c.i lewis' calculus of predicates

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C.I.Lewis's calculus of predicatesChris Swoyer aa Department of Philosophy, University of Oklahoma Norman, Oklahoma, U.S.A

To cite this Article Swoyer, Chris(1995) 'C.I.Lewis's calculus of predicates', History and Philosophy of Logic, 16: 1, 19 37To link to this Article: DOI: 10.1080/01445349508837238URL: http://dx.doi.org/10.1080/01445349508837238

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H I STO RY A N D PH I L O SO PH Y O F L O G I C, 16 (1995), 19-37

C. I. Lewis's Calculus of PredicatesCHRISSWOYER

Departm ent of Philosophy, University of Oklahoma N orman, Oklahoma 73019, U.S .A.Received 25 Septemb er 1993

In 1951 C. I. Lewis published a logic of general terms (or properties) that he called the calculus ofpredicates. Although this system is of less significance than Lewis's earlier work on propositionalmodal logic, it has considerable historical interest and does not deserve the almost total neglect it hasreceived. My aim here is to situate this system in the context of Lewis's earlier work and to examineseveral of its central features. After sketching the historical background, I present the syntax ofLewis's system, discuss his reasons for preferring it to quantified modal logic, and develop asemantics for it that is suggested by Lewis's informal discussion of his system together with hisgeneral views on meaning. I then discuss Lewis's sketchy extension of his system to includequantifiers and examine his claim that it can serve as a foundation for logic in general. I conclude bynoting two minor changes in CP that, from today's vantage point, would count as improvements.

In a series of work s, from t he yea r of his first published pap er in 1912 to th eappearance of Symbolic logic in 1932, C. I. Lewis developed several intensionallogics that laid the groundwork for modern propositional modal logic. In a lesswell-known paper, 'Notes on the Logic of Intension', published in a Festschriftfor his colleague Henry Sheffer in 1951, Lewis presented a logic of propositionalfunctions that he called the calculus of predicates. He believed that this systemcould provide a foundation for logic in general; in particular, he thought itafforded a way of combining modality and propositional functions that wassuperior to quantified modal logic. History has not vindicated Lewis's judgmentabout the importance of his system, and its intrinsic interest clearly is not asgreat as that of his earlier work. Still, Lewis's calculus of predicates goes wellbeyond the attempts of earlier thinkers to devise an intensional logic of generalterms; it shows how the architect of modern propositional modal logic wouldhave developed an alternative to quantified modal logic, and it prefiguresseveral recent logics of properties. Hence, Lewis's calculus of predicates is ofsufficient historical interest to deserve better than the almost total neglect it hasreceived. My aim here is to situate this system in the context of Lewis's earlierwork o n propositional mo dal logic and t o exam ine several of its central features.In $ 1 I sketch th e historical backg round of Lewis's calculus of predicates(CP, for short) and give an informal account of its central notions. In $2 Ipresent th e syntax of C P and in $ 3 consider why Lewis preferred this system toquantified modal logic. Althoug h the purely syntactical features of C P are not ofgreat interest, since they so closely parallel those of one of Lewis's propositionalmodal logics, S2, the sorts of interpretation s he envisioned for C P differ ininteresting and fundamental ways from those he had in mind for his proposi-tional modal logics. Accordingly, in $4 I supply CP with a semantics that issuggested by Lewis's informal discussion of his system (together with his generalviews on m eaning). In $5 I discuss Lewis's sketchy extension of C P to includequantifiers and in $6 I examine his claim that C P can serve as a found ation forlogic in general. In the final section I note two minor changes in C P that, f romtoday's vantage point, would count as improvements.

