditionals express the claim that situations where the 4 Introduction^ conditional's antecedent holds "result in", "produce" or (at least) are "temporally succeeded by" situations where the consequent holds. The causal / temporal transition between states is given frame-theoretic articulation via the g.i. frame-relations: a> 0 holds at a given point just in case every a-state "produces" a 0-state in the sense of bearing the given point's relation to some 0-state. As emphasized by Jennings, on the categorical view of conditionals, one situation may "produce" another only by virtue of being identical with it. In order to present Jennings' generalization of universal strict implication, suppose we have at hand a propositional language containing material implication, —*, and the familiar necessity operator, 0. Within this language, we represent the strict implication of 0 by a by the sentence 0(a—> (3). Now, suppose we have a "pre-accessibility" model .M = (W,V), where W is a nonempty set of possible worlds and V a valuation which assigns truth values to sentences relative to possible worlds according to the truth condition that Oct is true at a possible world w just in case a is true at all possible worlds in W. We may M express the truth of a at a possible world w by the familiar notation k a . It is w convenient to think of V as assigning "truth-sets" to the sentences of our language such that M the truth set of a, Eta Al is the set worlds w such that^a. w , Then the defining truth condition for strict implication may be taken to be: At kw 0 (a —0)^<=>^ cram Al c 1113EM Jennings proceeds by expressing the inclusion of II a.11.84 in [PI M by the quantificational scheme (VxE if an) (ly e ff [3]Dx = y . Then the truth-set inclusion representation of universal strict implication may then be expressed by the condition 0(a—* ^<=* (VxE [Calm ) (lye CI DE M ) x = y . 5 Introduction^ M is now "transformed" into a generalized inclusion model by generalizing role of identity to that of arbitrary binary relations determined by the choice of w. Taking the wedge to express the resulting notion of generalized strict implication, we obtain the truth condition M [U>l k a> (3 4.> (Vxe MI") (3y e [E(311M ) x R w y . w Here, of course, we think of )14 as a model based on a g.i. frame (W,R), where R is a function which assigns R w to w E W. Let's call the logic determined by the class of g.i. frames under truth condition [U>] JS 5 (for "generalized S5"). Now, within the setting JS 5 , the original "universal" necessity operator 0 may be recovered via the liebnizian definition that Da =df "la > 1. JS 5 is an extension of S5 in the sense that S5 is its unary fragment under the leibnizian defmition of the necessity operator. (See chapter 0 for a definition of the unary fragment of a dyadic logic). Further generalization is required to model relative necessity in this setting. So, an obvious question arises: how best to parallel the introduction of the accessibility relation into the g.i idiom and thereby allow for the semantic articulation of J/ for unary modal logics Z extending the base monadic modal logic K? (See Chapter 0 for a presentation of K.) David Lewis' presentation of the V-logics of comparative possibility (in [14], [15], [16]) provides two answers to this question. We can modify one to provide for an "accessibility-theoretic" relativization of the [U>] which maintains the philosophical spirit and the mathematical form of Jennings' generalization of universal strict implication. First, consider the following version of the comparison semantics presented in [16], where our Introduction^ 6 'a >13' stands for Lewis' '(3-<^, (read "(3 is at least as possible as a", or "a is as far- fetched as 13 is") Assume a propositional language with the wedge as a dyadic modal connective. Let W be a nonempty set (of "worlds") and let D be a function which assigns a set D(w) c W of "accessible" points to each w in W. Let R be a function that assigns a transitive, strongly-connected binary relation on D(w) to each w in W. Let hitit be a model on a frame (IV, D, R). Then Lewis' truth condition for the wedge may be presented thus: [L>] kw a > 13 <=> (vxE D(w)) [x [laM M (3yE I[1311 M )x In [15] Lewis considers the suggestion that rather than assuming the accessibility function D as an additional primitive, he might have let the comparison relation do the structural work of D(w) by relativizing the universal quantifier in condition [L>] to the field of v . In effect, R would come to assign a set of accessible points to a given point in the form of the field of that point's comparison relation. Lewis rejects this suggestion as "clumsy", but we shall take advantage of a related proposal which relativizes the universal quantifier in [L>] to the domain of 5..w. Recall the causal intuitions among those that motivated Jennings' generalization of universal strict implication. It seems reasonable, in evaluating the truth of a conditional which is asserted on roughly causal grounds, not to require every a-sate to produce a 13state, but rather only those a-state's which satisfy certain preconditions. For example, causal conditionals which are asserted against a background of theory might be committed only to the causal outcome of antecedent states that satisfy certain theory-global (as opposed to antecedent-relative) ceteris paribus conditions. Or some conceptually possible states might not be causally efficacious at all (perhaps because they are physically impossible). 7 Introduction^ In both cases, it seems appropriate to articulate such restrictions in terms of the states or events to which the causal relations apply, namely, those in the domain of the causal relation. We might want to say then that 13 on the condition that a" is true if every astate that "produces" anything (according to our theory) produces a I3-state. For any binary relation H, Let dom(H) stand for the domain of H, i.e., dom(R) =df {x I (3y) x Hy). We would then obtain the truth condition [R>] a>{3 <=> (VxE dom(R w ))[.,x€ Dm ), (3 E ViM M XR wy] , for models ,44 on a g.i. frame (W, R). The key virtue of this style of proposal is that it allows g.i. semantics in general, and comparison semantics in particular, to be viewed as a mathematical generalization of Kripkean semantics, as follows. Let a Kripke frame (14/, D) be given, where D is an accessibility function, that is, a function which assigns a subset of W to each point in W.. For each subset S of W, let =s be the identity relation on S. Note that dom(=s) is S itself. Assuming a model M , we can express the Kripkean truth condition for the strict implication of 0 by a at W E W by .1t v CI( a (3) <=› (Vx dom(=D(w)))[x llocEM (3y Eff 13B m ) X 7-1 D(w)Y1• Generalizing the role of identity to that of an arbitrary binary relation R w , determined by the choice of w on a generalized inclusion frame (W, R), we obtain [ll>], as desired. Jennings' original truth condition [U>) is not quite restored in the special case of "universality", viz. when dom(R w ) = W for all w in W, since [U>] does not require every frame relation to provide every point in W with a relatum. Under [U>], this requirement on g.i. frames corresponds to the principle 8 Introduction^ [N*) T>T. (or, equivalently, a > T). So, under [R>], we are stuck with [N*], and thus in the move from [U>] to [R>] we trade some of the semantic generality afforded by Jennings' analysis for greater flexibility in modelling unary modal logic. In this way, we come to simulate the Kripkean accessibility relation by the domains of g.i. frame relations. In effect, each g.i. frame F = (1 V , R) induces a unique Kripkean frame (IV, RF), where the accessibility relation RF is defined as the existential projection in y of the ternary relation R wx, y : wRFx 4:=>df (3ye W) xR w y (w, XE W). Thus I can unify two formerly unbridged semantic idioms. One immediate payoff is that many of the preservation concepts and results for Kripkean definability theory are automatically translatable into the model theory of dyadic modal logic under the uniform process of generalizing on the relation of "Kripkean identity" between worlds. In this essay, I use the terms "possible world", "frame-object", "point" and "state" interchangeably. It is not my intention to invoke a notion of possible world which is any more metaphysically robust than is the model theorist's mathematical point. Frame-objects should be thought of in any way that will yield a good interpretation. Prima facie candidates are: points in time, counterfactual situations, situation types, physical states or state types, events or event types, actual or possible global welfare distributions, states of a machine, etc. I want to play down the admittedly important task of finding significant interpretations for my relativization of the truth-set inclusion schema. Readers who are happy with the the Kripkean notion of metaphysical accessibility ([10], [11]) or Lewis' notion of comparative metaphysical accessibility ([15], [16]) are free to interpret [R>] as a notational variant [L>] intended to apply to the full range of g.i. models; no old Introduction^ 9 interpretations are lost in the move to [R>]. Those who are not will be assuaged by the fact that [12>] does not require such interpretations: the door is open for new, perhaps naturalistically acceptable, interpretations of the wedge. In Chapter 1, JS 5 is presented and semantically determined as the logic of the "universal" g.i. truth condition. Chapter 2 presents JK and various of its first degree extensions. In Chapter 3, semantic correspondences between these logics and first order sentences in a single ternary relation are established. In Chapter 4, further completeness theory is developed which serves as a basis in Chapters 5, 6 and 7 for model constructions that provide for the semantic determination of many of these logics. These constructions allow, in particular, for the semantic isolation of the order-theoretic principles employed in Lewis' V-logics. 10 Chapter 0^ Chapter 0: Logical Preliminaries. Most of our notational conventions and metatheoretic definitions are taken over from Goldblatt [5]. I shall observe Chellas [2] in the naming of modal principles and inference rules. We shall be considering a propositional language L(0) whose primitive basis consists of a countably infinite set = {Pi , P2, - ..} of propositional variables (or, atomic formulae), a propositional constant 1 (the falsum), a binary truth functional connective --> (material implication), and a binary modal connective > (the wedge). The set Fma(0) of formulae, or sentences, of our language is generated from this primitive basis according to the following scheme: _LE Fma(0) 0 g Fma(0) a, 0e Fma(0)^(a>f3)E Fma(0). As indicated 'a' ,^,^ '8' , . . . will be used to denote formulae. Parentheses will , often be omitted or replaced by informal scope-delimeting punctuation (such as brackets, braces, dots etc.) when it is convenient to do so. A formulae a is called first degree iff it contains no nested occurrences of '>', that is, iff no occurrence of '>' is in any subformula of a of the form (3>y. I'll assume a enumeration •:1) : w ---> Fma(0) of the language with the convention that '4)i' denotes the i th formula in the ordering. We introduce expressions for the familiar truth-functional constants and connectives into our metalanguage according to the standard abbreviatory definitions, as follows: - 1a=dfa * 1, T =df - avr3 =df 0, a/4i =df -IP) etc. The unary modal operators are introduced by the following definitions: Da =df la> 1 and o a =df - 11 Chapter 0 - 1(a>1). I shall make use of another defined unary modal operator: Da =df T > a. A convention used in the sequel is that in case n = 0, the expression '(ai A . . . n a n )' denotes T and '(a1 v v a n )' denotes I. Hence, '(al A ... A a n )--4 13 1 may be taken to denote the formula 13, and '13 —)(ai v v a n )' to denote 113, when n = 0. - A schema is a collection of uniform substitution instances of a given formulae. For example, by the schema Da-* a , we intend the set of formulae 0=0—>P :13E Fma(0)). A (dyadic) logic is a set A c Fma(0) such that: i) A includes all tautologies ii) A is closed under the rule modus ponens, [MP], i.e., a, a -313 E A^PE A. iii) A is closed under uniform substitution, [US], i.e., OCE A^a'e A, where a' results from uniformly substituting formulae for atomic formulae in a. If A is a logic, we write 'FA a' (a is a theorem of A) for 'ae A'. If F is a set of formulae, 'F FA a' ("a is deducible from r") means there are n E CO and al, . . a n E F such that. FA (ai A • A an) --> a 12 Chapter 0 Observe that tivially: r FA a iff there is a finite subset r of r such that r' FA a . A formula a is quasi-atomic if either cm 0 or a is 0>y for some (3,y€ Fma((). Let 04 be the set of quasi-atomic formulae. A valuation of Fma(0) is a function Val :^-4 T, F ). Since every formula can be constructed from members of 04L)^) using --> , every valuation Val may be extended to a unique assignment of truth-values to all formulae according to the truth tables for 1 and ; we shall identify Val with its truth-tabular extension. A formula a is a tautological consequence of formulae al, , an iff a is assigned T by every valuation of Fma(0) which assigns T to all the al, . ., a n . It is well known that every logic A is closed under the rule [RPL] FA al, • • • , an^FA a , where a is a tautological consequence of al, . • a n (cf. [2], p.3'7) . Hence, [RPL] will be used as a derived rule in the sequel. A set of formulae is A- consistent just in case F/A 1; otherwise it is A- inconsistent . A set of formulae is maximally A- consistent, or, A-maximal (sometimes: a A- maxi-set ) iff it is A-consistent and has only A-inconsistent proper extensions. Throughout the sequel, we wil assume as given, and make free and often implicit use of, the following well-known results concerning the existence and properties of A-maximal sets. The reader is referred to [2], pp.53-56, or [5], pp. 18-20 for further details. Let r be a maximally A-consistent set of formulae. Then: (1) a e F^iff^r F A a (2) A c r (3) T E r 13 Chapter 0 (4) a --) 00 (5) 1 0 r^iff^if a E F, then 13 E r r (6) r is closed under tautological consequence. Lindenbaum's Extension Lemma: Let F be a set of formulae. If r is A-consistent, then there exists a set A of formulae suchc that (i) r g A, and (ii) A is maximally Aconsisitent. Let MAXA be the set of all A-maximal sets. For any formula a, let I a1 A be the set of all A-maximal sets that have a as a member (the A superscript will often be omitted). For any set of formulae F, let Ir.1Abe { s E MAXA 1 F s s } . We'll also observe the standard convention that for any unary connective • (defined or otherwise): •{11 =df (*a I ae r) •(r) =df { a I •cm r} For example, Dm =df {Oa I ae r} and and On =df fa 1 0a€ F ) . By the unary fragment of Fma(0), denoted Fma(0) 0 , I intend the collection of formulae all of whose modal subformulae are of the the form Oa or 0 a. Fma(() 0 may be defined recursively as the least subset of Fma(() which includes 0, contains 1, and contains a-->13, -I a> _l_ and -1(a>.1) whenever it contains a and J3. The unary fragment of a dyadic logic A, denoted Ao, is defined as the restriction of A to the sublanguage Fma(0)0 , i.e., A° =df Fma(4 0)13 nA. The above definition of deducibility and [RPL] allows us to prove a "Deduction Theorem": rufal FA 13 <4.^r FA a--->13 . 14 Chapter 0 Let A be a dyadic logic. A is prenormal if it is closed under the rule schemata for "right" and "left" "monotonicity", [IRK FA a -43^FA D>7 --> a>7 , [RMT] FA a --0^FA 7>OC -4 7>P , and includes the schema of "disjunctivity", [DS]^FA (a>7 .n. 3 >7) -4 ((avP)>7 ) , together with the principle of "necessitation" for ^ : [N]^FA DT,^i.e.,^PA 1>1_ (equivalent to 1> a given [RMTD. A is normal if it is prenormal and includes the schema of necessitation for [N*]^FA 0 T,^i.e.,^FA T >T. (equivalent to a> T ginen [RM]si,). If A is (pre)normal and A includes the schema [M11^FA ^(a -) (3 ) -> (7>a .—>. 7>(3 ) , then A is called (pre)natural. Its easy to show that a logic which is closed under [1RM] and [RMT] is closed under the replacement of provable equivalents, allowing admission of the derived "Rule of Extensionality" for (pre) normal dyadic logics: [RE]^FA ai--->a'^FA (3<-->3' where 13' is obtained from 0 by replacing one or more occurrences of a by a'. 