Review: [untitled] Author(s): Carl G. Hempel Reviewed work(s): An Analysis of Knowledge and Valuation. by Clarence Irving Lewis Source: The Journal of Symbolic Logic, Vol. 13, No. 1, (Mar., 1948), pp. 40-45 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2268139 Accessed: 14/05/2008 17:16 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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TBB JOtJIUU.L OP SnrBoLlc LOGIC
Volume 13, Number 1, March 1948'
REVIEWS Numerical CrOSS references are to previous reviews in this JOURNAL or to A bibliography oj symbolic log c (this JOURNAL, vol. 1, pp. 121-218) or to Additions and corrections to the latter (this JOiURNAL, vol. 3, pp. 178-212). References beginning with a roman numeral are by volume and page to previous reviews (or to ,,·or ks previously reviewed). When necessary in connection with such references, a third number will be added in parentheses, indicating the position of the review on the page. Th us "III 157" will refer to a review beginning on page 157 vol. 3 of this JOURNAL (or to the publication reviewed); "III 157 (1)" and "III 157 (2)" will refer respectively to the first and second revie\\·s beginning on page 157 of vol. 3 (or to the publication there reviewed). References such as 7145, 1253 are to the entries so numbered in the Bibliography. Similar references preceded by the letter A or containing the fraction I or a decimal point (as A171, 70!1, 3827.1) are to the Additions and corrections. A reference followed by the letter A is a double reference to an entry of the same number in the Bibliography and in the Additions and Corrections. CLARENCE hVING LEWIS. An analysis of knowledge and valuation. The Paul Carus lectures, seventh series, 1945. The Open Court Publishing Company, La Salle, Ill., 1946, xxi + 567 pp. This volume is divided into three books, which are preceded by an introductory section (pp. 3-31). Book I (pp. 35-168) deals with the problems of meaning and analytic truth, Book II (pp. 171-362) with the foundations of empirical knowledge, Book III (pp. 365-554) with the theory of ethical and esthetic valuation. In consideration of the scope of this JOURNAL, the following discussion will be limited to the first two books, which, in an impressive system of closely reasoned analyses, make a substantial contribution to the clarifiation of basic issues in logic and epistemology. Book I begins with an exposition of Lewis's theory of meaning, which is a slightly modified form of an earlier version outlined in an article in 1943 (IX 28). The reader's attention is called to the discussion, of the ideas advanced in that article and in the book, by Church (IX 28), Baylis (X 106(2) and XIII 60(7», and Carnap (Meaning and necessity, Chicago 1947). The author attributes four different modes, or kinds, of meaning to every term, Le., to every "linguistic expression which names or applies to a thing or things, of some kind, actual or thought of" (p. 38). rrhe denotation or extension of a term is the class of all actual things to \vhich the term applies; each of the latter is said to be denoted by the term. The comprehension of a term includes, in addition to the elements of its extension, all nonactual, but consistently thinkable things to which the term would apply; each of the elements of the comprehension is said to be named by the term. (This seems an infelicitous terminological choice, which is at variance \vith customary usage and is li~kely to create confusion.) The signification of a term is that character whose presence in a thing is necessary and sufficient for the correct applicability of the term to the thing. Finally, a term B is said to be connoted by, or to be contained in the intension of, a term A, if by virtue of the "criterion in nlind" which determines the conditions of application for A, the term A applies to a thing only if B applies, too (pp. 43,44). Thus, e.g., the term "man" signifies humanity and connotes, among others, the terms "mammal," "vertebrate," and "biped." The intension or connotation of a term is the (infinite) conjunction of all the terms connoted by it. Certain perplexities arise from Lewis's admission, to the reahn of his logical entities, of non-existing, or non-actual~ things. Thus, e.g., it appears that he has to include not only physical objects, but also their properties and relations, as well as numbers, etc., and also ordered couples, triples, etc., of these entities, among the "things" which a term may denote. But how is one to distinguish between actual and non-actual things and to sepa rate, accordingly', the extension of a ternl applying to such entities from its intension? Lewis does not enter into the question of how a language countenancing non-actual things