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20 Chris Swoyer1. Background and motivation

1.1 . Lewis S propositional intensional logicsLewis's first paper in logic was published in 1912, just two years after theappearance of the f irst volume of Principia mathematica (P M ). Although he was

impressed by the scope and rigor of this work, he found its exclusive focus onextensional logic too restrictive [A, 13-15; L & P , 41.' In the propositional logicof P M , all comp oun d propositions are truth functions of simpler propositions,and so the system can only deal with relationships among propositions (like thematerial conditional) that hold in virtue of the truth values the propositionshap pen to have [e.g , 4201. Similarly, the predicate logic of P M only representsrelations among propositional functions that hold in virtue of the extensions thatthey happen to possess [e.g., MWO, 433ffl. Because of these restrictions, thesystem of P M fails to provide a satisfactory treatm ent of those logical relationsthat dep end o n the m eanings of the related expressions [SS L, 230ff; SM L ,440-4411. An d since many everyday inferences depe nd o n meaning relationsam ong propositions and terms, P M fails to do justice to the ' logic of ordinarydiscourse' [423; cf. R , 6591.Lewis's dissatisfaction with the extensional approa ch of P M e merg ed mostvividly in his polemics against what he took to be its identification of logicalimplication with the material conditional (symbolized by '3') . He began his veryfirst article on logic with a criticism of this treatment of implication and for thenext fifty years hammered away at the theme that it generates serious anomaliesthat came to be known as the paradoxes of material implication. For example,on this construal of implication a false proposition implies any proposition (thisis reflected in the fact that '- p 3 p 3 q) ' is a theorem of PM), and anyproposition implies a true proposition ( 'q 3 p 3 q)'). Indeed, in a paper of 1913Lewis listed thirty-five theorem s of P M w hose main connective is the materialconditional, b ut which are n ot plausible principles of implication [I T , 239-241;cf. ZAL, 351; SS L, 229-236; L & P , 6ff; S L , 85-89]. An d these perceivedanomalies stimulated Lewis to devise an intensional logic containing a relation ofstrict implication (symbolized by ' 4 ' ) hat behaves more like our ordinarynotion of logical implication than does the material conditional.Whitehead and Russell did sometimes gloss 'p 3 q' as 'p implies q', and thiscertainly encouraged Lewis's reaction. There is no obvious reason to follow theirusage, however, and Quine has even suggested that it is difficult to do sowithout the assistance of a use-mention confusion. For although 'only if ' (and'3') re bon a fide sentential connectives, ' implies' is a verb that connects nam esof statements. Hence, Quine contends, the claim that p implies q should bema de in the metalanguage (as "'p' implies 'q'") rather than in the objectlanguage, as Lewis tried to do (Quine 1953, $11).In a paper presented to the Association for Symbolic Logic in 1949, Lewisbrusquely declared that this 'superficial objection' could be countered byconstruing propositional variables as standing for that-clauses [SML , 4351, sothat we obviate the need for quotation marks by reading ' (p & q) +p ' as ' that pand t hat q implies that p ' . Still, there is no denying that Lewis was often carelessabout use and mention, and a more satisfying response by today's standards1 I will use the abbreviations in the left margin of th e bibliography in citing Lewis's works. Unlesscontext requires an abb reviation, however, references to 'Notes on the Logic of Intension' (NLI),which contains Lewis's presentation of his calculus of predicates, will simply be by page number.


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C. I. Lewis's Calculus of Predicates 21would begin by drawing a crisper distinction between object language andmetalanguage than he usually did. We could then hold that an object-languagesen ten ce 'a a1 is a theorem (b a > p [420-423; S M L , 441; cf. ASL, 415; C S I ,2421. By contrast, we can express paradigmatically logical claims like ' a mpliesa- directly in Lewis's systems as ' a