15 Chapter 0^ Since this essay presents dyadic modal logic as a generalization of monadic modal logic, it seems fitting to briefly present K, the smallest monadic modal logic admitting of a Kripkean semantics. Rather than repeating the standard definition of the language of monadic modal logic (as in [2], pp. 25-26, or [5], pp. 3-4), we will use the language Fma(0)0, since it is allready at hand. Then we can define K to be the smallest dyadic logic contained in Fma(0)0 which includes the axiom schemata [N] and [C]^(^an^O) 4 - I:1(a A (3) and is closed under the "Rule of Monotonicity" [RM]^F a-> (3^I- 0a-4013. Now let's table some semantic notions: A frame is a pair F = (W, R), where W is a nonempty set and R : W ----> 2W is a function which assigns a binary relation R w on W to each w E W. A 4)-model based on F is a triple .1Vi = (W, R, V) , with V:0--> 2W a function assigning to each atomic formula p €0 a subset V(p) of W. The prefix 'Ix' will generally be omitted. We extend V to valuations II DM : Fma(0)---> 2Wx W via the usual inductive clauses for 1 and --->, viz, M LLB =0 M ff(a-4(3)31 = (IV— m m if a 11 ) u 1113]] . In Chapter 1, V is extended under the following inductive clause for > : M^ m^A4 [U>]^ffa>133)^= 141 E W I (b5cE ffall ) (3y€ Q(3.11 ) x R wy) . { 16 Chapter 0^ so that we may briefly study the "universal" generalized inclusion. These models are called universal g. i. models. The remainder of the essay focuses on "relativized" generalized inclusion models obtained by the truth condition ^,^M , , = we WI Vx e dom(R w Axe Man^k By E Q131 )xR w y]l . [R>] Ila > 13B^t These models are called relativized g.i. models. In both semantic settings, we define the relation of a is true at a point w in a model M .A4, symbolically, k , by w .A4 .A4 4:=> cif w E Ea^(w € W) k a w The notion of truth on a model and validity are defined as usual: A formula a is true on a model M, denoted .,A4 M ka or k a, iff it is true at all points in M, i.e., if M k a for all w E W . w a is valid in a frame F = (W, R) , denoted F k a , iff .A4 k a for all models M based on F I shall often identify a frame F with the unique ternary relational structure (W, RF), where RF c W3 is given by RF (x,y , z) iff yR xz^(x, y, z E IV) Accordingly, I shall often identify a model ,A4 based on F with the triple (W,RF,V). In contexts where F is given, RFwill often be referred to simply as R. I will use expressions of the forms 'R x (y, z)', 'y R x z' , 'R (x, y , z)' , 'xR y, z' interchangeably. 17 Chapter 1 Chapter 1: Presenting JS 5 . In this chapter we present JS 5 and determine it as the logic of the "universal" g.i. truth condition [U>]. One of the themes of this and the next chapter is that when the unary modal operator is introduced by the leibnizian definition, the conditions that constitute normality in Kripkean modal logic are seen to be special cases of some fairly subtle properties of classical implication. For example, the axiom [K]^0(a -4 (3) -4 (^a—>^0) is an instance (modulo an application of [RE]) of the following left-downward monotonicity principle, to the effect that the wedge admits strengthening of antecedents under its defined strict implication: 0(a --> (3) -÷ (13>y —> ot>y) . (0(cc-43) --> ^(1(3--ria) by [RE], so 0(a --> (3) --> (-1a>1 -4 -1(3>1) by [1M], but this last is [K], by the definition of ^.) Note that, from a deductive point of view, [.ARM] stands to [1M] as the "Rule of Monotonicity" [RM]^I- a-->I3^I- ^a—>^13 stands to [K] in monadic modal logic. Indeed, in the derivation of [IM] from OM] below, [DS] plays the same role as does the modal "aggregation principle" [C]^(^aA^13) --> ^(an(3) in the derivation of [K] from [RM]. 18 Chapter 1^ Such principles as [1.M] relating the wedge to its defined strict implication are hard to isolate semantically when the study of generalized inclusion is restricted to comparison frames. The reflexivity and transitivity of comparison are particularly guilty offenders. Reflexivity of the frame relations R w forces enough of the classical subset representation on us to guarantee that the wedge preserves classical entailment, whence the validity on comparison frames of a "deduction theorem" for the wedge: [RD]^I- a 0^F a>13. - However, reflexivity corresponds to the principle of "Preservation of Strict implication" [PS]^El(a--> 13) ---> (a>13) . which is equivalent to closure under [RD] in some prenormal logics. Notice that [PS] entails [1M] given the transitivity of the wedge. Hence, [PS] is not separable from [1M] on comparison frames. Let JS 5 be the smallest prenatural logic which includes the S5 principles [5] and [T] below. Recalling Chapter 0, JS 5 may be presented as follows: [IRK^liss a—>13^1s5 py . >. a>y , — [RMi]^I-JS5a—> f3^I- y> a .-4. y>(3. ^J SS [DS]^lis5 (a>y .A. I:3>y) —> ((avI3)>y) [N]^IOT JS5 [MT]^FJs50(a-413) --> (y>a .—>. y >p) [5]^IJS5oa-400a 19 Chapter 1^ I-0a-> a 1 55 [1] The explicit inclusion of [N] is redundant, since [N] is an [RPL] consequence of [5] and [T]. Noting that [RM] is an [RPL] consequence of [1RM] and recalling that [K] is an [RE] consequence of [1M], to show that (JS 5 )° is S5, it suffices to show that [.I.M] is derivable: 0.^I05 -1(a--413)>1.--->. -1(a-->0)>y^[RMi], [RPL] 1. I0(a—> (3) --> ( -1(a --> r3)>y) JS5^ 2. I0(a JS5 3. F 1=1(a —>13) -5 ((3>7) --> ((avi3)>Y JS5^ 4. I0(a—> r3) -4 (0>y .-4. a>y ) JS5^ -4 0, df. 0 5) -4 (([3>'y) —> (( --7 (a ---> f3)v 13)>y)) 1, [DS], [RPL] )) 2, [RE] 3, [RPL] and [.ARM] The "Rule of Necessitation" [RN]^I- a^I- Da endows the wedge with the classical property that the negation of a logical truth implies a contradiction, whence, on the leibnizian definition, the necessity of logical truth. JS 5 's closure under [IRM] , respectively, [RMi] is guaranteed by [RPL], [RN] and [IM], respectively, [MT]. Therefore, JS 5 may be more elegantly characterized as the smallest dyadic logic with [T], [5] which is closed under [RN] and includes [DS] (a>7 .A. 13>y) -4 ((avf3)>y) [-I-M] o(a--->13) --> (0>Y .—>. a>y) [MT] 0(a—> I3) —> (y>a .-->. y> 13) . (Hence the explicit inclusion of [RMi] in the first characterization is redundant.) 20 Chapter 1^ Of importance in the sequel is the "preservation of diamonds" principle [PD]^li s5 (a>(3)^(0 a - ^ o(3) to the effect that the wedge preserves possibility: 0. F13>1 .-->. -1-10>J_ [RE], [RPL] 1. lis5^El((3 41) 0, df. El, df. i 2. I- DO —>.1) ---> Qa>(3)--> (a>1)) J S5 3. lis5^((a> (3) —> (a>_L)) 1, 2, [RPL] 4. I-^(a> fi) --> ( 1(a>1) —> 1(0>1)) 3, [RPL] 5. F.^(a> f3)^(o a ) 0(3) JS5 4, df. 0 , JS5 - JS5 - -- — At this point the reader may whish to recall from Chapter 0 (pp.15 - 16) the semantics of "universal" generalized inclusion, in particular, the definitons of frame and universal g.i. model . The proof of the semantic completeness of JS 5 with respect to the class of universal g.i. models requires a few preliminary results: Theorem (Scott's Rule) 1.0:^r F a^o[F] I Da - Proof: Suppose F F a. By the definition of deducibility (Chapter 0, pp. 11 - 12) there are n E Co and al E r (1^n) such that (ai A...A a n ) --> A . . A an) --> Da. so bt [RM], F 0(a1 Now generalized aggregation (repeated applications of [C]) yields 21 Chapter 1^ F (oai whence I- (^ai A. . . A A...A Dan ) --) 0(ai A... A an) Ela n ) --> Oa by [RPL]. Then we have the desired result that ^[r] F Oa by the definition of deducibility.^■ Separation Lemma 1.1: Let A be a prenatural logic. Let WE MAXA and a,13 be formulae such that a> 130 w. Let r = { 8 I (37)[ (8>y)E w & ^(w) F A y -413 ll. Then there is a UE MAXA with 0(w) g u such that aE u and Fnu = 0. Proof: Suppose that a>fle w. By Lindenbaum's Extension Lemma, it suffices to show that 0(w) u { al un [F] is A-consistent. Towards a contradiction, suppose not. Then by def. deducibility and deLA-inconsistency and [RPL], there are n 0 and S1, . . ., 8 n E r such that ^(w) F A (1) ('61A ... A -- 1811 ) —> —la ; by [RPL], ^(w) F A a ---) (Si v . . . v 8 n ), whence the choice of F gives yi, . . . , yn E Fma(0) such that (8i>yi)E w and 0(w)F Ayi —>I3 (1 S i 5_ n). Let 8n = Siv ... An and yi . yi v ... v yn . Then (1) yields, (2) w F A E(a--)8n), by Scott's Rule and the fact that ^[^(w)] g w. Now, repeated applications of [DS] and [RPL] yield (3)^FA (( 8 1 > Y)A — A( 5 n>Y)) -9 (8/1>r). 22 Chapter 1^ By [RPLJ, F A^[RMT] yields F A (81>yd (Si> r); since Si> yiE w , we have Si>rE w (1 S i S n). Then (Si> 'Yn) A^A on>yoe w , whence (3) yields the result On>yle w . Now, 0(w)I- Ayi—>p (1^n) yields 0(w)i- A r—>I3 by [RPL], whence we have w F A Exyn—*(3) by Scott's Rule, whereby [Mt] yields w F A (a>71 ) --> (a>(3). But by [1M], (2) yields w F A (8n>1) (a>yl) , SO W F A (On>r) --> (a> P). Since 811>r) e w, we have (a>13)E w, contrary to the hypothesis of the lemma. ■ Let A be a dyadic logic with [T] and [5]. Define a binary relation ^on MAXA by s t^iff^^(s) g r.^(s,tE MAXA) Lemma 1.2:^is an equivalence relation. Proof: The proof is routine: [T] insures that is reflexive, [5] that is euclidean. Hence is symmetric and transitive.^ Corollary: s t ^ ^(s) = ^(t).^ ■ ■ I shall now establish the completeness of JS 5 with respect to the class of universal g.i. models. Let A be a -prenatural logic which includes the S5 reduction principles. The strategy is to construct, for a given A-consistent set A, a -model MA which satisfies A in Chapter 1^ 23 the sense that MA contains a point where all of A members are true. The universe of MA will be constructed out of the traditional (i.e., Lemmon-Scott, cf. [13]) canonical universe MAXA by a proccess which may be thought of as the addition of extra "copies" of elements of MAXA. Hence, .84A will contain numerically distinct worlds which do not differ according to the sentences they verify. Invoking Lindenbaum's Lemma, let ME MAXA be a A-maximal extension of A. For each WE MAXA and a> 0 0 w as given in the hypothesis of the Separation Lemma (1.1), let 'w a>j' denote a fixed u given by that lemma. That is, when a> Pe w, then wa>0 E MAXA is given by 1.1. such that ^(w) c w a >p, a e w a> p and rn w oc >0 = 0, where F = [8 I (3y)[ (8>y)E w & ^(w) F A y--> p.• }}. Note that this choice of u given w and a> 0 does not require the Axiom of Choice, since we could specify u uniquely (in terms of our enumeration 0:a) --> Fma(0) from Chapter 0) as the 0—canonical maximally consistent extension of la } u 0(w) u —1[F] , which the proof of 1.1 shows to be Aconsistent. Recall from Chapter 0 that I ^(A*) I A =df {SE MAXA I ^(M) g s} . Define WA, the universe of MA, to be the following subset of I ^(0*) 1 A x (Fma(0)U f 0 1) : { (w, 0) I WE I D(0*) I A 1 U { (w a> 13, a>13) I WE 1 0(M)1 A & a>f3e w } . We will use the left and right projection operators LL), R(_) so that P = (L(P), R(P)) for any pair P E 10(A*)1 A x (Fma(0)L4 0 } ). The pairs Pe WA are treated as "copies" of the A-maximal sets L(P)e I ^(A*) I A . For the rest of this chapter, we will use the variables x, y, z to range over W. 24 Chapter 1^ Lemma 1.3: Let XE W. Then ^(L(x)) = ^(A*). Proof: If xe { (w, 0) :wE I 0(6.*)1 A 1 , then A*.---L(x), whence the desired result follows by the Corollary to 1.2. So assume that xe { (wa>p, a>13): w e I D(A*)I A & a> PE w } . Then L(x)=w cf>0 for some wE I ^(6,*) I A , whence ^(w) g w a>13 by Lemma 1.1 and the choice of w a>0. But ^(0*) c w and ^(w) c w oc >0 yields A*.--: w = w oc >o, so by the transitivity of ..-- (Lemma 1.2.), we have 0*----.w a>0. It follows by the Corollary to 1.2 that ^(w a>0) = ^(0*), as required. ■ The construction of M 6, is in two stages; the first stage guarantees that the left-toright direction of the truth condition [U>] holds on MA and the second that [U>] holds right-to-left. TF For XE WA, define a pair of binary relations R x, R X on WA by: T R x = (y, z)I (13a>(3)[y= (L(x) a> p,a>13)] & (38>y E L(x))(SE L(y) & ye L(z))} { R x = {(y, z) 1(3 a>I3)[y = (L(x) a> p,a>13)] & (38>y E L(X))[OE L(y) & ye L(z) & (3o L(z)] 1 _.> 2 WAxWA i s Then let RX =df RX l.) RX . Define a frame F = ( WA, R) where R: We a function which assigns RX to each x in WA. Lemma 1.4 : Let xe WA. Let 8>y be a formula. Then 8>ye L(X) <=> (bye WA)[8e L(y)^(3ze WA)(yE L(z) & yRx z)]. Proof:^. Let S>yE L(x). Let y E WA such that SE L(y). We need to show that there is a ZE WA such that yE L(z) and yRx z There are two cases: 25 Chapter 1^ i) (13 a>(3)[y = (L(x) a> p,a>13)]. Since x, y E WA, we have El(L(x)) g 111(L(y)) by Lemma 1.3, whence o [L(y)] c L(x). So SE L(y) yields 0SE L(x), whence 8> yE L(x) gives oyE L(x) by [PD]. So y>10 L(x), whence there is a UE MAXA such that ^(L(x)) S u and ye u by Lemma 1.1. Then 0(L(x))= O(0*) (from 1.3) yields ^(A*) g u, that is, UE 10(A * ) I A . Let z= (u, 0). Then zE WA by the definition of WA. So we have a ZE WA such that ye L(z). Then (y, z )E R. by the definition of that relation. This is sufficient. ii) (304)[Y=(L(x) ce>p,a>(3)]. Then L(x) ce>pe MAXA was chosen using Lemma 1.1 such that EXL(x))g L(x)ce>p, ae L(x)a>0 and rnL(x) ct>13 = 0, where r = {8 I (3y)[ 8>ye L(x) & ^(L(x)) F A y--)13 ] ). Since 8> yE L(x), it follows that ^(L(x))1/ A y--> 0, since otherwise SE I' by def. r, contrary to the fact that SE L(y) and 1 - nL(x) a>0 = 0. Now, XE WA gives ^(L(x)) = OW) by Lemma 1.3. So we have El(A*)W A y --) 0, whence 0(0*)u { y, -101 is Aconsistent. So by Lindenbaum's Lemma, there is a z= (L(z), 0) E WA such that yE L(z) and 13e L(z). Then we have . Suppose that (y,z)E R Fx by the def. R Fx as ' required. 8>ye L(x). We need to show that there is a yE WA with SE L(y) such that for all zE WA, yRx z ye L(z). Since XE WA, we have L(x)E I ^(A * )1 A , whence 8>ye L(x) yields (L(x)8,7,8>y)E WA by the definition of WA. Also, L(x) 5>y is given by 1.1 such that SE L(x)8,7. Then it suffices to show that L(x) 8>yRx z^ye L(z)^(zE WA) Towards this, let ZE WA such that L(x)8,1,Rx z. Then we have L(x) 5>yR x z by def. R x. Then the definition of R x ii nsures that ye L(z), as required to complete the proof of Lemma 1.4.^ ■ 26 Chapter 1^ Now, we can define our desired model Jtot A to be (WA,R , V), where V: d• --* 2 w e is defined by : XE V(p) .> pE L(x)^(xe W°). MA Then, as usual, V is lifted to a valuation II 11 ^of Fma(0) with the desired property that Truth Lemma (L5): II a 3MA = {xe WA I a e L(x)} . Proof: By induction on the complexity of formulae, using the properties of maximal consistent sets for —) and 1 and Lemma 1.4. for the wedge. ^■ Theorem 1.6 ("Determination of JS 5 "): lis5 a if, and only if, a is valid in all frames. Proof: Soundness: For any frame F, IF = fp! F 1 131 is a prenormal dyadic logic which includes S5, so JS 5 c EF , i.e., F a^Fa. Js5^•^• Completeness: Suppose that fis5a. Then -1a is JS 5 -consistent, so by Lindenbaum's Lemma there is a A * E MAX05 with { -ta} c A*, and thus, ae A*. Then taking JS 5 as the value of A in the construction of ,M °, the Truth Lemma guarantees that A4 (tea} falsifies a at A*, i.e., ,A4 (tea) V A,, a. Hence, ,A4 (-l a) V a, as required. ■ The above construction is admittedly messy. In Chapters 5, 6 and 7 more elegant constructions yielding uniform completeness results for extensions of normal logics are presented. However, the present construction illustrates in a straightforward fashion the apparent need, in modelling weak dyadic logics, to replace the traditional canonical universe by a "copy" in which it is not the case that distinct points are differentiable 27 Chapter 1 according to the formulae they contain. The constructions to come can be viewed as implementations of more sophisticated coloring schemes. Still, we can squeeze one more determination result out of the above construction. As noted in the introduction, necessitation for [N*]^T =df T >T corresponds under [U>] to the first order condition on g.i. frames (W , R) that dom(R,v ) = W for all w in W, or, Vx b'y 3z .R x y, z . To establish that JS 5 +N*, the smallest natural logic containing S5, is determined by the class of universal g.i. models whose frames satisfying this condition, we simply note that if A of the construction of MA includes [N*], then by Lemma 1.4, for every XE WA, (VyE WA)[T E L(y)^Oze WAXT E L(z) & YR x z)] • More on normal dyadic logics in the next chapter. 28 Chapter 2^ Chapter 2: Presenting JK and Some First Degree Extensions. In this chapter, I present JK, the weakest of the normal dyadic logics studied in this essay. Our main goal is to make some proof theoretic remarks concerning JK and some of its first degree extensions. Let JK be the smallest normal logic. Recalling Chapter 0, we have : [1RM] JK " —) 0^!if( (0>Y . - 4. a>Y) , I-A (y> a .-4. y>13). [RMi]^Ia-413 J1C^ [DS]^JK (a>y .A. (3>y) ---> ((a v [3)>y) DT [N]^I=df 1>1 JK^ [N1^JK © T^=df T>T . In Chapter 1, we saw that [RM] is an [RPL] consequence of [1RM], and that [K] is an [RE] consequence of [ LM], while [1M] is itself derivable by [RPL] and [RE] from [DS] and [1RM]. This suffices to show that (JK) ° is the monadic logic K. JK may also be characterized as the smallest dyadic logic which is closed under [RN] and [RMT] and includes [DS] (a>y .A. 0>y) --> ((a v (3)>y) [J,1■4] 1(a-> 13) -4 ((3^.—>. a>1') [N*] 1:11 =df T>T . Gone with U■411 is the principle 29 Chapter 2^ [PD]^I - (a>(3)---,(0a--> o (3) which functions as an important "consistency principle" in natural dyadic logics. Note that [PD] is "hypothetical syllogism" for the special case involving logically false consequents: (a>(3 .A. 13>J) -9 a>1 . Recall from Chapter 0 that a dyadic formula is called first degree just in case it contains no nested occurrences of >. Some first degree principles extending JK to be studied in this essay are: [CON] [PS] [PW] [SI] [TRIV] [TRIV 1] [AD] [C*] (Man 13 -4 o(ocA(3) [m1] Ea —) Da [1:132] 13 a [GI] (oa--4E1(3)—)(a>(3) [CP] (oa—f3)—>(a>13) ) -Oa 30 Chapter 2^ [NE]^(a>(0 v y) .A. 7>i) -+ (CC>(3) and, of course, [MT]. Here are some order-theoretic principles relevant to comparison semantics: [TR]^(a> 0) -4 ((3 >7 .-4. a>Y) [STR]^(a> (R v 7)) -4 (PS .