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might be built up in a rigorous nlanner; but the observation might be ventured here that when Lewis speaks of "consistently thinkable things," ,,~hat has to be consistently thinkable is rather a combination of attributes, or a conjunction of tenns signifying them. This suggests the possibility that all the contexts in the analysis of meaning for which Lewis would resort to the concept of comprehension, might be equally well dealt \vith in terms of the concept of intension, so that perhaps-as has been suggested by Carnap (loc. cit.)-the idea of comprehension might be dispensed with altogether. Lewis asserts that comprehension and connotation of a term vary inversely (p. 47), and he implies in particular that a term of universal comprehension has zero connotation (p. 48). But this latter point is not correct on his definition of connotation; for a term such as "red or not red" connotes infinitely many other terms, all of which again have universal cODlprehension. It seems, therefore, that Lewis's definition should be changed to read: The connotation of a term is the conjunction of all those terms which are connoted by it (in the sense stated above), but which are not connoted by every term. Lewis ascribes his four modes of meaning even to those words which are customarily classified as syncategorematic; indeed, he briefly expounds the thesis that "all words have meaning in the same general sense as those which are recognized as being terms" (p. 82). Thus, the phrase "The kennel where Harry's dog sleeps" is said to name something which is an instance of where-ness; and the indefinite article, according to Lewis, "signifies that property which is common to all instances of anything; the property of being a thing nameable by a general name or common noun" (p. 81)-an interpretation which it seenlS difficult to reconcile with the syntax of the indefinite article and with its ability to function in the manner of a universal or an existential quantifier. Generally speaking, it seems to this reviewer that Lewis's discussion of syncategorematics calls for supplementation and substantiation by a more fully developed systematic theory. For surely, the soundness of the proposed approach is not exclusively a matter of logical or empirical fact, nor can it be ascertained merely by probing our intentions or associations in using syncategorematic terms; rather, it is a matter of appraising the adequacy and fruitfulness of a system of concepts proposed as tools for semantic analysis. The same general comment might be applied to all of Lewis's theory of meaning: An adequate appraisal of the value and even of the possibility of the analytic approach advocated by him requires t.he construction of a formalized theory in accordance with it. Lewis extends his distinction of modes of meaning to an analysis of the meaning of statements, statement functions, propositions, and propositional functions. A statement, such as "Mary is making pies," is construed as an expression which asserts, or applies to the actual world, a corresponding proposition; the latter is a linguistic expression without assertive character, which may best be put into the form of indirect discourse or of a participial phrase, such as "that Mary is making pies," or "Mary making pies (now)." Thus, a proposition is construed as a term which is meaningfully applicable to any consistently thinkable world and which signifies a certain state of aft'airs-Buch as Mary making pieswhich any possible world may, or may not, exhibit. Accordingly, the c011l:prehension of a proposition contains all those among the possible worlds which e.xhibit the state of affairs it signifies. The extension of a proposition is then shown to be" the actual world" (strictly speaking, it would be the corresponding unit class) if the statement asserting the proposition is true; otherwise, its extension is "null or zero" (p. 52). The intemion of a proposition "comprises whatever the proposition entails: and it includes nothing else" (p. 55). "All the deducible consequences of a proposition, taken together, exhibit the intension of it discursively" (p. 56). However, as Lewis wishes to characterize analytic propositions 8.8 having zero intension (p. 57), whereas on the given definition, the intension of an analytic proposition would include the infinite class 'of all analytic propositions, it seems advisable to redefine his concept of the intension of a. proposition so as to comprise all those propositions which are entailed by it, but not by every proposition. As a consequence of Lewis's definition, a proposition has no assertive character and,thus cannot be qualified as true or false in the usual sense; Lewis himself uses the device of calling a proposition true if and only if the corresponding statement is true. Furthermore', &S has
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been pointed out by Church (loc. cit.), since a proposition is cOllstrued as a certain kind of linguistlic expression, such phrases as "John eating an apple" and" An apple being eaten by John" are not the same proposition. The interpretation of the different meanings of propositional functions is strictly analogous to that of predicate terms. Lewis is not very explicit, however, as to whether statements are to be considered as terms. While certain passages suggest an affirmative answer, Lewis does not indicate what the different meanings of a statement would be. On this subject he says merely: "The logical properties of statements are correlative with and derivative from those of the corresponding propositions" (p. 65). But the signification of a proposition, which is a certain state of affairs, can hardly be identified with that of the corresponding statement, which ascribes that state of affairs to the actual