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22 Chris Swoyerit is true in virtue of the meanings of the logical constants it contains [AKV,124-129; S M L , 440; cf. NLZ, 423; S L , 211-2131. Fu rth erm ore , logical relationslike implication and consistency hold among propositions whether or not wehave any rules for showing that they d o [A KV , 112-113; cf. SM L, 4321, and alogic will be 'adequate, from any reasonable point of view' only if it capturesthese independently existing relations [A KV , 128; cf. 165-1671. So in retrospec t,it appears that Lewis was groping toward an object-language counterpart of ametalinguistic semantic consequence relation (what we symbolize by 'P' ) , al -though he sometimes tended to conflate it with a syntactic relation of deduci-bility.Lewis held that there was no objective fact as to precisely which wordscounted as logical constants and, hence, that there was no principled differencebetw een tru ths of logic and ot he r analytic truths [A K V , 124-1291. In his earliestwork , he often wrote as though the axioms of P M were just plain false, itsprinciples of inference simply invalid [N A , 429-430; I T , 2421, but by the ea rlytwenties he had arrived at his "conceptual pragmatism", and he thenceforthmaintained that any consistent logic consists of claims that are analytically trueof its own logical constants. Hence, the choice among alternative logics can onlybe based on pragmatic considerations like 'simplicity or comprehensiveness oraccord with our most frequent purposes in inference' [ PW A L, 74; cf. PC A , 232;L & P , 12-13; A S L, 416-417; S L , 255-2621. But of course Lewis felt that hisown systems of strict implication, particularly S2, fared better on these countsthan their extensional rivals.1.2. Basic concepts of C P

Th e basic ide a behind C P is that propositional functions can have logicalproperties like consistency and can stand in logical relations like implication inthe same way that propositions can. Furthermore, the principles governing theselogical properties and re lations a re just th ose of Lewis's system of strictimplication for propositions.C P provides a formal development of Lewis's treatm ent of propositionalfunctions in several works written shortly before NLI, particularly An analysisof knowledge and valuation published five years earlier in 1946. Here andelsewhere Lewis typically employs Whitehead and Russell's terminology of'propositional functions', but in NLI he usually calls them 'predicative func-tions', and since my concern is with this paper, I shall follow suit. Russellsometimes treats propositional functions as linguistic entities, namely opensentences of the sort we now write as 'Fx', nd sometimes as properties, likeF-ness. However, Lewis is quite explicit that in his terminology both proposi-tions and predicative (i.e., propositional) functions are linguistic expressions[422; AKV, 651.On Lewis's account, the statement 'Socrates is human' asserts the proposi-tion 'that Socrates is human', and if we replace 'Socrates' in the proposition bythe variable 'x', we obtain the predicative ( i.e., propositional) function, ' that x ishuman' (which Lewis counts as equivalent to 'x being human' [425; 4221). InCP, Lewis forms predicative functions by writing a predicate followed by acircumflexed variable, as in ' ~ 2 ' . lthough logicians would now regard thecircumflex as a variable binding operator that binds the 'x' to produce a complexpredicate, Lewis views the 'x' here as free [427]. Hence, a predicative function


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C . I . Lewis's Calculus of Predicatesnot only works like a predicate, but is also a function, since it takes linguisticitems like 'Socrates' as argume nts and yields a proposition as a value [ 42 51 .~A s early as 1913, Lewis had stressed that just as one proposition can imply asecond, one predicative function can imply a second; moreover, the principlesgoverning the behaviour of the implication relations in the two cases are thoseof his calculus of strict implication [NA, 433; MAZ, 590-590; S S L , 320ffl. Forexample, 'Socrates is human and Socrates is mortal' implies 'Socrates is human'.However, the meaning of the name 'Socrates' plays no role in this, and so wecan also say that 'x is human and x is mortal' implies 'x is human'. Moregenerally, complex predicative functions contain logical constants whose mean-ings give rise to logical relationships among the predicative functions containingthe m. A nd the purpose of C P is to codify these relationships.

2. Formalization of Lewis's calculus of predicatesLewis's calculus of predicates is a formal system, and the best way toappreciate its details is to set it out formally. There are various gaps in Lewis'spresentation, but in most cases it is straightforward to see how he would havefilled them, and in this section I will present the syntax of CP without muchdiscussion (in $4 I will turn to the more difficult matter of semantics).