--.). a>(8 v y)) [CONNEX]^(a> (3) v ((3>y) [DIS]^(a>(13v7)) -4 (a>f3 .v. a>7) . A few proof theoretic remarks on the comparison principles: Both [IM] and [MT] are [RPL] consequences of [TR] ("transitivity") plus [PS]. Conversely, [TR] is an [RPL] consequence of [PW] ("preservation of the wedge") in the presence of either [MT] or [1M]. [STR] ("strong transitivity") yields [TR] by [RE]. But note that [STR] is obtainable from [TR] in the presence of Lewis's [DIS] by using [RMT]. Alternatively, [DIS] is obtained from [STR] in the presence of [CONNEX] as follows: Let A be a normal dyadic logic which includes [STR] and [CONNEX]. Then, By [STR] 0. FA (a>((3 v y)) A (0>7) 1. FA (a>(13 v y)) A (11>i3) .-*. («>(13 v 13)) By [STR] 2. FA (0>Y) v (Y>13) By [CONNEX] 3. }-n(a>(13 v Y))--3(«>(13 v p) .v. a>(7 v l)). 0, 1, 2 by [RPL] 4. FA(a>(13 v y))—qa>13 .v. a>Y). From 3 by [RE], .-9. (a> (Y V Y)) 31 Chapter 2 as required to show that A includes [DIS]. Surprisingly, [NE] is equivalent to [MT] : Let A be a natural dyadic logic. Then 0.^FA ^ 1 7 .- ^((13 V y) -> 0)^ - By [RM] 1. F A ^((0 \ '0 -43 ) -) (a>(3 v 7) .-). (a>13))^[Mt] 2. FA ^ny -4 (a> ((3 v 7) • 4 . ( a>(3))^From 0, 1 by [RPL] - 3.^FA (a>((3 v y) .A. y>1) -4 (a>13),^From 2, [RPL], df. ^ and [RE] since anis equivalent to 'pl. This suffices to show that A includes [NE]. Conversely, let A be a normal dyadic logic and assume that A includes [NE]. Then 0.^I-. A y>a .--). y>((an(3) v (aA 1(3)) - [RE], [RPL] 1.^FA (7>(an(3 .v. an -10)) A (an -1(3)>1. .->. y>(anri) [NE] 2. 3. FA (an -11)>1 .4-4. 0(a )13) -- FA ^(a-43)A(y>a) .-4. 7>(ocA(3) 4.^FA ^(a->13)A(y>a) .--). y> f3 [RPL], df. ^ and [RE] 0, 1, 2, [RPL] 3, [RMT], [RPL] which is sufficient to establish that A includes [MT]. Since [NE] is an [RE] consequence of [STR], it follows that [MT] is a consequence of [STR]. Note that the principle of "implicational reflexivity" [WIZ] a >a 32 Chapter 2^ is obtained directly from [CONNEX]. In (pre)normal dyadic logics, [PS] is equivalent to [WR]. To demonstrate: Given [PS], obtain El(a---) a) by [RPL] and [RN]; then a>a follows. Conversely, given (13>(3), assume D(a---> 0). Then (a> [3) follows by [1M]. Note that many of the comparison-theoretic principles are consequences of the "triviality" principles. For example, [CONNEX] follows from [TRIV1], and [STR] follows from [TRIV]. [C*], the "aggregation principle" for El is an instance of [AD]. [IB1] is a consequence of [MI] : Let A be a natural dyadic logic. Then 0. FA Oa —1:1(T--)a) By [RE] and [RPL] 1. FA T>T By [N*] 2. FA D(T--)a) * QT>T)--4(1 >a)) by [Mr] 3. FA 0a--->(T>cc) From 0, 1, 2 by [RPL] — - as required to show that A contains [1131]. Further, [GI] is equivalent to [IBI] : Let A include [1131]. Then: 0. FA a>1^a>I3 By [RMI], [RLP] 1. FA-1(a>f3)--) o a From 0 by [RPL] and df. 2. FA 013 --> (T>0) By [IB1] 3. FA 00 -> From 2. by [1RM] and [RLP]. 4. 1-A-1(a>13)^o (a> 0) By [RPL], Chapter 2 ^ 33 5. FA -1(a>(3) ---> (Oa A 0'(3)^From 1 and 4 by [RPL] 6. FA (0a —4E1(3) -4 (a>(3) ^From 5 by [RPL] and df. 0, as required to show that A includes [GI]. Conversely, let A include [GI]. Then: 0.^FA (0T A D(3) -- (T> (3)^By [GI] and [RPL] A ^(i) —* (T> ()^From 0 by df. 0, df. ^ 1. FA ( 1 ^1 2. FA 1:11-->(T>(3)^By [RMT], df. ^ and [RE] 3. FA ^1 v -Ca^[RPL] 4. FA ^f3 -- (T>(i)^From 1, 2, 3 by [RPL], - as required to show that A includes [II3 1]. [IB2] is equivalent to the elegant preservation principle (a> 0) --) (Da >^(i). — To see that [IB2] is a consequence of the given principle, suppose it. Then 0.^FA (T>(3)---)(01 --400)^By [ 1 B 2 ] - 1.^FA CID -----) DP^From 0 by [N], df. 0 and [RPL], as required to show that A includes [IB2]. Conversely, let A include [IB2]. Then: 0. FA ^a->^(T-xa) By [RM] and [RPL] 1. FA ^(T->a)-q(a>13)-->(T>(3)) By [1,1‘4] 2. FA Oa .-*. (a>(3)->cr>(3) From 0, 1 by [RPL] 34 Chapter 2 3.^FA (a>13)—“Da--3 0 (3) From 2 by [RPL], df. 4.^FA (a> I3)--> (Cla-40(3). From 3 by [IB2] and [RPL], El as required to show that A includes the given principle. Since I have occasion to refer to Lewis' V-logics, it might be helpful to present one of Lewis' axiomatizations of V, the smallest V-logic, and characterize it in our terms. In his [15], Lewis characterizes V as the smallest dyadic logic which contains [TR] (there called "Trans") and [CONNEX] and is closed under the Rule for Comparative Possibility : 1- v (a41)v... v(a>13 n ) (n 1) F v a —030./...4n ) Notice that [DIS] is a consequence of the Rule for Comparative Possibility in the presence of [TR]: 0.^1-v ((3v7) --*(0v7) ^ ^ [RPL] 0, Rule Comp. Poss. 1. 1-v (0v7)>0 .v. (13 v7)>7 2. 1-v a>(13v7) . -4 . ((i3 v -Y)>(3 .—>. a>0) [FR] 3. I-v a>((3vy) .—>. (((3vy)>y .—>. a>y)^[TR] 4. I-v a>((3v7) --> ((ct>(3) v (a>7)) ^ 1, 2, 3, [RPL] We can characterize V in our terms as the smallest natural logic containing [PS], [CONNEX], and [DIS]. Since [MI] is a consequence of [PS] in the presence of [TR], and [PS], equivalently [WR], is a consequence of [CONNEX], this logic may be denoted by 'JK[CONNEX] [DIS] [TR]'. 35 Chapter 2^ For convenience, we shall refer to JK[CONNEX][DISNTR] as A. To see that V is a sublogic of A, we need only show that the Rule of Comparative Possibility is a derived rule of A: Towards this goal, let n^1 and assume FA a --> (131v. . .v (3 1). Then 0. FA a -9 (Pi v. . .v f3„) Assumption 1. FA ^(a —> (Pi v. .v PO) 0 by [RN] 2. FA a>.(131 v. . .v (3 n ) 1 by [PS], [RPL] 3. FA . (a> (31) v... v (a> (3 n ) 2 by [DIS], [RPL], as required. Conversely, since [DIS] follows from the Rule for Comparative Possibility and [TR], to see that A is a sublogic of V, it suffices to show that JK is a sublogic of V. Now, [1RM] and [RMT] are obvious consequences the Rule of Comparative Possibility and [TR]. And [N] and [N*] are obvious consequences of the Rule of Comparative Possibility. Hence, it suffices to show that [DS] is derivable in V: 0. Fy 1. 1- v (av(3)>(avi3) 2. Fy (av(3) >a .v. (av(3)>13 1, [DIS], [RPL] 3. Fy (av(3) >a^(a>y.-4.^v(3)>Y) [TR] 4. Fy (avr3)>13 .-*. (13>T.—>. (ocv0)>Y) [TR] 5. Fy (a>'Y .A. (>Y)^(0/13)>Y 2, 3, 4, [RPL] (avO) ---> (av(3) [RPL] 0, Rule Comp. Pos. This concludes our presentation of the dyadic logics to be studied in this essay. 36 Chapter 3^ Chapter 3: Some Semantic Correspondences. This chapter establishes some purely model-theoretic correspondences between first degree extensions of JK and first order sentences in a single ternary relation R. Let L be a first order language (with identity) whose only nonlogical symbol is the ternary predicate symbol R. In accord with the convention of Chapter 0, we will write R x y z for R (x ,y , z). Thus, frames F = (W, R) are first order structures for L (or, Lstructures). The double turnstile will serve both its customary role of expressing truth and validity of dyadic formulae, as well as the concept of truth on an L-structure for Lsentences: when A is an L-sentence, we write F k A just in case A is true on F. The context will usually serve to disambiguate this double usage. Given a formula a and an L-sentence A, we say that a corresponds to A just in case, for all frames F , F k A a F a, that is, just in case a and A determine (in their own ways) the same class of L-structures. We shall let the informal expression 'yE dom(Rx )' denote the L-formula (3z)R x y z and '(dye dom(R x))A' denote the L-formula (Vy )(ye dom(R x) --> A). We associate an L-sentence with each of the first degree extensions of JK presented in the previous chapter: 0. [MI]^^(a—> 13) (y> a .—>. y>13) (Vx)(VY)R3z)(Rx z y) 1. [CON]^-ID 1 (Vx)(3Y)(3z) Rx y z -4 - yE dom(Rx)] 37 Chapter 3 2. [PS] ^ (a (1)^(a> 13) (Vx)(VyE dom(R))R xy y 3. [PW] (a> po ^(a -* (3) (Vx)(VA(Vz)(Rx z --> y =z) 4. [SI]^^(a-4 0) +4 (a> f3) (Vx)(VyE dom(R))(Vz)(Rx y z<-> y=z) 5.[TRIV1]^(a—> 0) -4 (a> [3) (Vx)(dy)(ye dom(Rx) (y=x 6. [TRW] A RX X x)) (a>13) <-4 (a (dx)( 5IY)(Vz)(Rx Y z <->x=y=z) 7.[AD]^((a> 13) A (a>y)) -4 (a> (f3Ay)) (Vx)(VyE dom(R x ))(3!z)(Rxy z) 8. [C*]^(aotA 00)^©(a n(3) (Vx)(VyE dom(Rx))(3z)(Rxy z n (3u )(k/f)(Rxu z' <-> z=z') 9. [IB1] ^ a 4 ma - (dx)(klyE dom(Rx))(3zE dom(R))Rx y z 10. [1B2] a -->Cla (Vx)(Vy€ dom(Rx))(3u)(\lz)(R x u z H y =z) 11. [CP] (0a-><>(3) -> (a>13) (Vx)(biy, zE dom(Rx))Rxy z 38 Chapter 3^ 12. [STRJ ^ (a>(3vy))—> (p>8 .-->. a>(8 vy)) (Vx)(VY)R3z)(Rxz y)---)yE dom(Rx)] A (V x)(Vy)(V z)(V u)[(Rxy z) -4 (Rxzu-4RxY 01 13. [CONNEX]^(a> (3) v (0>y) (Vx)(Vy,ze dom(Rx))(Rxy z v RxzY) 14. [DIS] ^ (a> ((lvy)) —) ((a>f3) v (a>y)) (Vx)(Vy,ze dom(Rx))[(Vu)(Rxyu -3 Rxz u) v (V u)(R xz u -4 12xY14)1 First order principle 0. will be refered to as "pointwise seriality" and is equivalent to the condition that the range of Rx be a subset of the domain of Rx. Pointwise seriality is a feature of comparison frames (indeed, of any frame where the Rx are reflexive), and, in particular, of Lewis' frames for his V-logics. Note that first order principle 14., here called "mutuality", is a property of connected, transitive relations. Logics containing the associated dyadic principle [DIS] will be called "transparent", since [DIS] is one of the features of material implication which is not shared by strict implication. As we shall see, although [DIS] is undesirable as a principle governing implication, it plays an important role in simplifying the presentation of dyadic logics (and, in particular, of V-logics). Theorem 3.0: Dyadic principles 0- 7, 9 - 14 inclusive correspond to their associated first order principles. Proof: By the following lemmata: Let F = (W, R) be a frame. Then, 0. F k (V x)(VY)[( 3 z)(Rxz y) ---> y E dom(Rx)] <=> F k o(a > 13) > (y> a .--->. 7> () • — — 39 Chapter 3^ Proof of 0:^Assume that F (Vx)(VY)[(3z)(Rxzy)—> y.€ .dom(Rx)]. Note that this is equivalent to the condition that mg(Rx) S dom(Rx). To see that F ^(a^(3)^(y>a .-->. y>13), let V be a valuation and xe W such that (W, R,^6 ^(a—)p), y>a. Claim: (W, R, V) 6 y> 13. Pf.: Let y E dom(R x) such that (W,R,V) y. We want to find a 13-point that y bears Rx to. By assumption, there is a u such that Rxy u and (W,R ,V) 6 cc. Since (W,R,V) D(a —> (s), it follows that (W,R,V) (3. Hence, u is our desired (3-point . This is sufficient. Suppose that F^x)(V y)[(3z)(R x z y) —4 y E dom(R x )] Let Pl,P2,P3E distinct. By assumption, there are x,y ,ze W such that Rxy z but ze dom(Rx). Let V be a valuation based on F such that all of the following hold: i) V(pi) = {z} ii) 17(P2) = 0 iii) V(P3) = {y}. By i), there is no UE dom(R x) such that (W,R , V) 6 pi. Hence, we have (W,R,V) 4 El(Pi —> P2). By iii), y is the only p3-point in dom(R x), whence Rxy z and i) yield y in dom(Rx ) is a P3-point. But by ii), there is no p2-point (W,R,V) P3>Pi. By^ for y to bear in Rx to, whence (W,R ,^bk P3 >P2. This suffices to establish that (W,R , V) ix ^—>p2) (p3>pl .—>. p3>p2), and completes the proof of Lemma O. ■ 1. F k (Vx)(3Y)(3z)Rxy z^<=>^F^ ID _L. -- Proof of 1: Trivial. 2. F k (Vx)(VyE dom(Rx))Rxyy <=> F^---> (a> (3) • 40 Chapter 3 Proof of 2: . Assume that F (V x)(Vy F E dom(Rx ))Rxyy. We need to show that k 1:1(a—)(3)—)(a>(3). Towards this goal, let V be a valuation and XE W such that 6 ^(a—> 13). It suffices to show that (W,R, V) 6 a> 0. Towards this, let yE dom(R) such that (W,R, V) a. Since (W,R, V) 6 ^(a —> (3), it follows that (W,R,V) 13. But by assumption, Rxyy. Hence, (W,R,V) 6 a>13, as required. (W,R, V) . Suppose that F bk (Vx)(Vy€ dom(R x ))Rx yy. We need to find an instance of [PS] which is not valid on F. Towards this, let p1,P2€ O. By assumption, there is an XE W and a yE dom(Rx) such that not Rx yy. Now we let V be a valuation based on F such that both of the following hold: i) v(pi) = {y} ii) v(p2)= fyl By i) and ii) P1 -P2 is true under V everywhere in dom(R x), whence (W,R ,V) ^(p1 —q, 2). Further, y E dom(R x) such that yE V(p1) but for all z such that R x u z, zi V(p2). Hence (W,R , V) ik P1 >P2. Then (W,R,V) 14 1=(P 1 -->P2) -4 (P1> P2), as required to complete the proof of Lemma 2.^ ■ 3.^F k (V 4(9' y)(V z)(R x y z -- y= z) <=:. F k (a>(3) —> o(a->(3). Proof of 3.:^. Assume that F k (V x)(Vy)(V z)(R x y z —4 y = z). We need to show that F k (a>f3) ---> ^(a—> 13). Towards this goal, let V be a valuation and XE W such that (W,R,V) 6 a> f3 . It suffices to show that (W,R,V) 0(a —>13): Let yE dom(Rx) such that (W,R,V)^a. Since (W,R, V) R x y z and (W,R,V) k 6 a> 13, it follows that there is a z such that 0. Then by assumption, y is such a z. Hence, (W,R , V) a —) 13. Since y E dom(R x) was arbitrary, it follows that (W,R , V)^Cl(a --) 13), as required. 41 Chapter 3 . Suppose that F V (V x)(V y)(V z)(R x y z —4y= z). We need to find an instance of [PW] which is not valid on F. Towards this, let p1,P2E 0 distinct. By assumption, there are x,y,zeW such that Rxy z but y #z. Let V be a valuation based on F such that both of the following hold: i) V(Pi) = {y} ii) V(P2) = rneRx) — {A Claim 1: (W,R,V) 6 pi >p2. Pf.: By i), y is the only pi-point in dom(R x). By assumption, Rxyz . And by ii), ZE V(p2), since ze rng(R x) and y# z. This is sufficient to establish the Claim. Claim 2: (W,R,V) IA 1:3(pi -4/02). Pf.: Since yE dom(R x) and ye V(pi) it suffices to observe that, by ii), y 0 V(p2). From Claims 1 and 2, we have (W,R , V) vIx ( nv- 1> P2) --> D(P1 -4 /32), as required to complete the proof of Lemma 3.^ 4. F ■ k (Vx)(Vy E dom(R))(Vz)(Rxyz <--> y=z) <=> F k 0(a-0) <—> (oc>(3)• Proof of 4: By 2 and 3 above.^ 5. F = (Vx)(Vy)(yE dom(Rx) -4 (Y=x Proof of 5: . Assume that F A Rxxx)) <=> F ■ k (a -4 0) —) (a> (3). = (Vx)(Vy)(yE dom(Rx) -4 (y=x A Rx xx)). We need to show that F k (a —>I3) —4 (a>13). Towards this goal, let V be a valuation and XE W such that (W,R , V) 6 a —>13. To see that (W,R, V) 6 a > 0, let yE dom(R x) such that ( V ,R ,V) a. Then by assumption, y=x and R x x x. From y=x it follows that (W, R, V) R. Then Rxy y yields (W,R, V) 4 a> f3, as required. 42 Chapter 3 . Suppose that F ik (Vx)(Vy)(yE dom(R x ) -4 (y=x A Rxxx)). We need to find an instance of [TRIV11 which is not valid on F. Towards this, let pi,p2e 0 distinct. Then by assumption, there are xe W and yE dom(Rx) such that either x #y or not R x x x. We separate these two cases: Case 1: x *y. Let V be a valuation based on F such that the following hold: i) V(P1) = {Y} ii) V(p2) = 0. By i), x*y yields (W,R ,V) 4 P1-4P2 . But i) and ii) yield (W ,R ,V) bk P1 >P2, since yE dom(R x ). So (W,R,V)iik (P1 —>P2) - (Pi >P2). Case 2: not Rxx x. From Case 1, we may assume that x=y. Let V be a valuation based on F such that the following hold: i) V(pi) = {x) ii) V(pz) = (x). By i) and ii) we have (W,R ,V) 4 P 1 -- )1,2. But i), ii) and Rx x x yield (W,R , V) Pi>p2, since XE dom(Rx). So (W,R,V) ix (13 1 —>P2) --> (P1>P2)• In both cases, then, we have the desired falsifying valuation V. This completes the proof of Lemma 5.^ ■ 6. F k (V x)(V y)(V z)(Rxy z i 4 x =y =z) < F ^(a>13) +4 (a --)0.). Proof: Similar to 5. 7. F k (d x)(V y € dom(R x))(3!z )(R xy z) <#. F k ((a>0) A (a>7) —> a>(i3 AY)). Proof of 7: . Assume that F k (Vx)(VyE dom(R x))(3!z )(R xyz). We need to show that F k ((a>() A (0C> y) .-* . a> 03 A y). Towards this goal, let V be a valuation and 43 Chapter 3 xe W such that (W,R, v) 6 (a>(3)A(a>y). To see that (W,R, V) j a> ((3 A y), let y E dom(Rx) such that (W,R , V) a. Since (W,R , V) 6 a >13, a> y, it follows that there are z, z' e W with Rxyz, RxY z", (W ,R ,V) 613 and (W ,R ,V) k. y. Then by assumption, z=z', whence (W,R, V) kz f3Ay. Since y was an arbitrary a-point in dom(Rx), it follows that (W,R , V) 6 a> (J3 A y), as required. . Suppose that F ik (Vx)(VyE dOM(Rx ))(3!Z )(RxyZ). We need to find an instance of [AD] which is not valid on F. Towards this, let pi,p2,p3E 0 all distinct. By assumption, there are x,zi,z2E W and y E dom(Rx) such that Rxyzi andRxyz2 but zi #z2. Let V be a valuation based on F such that the following hold: i) V(Pi) = {y) ii) V(p2) = { z i ) iii) V(p3) = { z2) . Claim 1: (W ,R , V) 6 (P i> P2) UE A (Pl> V(pi). Then by i), u= y. By ii), ZiE p 3). Pf: Let u E dom(Rx) such that V(pi) (i= 2, 3). By assumption, R x u zi (i=2, 3). This suffices to establish the Claim. Claim 2: (W,R , V) lit pi > (p2Ap3). Pf: Since y E dom(R x) with yE V(p 1), it suffices to observe that, by ii) and iii), there is no UE W such that UE V(p2)nV(p3). From Claims 1 and 2, we have (W,R, V) ix (pi>p2)A(p1>p3).--3•p1> (P2Ap3), as required to complete the proof of Lemma 7.^ ■ 8. F k (Vx)(Vye dom(Rx))(3zE dom(R))Rxy z <#. F k Cla ---> Da. Proof of 8: . Assume that F k (Vx)(Vye dom(Rx))(3zE dom(R))Rxy z. To see that F k Da -4 cla, let V be a valuation and X E W such that (W,R , V) Oa. Let y E dom(Rx). By assumption, y bears R x to some UE dom(Rx). Since (W,R, V) Ea, it follows that (W,R,V) 6 a. Hence, (W,R, V) 6 T>a. This is sufficient. 