world; and again, the intension of a proposition consists of propositions, while that of a statement would presumably consist of statements. Thus, the question of the correlation between the meanings of a proposition and those-if any-of the corresponding statements remains unclear. Analytic propositions are characterized as those which apply to every possible world, and which have, accordingly, universal comprehension and zero intension. Consequently, any two analytic propositions have the same meaning in all four modes discussed so far. But not all analytic statements are commonly regarded as synonymous. To account for the idea of synonymy here involved, Lewis introduces some new' concepts of considerable interest. He calls two expressions analytically comparable if they can be analyzed into complexes of constituents in such a way that the complexes have the same syntactical structure, and the constituents occurring at corresponding places in the two complexes have the same intension, which must be neither zero nor universal. Synonymy of analytic or contradictory expressions is then defined as analytic comparability, while synonymy for all other expressions is-perhaps not too fortunately (cf. CarIlsp, loco cit., p. 6l)-construed as simple identity of intensions, rather than as analytic conlparability. The meaning of an expression as determined by the intensions of its constituents and their syntactical relations is referred to as its analytic meaning, in contradistinction to its holophrastic meaning, Le., its intension as a whole expression. Lewis then presents in considerable detail the theses that analytic truths are those which are certifiable by reference to the relations among the intensions of its constituents, and that all a priori truths are analytic. Against the "conventionalist" view, which would construe analyticity as dependent on, or relative to, the linguistic conventions which underly the definitions and the rules of logical inference invoked in ascertaining analytic truth, Lewis argues that what is subject to convention is merely our system of symbolic devices for referring to meanings, but not those meanings themselves, and that the relations among the latter are" absolute and eternally fixed, regardless of any convention" (p. 109). Furthermore, he urges that, rather than being accepted by mere convention, the definitions and the presumptive principles of logic which might be resorted to in establishing analytic truths require a test of soundness which amounts precisely to exhibiting them as analytic. In regard to definitions, this thesis is a consequence of Lewis's view that the definitions essential for proofs of analyticity are of the explicative type, Le., that they declare an equivalence of meaning among terms which already possess meanings, and that definitions of this sort are true, and indeed analytic if c, they equate expressions whose equivalence of intensional meaning is a fact" (p. 97). But it is at least questionable whether the meanings of all terms are so completely fixed as to render the requisite decisions unequivocally possible. Rather, it seems that the exact meaning even of familiar terms, and thus the analyticity of certain explicative statements, will in part have to be determined by convention. Furthermore, Lewis's own definition o~ intensions as consisting of linguistic expressions makes the meaning of a term dependent upon the means of expression available in the given language; and if a language be considered inadequate unless all meaning relations can be expressed in it, then there can be no adequate language which is consistent, as is shown by the results of Gadel, Tarski, and others. Lewis goes on to distinguish two interpretations of intensional meaning, both of which are said to be compatible with what has been said on the subject 80 far. The linguistic
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meaning of an expression is determined by its logical relations to t.he other terms of the language; it can be known without knowledge of the semantics and pragmatics of the language. The sense meaning of an expression is roughly what is commonly called its operational meaning; more precisely, it is "the criterion in 1nind, by reference to which one is able to apply or refuse to apply the expression in question in the case of presented, or imagined, things or situations" (p. 133). The criterion in question is construed as referring, in every ca.se, to data \vhich may be apprehended in direct experience. Le\\yis argues that the decision a,s to the analyticity of propositions containing empirical terms ultilnately is provided by an "experiment in imagination" (p. 151) of trying to combine the sense Ineanings of the empirical terms in the manner dictated by t.he logical terms and the syntax of the proposition. This kind of experiment would reveal, for example, that" All squares being rectangles" is universally applicable, and thus analytic (p. 152). This method of connecting analyticity \\yith sense meaning, however, raises serious problems. Even if it is possible to avoid, in the suggested approach, a subjectivistic interpretation of analyticity, which is repudiat.ed by the author, there remains a logical difficulty: If the sense meaning of an expression E is construed