2.1. Syntax of C PIn NLI, Lewis uses his standard symbol '0'o represent self-consistency,i.e., logical possibility, but he departs from earlier works in speaking ofentailment (rather than of strict implication), and he represents it by the symbol

'K' ather than '< ' . I will adopt Lewis's numbering of definitions andpostulates, but to enhance readability I will replace his use of dots aspunctuation marks by parentheses, his use of the dot for conjuction by '&', andindicate definitions with the symbol i=d f ' . Lewis represents necessity with thestring of symbols '-0-',ut I will follow Barcan 1946 in using the nowstandard 'El' as an abbreviation for this. Finally, I will use ' a ' , p' , 'y ' , '6 ' and'a' as metalinguistic variables.2 .2 . Primitive vocabulary

We begin with the primitive vocabulary of the language of CP:Predicates: q , , , with or without positive integer subscriptsVariable: xLogical constants: -, &, 0

A

Syncategorematic expressions: , (, )2.3. Formation rules and definitionsAlthough Lewis does not list formation rules for C P , he clearly presupposessome equivalent of the following recursive definition of a yredicative function : If

3 Lewis 's notation is similar to that of P M , where ' ( @ ? . - y R) ' s tands for the complex propertybeing q- but not y ~ . os t wri ters now would regard 'q-2' as a complex term formed by abs trac t ion.In the early thir t ies Church introduced the now famil iar ? nota t ion fo r funct ional abs trac t ion, andby the mid-fort ies Carnap had adapted i t to form complex predicates that he t rea ted as names ofproperties. Lewis 's treatment of the circumflexed variable as free lead5 to difficulties ininterpreting his overall account, but since it has relatively little effect on the main aspects of hissys tem. I will defer discussion of this to $7.2 .


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24 Chris Swoyern- is a pred icate , then 'n-21 is a(n atom ic) predicative functio n, and if a an d /3are predicative functions, then '-a1, r ( a & /3), and 'Od are also predicativefunctions. Moreover, only expressions generated by a finite number of applica-tions of these rules are p redicative functions.The additional truth-functional connectives 'v' , '>', and '= ' are introducedby standard contextual definitions, and three further intensional connectives aredefined as follows [425]:(11.02) q 2 < ~2 =d f - O ( @ & - ~ 2 ) (Entailment is officially defined interms of possibility)(11.03) qx E qx d f ( p 2 < V2) & (Strict, or logical, equivalence is two-

(@ < @ ) way entailment)(11.02) 1712 o V2 = d f - (@ < -q&) (Propositional functions are consistentjust in case on e does not entail thenegation of the other; thus 0 ~ 2 'is equivalent to rO(qfi& ~ 2 ) ~ )For readability I will omit unnecessary parentheses, and when issues of use andmention are not at stake I will follow the convention of autonymous use, lettingeach simple expression stand for itself and each juxtaposition of expressionsstand for their concatenation.2.4. Postulates

Lewis writes 1 a to assert a ; in formal contexts this amou nts to the claimthat a is a theorem of CP. Dropping two redundant postulates, the axioms ofC P are [425]:

These axioms exactly parallel Lewis's axioms for propositional S2 [SL, Ch.61.Axiom 19.01 is Lewis's consistency postula te; it is the distinguishing axiom of S2,telling us that only a consistent predicative function can be a conjunct of aconsistent, conjunctive predicative function.2.5. Rules of inference

Lewis says that 'the usual operations of proof are assumed' 14261, andadapting his standard rules for propositional S2 we have:Conjuction: If I- N and 1 3, then k a & /i.Strict modus ponens: If k N and 1 ai3, then k P.Substitution of strict equivalents: If 1 a,1 y = A, and a differs from 13 atmost in having y in one or more places where /3 has 6, then k 0.Rule of uniform substitution: the result of uniformly replacing any predicativefunction in a theo rem by ano ther predicative fu nction is itself a the ore m .