44 Chapter 3 . Suppose that F V (V x)(V ye dom(R x ))(3zE dom(R x))R x y z. Let p E 0. By assumption, there are xE W, ye dom(R x) such that y does not bear Rx to any ZE dom(R x ). Let V be a valuation based on F such that V(p)=dom(Rx). Then clearly, (W,R ,V) Op. Further, y does not bear R x to any point in V(p), whence (W,R , V) T >p. This suffices to show that (W,R,V) bk Oa -*Da. ■ 9. F k (V4(`9/yE dom(Rx))(30(Vz)(Rxuz <-) y=z) t=> F k ©a—)Da. Proof of 9: =. Assume that F H3(Vx)(Vy e dom(R x))(3u)(Vz)(Rx u z <--->y=z). To see that F kaa -40a, let V be a valuation and XE W such that (W,R , V) ®a. Let y E dom(Rx). By assumption, y is the unique R x-successor of some (W,R,V) UE dom(R x). Since 6 T> a, it follows that (W,R,V) a. Hence, (W,R,V) kx Oa. . Suppose that F 1(kf x)(Vy e dom(Rx ))(3u)(Vz)(Rx uz <-3 y=z). We need to find an instance of [IB2] which is not valid on F. Towards this, let pE 0. By assumption, there are XE W and ye dom(R x) such that y is not the unique R x-successor of any UE W. There are two cases: Case 1: yo rtig(Rx). Then let V be a valuation based on F such that V(p)= nig(Rx). Then clearly, (W,R , V) 6 T>p but (W,R , V) bk Op. Case 2: Otherwise. Then let V be a valuation based on F such that V(p)=rng(R x ) - {y 1 . Claim: (W,R, V) 6 T>p . Pf.: Let uE dom(R x ). Since y is not the unique R x successor of u, there is a ZE rng(R x )- fy 1 such that Rx u z. By the choice of V, we then have (W,R , V) kz p . Since u was an arbitrary element of dom(R x ), this suffices to establish the Claim. 45 Chapter 3 Since ye dom(R x) but ye V(p), we have (W,R , V) Li Op. In both cases, then, we have a valuation V based on F such that (W,R, V) lk Da —1:1a. This completes the proof of the Lemma. ■ 10. F k (Vx)(Vy,ze dom(Rx)) Rxyz^<=> F 1 (0a > 00) > (a>(3). — — Proof of 10:^Assume that F k (Vx)(Vy, ze dom(Rx))Rxy z. To see that F (0 a 0 0) 4 (a> (3), let V be a valuation and xe W such that (W,R , V) k <>a ---> 0 0. - Let ye dom(R x) such that (W,R , V) a. Then (W,R, V) o a , so by assumption (W,R ,V) 4 o 0, that is, there is a ue dom(R x) such that (W,R, V) 6 0. By assumption, R x y u. Since y was chosen as an arbitrary a-point in dom(R x ), it follows that (W ,R ,V) 4 a>0. Suppose that F ik(V x) (V y, Z E dom(R x ))R x y z. Let p 1,p 2 E 0 distinct. By assumption, there are xe W and y,zE dom(R x) such that y does not bear Rx to z. Let V be a valuation based on F such that both of the following hold: i) 17(P1) = tY1 ii) V(p2) = {z) • Then clearly, (W,R, V) I Opi, 0p2, whence (W,R,V) l Opl-* 0/32. Further, y is a Pi Point which does not bear Rx to any point in “P2), whence (W,R ,^kk P1>P2. This - suffices to show that (W,R , V) IA (0 a 0(3) ---> (a> .^■ 11. F (vx)(vY)1(3z)(Rxz y) -4 y E dom(Rx)] (V x)(Vy)(V z)(V u)[Rxy z (R x z u —> R x y u)] <=> F 1 (a>(0vy))^(0>8 .-->. a>(Svy))• Proof of 11.:^Assume that 46 Chapter 3^ (a) F (Vx)(VY)R3z)(Rx zy) ye dom(Rx)] and (b) F (Vx)(Vy)(Vz)(Vu)[Rxy z To see that F (a>(f3vy))^((3>8 - 4 (Rx Z U Rxy u)). .- ). a> (8v y)), let V be a valuation and XE W such that (W, R, V) 6 a>a3vy), 13>8. Claim: (W,R, V) 6 a> (8vy). Pf.: Let ye dom(Rx) such that (W,R, V) a. We need to find a z such that Rxy z and (W,R, V) kz 8 vy. By assumption, there is a u such that Rxy u and (W,R,V) 6 13vy. Then one of 13 or y holds at u. If y holds, then z is our desired (8v y)-point u. So assume that (W,R,V) dom(Rx), so UE dom(Rx). Since (W,R,V) 6 13. By assumption (a)., mg(R x) 6 ((3>8), it follows that there is a z such that Rx u z and (W,R, V) kz 8. Then Rxyu and Rx uz yield Rx y z by assumption (b). Hence, z is our desired (6vy)-point. Suppose one of (a) or (b) above fail. It suffices to find an instance of [TR] which is not valid on F. Let pi,p2,p3E^all distinct. Claim: 14 (P1>P2) (P2>P3^P1 >P3). Pf•: There are two cases: Case 1: (a) fails. By assumption, there are x,y, ZE W such that R xy z but ze dom(Rx). Let V be a valuation based on F such that all of the following hold: i) v(pi) = fyl ii) v(p2) = (z} iii) V(p3) = 0. By i) and ii), Rxy z ensures that (W,R,^6 pl> p2. By ii), there are no p2-points in dom(Rx), so (W,R,^6P2>P3. By i), y is a pi-point in dom(Rx). But by iii), there is no p3-point for y to bear Rx to, whence (W,R, V) it pl>p3, as required. 47 Chapter 3 Case 2: (b) fails. By assumption, there are x,y,z,u E W such that Rx y z and R x z u but not Rxy u. Let V be a valuation based on F such that all of the following hold: i) v(pi) = fyl ii) V(P2) = {z} iii) v(p 3 ) = { u ) . By i) and ii), R x y z ensures that (W,R ,^6 P1> P2 By ii), Rxz u ensures that (W ,R,V) 6p2>p3. By^y is a pi-point in dom(R x ). But by iii), not Rxy u ensures that y bears Rx to no P3-points, whence (W,R , ^It P1> P3. This suffices to establish the claim, and completes the proof of Lemma 11. ^ ■ 12. F k (Vx)(Vy,ze dom(Rx))(Rxyz v Rxzy)^a^F k (a>13)v 03>7)• Proof of 12:^Assume that F k (V x)(Vy,z E dom(Rx))(Rxyz v Rxzy). To see that F (a> (3) v((3>7), let V be a valuation and XE W such that (W,R, V) ik a> P. Claim: (W,R,V) 613>a. Pf.: Let yE dom(R x) such that (W,R,V) (3. We want to find an a-point that y bears Rx to. By the assumption that (W,R , V) a> an a-point ue dom(R x) such that not Rx uz for all ZE if 0, there is n, and hence, not Rx uy. Then by the assumption that Rx is strongly connected on its domain, we have Rxy u . Hence, u is our desired a-point, and the Claim is proved. Suppose that F V(Vx)(Vy,ZE dom(R x ))(Rx y z v R x z y). Let p1,p2E 0 distinct. By assumption, there are x,yE dom(Rx ) such that neither R x y z nor R x z y. Let V be a valuation based on F such that both of the following hold: i) v(pi) = { y } ii) v(p2) = f zl. 48 Chapter 3 By i), there is a pi-point, namely y, in dom(R x), which, because of not Rxy z and ii), is not R x -related to any pa-points. Hence, we have (W,R,V) bk p 1 > P2. By symmetric reasoning, (W,R, V) 14p 2 >p i . This suffices to establish that (W,R, V) ik (P1>P2) v (P2>P1), and completes the proof of Lemma 12. ^ ■ 13. F k (Vx)(Vy,ze dom(Rx))[eclu)(Rxyu -4 Rx zu) v (Vu)(Rx zu ---> RxYu)1 <=>^F k ((a> (0 vy)) —> ((a> (3 v (a> y)) Proof of 13.:^. Assume that u)(Rxzu x zu --> R x y u)]. F k (V x)(VY,z€ dom(R x ))[(Vu)(Rxyu -4 Rx zu) v (Vu)(R To see that F k (oc>((3vy)) -+ (a> 0 .v. a>y), let V be a valuation and XE W such that (W ,R ,V) 6 a> ((3 v y) and also (W,R , V) K a> 0. Then it suffices to show that (W,R, V) 6 a>y. Towards this goal, let yE dom(R x) such that (W,R,V) I a. Claim: There is a y-point s such that Rxy s . Pf.: Suppose not. Then, y bears R x to some f3-point z, since (W,R, V) 6 a>((3vy). By the assumption that (W,R, V) it a> 13, there is an a-point ue dom(R x) such that not Rx uz. Again, since (W,R , V) a> ((3 v y), there is a y-point g such that Rx ug. Now, z witnesses that it is not the case that u is RA-related to everything that y is, so by the assumption that Rx is "mutual" on its domain, we know that y is RA-related to everything that u is. Hence, we have Rx y g. Taking g as our desired s, we are done. Since y was an arbitrary a-point in dom(R x), the desired result that (W,R, V) 6 a > y follows from the Claim. . Suppose that 49 Chapter 3 F (Vx)(Vy,zE dom(Rx))[(Vu)(Rxyu ---)R x zu) v (Vu)(Rxzu --) Rxyu)]. Let pi,p2,p3E fi distinct. By assumption, there are XE W and y,z e dom(Rx ) such that neither (V u)(Rxy u Rx zu) nor (V u)(Rx z u —) Rx y u). So there are ui,u2E W such that Rxy ui but not Rx z ui and Rx z u2 but not Rx y u2. Let V be a valuation based on F such that all of the following hold: i) V(pi) = (y, z) V(P2) =^} iii) v(P3) = (1421. Then every pi-point in dom(R x) is R x -related to a (p2v p3)-point. Hence, we have (W ,R , 6pi> (p2vp3). But by i) and iii), y is api-point that does not bear Rx to any p3-point. And similarily, by i) and ii), z is a p i-point that doesn't bear Rx to any p2-point. Hence, (W,R , ikpi>p2 and (W,R , V) p >p3. This suffices to establish that (W ,R,V) ii(13 1>P2) v (P1>P3), and completes the proof of Lemma 13. ^ ■ Lemma 3.1: F k (Vx)(VyE dom(R x ))(3z)(Rxyz n (Bu )(Vz')(Rx uz' H z =z)) F^( anci(3)--> Ei(aA (3). Proof: Assume that F k (Vx)(VyE dom(Rx))(30(Rxyz A (3u )(Vf)(Rxuz' H z =z')). To see that (W,R , V) Let V be a valuation and WE W such that (W,R ,^]a A^To (a A (3), let yE dom(R x). Then, by assumption, there is a ZE W such that Rxy z and z is the unique R x-successor of some UE W. Since (W,R , V) 6 T > a, T > it follows that ^ 50 Chapter 3^ (W,R, V) kz a AO. Since y was an arbitrary element of dom(R x), we have shown that (W ,R ,V) 6 T>(aAf3), as required.^■ The converse of Lemma 3.1 does not hold, as we will now show by constructing a g.i. frame on which [C*] is valid but first order principle 8 is false. Let IR be the set of real nunbers and N g IR be the set of natural numbers. Let Pinf(N) be the set of all infinite subsets of N. Since IR has the same cardnality as Pinj(N), let •:1) :IR --> Pinf(N) be an injective, onto map. Let F = OR ,R) be a ternary relational frame, where R g IR x IR x N is given by R= {(x,y, 4 I x = 0, Z E Claim: F k (a OCA El 13 ) -4 0 (aA13) Proof: Suppose there is valuation V and an xE R such that .M= (F,V) It a(ccA (3). Then there is a y E dom(Rx) such that (#) .A4^.A4 R x yz .. zoffall^or z 0 II(311^(zE R). Since dom(R x,) = 0 (w x), it follows that x =0 by def. R. Then by (#), we can partition M^m 4)(y) into two sets A, B such that EMI n B = 0, CPI n A = 0. Since 0(y) is infinite, one of A or B must be infinite. Without loss of generality, suppose A is infinite. Then, since 4) : R --> Pi ni(N) is an onto map and AEPi nf(N), there is a w € III such that 4)(w) = M A. Hence, by def. R and the fact that [IP n A = 0, we have R x w z^z 0 EP M^ (z E R ). Since dom(R x ) = IR , W E dom(R x ). Hence, ,i4 it El 13, whence M bk Ei a A op, as required to show that F k (0 a A0(3)--> el(a A(3) .^■ Chapter 3^ 51 Since every point in F's universe is related to an infinity of points, no point in F's universe is the unique successor of any point in F's universe. It follows that no point in dom(R x) bears R x to any point which is the unique R x -successor of some point in dom(Rx). Thus, F fails to satisfy first order principle 8. Note that the correspondences proven in this chapter are purely model-theoretic, that is, they deal with a notion of definability for classes of ternary relational structures which is completely independent of proof-theoretic considerations. The determination results to come in Chapter 4 provide for the proof-theoretic characterization of many these classes of structures. Chapter 4^ 52 Chapter 4: Prestudy For Model Constructions. In Chapter 1 we saw how the construction of "universal" models for JS 5 consistent sets of formulae seemed to require a universe somewhat larger than the Lemmon-Scott canonical universe of maximally consisitent sets. In particular, we had the misfortune of witnessing a rather ugly "coloring" scheme for expanding MAXjs 5 by adding multiple "copies" of certain JS 5 maxi-sets. In the next chapter, we present a far more elegant construction technique which offers a high degree of uniformity in modelling extensions of JK. In chapter 6, the construction is tailored to model dyadic logics that contain the schema of strong transitivity [STR]. The universes of the models presented in the next two chapters are constructed "on top of" the canonical universe of maximally consistent sets. In preparation, we devote the present chapter to the development of some theory which plays the same role for normal dyadic logic as the study of Henkin sets play in the canonical construction. In this and the next two chapters, the standard results concerning the properties of maximal consistent sets (presented in Chapter 0, pp. 12-13) will be assumed and used without special mention. For the rest of this chapter, let A be a normal dyadic logic. Let w be an element of MA X A. In general, we will use 'u' , ,'x' , . . . (and, of course, 'w') to denote elements of MAXA. Weak Separation Lemma (4.0): Let a, be formulae such that a> 13 w. Let F = { S I (3y)(5>ye w & 1- A y--0) } Then there is a u E MAXA with ^(w) g u such that cce u and Fnu = 0. Proof: Suppose that a> [3o w and, towards a contradiction, for all u OW, a E U Fnu #0. By the definition of deducibility, there are nO and 51, ^, O n E F s.t. 53 Chapter 4^ ^ (w) F A a-(81 v...v 8n ). Let Eon = Oi v...v ö n . Then by Scott's Rule, (*)^w F A 0(a —*8 11 ). Let yi,...,yn be the formulae given by the definition of r s.t. 8i> yi E w and F A yi .—> 3 (1 <_i <_n). Let 14 = yi v...v yn . Repeated applications of [DS] and [RPL] yield (#)^FA ((81>1+i ) A—A ( 8n>711 )) -- (S n >in ) • By [RPL], F A y;> yn , SO [RMT] yields F A (8i> yi) -- (8i> yn), whence 8i> yi€ w gives 81> yn E W (1 5 i 5 n). Then (51> yn) A . . . A (8 n > yn) e w , whence (#) yields 5n> yre e w . Now, F A yn --) 0 by the choice of the yi , so [RMi] yields I-A(a>r) ---> (a>(3). But by [1M], (*) yields w F A (8n>'yn) -4 (a>yn) , SO w F A (8/2>r) -4 (a>(3), whence 8n > y)E w gives a> 0 E w, contrary to the hypothesis of the lemma. ■ Corollary: Let a, 13 be formulae such that a>00 w. Then there is a u E MAXA with ^ (w) c u such that a EU and 8 0 u for all 8 such that 5> [3 E w. 54 Chapter 4^ Proof: Let u be given by the Lemma and 8 a formula such that 8>(3E w. Since F A 13 —> 13, it follows from the Lemma that Se u.^ ■ As in the model construction of Chapter 1, the "canonical" universe of the models constructed in Chapters 5, 6, and 7 will be populated by pairs whose first element is a maximal consistent set of formulae. However, the second element of one of our "pair worlds" will be a set of formulae which bears an intimate logical relationship to the first element. When considering pairs X of subsets of Fma(0), we again use the "projection operators" L(_), RL) so that X .(L(X),R(X)). Let X be a pair of subsets of Fma(0) such that L(X)E MAXA Definition: X is a cut around w iff (CO) a>EtnE w^(aE L(X)^{8i 115.i Srz)) 4 R(X)) (n> 0), where 8/1 is the n-termed disjunction 81v...v 8 n . If in addition X satisfies the condition: (C1)^_LE R(X), then X is said to be initial. Let Cut(w) be the set of all cuts around w. Note that satisfaction of (CO) is equivalent to satisfaction of the condition (CO') F A 13 ----> (81 v. . .v 8n ) & a >13 E W & 81,^, 8 n E R(X)^coz L(X) (n >0) because of [RMi]: Assume X satisfies (CO) and let n> 0 such that X satisfies the antecedent of (C0'). Then F A 13 --> (81v...v 8 n ) plus a>13 E w yields a> (81 v. . .v 8n ) E w 55 Chapter 4^ by [RMT]. So the desired result that ae L(X) follows from (CO). Conversely, assume X satisfies (CO') and let n > 0 and a, 81, ... , 8 n E Fma(1) such that X satisfies the antecedent of (CO) and f si 11 5.i n) } c R(X). By [RPL], 1- A (81v. . .v 8 n) --) (81v.. .v 8 n ), so (CO') plus a>(,,Riv...v8 n)e w yields ao L(X) as required. Following Lewis [15], I'll write "Sin S2" for "Si nS2# 0" (Si,S2 g MAXA). Lemma 4.1: Let a> 0 E w and X E Cut(w) with 0(w) c L(X). Then ae L(X) ^ 10 1 6) I 1[R00]1. - Proof: Assume the hypothesis and that 1[31 nI i[R(X)]I=0. Then there are n. 0 and - 81,..., NE R(X) such that F A 11—) (81v...v8n) • If n =0, then [RMi] yields a>_LE w, whence ^(w) g L(X) gives a€ L(X), as required to establish the contrapositive of the Lemma. On the other hand, if n > 0, then a> (3 EW yields ae L(X) by condition (C0'). In either case, the contrapositive of the Lemma is established. ■ Corollary: Let a>13€ w and X E Cut(w) with ^(w) g L(X). Then, 1 1[R(X)] I is - nonempty. Proof: Since A is normal, A contains [N*] by def. normal. Hence, T> T E W. Since T E L(X), Lemma 4.1 yields I TI rz. 1 -1 [R(X)] I. That is, I n [R(X)] i is nonempty, as ) required. ■ 56 Chapter 4^ Lemma 4.2: Let a> (3 0 w . Then there is an initial XE Cut(w) such that ae L(X) and [3 E R(X). Proof: Suppose that a> (3 e w. Let xe MAXA be given by Weak Separation such that 0(w) gx, a E x and ye x for all y such that y>13e w . Let X= (x,{1,13}) . Claim: X is an initial cut around w. Pf.: (Cl) is satisfied by the definition of R(X). To see that X satisfies (CO), let ye Fma(0), n > 0 and 81, ... , S n e R(X) such that Y> (Si v• • •v 8 n )e w. Since {St, • • • , 8,) c { 1 , 0), by [RE], we have either y>le w or y> (3 E w. In the first case, ye x by the choice of x, that is, by 0(w) gx. In the second case, y> [3 E w, so again yo x by the choice of x. This establishes the Claim. From the Claim, we have XE Cut(w) with the desired properties. ^■ Let XE Cut(w). Say X is maximal at w, or maximal@ w, iff for all sets F of formulae properly extending R(X), (L(X),F) is not a cut around w. Perhaps more explicitly, iff: F D R(X)^(L(X), no Cut(w)^(F c Fma(0)). X will sometimes be called a w-point if it is maximal at w. Let MCut(w) =df { XE Cut(w)I X is maximal@w}. That is, MCut(w) is the set of all w-points. 4.3 (Extension Lemma): Let X1E Cut(w). Then there is an X2E MCut(w) with L(X2)=L(X1) and R(Xi) g R(X2)• Proof: X2 is obtained by simply closing R(X1) under (CO). However, its useful to lay down a uniform procedure for doing this: Recall that 4 denotes the ith formulae in the standard enumeration of Fma(4)) assumed in Chapter 0. Let X2 be (L(X1), F) where F 57 Chapter 4^ is defined as the union of the sequence { Fi I i < w } of subsets of Fma(0) given inductively by: Kt) ro = ^ R (Xi ) (1)^= rl u^if for all n 0, 81, , O n e Fi and all ae Fma(0), a>(41 v Siv ...v On) E W^L(X1). otherwise. By assumption, (L(X1), F0) is a cut around w. And the set of XE Cut(w) with L(X) = L(Xi) is closed under (1). So (L(Xi), Fi)e Cut(w) for all Fi in {Fi I i< co }. To see that X2 is a cut around w, suppose (L(Xi), F) fails to satisfy (CO). Then there are n>0, 01, ..., On e r and a€ Fma(0) such that a> (Si v...v 8n ) E w and a E L(Xi). Let m< co be the least number such that all of Si, ..., O n e Fm . Then (L(Xi), F m ) fails { 1), contrary to the above. To see that X2 is maximal@w, let r CFma(0) such that D F. Let j< co such that (1:0i E r'— r. Since (1:•jo rp. i , by { 1) there are n>_ 0, 81 ..... On E rj , and a E Fma(0), such that a> (4)i v Si - v...v S n )E w but a e L(X1). Since ri cr, we have {4)j, 81, • • O n C F'. Then (L(Xi), I ) fails (CO), whence (L(X1), r) - ■ Cut(w).