as representing a necessary and sufficient condition for the correct applicability of E, then clearly most ternlS and propositions of empirical science-especially all the theoretical constructs-have no sense meaning. And if-as most of Lewis's formulations suggest-it is to be understood as representing not a necessary and sufficient condition, but rather, say, a condition whose fulfillment makes its correct applicability highly probable, then the logical relationships between the sense meanings of two expressions will not generally be paralleled by the same relationships between the two expressions themselves; hence the logical relationships among sense meanings cannot serve as a criterion of analyticity. The same difficulty is encountered if the sense meaning of a statement is construed, as I.Jewis does at some places, as" an unlimited number of predictions of possible future confirmations" (p. 192); besides, this interpretation confers upon the sense meaning of a statement a dependence upon time which is alien to the customary notion of intension.-l'hese observations seem to indicate that, apart from its unsuitability to explicate analyticity, the concept of sense meaning does not provide, as Lewis maintains, an interpretation of his earlier concept of intension. The concept of sense meaning occupies a central position in the penetrating analysis of the foundations of enlpirical knowledge which Lewis presents in Book II of his work. Discussion will here have to be focussed on those aspects of this analysis \\yhich are particularly relevant to logic, to the exclusion of much material which is of great interest for epistemology. The sense meaning of any verifiable statenlent of objective fact is said to be exhibitable in an inexhaustible set of judgments to the effect" that a certain enlpirical eventuation will ensue if a certain mode of action be adopted" (p. 211). rrhe antecedent and the consequent of a hypothetical judgment of this kind are conceived of as being formulated in "expressive language," which refers to "a directly presented or presentable content of experience" (p.179). Conditional judgments of this kind are said to be capable of "decisive and complete verification or falsificat.ion" (p. 181), and are therefore called ter1ninating judg1nents. Lewis devotes special attention to the nature of "the 'if-then' relation in terminating judgments." He points out that it is not the relation of logical deducibility; and since the 'if-then' in question can and must be applied also in counterfactual instances, i.e., in cases where t.he test conditions referred to in the antecedent are not realized: he is able to show readily that neither formal nor material implication is signified by the expression under consideration. Lewis mentions that the latter has the same connotation as the 'if-then'" in, natural laws and says that it signifies natural, or real, connection; he does not, however, suggest an analysis in terms of concepts \vhich are luore clearly understood. Next, the author turns to a closer investigation of the relationship between the" nonterminating" empirical statements of scientific and everyday discourse and the terminating judgments which formulate the ultimate empirical basis for their validation. On very plausible grounds, he argues that his original simple thesis of the strict translatability of each statement of the former kind into an infinite class of terminating judgments ~pp. 181,
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193) h~ to be replaced by the weaker assertion that the two kinds of statements are connected by a network of probability relations (p. 237). In his detailed analysis of the structure of lthis network, the author relies rather extensively on the following principle: "When the probability that if 'P' be false, 'E' also will be false, approximates to certainty, the assurance of' P' itself will approximate to certainty when' E' is found true" (p. 238). This rule, however, does not generally hold, as is shown by the case where' P' stands for 'x is a billionaire,' and' E' for' x is an a.lbino.' The mathematical proof of the rule, as given on p. 238, turns on the assumption that the fraction WK/(WK + (1 - W)(1 - N», in which W, K, N are certain probabilities, "will be nearer to unity according as W is nearer to unity or N is nearer to unity"; actually, however, for given Wand N, however close to (but not identical with) unity, the ratio may still be as close to 0 as desired, if only K is sufficiently small.-While this point does not affect Lewis's position decisively, it necessitates a more complex account of the probability structure of empirical knowledge. Subsequently, Lewis presents his views on the nature of probability. In regard to the aprioristic approach, he rejects, in effect, the Principle of Indifference as a criterion of equiprobability; in regard to the "empirical theory" of probabilities as objective frequencies, bis fundamental criticism is that on this interpretation, a probability statement can never be certain, but only probable; and the assertion of this latter probability, if interpreted in accordance with the empirical conception, is itself merely probable, and so forth; so that, for the problem of the cognitive status of empirical beliefs, the empirical theory of probability "would not provide a solution but only the beginning of s, perpetual stutter" (p.289). Lewis then outlines his own conception of probability, which is aprioristic in character, but avoids reliance on the Principle of Indifference and incorporates what Lewis considers the sound core of the empirical theory. In effect, he construes probability statements as assertions of the type, "Relative to data' D', the probability is a/b that c, which is an instance of the property 1/1, will also have the property