The notions of a derivation and theorem are defined in the usual way, and the


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C. I. Lewis's Calculus of Predicates 25techniques of S L (136ff) can be used to show that an extensional logic ofpredicative functions that is isomorphic to classical propositional logic is asubsystem of C P (so that, e .g . , -y,9 3 9 9 3 ~ 9 )s a theorem of CP).Lewis does not supply C P with a formal semantics, but instead conveys theintended meanings of its formulae by paraphrasing them in English. Thus, hereads axiom 11.1, (rp9 & ~ 9 ) (I@ & y,?), as 'that x is y, and x is V entails thatx is y j and x is rp' [427] (he omits the ' that' before the second conjunct of eachconjunct ive expression) . Lett ing ' ~ 9 ' e 'that x is red ' and ~ 9 'that x feelsdam p', this says ' that x is red and x feels damp entails that x feels dam p and x isred' [ i b i d . ] .And with obvious adjustments, 11.3-11.7 can be read in similarways.

3. CP vs . quantified modal logicLewis explicitly advances C P as an improvement on quantified modal logic,and in view of the subsequent success of the latter, it is natural to ask why hepreferred a quantifier-free alternative. After all, he knew something aboutquantified modal systems, having briefly explored one as early as 1918 [ S S L ,320-324; cf. S L , ch.91. Mo reover , just five years before the appea rance of N L I ,Barcan had published a detailed, quantified version of Lewis's S2 in the Journalof symbolic logic, and three months later Carnap had published a quantifiedversion of S5 in the same journal. As far as I know, Lewis never discussesCarnap's paper. However, in N L I he suggests that his calculus of predicates is abetter approach to an intensional logic of propositional functions than Barcan'squantified modal logic, because her system is 'unnecessarily complex and opento objections on other grounds as well' , though he does not tell us what theseobjections are [424]. Eight years later, in Appendix I11 of the second edition of

SL, he is even more abrupt, saying of Barcan's approach that ' it happens that Ihold certain logical convictions in the light of which I should prefer to approachthe logic of propositional functions in a different way', but he does not sharethese convictions with us [Ap. 5081.Although Lewis never elaborates on his reasons for preferring a system likeC P over quantified modal logic, there are several considerations that almostcertainly played a role in his thinking. Firstly not only did quantified modal logiclack a semantics when Lewis wrote, but only Barcan's and Carnap's syntactictreatments of it were available. Hence, he could not have foreseen the powerand flexibility of today's quantified systems. Secondly (as we shall see in 6),Lewis believes that C P 'could be taken as the basic branch of logic in general',holding that propositional logic could be developed in it, and that with relativelyminor supplementation modal quantification theory could be too [429]. Thirdly,Lewis's sketch of CP is much simpler than Barcan's presentation of quantifiedS2, although this is partly so because she includes various details, e.g . ,formation rules, an explicit formulation of inference rules, and a treatment ofpolyadic propositional functions, tha t Lewis omits.44 In earlier works Lewis discusses the syntactic and the semantic features of polyadic predicativefunctions in some detail [AKV, Ch. 11; SL, 105ffl . H e stresses that NLI only contains an outlineof his system [424], and a fuller exposition would almost certainly have included relationalpredicative functions. Adding these to the syntax of his system (as well as to the semantics for i tpresented below) is straightforward, but it would reduce the net difference in simplicity betweenLewis's and Barcan's accounts.


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Chris SwoyerBut I think the most important reason why Lewis favoured C P over itsquantificational competitors had to do with his views about the roles quantifierscould legitimately play in intensional contexts. We can represent the fact that

'qx^& I)?' mplies 'q2' in Lewis's system with the formula (a) '(q? & I)2)

c.i lewis' calculus of predicates

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