^ Corollary 1: Let x E MAXA. Then there is an X = , R(X)) in MCut(w). Proof: X1= (x, 0) is a cut around w. Then the Extension Lemma (4.3) guarantees a maximal@w X2 = (x , R(X2)) in Cut(w).^ ■ Corollary 2: Let ^(w) c x. Then there is an initial X=(x, R(X)) in MCut(w). Proof: ^(w) C x ensures that X1= (x , {I}) is a cut around w. Then the Extension Lemma (4.3) guarantees an X2=(x , R(X2)) in MCut(w) with {1} c R(X2). So X2 is initial. ■ 58 Chapter 4^ Lemma 4.4: Let X= (x, R(X)) in MCut(w) with ^(w) cx. Then Xis initial. Proof: Towards a contradiction, suppose _Le R(X). Then R(X) u { I) properly extends R(X), whence X maximal@w entails that (L(X), R(X)u { 1) Cut(w). In other words, (L(X1), R(X)u {11) fails (CO). So, there are n.0, 81, ... , 8 n E R(X) and a aeFma((k) s.t. : a>(1v 81v...v8n )e w and aex. If n=0, then a>_LE w, whence ^(w) cx contradicts aex. If, on the other hand, n>0, then [RMT] yields a>(8,1 v...v 8n ) E w, whence (CO) contradicts ae x. This completes the proof of the Lemma. ■ Lemma 4.5: Let XE MCut(w), and let a, Pe Fma(0) such that F A a-+ P. Then 13E R(X)^a e R(X). Proof: Assume the hypothesis and ae R(X). Since XE Cut(w) is maximal@w, there are ye L(X), n>0, and 81, ... , O n e R(X) such that (*)^y>(av81v...v8n)e w, by (CO). Since 1- A a -4 0, we have F A (a v 81 v. . . v 8 n ) -4 ((3 v Si v. . . v 8 n ) by [RPL], whence (*) yields y>(13v. Si v. . v8 n )e w by [RMT]. Then fle R(X) by (CO), as required. ■ Corollary: Let XE MCut(w), and let a, 0 E Fma(0) such that F A a<---> P. Then 13 E R(X)^4:=>^a E R(X). Proof: By Lemma 4.5.^ ■ 59 Chapter 4^ The definition of a cut around(w) may be simplified by taking n to be 1 always in the special case of transparent dyadic logics. This fact is expressed as the following: Lemma 4.6 (Transparency Lemma): Let A be transparent, that is, let A include Lewis' schema [DIS] (a>([3vy)) ((a>p) v (a>7))• Let X be a pair of subsets of Fma(0). Then X satisfies (CO) just in case it satisfies the condition (COT)^a>8€ w & Se R(X) .^ae L(X). Proof: Since (COT) is a special case of (CO), satisfaction of the latter implies satisfaction of the former. Conversely, asume that X satisfies (COT) and the antecedent of (CO). Let a E Fma(0), n > 0 and Si, R(X) such that a>(Siv^v8n )E W. Repeated applications of [DIS] yield (a>81) v v (a>8 12 )e w, whence a>81€ w for some 1 S i . tz. Then (COT) yields ae L(X), as required. ■ Lemma 4.7: Let A be transparent, X1€ Cut(w) and = V (R(X) I XE Cut(w) & L(X)=L(X1)}. Then (L(X1), 1")E Cut(w). 60 Chapter 4^ Proof: By the Transparency Lemma, it suffices to show that (L(Xi), F) satisfies (COT): Let ae Fma(0) and Se I' such that (a>5)e w. Then there is a XE Cut(w) with L(X) = L(Xi) and SE R(X). Then (COT) yields ae L(X)=L(Xi), as required. ■ Lemma 4.8: Let A be transparent and Xie MCut(w). Let EXT(Xi) = R(X) I XE Cut(w) & L(X)=L(X1)). Then R(X1) = EXT(X1). Proof: (L(Xi.), EXT(X1))E Cut(w) by Lemma 4.7, so by definition EXT(Xi) includes R(X1). Conversely, let yE EXT(X1). Then there is an XE Cut(w) with L(X) = 1-(Xi) and yE R(X). Claim: ye R(X1). K.: Suppose not. Then R(Xi)u Y } properly extends R(Xi), whence Xi maximal@w gives (UX1),R(Xi)u {Y})e Cut(w). In other words, (L(Xi), R(Xi)u {y)) fails (CO). Then, A being transparent, (L(Xi), R(Xi)u {y}) fails (COT) by the Transparency Lemma (4.6), that is, there is an a> Se w such that Se R(Xi) u tyl but a E L(Xi). Now, 6e R(X1), since otherwise a> Se w with ae L(Xi) entails that Xi fails 5 = y. (COT), and hence, by the Transparency Lemma, (CO), contrary to Xi E Cut(w). So, But since XE Cut(w), a>ye w and ye R(X), we have oce L00= L(Xi) by (CO). This contradiction establishes the Claim. By the Claim, EXT(Xi) a R(Xi.), as required to establish the Lemma. ■ Corollary 1: Let A be transparent and Xi,X2E MCut(w) such that L(X1)=1-(X2). Then Xi = X2. Proof: Immediately from Lemma 4.8.^ ■ 61 Chapter 4^ Corollary 2: Let A be transparent and XE MAXA. Then there is a unique XE MCut(w) such that L(X)=x. Proof: By Corollary 1 to the Extension Lemma (4.3) and Corollary 1 to 4.8. ^■ Let F g Fma(0). Say that F is w-closed iff ^3>(Yi v • •.v YOE w & Il • • Yn E r - , .. 13E r^(n >0). Lemma 4.9 (Lemma on w-closure): Let A be transparent and let r be w-closed and either a) F g n[L(X)] or b) -- XE i[F] c L(X). Then Proof: Assume the hypothesis of the lemma. Let ye r. MCut(w). Let r cR(X). To see that yE R(X), suppose otherwise. Then R(X) u {y} properly extends R(X), whence X maximal@w entails that (L(X),R(X)u {T})O Cut(w), i.e., (L(X1), R(X)u (1 1 }) fails (CO). Then, by the Transparency Lemma (4.6), (L(X), R(X) u y} ) fails (COT), that is, there is an a> 5 E W { such that SE R(X)u {y} but a € L(X). Now, So R(X), since otherwise cc>5 E w with a E L(X) entails that X fails (COT), and hence, by the Transparency Lemma, (CO), contrary to XE Cut(w). So, 8 = 'y. Hence, both (a> y)E w and a E L(X). Since yE F, the assumption that r is w-closed yields aE F. In case a), then, a= -113 for some f3E L(X), whence ao L(X) since L(X) is A-consisitent, and in case b), iaE L(X), so once again, ao L(X). In both cases, we have a contradiction.^■ Lemma 4.10 (Lemma on Right Closure): Let A contain the schema of general transitivity [GTR]^(a> (0 vy)) --. (ps .-4. oc>(Svy))• Let XE MCut(w). Then R(X) is w-closed. Proof: Let f3E Fma(0), m >0 and yi, • • •• I'M E R(X) such that 62 Chapter 4^ 0>(yi V...Vym)E W To see that 13E R(X), suppose otherwise. Then R(X) L.) {13} properly extends R(X), whence (L(X1), ROOL) {y} ) fails (CO), since X maximal@w. It follows that there are OGG L(X), n _1:1 and Si, , S n E R(X) u DI such that a> (Si v...v 8 n )e w. We may assume that [1E [8 1 ..... s n } since otherwise { 81, , On} c R(X), whence X fails (CO), contrary to XE Cut(w). Then, by [RE], there is no loss in generality in assuming that (3 = Hence, we have a>((3 v 82v...vO n )E w. Then by [GTR], we have a>(yiv...vym v 82v...v 8 n )E w. Since { yi, , ym , 82,^, S n g R(X) and XE Cut(w), by (CO) we have coz L(X), contrary to OCE L(X) from above (that is, by devine intervention). Q.E.D. Lemma. ■ Lemma 4.11: Let Ti CFma(0) such that I[Fi] is A-consistent and let r2 g r i such – that I 2 is w-closed. Then there is an - XE MCut(w) with n[rd S L(X) and F2 c R(X). Proof: Assume the hypothesis. Since 1[r1] is A-consistent, there is a ze MAXA with – g z. Claim: (z, T2 )E Cut(w). Pf.: We need to check that (z, 1 2 ) satisfies (CO). Let - OCE Fma(0), n >0 and Si— s n E r2 such that a>(Oiv ...v5n )e w. Since F2 is w-closed, it follows that a E r2 c F 1 . Since n[1 1] g_ z, the desired result - that ow z follows immediately. Hence, (z, r2 )E Cut(x). Q.E.D. Claim. From the Claim, the Extension Lemma (4.3) gives an XE MCut(w) with z = L(X) and r2 C R(X). Hence, we have our XE MCut(w) with the desired properties. ^■ 63 Chapter 4^ Lemma 4.12: Let A be natural, i.e., contain [MI]. Let XE MCut(w) initial. Then {131f3>IE w}R(X). Proof: Assume the hypothesis. Towards a contradiction, let 13e Fma(() such that 0> 1 Ew but Pe R(X). Since X is maximal@w, there are n 0, 81, , 5 n E R(X) and an a eFma((h) such that a>((3 v s i v...v 5n ) E W and a€L(X). Recall that since [MT] s A, all members of [NE] FA (a>((3v8) .A. (3>1) --> (a>8) are also members of A. Hence PIE w yields a> (Si v...v 8 n )e w. Then, if n = 0, we have w, whence 0(w) S L(X) yields asp L(X), a contradiction. If, on the other hand, n > 0, then a E L(X) means that X fails (CO), contrary to assumption that X E Cut(w). This establishes the Lemma. ■ Lemma 4.13: Let A contain the schema [PS]^0(a-4 (3) -4 (cc> (3) and let XE Cut(w). Then 1 [R(X)] S L(X). - Proof: Let -iaE i[R(X)]. Then, by [WR], equivalent to [PS] given (pre)normality, we -- have (a>a)E w. Then ocE R(X) gives ae L(X) by (CO), so Tee L(X), as required - to establish the Lemma. ■ The proofs of 4.14 - 4.16 are aided by the following observations: Let A, r be sets of formulae. 64 Chapter 4^ Observation 1: (i) -1(r)v-i[r]CA^r g 1(e)u i[A], and (ii) AD-1(r)U-1[11^-I(A)Un[A]DF. - - To see (i), suppose --i(r)un[r] g A and let aE r. Then la belongs to 1 [r ], and - - hence to A. Then a e , (A ), whence a E 1(A ) u 1[A], as required to show that r c - - --1(e)u-i[e]. To see (ii), suppose AD 1(r)u [r]. Then by (i), r c I (A ) u I [A]. Hence, to show 7 -- -- , (A )u i[A] Dr, it suffices to find a formula belonging to (n (A )L.)7 [A]) — r. Let a E - A-(--i(r)u-i[r ] ). Since a e 1 (r), -men - But la E 1[A]. Hence, we have 7 a E - - ( I (A ) u I [A]) - r, as required. - - Observation 2: If 1[A] is A-consistent, then n (A )u 7 [A] is. - First, note that ae n (A )^na€ A^ 1 1 otE "i [A]. - - Suppose n (A ) u n [A] is not A-consistent. Then there are ali..., a n E 1(A ) such that - [A] u { al, ... , a n } is not A-consistent. Then by [RPL], - 1[A]u{ Ina', ... , -1-ta n } - is not A-consistent either. But by the above note, - f - 1 nal, ..., 1 n an 1 - 1[A] is not A-consistent. Lemma 4.14: Let A contain the schema of adjunctivity: [AD]^((a>13) A (a>y )) --> (a>03Ay))• Let XE MCut(w) be initial. Then n [R(X)] u I (R(X))e MAXA. - c 7 [A], whence 65 Chapter 4^ Proof: Since X is initial, we have ^(w) g R(X) by (CO). So by Lemma 4.1, 1 [R(X)] - is A-consistent. Then by Observation 2 above, 1[R(X)] u (R(X)) is A-consistent. -- Claim: Let r_cFma(0) such that rD --1[R(x)]u (R(X)). Then r is not Aconsistent. Pf.: By assumption there is a 13E r such that [3( 1[R(X)], 1(R(X)). - - Towards a contradiction, suppose F is A-consistent. Then 111e F. It follows that neither - - 13 nor Ifs belongs to R(X). Since X is maximal@w, by (C1) there are al, a2E - L(X), m, n ?.0 and Si,•, 8 m , , yn e R(X) such that a) cci > (0 v8m) E w^and b) cc2>( 1(3vyn) E W , - where 8m =df Si v ...v 8m and r =df Yi v v yn . Then by OK, we have (ai na2)> (0 v 5m), (aina2) >^v Tn) E w, whence [AD] yields ( 4 )^(al Aa2)>Q(3V8M)A(_1PVTn)) E W. Now, the formula ((3 v 5m) A (3 yr) is propositionally equivalent to the disjunction of the following three generalized disjuctions V{Oinyil 15d Sm,^V{PAyil 1.jn}; V{ 113AS 1 l - Note that each generalized disjuction propositionally entails the formula Smvyn, whence (*) yields (#) ^ (ai a2)> (Snivr) E W by [RMT]. Now, ai,a2€ L(X) means that ain a2E L(X). Then Si,yjE R(X) (1 m , 1j.n) and (#) contradict the fact that XE Cut(x) satisfies (CO). Q.E.D. Claim 66 Chapter 4^ From the Claim, we have 1[R(X)] kJ 1(R(X)) maximally A-consistent, as required. ■ - - Lemma 4.15: Let A contain the aggregation schema for [C*]^(Ela A 13)^1:1 03cA(3). Let XE Cut(w) be initial. Then there are u, z E MAXA such that (u, 1(z) u i[z]) is -- - initial and belongs to MCut(w) and i[R(X)] z. -- Proof: Recall from Chapter 0 that (w) =df{ aE Fma(0) IT>aE w}. Claim 1: El(w)u I[R(X)} is A-consistent. Pf.: Suppose otherwise. Then there are - m , n 0, al,^am E 0(w) and Si,^8nE R(X) such that (# ^ ) F A (ain .A a m ) --> (81v But^amE (w) yields El(cci .v8 n A • • -A am)E w ). by repeated applications of [C*]. So [RMT] yields Ei(81 v . . .v 8, 1 )E w. But then 81, .., 8n E R(X) and (#) entail that TE L(X) by (C0'). This contradicts L(X)E MAXA, and establishes the Claim. From Claim 1, let z E MAXA with 0(w) u [R(X)] c z. Define F =df { aE Fma(0)1 (3 8)(80 z & a>SE w)} Claim 2: Let ae F and PE Fma(0) such that F A (3--* a. Then OE F. Pf.: By the definition of r, there is a So z s.t. a> 8 E W. Then F A --> a yields 13> 5E w b y [RMT], whence [3 E r by definition. Claim 3: To r. Pf.: TE r implies there is 80 z such that T> SE w, contrary to the fact that El(w) c z by the choice of z. 67 Chapter 4^ Claim 4: 1 [11 is A-consistent. Pf.: Suppose not. Then n 0, al, ..., a n E F - such that (#) ^ I- A aiv ...va n , whence by def. of r there are 81 , ..., 8,1 0 z such that ai> Sie w (1 i n). This last yields ai> (Si v . . .v O n ) E w (1 5..i rt) by [RMT], whereby repeated applications of [DS] yield (*) (aiv ...va n )>(Siv ...v8„)E w . Now, (#) yields F A T--->(ociv . ..va n ) by [RPL], so (*) gives T> (81v ...v8n )e w by [1.RM]. But 81, . . ., S n 0 z means Si v . . .v S n e z (since ze MA XA), so T E F by the definition of F, contrary to Claim 3. QED Claim 4. By Claim 4, let u E MAXA with 1[r] c u. - Claim 5: ^(w) C u. Pf.: It suffices to show that ^(w) c 1 [F], for which it is -- sufficient to show that: a>_Lew^a E F^(a E Fma(0)). So let aE Fma(0) with a>.I_E w. Recall F = [ cce Fma(0) I (38 e z) a> SE w } . 68 Chapter 4^ Then the desired result follows from the fact that le z. QED Claim 5. Let u = (u, 1(z) u 1 [z]). - -- Claim 6: u is an initial cut around w. Pf.: Since -11=1-4 1 E z, 1 E 1(Z) ..0 R(U) - by def. 1(z). So U satisfies (C1), whence U is initial. To see that ti0 Cut(w), we need -- to check that U satisfies (CO). Towards this goal, let ae Fma((), n > 0 and Si, • • ., 8n E R(u) such that a> (S1 . . .v S n )E w. Since Si, . . ., O n e 1 (z) u I [z] and z is A- consistent, we have Si, • .., On 0 z, whence Si v . • Ai 8,2 0 z, -- so (X E F by definition. Since I[E] g u, the desired result that ao u follows immediately. QED Claim 6 - Claim 7: U is maximal@w. Pf.: Let A g Fma(0) s.t. AD 1(z) u 1 [z]. We need - - to show that (u, A)0 Cut(w). Towards a contradiction, suppose (u, A)e Cut(w). Since A D 1(z) u 1 [z], by Observation 1 above, 1(A) u I [A] DZ . Since z is maximally A - - - - consistent, it follows that 1(A) u 1[A] is not A-consistent. Then by Observation 2 above, - -- - I [A] is not A-consistent. Since D(w) g u, this contradicts the Corollary to Lemma 4.1. QED Claim 7. From Claims 6 and 7, Lie MCut(w) and is initial. And 1 [R(X)] g z by the choice • of z. This completes the proof of Lemma 4.15. ^ - Lemma 4.16: Let A contain [II32]^oa--> Da . Let XE Cut(w) be initial. Then there is u E MAXA such that (u, 1 (L(X)) u 7 [L(X)]) - is initial and belongs to MCut(w). Proof: Let F = f ctE Fma(0) I (38)(80 WO & a> SE w)l- 69 Chapter 4^ Claim 1: Let aE F and 3E Fma(0) such that 1- A (3—>a. Then PE F. Pf.: By the definition of F, there is a Se L(X) s.t. a>SE w . Then F A P-->a yields (3>SE w b y [RMT], whence pe r by def. F. Claim 2: 1[r] is A-consistent. Pf.: Suppose not. Then there are n ?_0, al, ...,an E F – such that (#)^FA aiv ...va n whence by def. F there are Si, . . ., 8 n 0 L(X) such that ai> SiE w (1 5_ i n). By [RivIt], we have ai>(81 v . . .v 8 n )E w (1 _i5_n), whence [DS] yields (*)^(al v . . .v a n ) >(8 1 v . . .vS n) e W . Now, (#) yields I-A T-4 (ai v . . .v a n ) by [RPL], so (*) yields T>(Siv ...vSn )E w by [.ARM]. Then [IB2] yields ^(Si v . . .v Sn )E w. But X initial gives ^(w) g L(X) by (C1), so Si v . . .V8nE L(X). This contradicts Si, ..., 5 n e L(X) and establishes Claim 2. From Claim 2, let u E MAXA with I [r] c u. -- Claim 3: ^(w) g u. Pf.: It suffices to show that ^(w) g n[r], for which it is in turn sufficient to show that: a>1. E W ^ So let a E Fma(0) with a > 1E w. Recall a E F^(a E Fma(0)). 70 Chapter 4^ F = t cc€ Ftna(() I (38) (8o L(X) & a> Se w) } - Then the desired result follows from the fact that _Lo L(x). QED Claim 3 Let u = (u, 1(1,(X)) 1[L(X)]). - — Claim 4: U is an initial cut around w. Pf.: Since L(X) is A-maximal, 11= 1 ---> - E L(X), whence le 0(X)), so U is initial. To see that U E Cut(w), we need to check - that U satisfies (CO). So let ae Fma((), n > 0 and 51, ..., 8n E "[WO] {1} such that a>(81v ...v5n )o w. Now, 51,^8 n E 1 (L(X))^[WO] and L(X) E MAXA means that Si v . .v 8 n L(X), so a e F by def. F. Since 1 [F] c u, the desired result that a u follows - immediately. QED Claim 4. Claim 5: u is maximal@w. Pf.: Let A c Fma(0) such that ADR(u). We need to show that (u , A) Cut(w). Towards a contradiction, suppose (u, 0)E Cut(w). Since A (L(X)) L.) [L (X)] , by Observation 1 above, I [A] u 1(A)DL(X). But L(X) is - - maximally A-consistent, so I [A] u 1(A) is not A-consistent. Then by Observation 2 -- - above, 1[A] is not A-consistent. Since Claim 3 ensures that ^(w) C u, this contradicts the - Corollary to Lemma 4.1. QED Claim 7. From Claims 4 and 5, u = (u, 1(L(X))u I[L(X)])e MCut(w) is initial, as required - - to complete the proof of Lemma 4.16.