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Dlents of science; but it might be viewed as casting doubt upon the suitability of that empirical concept for an adequate interpretation of the so-cal~ed probability of theories on the basis of given evidence. Lewis's suggestion that a concept of a purely logical character is needed for this purpose is in agreement with the ideas of Keynes, Jeffreys, Carnap, and others; and his further point that this logical probability should be interpretable as an estimate of a frequency accords well with the interpretation explicitly suggested by Carnap for the logical concept of probability (cf. XI 19(1), p. 528, and XII 104(1). However, the logical concept of probability envisaged by Lewis does not explicate that meaning of probability which underlies the statistical probability statements of science; for these are clearly factual in character. Therefore, there appears to be room and need for both the empirical and the logical concept of probability. (3) One of the most puzzling aspects of Lewis's treatment of probability is his concept of categorical probability statement. For hypothetical probability statements \vhich, on the basis of data (D', assert a probability value alb exceeding 1/2, Lewis uses the abbreviation "If D, then it is probable that P" (p. 318). In contradistinction to this kind of hypothetical probability statement, a categorical probability statement is then defined as having the form "If D, then it is probable that P, and D is the fact." Repeatedly, however, this statement is rendered simply by the phrase" P is probable," which is said to be a "categorical probability conclusion" that follows from the data' D', by virtue of "the principles of probability or rules of induction," in a manner analogous to inference by modus ponens in deductive logic (pp. 319,321). Lewis adds that this categorical conclusion "is still relative to the grounds of judgment," and that it might be construed as referring to "all the available evidence which is pertinent" (p. 319); but this characterization seems irreconcilable with the previous interpretation of categorical probability statements as conclusions derived from data which presumably may be chosen 88 narrowly as one pleases. Thus, the matter remains obscure. Lewis himself expresses the opinion that categorical statements of probability represent a fundamental and independent category of knowledge whose meaning cannot be explicated by reduction to factualities (p. 320). The obscurities here referred to affect seriously certain phases of Lewis's inquiry, including his formal analysis of the probability structure of empirical knowledge (pp. 247-253), in which he uses a special notation, '(h)X', for "In all probability, X." (4) Lewis's discussion provides no full explication of the intended concept of probability: no general rules are given for the determination of probabilities and reliabilities, and there is no way of ascertaining to what extent Lewis's concept of probability would satisfy the principles of probability theory. The constructive part of his observations is therefore programmatic in character, and a final evaluation has to wait for the presentation of a CARL G. HEMPEL rigorous formalized theory which embodies Lewis's ideas. ANDRZEJ Mos'rowsKI. Axiom of choice for finite sets. Fundamenta mathematicae, vol. 33 (1945), pp.137-168. As is well known, the axiom of choice is equivalent to the following proposition: "For every class K of non-empty sets there is a 'choice function' ~(X) = ~i:(X) such that, for all X EK, ~(X) is defined and ~(X) EX." The aim of this paper is the study of the mutual dependence or independence of the particular cases of this axiom in which K is restricted to be a class of finite sets of given cardinal number. (Dependence is meant relatively to the Zernlelo postulate system of set theory, as formulated exactly by Quine (II 51 (3)).) For every positive integer n, let [n] denote the proposition arising from the above formulation of the axiom of choice by replacing the clause" for every class K of sets" by "for every class K whose element.s are sets of cardinal number n." For every finite set Z == tfl,} , n2 , , nk} of positive integers, let [Zl denote the conjunction of the propositions
[nl], [n2],
, [nk].
By means of group-theoretic (as ,veIl as set-theoretic and metamathematical) considerations, Mostowski gives on the one hand a sufficient condition, on the other hand a necessary condition under which a proposition [Z] implies the proposition [n]. In order to facilitate a clear reformulation of these conditions, let us, for any group G, call the set of positive integers i such that G has a proper subgroup of index i, the index set of G; and the set of the