^ ■ Lemma 4.17: Let A contain the schema [CONNEX]^(a>13) v ((3>a). and let X1,X2o Cut(w). Then either i[R(Xi)] c L(X2) or 1[R(X2)] c L(X1). -- - 71 Chapter 4 Proof.: Towards a contradiction, suppose that neither a) 1[R(X1)] g L(X2)^nor^b) i[R(X2)] c L(X1). - - Then there are formulae a and (3 such that i) aE R(X1) and nag L(X2)^(from a), ii) f3E R(X2) and 10 L(X1)^(from b). - By [CONNEX], either a> f3 E w or p> a E w. Without loss of generality, assume a> (3E w. Then OE R(X2) from ii) yields ae L(X2) by (CO). This contradicts i), since L(X2) is complete. The other case is symmetric. This completes the proof of Lemma 4.17. ■ Lemma 4.18: Let A be transparent, that is, let A include Lewis' schema [DIS]^(a> (13 v 7)) —* ((a> f3)v(a>7)) and let X1,X2E MCut(w). Then either R(X1) g R(X2) or R(X2) g R(X1). Proof: Towards a contradiction, suppose that neither R(X1) g R(X2) nor R(X2) g R(X1). Then there are formulae a and 13 such that both (i) aE R(X1) and ag R(X2), and (ii) (3E R(x2) and 13 E R(x 1 ). Since X1, X2 are maximal@x, by the Transparency Lemma (4.6.) there are formulae a' and (3' such that both 1) a'> aE x and a'E L(X2), (from (i)) , and 72 Chapter 4 2) 0'>13E X and f3'E L(Xi) (from (ii))- Then [RMT] yields a'> (a4), 13 >(av13) E X , whence [DS] yields 1 (aivii')>(avf3) EX. It follows by [DIS] that either (a'v[3')> aE x or (a ' v (3 )> PE X. Assume, without 1 loss of generality, that (a'vf3')> a E x. Then f3'>a E x by [.ARM], whence aE R(X1) from (i) above yields 134 L(Xi) by (CO). This contradicts 2) above, and, since the other case is symmetric, establishes Lemma 4.18. ■ Lemma 4.19: Let A contain the schema [TRIV1]^(a--43)—>(a>0), and let XE Cut(w) initial. Then L(X)=w. Proof: Let az w. Since [TRIV1] yields the principle [Tc ]^a—>Ela we know that ^aE w. Since Xis initial, ^(w) c L(X) by (CO), whence ae L(X). We have shown that w g L(X), so w, L(X)E MAXA yields w = L(X). QED 4.19. ■ Lemma 4.20: Let A contain the schema [CP]^(occ-->0(3 )--“a>(3), and let XE Cut(w) initial. Then R(X) c {13 113>1E w } . Proof: Let 13€ Fma(0) such that p>10 w. Then we have 0 DE w, whence [RPL] yields oT -4o13Ew, whereby [CP] yields T> (3E w. But by (CO), X initial means that 73 Chapter 4^ ^(w) C L(X), so Lemma 4.1 gives 1131 r... I -1 { R ( -)0 ] I . Hence 130 R(X), as required to establish the Lemma.^ This concludes our prestudy to the completeness constructions. ■ 74 Chapter 5^ Chapter 5: Model Construction for Normal Dyadic Logic. Let A be a normal dyadic logic. In this chapter, we construct a "canonical" model 4A for A. The universe WA for „A4A is given by: WA = { Xe MCut(w)I wE MAXA . So, our canonical universe is populated by w-points for arbitrary w in MAXA. I will observe the convention that members X of WA are treated as "copies" of L(X) in MAXA. In particular, I'll think of a formula a as "true at" X just in case aE L(X). As demonstrated, I use 'X', 'X1', 'X2', etc., to range over the set of all cuts. However, I will use outlined lowercase variables: x , y , z ,..., etc., to range over WA, according to the following convention: When x, y, Z, ... in WA are given, 'x', 'y', 'z', ... denote the A-maximal sets of which x, y, z,... are respective copies. (Hence, the outline notation encodes the left projection operator over w-points.) In accord with these conventions, I'll write 'ae x' just in case ae x (a E Fma(0)). Then we can extend the standard semantical definitions to apply to sets of "worlds" in WA in the obvious way. I will use the outline convention to take over previous notation whenever possible. For example the canonical truth set of a in MA , denoted 0 a 0 A , can be defined as (ye WA lye lal A l. Continuing our construction, let A R x = { (y, z) I y,z E MCut(x) & y is initial & i[R(y)] c L(z)} (xE WA), - RA :^2wAxwA defined by: RA(x) =df RxA (X E WA) , FA = (WA, RA) , 75 Chapter 5^ VA : Fma(0) ----> 2 wA defined by: ^ VA (P) =df 11P0 A (/) E 0) , M A = (WA RA vA) . , , A Proposition 5.0: dom(R x ) = {X EMCut(x) I X is initial) (x E WA). A A Proof: Let xe WA. Then def. R x ensures the inclusion of dom(R x ) in the set of initial x-points. Conversely, let XE MCut(x) initial. Then by the Corollary to Lemma 4.1 there is a ze MAXA such that -1[R(X)] g z, whence Corollary 1 to the Extension Lemma gives a Z E MCut(x) A with z =L(z). Hence, XE dom(R x ), as required.^■ Proposition 5.1: IL(y) I y E dOM(R xA)) = I ^(x) I^(x E WA). A Proof: Let x € WA and ye dom(R x ). Then y is initial by Proposition 5.0, whence ^(x) cL(y) by (C1). Conversely, let 0(x) gy. Then there is an initial y= (y, R(y)E A MCut(x) by Corollary 1 to Lemma 4.3, whereby Proposition 5.0 yields y e dom(R x ), as required.^ ■ Lemma 5.2 (The Fundamental Theorem of Dyadic Logic): Let x E WA and a>13e Fma(0). Then, a>13 E X^<=>^(Vy E A A , dom(Rx ))[oce y^(3zE WA) (Pe Z& YR x z)J • Proof:^. Let a>f3e x and y E A dom(R x ) such that ae y. We need to find a A Z E WA A with (3e z and yR x z. Now, y E dom(Rx ) yields ^(x) c y by Proposition 5.1, whence ae y with a> (3 EX gives 1131 6)1 "[WY)] I by Lemma 4.1. So there is a zE MAXA with ( { (3) u I[R(y)] c z. Let our desired z be given as (z, R(z))E MCut(x) by Corollary 1 to - A the Extension Lemma (4.3). Since 1[R(y)] gL(z), the definition of R x ensures that - A yR x z. Since f3e z, we are done. 76 Chapter 5^ A Let a>13 x. We need to show that there is a y e dom(R x ) with a e y but for all A Z E WA, yR x z implies 00 Z. By Lemma 4.2, a> 13 x yields an initial XE Cut(w) with ^(x) C L(X) such that aE L(X) and 13e R(X). By the Extension Lemma, we may assume that Xis maximal@x. So XE MCut(x). Then by Proposition 5.0, XE dom(R x ). A So let X be our desired y. Clearly, ae y. Now let z E WA such that yR x z. Then by A the definition of R x' -1 [R(y)] c z. But 13 E R(y), so 0* z, as required to complete the proof of Lemma 5.2.^ ■ Having established the Fundamental Theorem for normal A, the way is now clear to show that when A contains various combinations of the first degree dyadic principles presented in Chapter 3, M A satisfies the associated first order sentences. However, lets first consider the strategy embodied in our construction of MA and its limitations. A brief study of right-to-left direction of the proof of The Fundamental Theorem will convince the reader that the purpose of populating WA with multiple copies of A-maximal sets is to ensure that the domains of the canonical frame relations R R are "large enough" to allow for the "falsification" of wedge statements not belonging to L(x). One limitation of this strategy is that dyadic principles whose first order correspondents express the unique existence of certain kinds of relata, e.g., [AD], [C*], [IB1], etc., seem to require the auxiliary assumption of transparency in order for M A to meet these structural criteria. Its not hard to explain why this is the case. Take, for example, [AD], which was shown in Chapter 3 to correspond to the frame requirement that frame relations be functional. Now, it is interesting to look ahead to A Lemma 5.8 to see that, in the presence of [AD], the canonical frame relations Rx are indeed A "functional up to identity of left projections", i.e., any two R x -relata of a given point in WA are "copies" of the same A-maximal set. Hence, the strategy of allowing multiple copies of a given A-maximal set, which is essential to giving the domains of canonical 77 Chapter 5^ frame relations the right structure, frustrates our ablity to "control" the range of those relations. Now, by Corrollary 2 to Lemma 4.8, when A is transparent, the identity of the right element of a point in WA is uniquely determined by the choice of the left element. A Thus, when A is assumed to be transparent, the canonical frame relations R R are genuinely functional in the presence of [AD]. (Indeed, one advantage of our construction is that when A is transparent, the canonical universe WA coincides, up to isomorphism, with the traditional Lemmon-Scott canonical universe.) Similar remarks can be made for [C*] and [IB1]. Now, one way to avoid relying on transparency would be to change the definiton of A R x so that for each A-maximal set y required to be the left element of a point in the range A of R x' a unique representative be chosen (from among the points in WA having y as their left element) to play that role. This strategy is developed in Chapter 7. For the rest of this chapter, let xe WA. A A Lemma 5.3: Let A be natural. Then rng(R x ) c dom(Rx ). A Proof: Let ze rng(R x ). Then z E MCut(x) and there is an initial yE MCut(x) such A that yR x z. Since y is initial, ^(x) cL(y) by (C1). Also, by Lemma 4.12, we have {13113>le x} C R(y). A Claim: z is initial. Pf.: We need to show that z satisfies (C1). Since yR x z , -1[R(y)] c L(z). From Claim 1, { 0 I ((3> 1) e x } c R(y), whence ^(x) c L(z). Then by Lemma 4.4, p. 58, z is initial, as required to complete the proof of the Claim. By the Corollary to Lemma 4.1, there is a ZE MAXA such that -1{R(z)) c z'. Let z ° = (z', R(z)) be given maximal@x by Corollary 1 to the Extension Lemma. Since the Claim A A gives z initial, we have (z,z°) ER x ' whence z € dom(R x ), as required to complete the proof of Lemma 5.3. ^ ■ 78 Chapter 5 Lemma 5.4: Let A contain [CON]^--CI, equivalently, --1(T>1). A Then Rx is nonempty. Proof: Suppose that F A 1(T>1). Then —1(T>1)E x yields a ye dom(R xA) by the rightto-left direction of Lemma 5.0. QED Lemma 5.4.^■ Lemma 5.5: Let A contain [PS]^El(a -4 13) --> (a> 13) . A^A A^ Then Rx is reflexive on its domain, i.e., (VyE dom(R x ))yR x y . A Proof: Let y Edom(R x ). Then y is initial. Since ye Cut(x), [R (y )] c L(y) by Lemma A A 4.13. Hence, y initial ensures that yR x y by def. R . This completes the proof of Lemma 5.5.^ ■ Lemma 5.6: Let A be transparent and contain the schema [PW]^(a> ) ^(oc -4 (3) A A Then Rx is "at most reflexive", i.e., (Vyi, y2) (If1R x 192^1y1 = y2). A Proof: Let yi,y 2 E W A such that yiR x y2. We need to show that yi= y2. Since A yiE dom(R x ), we have ^(x) g L(y1) by Proposition 5.1. Claim: Lcy2) g L(n). A Pf.: Let 13E 142). Since yiR xy2, we know that --i[R(Y1)] g 1(1y2), whence I3e R(y1). Since A is transparent and yi is maximal@x, by (COT) there is ae L(y1) such that a>13E X. Then by [PW], we have 0(a—d3)e x, 79 Chapter 5^ whence 12(x)u { a} c 141) yields 13E L(y 1 ), as required to establish the Claim. Since L(Y1),L(Y2)E MAXA, the Claim yields uy2)=L(y1). Then then X11 =Y2 follows by Corollary 1 to Lemma 4.8 given the assumption that A is transparent. This completes the proof of Lemma 5.6.^ ■ Lemma 5.7: Let A be transparent, that is, let A include Lewis' schema [DIS]^(a> (13 v 7)) —, ((a> 0) v (a>7)). and contain and the schema [TRIV1]^(a .-413) -4 (a> p ) A A A ^) Then, x E dom(Rx )^[xRx x & (Vy)(y E dom(Rx ^y =x)] . Proof: Assume xE dom(R xA). Since [TRIV1] yields [WR], equivalently, [PS], we have A xR xx by Lemma 5.5. So it suffices to establish (Vy)(yE A )^y=x). dOM(R x A Towards this goal, let ye dom(R x ). Then y is initial by Proposition 5.0, whence L(y)= x by Lemma 4.19. Then the assumption that A is transparent yields x= y by Corollary 1 to Lemma 4.8. This establishes Lemma 5.7. ■ Lemma 5.8: Let A contain the schema of adjunctivity: [AD]^((a>13) A (a>7)) —> (a>(DAY))• A^A^A Let zi, z2 E WA and yE dom(R x ) such that yR xz i and yR x z2. Then L(z1)=L(z2). 80 Chapter 5^ Proof: Since ye MCut(x) is initial, by Lemma 4.14 -1 [R(y)] u (R(y))E MAXA. Since --i[R(y)] g L(zi) and L(zi)E MAXA, it follows that "I[R(y)]u , (R(y)) g L(zi) (i= 1, 2). Hence L(z1), L(z2)E MAXA yields the desired result that L(z1)=1-(z2). ■ Lemma 5.9: Let A be transparent and contain the schema [AD]. Then R )At is functional, i.e., A A (VyE dom(R x ))(3! z)yR x Z. A A A Proof: Let z, z'e WA and ye dom(R x ) such that yR x z and yR x z . We want to show that z=z ° . By Lemma 5.8, L(z)=L(e). Then transparency yields the desired result by Corollary 1 to Lemma 4.8. ■ Lemma 5.10: Let A be transparent and contain the aggregation schema for [C*]^A (3) -5 ri(a A f3). A A A Then every point in dom(R x ) bears R x to a point in WA which is the unique Rx -successor A of some point in dom(Rx ). Symbolically, x e <=> Z = z g )]. (VyE dom(R x ))( 3 z)[YR Ax z & ( 3 ue dOln(Rx))(Ve) (UR A A^A Proof: Let ye dom(R x ). Then by def. R x , y E MCut(x) is initial. By Lemma 4.15, there are u, z E MAXA such that u =(u, -1 (z)U -1 [z])E MCut(w) is initial and —1[R(y)] g z. A Since UE MCut(x) is initial, u E dom(R x ) by Proposition 5.0. Let z = (z, R(z)) be given maximal@x by Corollary 1 to the Extension Lemma (4.3). Now, to establish the Lemma, we need to show that all of the following hold: A^A^, A a) yR x z^b) uR x z^c) (Vz)(uR x z °^z =z°). A Since —1[R(y)] g z by the choice of z, it follows by def. that yR x z , whence a) is established. ^ 81 Chapter 5^ A For b), it suffices by def. R x to show -1[R(u)] g L(z). So assume a = -113E -1 [R(u)]. Then 0 E R(u) = -1(z) u --1 [z]. If DE 1(z), then a E Z. On the other hand, if 13 E --I [z], - then (3 = --ly for some yE z, whence again a. --rlye z., since z E MAXA. Hence, b) is established ). A 0 x To show c), let z a e WA such that uR xz . Then --1 [R(u)] g L(e) by the definition of RR. Claim: L(z)=L(e). Pf.: Since L(z),L(z ° )E MAXA, it suffices to show that L(z) g L(z'). Now R(u). -1(z) k..) -1[z], so —I [z] g R(u), that is, --1[1.(z)] g R(u), whence -i[R(u)] g L(e) yields -1 -1[L(z)] gL(e). So, L(z°)E MAXA yields L(z) c L(z ° ), as required. QED Claim . From the Claim, we have z ° =z, by Corollary 1 to Lemma 4.8, as required to complete the proof of Lemma 5.10.^ ■ Lemma 5.11: Let A contain [IB1]^Da.--> Da . A Then (V y E dOM(R x A ))( 3 Z E dOM(R x )) A yR x z . A ^A^ Proof: Let ye dom(R ). Towards a contradiction, suppose that zE dom(R ) implies x x -1 [R(y)] ct L(z) (z e WA). By Proposition 5.2: { L(y) I y E dOM(R xA)/ = I O(T)I. Then it follows that El(x) g z^—1 [ R(Y ) ] '4 z ( ZE MAXA). That is, fl(x)ui[R(y)] is not A-consistent. So there are m, n< co , a 1, . . . , a m E ^(X) and Si_ S n E R(y) such that 82 Chapter 5 FA (al ...na m ) —) (81v ... A n ) . Then by [RM] F A ^(a1 A ... Aam)^0(51V vOn) • Since al, , a m E ^(x), [C] yields ^(ai ... na nd E x, whence ^(81v vO n )e x. Then by [1131], 3 (81 V ... VO n )E X , i.e., T > (Si v^v 8 n ) e x. Then by Lemma 4.1, we have 1(8 1 v ... v8,01 6) I [R(y)] , contrary to 61,. Sn e R(y). This completes the proof of Lemma 5.11.^ ■ Lemma 5.12: Let A be transparent and contain [1132]^Cia- Oa . A A A Then every point in dom(Rx ) is the unique R x -successor of some point in dom(R x ). A A „^ A ))( 3 U E dOM(R x ))(VZ)(URe a y =z) Symbolically, (V y E dOM(R x A Proof: Let yE dom(R x ). Since y E Cut(x) is initial, there isu E MAXA such that u= (L(y)) L.) [L(y)DE MCut(w) initial by Lemma 4.16, whence u e dom(R xA ) by (u, --1^ Proposition 5.0. To establish the Lemma, we need to show that A a) uR x y^and b) (Vz)(uR Ax z^y =z). A^A Since -i[R(u)] c L(y) by the choice of u, it follows by the definition of ; that uR x y . A So we need only concern ourselves with b). Let z be any in WA such that uR x z. Then -I[R(u)] c L(z) by the definition of R. A 83 Chapter 5^ Claim: L(y)=L(z). Pf.: Since L(y),L(z)E MAXA, it suffices to show that L(y) L(z). Now, R(u)= - (L(y))u -1[1.(y)], whence --1[L(y)] c R(u). Since -i[R(u)] c L(z), -1 -1[L(y)] c L(z), whence L(z)E MAXA yields L(y) c L(z), as required. QED Claim. Since L(y)=L(z), we have y=z by Corollary 1 to Lemma 4.8, as required to complete the proof of Lemma 5.12.^ ■ Lemma 5.13: Let A contain the schema [CP]^(o a o )^(a>13) A A^ ^A Then Rx is complete on its domain, i.e., (Vyi, y2E dOM(Rx ))(y iR x y2) . A A Proof: Let y1,y2 E dom(R x ). By def. Rx , it suffices to show that -1[R(Ifi)] c L(y2)• A Since yiE dom(Rx), yl is initial, whence R(yi).c {13 I (3> 1 E x} by Lemma 4.20. Then i[R(y1)] g ^(x). But Proposition 5.1 gives ^(x) g_ 142), so I [R(yl)] g 1(y2), - as required to complete the proof of Lemma 5.13. ■ Lemma 5.14: Let A be transparent and contain the schema of strong transitivity: [STR]^a>((3 v 7)^((R>s) .—>. a>(Svy))• ^A A ^A A Then Rx is transitive, i.e., eglyi, y2, y3) klf iR x Y2 & y2R x y3^y1RxY3)• A A Proof: Let y i , y2, y3 e WA. Assume yiR x y2 and y2R x y3. Claim: R(y1) c R(y2). Pf.: Since yl E MCut(x), we have R(y1) x-closed by the Lemma on Right Closure (4.10, p. 63). Since y1R Ax y2, we have Then by Lemma 4.9, p. 63, R(y 1 ) g R(y 2 ), as required. —1 [R(y1)] C 1-(Y2)• 84 Chapter 5^ A A A Since yiR x y2 entails that yi is initial, to show that yiR x y3, it suffices by def. R x to show that [R(y1)] g L(Y3). But from the Claim we have --i[R(Y1)] c -1 [R(Y2)]. Moreover, y2RY3 gives -I [R(y2)] c 143). Hence, the desired result is immediate. ■ Lemma 5.15: Let A contain the schema [CONNEX]^(a> (3) v ((3>a) A A A A Then Rx is connected on its domain, i.e., (Vyi, y2€ dom(R x )AY iR x Y2^Y2R x Y1)• A Proof: Let y1,y2E dom(Rx ). Since y1,y2E Cut(x), by Lemma 4.17, either -1[R(Y1)] c L(y2) or "I [R(y2)] g um. Since yi and y2 are both initial, the desired result that either A A y2R xyi Y1R02^ follows immediately by the def. R.. This completes the proof of Lemma 5.15.^ ■ A Lemma 5.16: Let A be transparent. Then Rx is mutual over its domain, i.e., (Vyi, y2 E dOM(R x A )) ((VZ)(Y1RxZ^y2Rxz) or ((Vz)(Y2R x Z^YiR x z))• A Proof: Let y1,y2E dom(R x ). Since y1,y2E MCut(x), by Lemma 4.18, we have either R(y1) c R(y2) or R(y2) c R(y1). We need to show that either A A a) (VZ)(YiR x Z^y2Rx z) or A^A b) (Vz)(Y2Rxz^yiRxz) holds. Assume, without loss of generality, that R(y 1 ) S R(y2 ). To see that b) follows, let A A Z E WA such that y2R x z- Then [R(1172)] cL(z) by def. R x , whence A^A -1[R(Y1)] cL(z) by assumption. Since yi is initial, the definition of R x yields yiR x z, as required. The other case yields a) in symmetric fashion. ^■ 85 Chapter 5^ We are now in a good position to establish some determination results. But first, let's recall from the introduction (and Chapter 0) the semantics of "relativised" generalized .A4 : inclusion. We extend valuations V for models ,A4 = (W, R) to assignments if E Fma(0)--> 2Wx W by the usual inductive clauses for —> ,1 and, for the wedge, .A4 m^ (3yE III-311 M ) x R wy]• [R>]^.1,, a>f3 <=>^(Vx€ dom(R w ))[xE [ta Now, VA is lifted to a valuation II E A of Fma(0) with the desired property that: 5.17 (Truth Lemma): if a E A = {we WA I ae w} . Proof: As usual, by induction on the formation of a, using the definition of VA for the case a=pE 1, the properties of maximal consistent sets for the the cases a=1, a =13—>y, and Lemma 5.2 for the case a = 0>y. ■ Corollary: MA determines A, i.e., for all formulae a. MA a <=:, Fh a . Proof: FA a <=> a belongs to all members of MAXA <=> (by def. WA) for all w E WA, a e^<=> (by Lemma 5.17) if ca A= WA.^■ Theorem 5.18 (Determination of JK): JK a if, and only if, a is valid in all g.i. frames. Proof: Soundness: For any frame F, EF = IpIF H3} is a normal dyadic logic, so A c EF , i.e., Fpc a^Fka. Completeness: If liK a then, by Corollary to the Truth Lemma, a is false on M K , whence a is not valid in the frame FJK.^ ■ 86 Chapter 5^ Theorem 5.19: Let A be a normal logic. i) If A includes any combination of the schemata 0, 1, 2, 8, 9, 11, 13, 14, given in Chapter 3, then A determines the class of frames which satisfy the associated combination of first order principles. Proof: Soundness: By Theorem 3 and Lemma 3.1. Completeness: By Lemmata 5.3, 5.4, 5.5, 5.10, 5.11, 5.13, 5.15, 5.16. ■ ii) Let A be transparent. Then, if A includes any combination of the schemata 0, 1, 2, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14. inclusive given in Chapter 3, then A determines the class of "mutual" frames which satisfy the associated combination of first order principles. Proof: Soundness: By Theorem 3.0 and Lemma 3.1. Completeness: By Lemmata 5.3, 5.4, 5.5, 5.6, 5.7, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16.^ ■ This completes (no pun intended) our general study of classes of g.i frames determined by first degree extensions of JK. In the next chapter, we focus on a large subclass of comparison frames, those determined by first degree extensions of JKSTR. 87 Chapter 6 Chapter 6: Model Construction for Normal Logics Containing STR. In this chapter, we shall see that JK[STR] emerges as a natural base logic. Let I be a normal dyadic logic containing [STR]. Recall from Chapter 2 that this means that I is natural, since [Mt] is a consequence of [STR]. Let's now construct a "canonical" model ME for I. The universe WE for ,A4 E is given by: WE = { Xe MCut(w)I X is initial & w e MAXE} . Thus, unlike the construction of the previous chapter, our canonical I-universe is populated by initial w-points for arbitrary we MAXE. In the sequel, u, V. w, x, y, z range over WE. We shall observe the conventions of the previous chapter regarding the interpretation of cuts as "worlds". Set: Z R x = { (if , z)I y ,z E MCWX) & I[R(y)j g L(z) & R(y) c R(z)} (XE WE), - E RE : WE ---) 2wExwE defined by: RE(X) =dfR x (x e WE) , FE= (WE , RE) , VI : Fma(40) ---> 2 wE defined by: V1 (p) =df OpI E^(p E 0) , ME_ (WE, RE, VE) • I Proposition 6.0: dom(R x ) = { XE MCut(x) I X is initial }^(x E WE). I Proof: Let XE WE. Clearly, dom(R x ) g { XE MCut(x) IX is initial} by the definition of WE. Conversely, let X1 E MCut(x) initial. We know that --1 [R(Xi)] is I-consistent 88 Chapter 6^ by the Corollary to Lemma 4.1. Also, since E contains [STR], R(X1) is x-closed by the Lemma On Right Closure (4.10). Then by Lemma 4.11, there is an X2E MCut(w) with - 1[R(X1)] c L(X2) and R(X1) c R(X2). Since J_E R(X1), we know that X2 is initial. z^E^ z Then the definition of R x ensures that XiRxX2. Thus X1 E dom(R x ), as required. ■ Proposition 6.1: {L(y) I y E dom(RxE )) = I ^(x) I^(x E WE). Proof: Similar to the proof of Proposition 5.1, using Proposition 6.0. in place of 5.0. ■ Lemma 6.2: Let x E WE and a>f3E Fma(0). Then E E^ y^(3z E WE ) ( PE z & YR x Z)J • a>13€ X <=> (Vy e dom(R x MaE z Proof: =. Let a> Pe x and ye dom(R x) such that ae y. We need to find a zE WE E with (3E z and yR x z. Now, R(y) is x-closed by the Lemma On Right Closure (4.10), E since E contains [STR]. Also, since ye dom(R x ), by Proposition 6.1, ^(x) Cy, folu i[R(y)] XE MCut(w) with -I [ { -1131 uR(y)] whence ae y with a>I3E x gives 1011 -1 [R(Y)] 1 by Lemma 4.1. So is E-consistent. Then by Lemma 4.11, there is an - c L(X) and R(y) c R(X). Since y is initial, it follows that X is initial. So let X E WE be our desired z . Clearly, -- I [R(y)] c L(z); furthermore, R(y) c R(z). Then the E definition of R x gives yR x z . Since - 1 -1(3E L(X), (3E z, and we are done. E . Let oc>130 x. We need to show that there is a ye dom(Rx ) with ae y but for all Z E WE, E^ yR x z i mpl i es pe z. By Lemma 4.2, a>pex yields an initial XE Cut(w) with ^(x) c L(X) such that OCE L(X) and [3e R(X). By the Extension Lemma (4.3), we may assume that X is maximal@x. Since XE MCut(x) initial, Proposition 6.0 ensures XE dom(R x ). So let's take X E^ as our desired y. Clearly, ae y. Now let ZE WE such z that yR x z. Then by the definition of R x , i[R(y)] c z. But (3 E R(y), so 0 e z, as - required to complete the proof of Lemma 6.2. ^ ■ 89 Chapter 6^ For the rest of this chapter, let xE WE. Since we are here interested in modelling extensions of JK[STR], we need to show that „M E satisfies the structural condition corresponding to [STR] given in Chapter 3, viz., that 1) the range of R is included in the domain of that relation, and 2) R x is transitive. Lemma 6.3: mg(R x ) c dom(Rx ). Proof: Let Z E mg(R x ). Then Z E MC ut(x) initial, so by Proposition 6.0, Z E dom(R x), as required.^ ■ E Lemma 6.4: R x is transitive. Proof: Let yi,y2, y3 E WE. Assume that 0 Y1Rx Y2 and ii) y2R x y3. To show that Z ,y 1R ^it suffices, by the definition of R x , to show that both of the following hold: a) -' [R(Y1)] c L(y3),^and^b) R(S91) c R(Y3). From i), the definition of R x yields R(y 1 ) c R(y2 ), while from ii), R x yields -1 [R(192)] c L(y3). Hence a) is immediate. Furthermore, ii) yields R(y2) c R()y3), whence b) follows. This completes the proof of Lemma 6.4. ■ We now procceed to show that when E contains any combination of the first degree dyadic principles presented in Chapter 3, M E satisfies the associated combination of first order sentences. Lemma 6.5: Let E contain [CON]. Then R x is nonempty. Proof: Suppose that 1- E -C11. Then --i(T>1.)E x yields a pair (y, y°)E R x by the right-to-left direction of Lemma 6.0. This is sufficient. ^ ■ Chapter 6 90 Lemma 6.6: Let I contain [PS]^0(a--->(3 ) —> (a > (3) . E Then Rx is reflexive on its domain, i.e., (VyE dOM(R xI ))yR zx y . E Proof: Let ye dom(R x ). Since ye Cut(x), we have --I [R(y)] c L(y) from Lemma 4.13. E^z Then since ye MCut(x) initial and R(y) c R(y), we have yR xy by the def. R x . ■ Lemma 6.7: Let Z contain the schema [SI]^^(a —)(3) <—> (a> J3). E^z Then (thy E dom(R x ))(Vz)(YR x z <=> y =z). Proof: First we note that [SI] yields [AD], as follows. From [SI], [RPL] yields I z ((a> 0) A (a>y)) .--->. ^(a —> 13) A 0(0C—yy) , - whence [C] yields FE ((a>I3) A (a>y)) --> D(a —> f3 .A. a--->Y) by [RPL], whereby [RE] yields FE ((a>f3)A (a>y)) —> ^(a—> (OAT)) whence [SI] yields 1-"z ((a>P) A (CE>Y)) --> (a>((3AY)) E by [RPL], as required. Now, to establish the Lemma, let yE dom(Rx ). We need to z show that: (Vz)(yR x z 4#› y = z ). 91 Chapter 6 Since E is "adjunctive", R x is functional by Lemma 6.7. Hence, it suffices to show that yR xy. But [PS] is a consequence of [SI], so the desired result follows from Lemma 6.6. ■ Lemma 6.8: Let E be transparent, that is, let E include [DIS]^(a> vy)) ((a> (3) v (a> y)). Suppose further that E contains the schema [TRIV1]^(a---)(3)-->(a>p). A A A Then x E dom(R x )^[xRx x & (Vy)(y € dom(R x )^y=x)i . A A Proof: xe dom(R x ). Since [TRIV1] yields [WR], equivalently, [PS], we have xRx x by Lemma 6.6. So it suffices to establish (Vy)(y e dom(R)d^y=x). A Towards this goal, let ye dom(Rx). Then y is initial by Proposition 6.0, whence L(y)=x by Lemma 4.19. Then the assumption that A is transparent yields x=y by Corollary 1 to Lemma 4.8. ■ Lemma 6.9: Let E contain the schema [TRIV]^(a>13)< 4 (a -)(3). - Then (yR x z <=> x=y =z) Proof: Let y,z E - (y, z E W E ). WE. First we note that the assumption that E includes [TRIV] entails that E is transparent and includes [PS] and [TRIV1]. 92 Chapter 6 . Suppose yR xz. Since I includes [TRIV1], we have L(x)=L(y) by Lemma 4.19. Then, since I is transparent, we have x= yby Corollary 1 to Lemma 4.8. Hence, it suffices to show that y=z, for which it in turn suffices by Corollary 1 of Lemma 4.8 to E E^ show that L(y)=L(z). Since yR x z we have 1 [R(y)] cL(z) by def. R x , and since - L(z)e MAXE, we have: (#)^-i[R(y)]un(R(y)) c L(z). Since I includes [PS], we have I [R(y)] gL(y) by Lemma 4.13, whence L(y)E MAXE -- yields (*)^-T[R(y)]un(R(y)) c L(y)• But since I contains [AD], i[R(y)]u 1 (R(y)) e MAXE by Lemma 4.14. Then (#) and - - (*) yield the desired result that L(z) = L(y) by virtue of L(z), L(y)E MAXE. E =. It suffices to show that xR xx. Since x is initial and R(x) g R(x), it suffices by def. E R x to show that a) xE MCut(x) and b) -i [R(x)] c L(x). Now, b) follows from the fact that I includes [PS] by Lemma 4.13. So it suffices to show a). Claim 1: xE Cut(x). Pf.: Since I is transparent, it suffices to show that x satisfies (COT). Towards this goal, let ae Fma(0) and Se R(x) such that a>SE L(x). This last yields a ---> SE L(x) by [TRIV1]. As we have seen, I [R(x)] C L(x), whence SE R(x) - yields n•SE L(x), whereby 'GEE L(x), i.e., a€ L(x), as required to complete the proof of Claim 1. Given Claim 1, to prove a), it suffices to show that x is maximal@x. Since Xe WE, it follows from def. WE that x is initial and there is a UEMAXE such that xE MCut(u). E Then by Proposition 6.0, xE dom(R U ), whence XE ^(u) by Proposition 6.1. Now, observe that since I contains [TRIV], I also contains 93 Chapter 6^ [n]^0a4->a. Claim 2: Let a> E u. Then a>13e L(x). By assumption, since a> Eu we have 0(a>(3)e u by [n], whence xE 0(u) yields a>13E L(x), as required. Claim 3: x is maximal@x. Pf.: Let r g Fma(0) properly extend R(x). We need to show that (L(x),^Cut(x). Since Z is transparent, it suffices to show that (L(x), F) fails (COT) with respect to x, i.e., that there is an otE L(x) and a SE R(x) such that a>(3 Ex. Now, since xE MCut(u), x is maximal@u, whence, by def. maximal@u, (L(x), r)f6 Cut(u). Then (L(x), F) fails (COT) with respect to u, i.e., there is an aE L(x) and a SE R(X) such that a> f3E u. But by Claim 2, then a> Ex, as required. From Claims 1 and 3, we have XE MCut(x), and a) is established. This completes the proof of Lemma 6.9.^ ■ Lemma 6.10: Let E contain the schema of adjunctivity: [AD]^((a> 13) A (a>y)) 3 (a>(3Ay)). — E Then Rx is functional. Proof: Let Zl,Z2 E WE and y E dom(R) such that yR:g1 and yRxz2. We want to show that z1=Z2. By assumption, y, z 1, Z2 E MCut(x) initial with --1[R(y)] c L(zi) and R(y) c R(z1) (i =1, 2). Since y, z 1, Z2 E MCut(w) initial, we have (*)^—1[R(U)]U-1(R(U))E MAXE^E { y,z1,z2} by Lemma 4.14. It follows that if wl, w2 E MAXz such that 1 [R(y)] c wi (i =1, 2), - then wi =w2. Suppose that -1[R(19)]C wi. Then WiE MAXz yields 7 (WY)) g Wi, so we have -1 [R(y)]u (R(y)) c wi, whence wi = --1[R(y)] u (R(y)) (i = 1 , 2). Hence, 94 chapter 6 - 1[R(Y)] c L(zi) (i =1,2) entails that L(z1)=L(z2). So we need only show that R(Z1) = R(z2). Now R(y) c R(zi), so we know that (#)^--I [R(y)] u 1 (R(y)) c I [R(zi)] u 1 (R(zi))^(i =1,2). -- - -- Then it follows from (*) and (#) that ($)^--1 [R(y)] u n (R(y)) = n [R(z1)] u 1 (R(zi))^( i =1,2). - Claim: R(z1)=R(z2). Pf.: Let aE R(z1). Then la E I [R(z2)]u 7 (R(z2)) by - - ($), whence one of a, 1 la belongs to R(z2). However, by the Corollary to Lemma - - 4.5, we have aE R(X)^<=>^ 1 1ccE R(X). - -- Thus, ocER(z2). We have shown that R(z1) c R(z2). But symmetric reasoning also yields R(z2) c R(z1). Hence R(z1)=R(z2) , as required. So we have shown that zi =Z2. Hence, Lemma 6.10 is established. ^■ Lemma 6.11: Let I contain the aggregation schema for [C*] ( El t3 anci(3) -p Ei(an(). z z^z^ Then every point in dom(Rx ) bears R x to a point in WE which is the unique R x -successor E of some point in dom(R x ). Symbolically, (VyE doril(R xE ))(3z)[yR Ex z & (3LBE d0111(4))(Vz0(d14.Z ° <1=> Z =e)]. Z^ z Proof: Let yEdom(R x ). Then by the definition of R x , ye MCut(x) is initial. By Lemma 4A5, there areu, z E MAXE such that u= (U, 7 (Z) U 7 [Z]) is initial, belongs to 95 Chapter 6 MCut(w) and --i[R(y)]cz. Since LAE MCut(x) initial, UE dom(Rx ) by Proposition 6.0. Let z = (z , 1(z) u 1[z]). - - Claim 1: ZE Malt(X) is initial. Pf.: Since _LE 1(z), we have _LE R(Z) by def. z. - So z satisfies (C1). To see that z satisfies (CO) (and hence is an initial cut around x), let aE Fma(0), n>0 and 81, . . ., 8 n E ( - 1(Z) Un[z]) such that a>(Siv...v8n)e x. Since I contains [STR] and uE MCut(x), we have that R(u) = 1(z) u '[z] is x-closed - - by Lemma 4.10. Hence, a E i (z) u 1 [z] by the definition of x-closure. Since - z E MAX, it follows that cco z, as required to establish that z satisfies (CO). To see that z is maximal@x, let A g Fma(0) s.t AD ( - 1(Z) U 1[Z]). - We need to show that (z,A) 0 Cut(x). Towards a contradiction, suppose (z, A)e Cut(x). Since AD 1(z), there is a PE - A s.t. 100 z; thus, since ze MAXE, 13E z. But then A D n[z] yields ilk A, whence - - both 13, 1 10E 1[i\]. So 1[0] is not I-consistent. Since i(z)u n [z] is x-closed and , - - - - - .LE 7(z), it follows that [13113>J_Ex} g 1(z)u I[z], whence 1:1(x)g z by the fact that - ZE - - MAXE. This contradicts the Corollary to Lemma 4.1 and establishes the Claim. Now, to establish the Lemma, we need to show that all of the following hold: E^E a) yR x z^b) uR x z^c) (Vz0(uR Ix e^z =z°). We have ' [R(y)] c L(z) by the choice of z, so n (L(z)) c R(z) yields R(y) c R(z), E whence a) is established by def. R x . To see that b) holds, recall that R(u) = 1(z) l..) 7 [Z] . Then 1[R(u)] c z =L(z), since - ZE - A MAXE. Hence, by def. R x it suffices to show that R(u) c R(z). But R(u) = 1(z) L) [z] = R(z), - so we have the desired result. 96 Chapter 6^ To show c), let telii/ such that uR xE z ° . Then -i[R(u)] g L(e) and R(u) g R(V) by the definition of R x . Claim 2: L(z) = L(z ° ). Pf.: Since L(z), L(z ° )E MAXE, it suffices to show that L(z) g L(z'). Now R(u)= (z)u --1[2], so -i[L(z)] c R(u), whence -1 [R(u)] g L(e) yields -1 1[L(Z)] gL(e), whereby L(z ° )E MAXI entails L(z) c L(e), as required. - Claim 3: R(z') c R(u). Pf.: Towards a contradiction, suppose that ae R(z') but a R(u)=df -i(z)u -1[z]. The latter yields aE z (since ze MAXE), whence 'ae R(u), whereby R(u) g R(V) gives -1 ae R(e). Hence -1 oc,a E R(e), so - 1 [R(z')] is not I-consistent . But z' is an initial cut around x, so ^(x) c z by (CO). This contradicts the Corollary to Lemma 4.1 and establishes Claim 3. From Claim 3, R(u)g R(e) gives R(u)=R(e), whence R(z)=R(u) yields R(z)= R(z'). Then 1=z follows by Claim 2. This establishes c), as required to complete the proof of Lemma 6.11. ■ Lemma 6.12: Let E be transparent and contain [IB2]^na—>^ . Then every point in dom(R x ) is the unique R x -successor of some point in dom(R x ). Symbolically, (fy E dOM(R x ))(3U E dom(R x ))(Vz) (uR x z <=> y =z). Proof: Let yE dom(R x/ ). Then y E Cut(x) is initial by Proposition 6.0, whence Lemma 4.16 gives a u E MAXI such that u =(u, 1(14) [L(yD is initial and belongs to - MCut(x). So uE dom(R x ) by Proposition 6.0, whence to establish Lemma 6.12 it suffices to show that a) uR x y ^ and b) (Vz) (uR Ex z^y 97 Chapter 6 Since i[R(u)] gL(y) by the choice of u, for a) it suffices by def.R to show that R(u) c R(y). Since u E MCut(x), R(u) is x-closed by the Lemma on Right Closure (4.10). Then by Lemma 4.9, we have the desired result that R(u) g R(y). To establish b), let ZE WE such that uR x z. Then -, [R(u)] c L(z) by the definition of R x . Claim: L(y)=L(z). Pf.: Since L(y),L(z)EMAXA, it suffices to show that L(y) L(z). Now R(u)--.dr(L(y)) u 1 [L(y)], so 1[L(y)] c R(u), whence n[R(1.1)] c L(z) - - yields -1-1[14)] c L(Z), whereby L(z) E MAXA entails L(y) g. L(z), as required. The Claim yields y= z by Corollary 1 to Lemma 4.8, as required to complete the proof of Lemma 6.12.^ ■ Lemma 6.13: Let E contain the schema [CP]^(oa--><>0)--> (a>(3). \ z Then Rx is complete on its domain, i.e., (Vyi,y2E dOM(Rx)lkylR 02) • Proof: First, let us observe that the "Rule of Monotonicity for Diamond" [RMo]^F a >13^I °a 4 o — - - is an easy consequence of [IRM] given [RPL] and the Df. 0 . Now, to prove 6.13, let Y1, Y2 E dOM(R x). It suffices to show that -1[R(191)] cL(y2) and R(y 1) R(y2) by def. R E x• Claim 1: E includes [DIS]. Pf.: By [PD] we have FE (ct>(13vy)) > (oa >o(13vy)) — Now, [RMO] yields — ^ 98 Chapter 6^ 1-z 0 (Pvy) ---* (013v oy), so by [RPL] we have FE (cit>(13vy)) * (occ--4 (0 Ovoy))• — But 0 a --) (013v 0 y) is propositionally equivalent to (0 a--4 o(3)v(oa--)oy), so [RPL] and [CP] yield FE (a>(Ov1)) --+ ((a>13 ) v (a>Y)), as required to establish Claim 1. From Lemma 4.20 we know that R(Y1) g (13 I PIE x ), i.e., -1 [R(Y1)] CI:7(x). But Proposition 6.1 gives ^(x) g L(y2 ), so -, [R( y1 )] gL(y2). Claim 2: R(y1) c R(y2). Pf.: Proposition 6.1 gives ^(x) g L(191), whence we know from the Corollary to Lemma 4.1 that -1 [R(y1)] is E-consistent. Then, since E contains [STR], R(y1) is x-closed by the Lemma On Right Closure (4.10). Since -1[R(Y1)] c L(y2), it follows that R(y1) c " 1 [L(y2)]. Since E is transparent by Claim 1, by the Lemma on w-closure (4.9, p.61), we have the desired result that R(y 1 ) c R(192)• This completes the proof of lemma 6.13.^ Lemma 6.14: Let E contain the schema [CONNEX]^(a>13)v (Pa) E Then Rx is connected on its domain, i.e., E ^E^E (\iyi,y2E dom(Rx ))(y1R x Y2 or y2R x yl) • ■ 99 Chapter 6^ Proof: Let yi, y2 E dom(R x ). By Lemma 4. 14 either -i [R(y1)] c L(92) or -1 [R(Y2)] CL(yi). Suppose, without loss of generality, that -1[R(111)] cL(y2). I shall show that ^ E E yiR x y 2. Since y 1 , y2e MCut(x) initial, it suffices, by def.R x' to show that R(Y1)gR(Y2). Since I contains [STR], R(y1) is x-closed by the Lemma On Right Closure (4.10). Now, we know from Chapter 3 that I is transparent, since [CONNEX] and [STR] yield [DIS]. Then by the Lemma on w-closure (4.9), -i [R(y1)] c L(y2) yields the desired result that R(y1) cR(y2). Since the case -I [R(y2)] gL(191) is treated symmetrically, this completes the proof of Lemma 6.14. ■ E Lemma 6.15: Let I be transparent. Then R x is mutual over its domain, i.e., E^E^E^E^E , (gf y 1 ,y2 E dOM(Rx))0,Z)(Y1RxZ ^y2Rxz) or ((Vz)(192R x z^YiR x z))• Proof: Let y1,y2E dom(R x ). We need to show that either E^E E a) (Vz)(YiRxZ^y2R x z) or b) (Vz)(Y2R x Z^yiRxz) E holds. By def. Rx , Y 1, Y2 E MCut(x), whence either R(y1) c R(y2) or R(y2) g R(Yi) by Lemma 4.18. Assume, without loss of generality, that R(y1) c R(y2). To see that b) follows, let ZE WE such that y2R xz. Then we have i) -1 [R(Y2)] c L(z)^and^ii) R(y2) g R(z) by the definition of R x . Since R(yt) g R(y2), we have 7 [R(y1)] g L(z) by i) and E^E R(y1) c R(z) by ii). So by the definition of R x , we have yiR x z, as required to establish b). In the case where R(y2) g R(y1), we obtain a) in symmetric fashion. This completes the proof of Lemma 6.15.^ ■ Now for some determination results. Again, we recall the truth condition for the wedge: 100 Chapter 6^ A4 [R>]^k a> 3 44. (Vx E dom(R w))[xE ffa] ^(3y e Q(3D w A4 ) xR,, y ). Recall also the definition of our model ME = (WE, RE, VI) at the beginning of the chapter. VE is canonically extended to a valuation IE II E of Fma(0) with the desired property that: 6.16 (Truth Lemma): if a D E= fwe WE I ocew). Proof: As usual, by induction on the formation of a, using the definition of VI for the case a =p E 0, the properties of maximal consistent sets for the the cases a = 1 , a = (3 -*y , and Lemma 6.2 for the case a= [3>y . ■ Corollary: ME determines E, i.e., for all formulae a, ,A4.E k cc^FE a . Proof: FE a <=> a belongs to all members of MAXE •=> (by def. WE) ae w^E WE ) 4* (by Lemma 6.16) Q a11 = WE.^ ■ Theorem 6.17 ("Determination of JKSTR"): Let C be the class of g.i. frames F = (W, R) that are "pointwise serial" and "transitive", i.e., such that 1) dom(R w ) c rng(R w ) (we W) 2) R w is transitive (WE IV). Then, I-^a jKsTR <=> C^a. 101 Chapter 6 Proof: Soundness: For any frame FE C, X,F = 1131F k 13} is a natural dyadic logic. And by Lemma 12 of Theorem 3.0, EF includes [STR]. Hence, JKSTR .c ZF , i.e., I-^a^ F a. jKsTR Completeness: If Iticsma then, by Corollary to the Truth Lemma, a is false on M ASTR , ■ whence a is not valid in the frame FJKSTR.^ Theorem 6.17: Let E be a normal dyadic logic with [STR]. Then i) If E includes any combination of the schemata 1, 2, 4, 6, 7, 8, 11, 13, 14 given in Chapter 3, then E determines the class of transitive, pointwise serial frames which satisfy the associated combination of first order principles. Proof: Soundness: By Theorem 3.0 and Lemma 3.1. Completeness: By Lemmata 6.5, 6.6, 6.7, 6.9, 6.10, 6.11, 6.13, 6.14, 6.15, ■ respectively.^ ii): If I is transparent and includes any of the schemata 1, 2, 4, 5, 6, 7, 8, 10, 11, 13 given in Chapter 3, then E determines the class of mutual, transitive, and pointwise serial frames which satisfy to the associated combination of first order principles. Proof: Soundness: By Theorem 3.0 and Lemma 3.1. Completeness: By Lemma 6.15 and Lemmata 6.5, 6.6, 6.7, 6.8, 6.9, 6.10., 6.11, 6.12, 6.13, 6.14, respectively.^ ■ 102 Chapter 7 Chapter 7: Model Construction for Natural Dyadic Logic. Let A be a natural dyadic logic. In this chapter, we construct a model MA of A which will afford us some determination results for natural logics which have so far eluded us. The model MA = (W, R, V) has universe W given by: W = {XE CUt(W)I W E MAXA } • Note that unlike the previous two constructions, MA has no maximality conditions placed on the cuts populating its universe. Let: x, y, z,... etc. range over W according to our previous conventions. Let w E MAXA and X1, X2€ Cut(w). I shall write 'X1 g r X2' whenever L(X1) = L(X2) and R(X1) c R(X2). Recall from Chapter 4 that in the proof of the Extension Lemma (4.3) we presented a uniform procedure which, for a given maximally A-consistent set w andXie Cut(w), produced an X2€ MCut(w) such that R(X1) c R(X2). Since X2 is uniquely determined by the choice of w, X1 and the enumeration 4) of Fma((), and since we hold 4) fixed, we may use the functional notation 'E w (X1) to denote X2. Let z E MAXA with ^(w) g z. 1 By the argument of Corollary 2 of the Extension Lemma (4.3), (z, {1.}) is a cut around w, and E w ((z , UWE MCut(w) is initial. Let E w ((z, fl ))) be denoted 'w[zr . The point here is that w[z] is an initial element of MCut(w) which is uniquely determined by the choice of ze MAXA. ^ ^ 103 Chapter 7 Let: Rx = {(y,x[z]) I yE Cut(x) & Z E MAXA & -1[R(Ex(10)] c z^W), WxW R : W —> 2 defined by: R(x) =dfR x^E W) , F= (141 , R) , V : Frna((b)^2W defined by:V(p) =cif 11P D A (p E^, M A = v R v) . Proposition 7.0: Let x,y E W. Let Z E MAXA such that -1 [R(Ex(Y))] -C z. Then Claim 1: (13 I 13>lEx} cR(Ex(Y)). Pf.: Towards a contradiction, let Pe Fma(0) such that 13>lex but fie R(E x (y)). Since Ex(Y) is maximal@x, there are n 0, 61, 8n E R(Ex(Y)) and an aE Fma(0) s.t. b) ae L(E x (y)). a) a> ((3vR-iv ...v 8n )E x^and Recall that owing to [MTh [NE] I- A (a> (8 v^.A. 13>i) --> (a>8). Hence, a> (81 v...v^E X follows from a) and 13>1 E X. Then, if n = 0, we have a>_Le x, whence ^(x) c L(E x (y)) yields a L(E x(y)), a contradiction. If, on the other hand, n > 0, then E x(y) fails (CO) because of b), contrary to the fact that E x (y)E Cut(x). This establishes Claim 1. Since by assumption -i[R(E x (y))} g z, Claim 1 yields ^(x) c L(z), as required to complete the proof of Proposition 7.0.^ ■ Chapter 7^ 104 Proposition 7.1: dom(Rx) = {XE Cut(x) I Xis initial} (x E IV). Proof: Let xe W. Then the definition of W ensures the inclusion of dom(R x ) in the set of initial cuts around x. Conversely, let X e Cut(x) initial. Then ^(x) c L(X) by (C1), whence ^(x) s L(E x (X)). Then the Corollary to Lemma 4.1 gives a ze MAXA such that n[R(E x(X))) C z. Then (X,w[z])e R x by def.R x , so XE dom(R x ), as required. ■ Proposition 7.2: L(y) I y E dom(Rx) } = 1:1 (41 (x E W). { Proof: Let XE W and ye dom(R x ). Then y is initial, so ^(x) c L(y) by (C1). Conversely, let y E MAXA such that ^(x) Cy. Then there is an initial y = (y, R(y)) E Cut(x). By Proposition 7.1, yE dom(R x ), as required. ■ Lemma 7.3: Let XE W and a> 0 eFma(0). Then cc> 0 e (dye dom(R x ))[aE y (azE W) (Pe z and yR x z)]. Proof: Let a> 0e x and ye dom(R x ) such that otE y. We need to find a ZE W with 0E Z and yR x z. Now, since yE dom(R x ), by Proposition 7.1, ^(x) CL(y), whence ^(x) cL(Ex(Y)). Then ote L(Ex(Y))with a>0 ex gives 101 (6) I i [R(E x (y))] by Lemma 4.1. So there is a z E MAXA with { u [R(Ex(Y))] C z. Let our desired z be x[z]. Since -1[R(Ex(Y))] c L(z), the definition of R x ensures that yR x z. Since E Z, we have 0e z, as required. Let a>f3ex. We need to show that there is a yE dom(R x ) with aE y but for all ZE W, YRxZ implies fle z. Since a>(3 x, there is an initial XECut(w) with ^(x) CL(X) such that a€ L(X) and 0e R(X) by Lemma 4.2. Since X is initial, X E dom(R x ) by Proposition 7.1. X is our desired y. Clearly, cite y, so we need only show that 105 Chapter 7 yRxz^13 0 z^( z E W) So let zE W such that yR xz. Then by the definition of R x , -I [R(Ex(Y))] g L(z). But De R(y) c R(E x (y)), so 'OE L(z), whence 13e z, as required to complete the proof of Lemma 7.3. ■ For the rest of this chapter, let xe W. Since we are here interested in modelling extensions of JICMT, we need to show that Alt A satisfies the structural condition corresponding to [MT] given in Chapter 3, viz., that the range of R x is included in the domain of that relation: Lemma 7.4: rng(R x ) c dom(Rx). Proof: Let ze rng(R x ). Then ze Cut(x) initial, so by 7.1, ze dom(R x ).^■ Lemma 7.5.: Let A contain [CON]^-ID I, equivalently, -1(T>1). Then Rx is nonempty. Proof: By Lemma 7.3.^ ■ Lemma 7.6: Let A contain the schema of adjunctivity: [AD]^((a>13) A (a>y)) > (a >([ A y))• — Then Rx is functional, i.e., (VyE dom(Rx))(3!z) yR x z . Proof: Let z, ee W and ye dom(Rx) such that yRxz and yRxe. We want to show that z= z ° . Since yRxz and yRxz ° , by def.Rx, we have n [R(E x (y))] g L(z), L(z ° ). Claim: L(z) = L(z ° ). Pf: Since E x (y)e MCut(w) is initial, we have 106 Chapter 7 (#) n [R(Ex(Y))] un (R(Ex(Y))e MAXA by Lemma 4 14. Since we have "[R(Ex(Y))1C L(zi) and L(zi) E AXA, it follows that n [R(Ex(Y))1u "(R(Ex(Y)) c L(Zi) (i = 1, 2). Then (#) yields the desired result that L(z)=L(e). From the Claim, we have x[L(z)] =x[L(z ° )], whence the desired result that z= z ° ■ follows by def. Rx . Lemma 7.7: Let A contain the aggregation schema for [C*] El an El D13) --> el(a Al3). Then every point in dom(Rx) bears Rx to a point in W which is the unique Rx-successor of some point in dom(Rx). Symbolically, (VyE dom(Rx))(3 z)[yRxz & (3ue dom(Rx))(Vz°) (URxe .4* Z = Z ° )]• Proof: Let ye dom(Rx). Since E x (y)E Cut(x) is initial, by Lemma 4.15, there are u,z E MAXA such that (u, -1(z)u -I [z])e MCut(x) initial and -1 [R(Ex(Y))] C z . Let =df (14, I (Z)U I [Z]). — — U By Proposition 7.1, u E dom(Rx). Let z =x[z]. Now, to establish the Lemma, we need to show that all of the following hold: a) yRxz b) URxZ c) (gf z ° ) (URxz ° z =z ° ). -1[12(E x(y))] g z, whence yRxzby def. Rx. Since R(u) = -1(z) u -1[z] and ze MAXA, we have --1[R(u)] g z =L(z). Hence, b) follows by def. Rx. To show c), let z ° E W such that URxe. Then -1[R(u)] g L(z ° ) by def. Rx. 107 Chapter 7 Claim: z c L(e). Pf: Let aE z. Since R(u) =1(z)u n [z], we have naE R(u). Then, since -1[R(u)] CL(Z 1 ), -1 -1 ae L(z'), whence ocE L(V) follows from L(z°)E MAXA, as required. Then z,L(e)E MAXA give z =L(e) by the Claim. But by def. Rx, z°=x[L(e)], i.e., z ° =x[z]. Then it follows that e=x[z] = z, as required to complete the proof of the Lemma 7.7. ■ Lemma 7.8: Let A be transparent, that is, let A include Lewis' schema [DIS]^(a> (I3 v y)) -4 ((a> 13) v (a>y))• Then Rx is mutual over its domain, i.e., (Vyi, y2E dom(Rx))((Vz)(YiRxz y2Rxz) or (Vz)(Y2RXZ YiRxz))• Proof: Let y1, y2 E dom(Rx). Now, Ex(n),Ex(Y2)E MCut(w), so by Lemma 4.18, we have that either R(Ex(Y1)) c R(E x(y2)) or R(Ex(y2)) g R(Ex(Y1)). To establish the Lemma, we need to show that one of a) (Vz)(YiRxZ y2Rxz) b) (Vz)(Y2Rx yiRxz) holds. Assume, without loss of generality, that R(Ex(Y1)) g R(E x (y2)). To see that b) follows, let ZE W be such that y2Rxz. Then, by def. Rx, z =x[L(z)J and n [R(E x (y2))] c L(z). Then by assumption, -' [R(Ex(Y1))1 g L(z). Then the definition of Rx yields yiRxz, as required. The other case yields a) in symmetric fashion. This completes the proof of Lemma 7.8. ■ As before, V is lifted to a valuation I 3] A of Fma(0) with the desired property that: 108 Chapter 7^ 7.9 (Truth Lemma): II a B A = {we W I ae w } Proof: As before, by induction on the formation of a, using the definition of V for the case a = p E 0, the properties of maximal consistent sets for the the cases a =1, a = 13 -4y, and Lemma 7.3 for the case a =13 > y. ■ Corollary: MA determines A, i.e., for all formulae a, MA ka <=> Fla. Proof: I Aa^a belongs to all members of MAXA - <=> (by def. W) ae w (W E W) <=> (by Lemma 7.17) QaBA= W. ^■ Let C be the class of g.i. frames F = (W, R) that are "pointwise serial" ", i.e., such that dom(R w ) c mg(R w ) (WE W). Then: Theorem 7.10: Let A be a natural logic. i) If A includes any combination of the schemata 1, 7, 8, 14, given in Chapter 3, then A determines the class of frames FE C such that F satisfies the associated combination of first order principles. Proof: Soundness: By Theorem 3.0 and Lemma 3.1. Completeness: By Lemmata 7.5, 7.6, 7.7, 7.8. ^ This completes Chapter 7, and our completeness study. ■ 109 Bibliography Bibliography. [0]^Äqvist, Lennart. An Introduction to Deontic Logic and the Theory of Normative Systems. Bibliopolis, Indices IV (Monographs in Philosophical Logic and Formal Linguistics), 1987. [1] Chellas, Brian, F. 'Basic Conditional Logic'. Journal of Philosophical Logic 4, (1975), 133-153. [2] Chellas, Brian, F. Modal Logic: An Introduction. Cambridge University Press, 1980. [3] van. Fraassen, Bas C. 'The Logic of Conditional Obligation', Journal of Philosophical Logic 1 (1972), 417 - 438. [4] ^'Values and the Heart's Command', Journal of Philosophy (1973), 5 - 19. [5] Goldblatt, R. I. Logics of Time and Computation. CSLI Lecture Notes No. 7, Center for the Study of Language and Information, Stanford University, 1987. [6] Jennings, R.E., 'Utilitarian Semantics for Deontic Logic;', Journal of Philosophical Logic 3 (1974), 445 - 456. 'Aspects of the Analysis of Entailment'. Unpublished monograph, [7] 1984. [8] Leibnizian Semantics: An Essay in Monotonic Modal Logic, Unpublished monograph, 1984. [9] 'The Natural Conditional'. Unpublished monograph, 1986. Bibliography 110 [10] Kripke, S. 'A Completeness Theorem in Modal Logic', J. Symbolic Logic 24, 114. ^ "Semantic Analysis of Modal Logic I. Normal Modal Calculi", Zeitschrift far Mathematische Logik and Grunglagen der Mathematike 9 (1963), 67 - 96. [12] Leibniz, G. W. von. "First Truths" in Opuscules et Fragments Inëdit de Leibniz, ed. Louis Coutourat. Paris: Felix Alcan, 1903. [13] Lemmon, E. J. and Scott, Dana S. Intensional Logic, preliminary draft of initial chapters by E. J. Lemmon, 1966, Nowadays available as An Introduction to Modal Logic (edited by Krister Segerberg). American Philosophical Quarterly Monograph No. 11, Blackwell, 1977. [14] ^ 'Completeness and Decidability of Three Logics of Counterfactual Conditionals', Theoria 37 (1971), 74 - 85. [15] ^ Counterfactuals. Oxford: Blackwell, 1973. [16] Lewis, D.K., "Counterfactuals and Comparative Possibility". Journal of Philosophical Logic 2 (1973), 418-446. [17] Nute, Donald, Topics in Conditional Logic, Reidel, Dordrecht, 1980. [18] 'Conditional Logic', in Handbook of Philosophical Logic II. D. Gabbay and F. Guenthner, eds. (Synthese Library), Dordrecht and Boston. D. Reidel Publ. Co. (1984), 387- 439. Bibliogaph.y_ [19] Scott, D. 'Advice On Modal Logic', in Philosophical Problems in Logic, Some Recent Developments. K. Lambert ed. (Synthese Library), Dordrecht and Boston. D. Reidel Publ. Co. (1971), 143-173. [20] Smith, T.V., and Marjorie Grene, eds. From Descartes to Kant. Chicago: University of Chicago, 1940. 111
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An essay in natural modal logic Apostoli, Peter J. 1992
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Title | An essay in natural modal logic |
Creator |
Apostoli, Peter J. |
Date Issued | 1992 |
Description | A generalized inclusion (g.i.) frame consists of a set of points (or "worlds") W and an assignment of a binary relation Rw on W to each point w in W. generalized inclusion frames whose Rw are partial orders are called comparison frames. Conditional logics of various comparative notions, for example, Lewis's V-logic of comparative possibility and utilitarian accounts of conditional obligation, model the dyadic modal operator > on comparison frames according to (what amounts to) the following truth condition: oc>13"holds at w" if every point in the truth set of a bears Rw to some point where holds. In this essay I provide a relational frame theory which embraces both accessibility semantics and g.i. semantics as special cases. This goal is achieved via a philosophically significant generalization of universal strict implication which does not assume accessibility as a primitive. Within this very general setting, I provide the first axiomatization of the dyadic modal logic corresponding to the class of all g.i. frames. Various correspondences between dyadic logics and first order definable subclasses of the class of g.i. frames are established. Finally, some general model constructions are developed which allow uniform completeness proofs for important sublogics of Lewis' V. |
Extent | 4055574 bytes |
Subject |
Modality (Logic) Semantics (Philosophy) |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0098835 |
URI | http://hdl.handle.net/2429/2277 |
Degree |
Doctor of Philosophy - PhD |
Program |
Philosophy |
Affiliation |
Arts, Faculty of Philosophy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1993-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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