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Journal of Philosophy, Inc.

More about `About'Author(s): Hilary Putnam and J. S. UllianReviewed work(s):Source: The Journal of Philosophy, Vol. 62, No. 12 (Jun. 10, 1965), pp. 305-310Published by: Journal of Philosophy, Inc.Stable URL: http://www.jstor.org/stable/2023637 .

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VOLUMELXII, No. 12 JUNE 10, 1965

THE JOURNAL OF PHILOSOPHYMORE ABOUT 'ABOUT'JNan earlier note, published in Mind,' a question raised by Good-man2 was answered; it was shown that if a statement S isabsolutely about k the negation of S must be also. The solutiongiveni applied directly to only a special class of cases, and so was

open to misunderstanding. In the present note we extend theearlier solution and then provide an alternative treatment which isfelt to have independent interest.The relevant portion of Goodman's explication runs as follows:S is absolutely boutk if andonly if somestatement T followsfromS differentiallywith respect to k . .. [where] a statement T followsj rom S differentiallywithrespect to kif T contains an expressiondesignatingk andfollowslogicallyfromS,while no generalization of T with respect to any part of that expressionalsofollowslogically from S (p. 7).In CG it was noted that S can always be expressed set-theoretically,with membership signified by a two-place predicate 'E'.3 Then itwas shown that if S implies a schema T without implying (y) Tthere is a schema R such that -S implies R without implying (y)R.In the proof given it was apparent that R could always be takenas -S itself. Further, it appeared to be a consequence of theanalysis offered that S could not be absolutely about k withoutmentioning k-in direct conflict with Goodman's announcedintentions.Goodman's example of a statement absolutely about somethingthat it does not mention is 'Cows are animals', taken to be absolutelyabout noncows. Now CG, with variables alone counted as desig-nating expressions, took the predicate 'E' to be governed by nospecial axioins, which amounted to the exclusion of all set-theoretic

1 J. S. Ullian, "Corollaryto Goodman'sExplicationof 'About',"Mind, 71,284 (October, 1962): 545. In what follows this paper will be referred to as CG.2 Nelson Goodman,"About," M1ind,70, 277 (January, 1961): 1-24.3 Clearly,precisetreatment of the problemat handrequires hat attention bedirected to formalrepresentationsof the sentences in question,e.g., their "trans-lations"into predicatecalculus. The analysis (bothGoodman'sandours)appliesprecisely to such formalizations,and so with as much success to sentences ofnaturallanguage as there is successin achieving their formalrepresentation.

305? Copyright1965 by Journal of Philosophy, Inc.

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306 THE JOURNAL OF PHILOSOPHYprinciples-even those involving the Boolean operations-from thelogical apparatus considered to be at hand. Given a free variablewhose designation was taken to be the class of cows, there was noready means for referring to the class of noncows; so the requisiteinference from 'Cows are animals' was not forthcoming. If theanalysis of CG is to lend itself to this more general case (moregeneral in its broader construal of what is to count as logical infer-ence) it must be shown that the argument of CG can be extendedto hold when the Boolean laws are counted as part of the logicalapparatus-more precisely, when Boolean operations upon whatare counted as terms yield what are again counted as terms ap-propriately governed. In showing this we will be upholding thepromise of CG's last paragraph.4To this end we first supplement the analysis of CO as follows:Again take variables as terms, but now allow Boolean combina-tions of terms to count as terms as well. Adopt axiom sche-mata (or definition schemata) to make 'E' conform to the Booleanlaws. For example, where 'r', 's', and 't' stand for terms, onemight adopt (a) Er -- Ers, (b) Er(s ' t) (Ers V Ert), and(c) Er(svt) (Ers-Ert). Now 'Cows are animals' may berendered '(x) (Exy D Exz)', and with the aid of two instancesof (a) we may derive by quantificational logic '(x) (Ex2 DEx)',containing the term 'g' which designates the class of noncows.'(y) (x) (Ex2 D Exg)' can clearly not be so derived; so 'Cows areanimals' turns out, as desired, to be absolutely about noncows.To extend the argument of CG to cover the present case-or anyparallel case where the terms are built by operations upon variablesit will suffice to show that if S implies a schema T that contains aterm t built from the variables y, ... , y. and S implies (y,)T forno i from 1 to n, then there is a schema R containing t such that -Simplies R while S implies (yi)R for no i from 1 to n. But thisis established by taking R as T D -S. Then R contains t, sinceT does, and R is clearly implied by S. In fact, since S implies T,R is equivalent to -S. So, if S implied (yi)R for some i, then-S would imply (yj)-S, and, by the argument of CG, S wouldimply (yi)T, violating our hypothesis.In the development just outlined it is expressions built fromvariables that are taken as designating; in CG variables alone servein this capacity. Now an alternative treatment is forthcoming ifwe vary in another direction the stock of expressions taken asdesignating. Let us think of variables and predicates alike as

4 "The argumentcan be extendedto cases in which logical truth (and hencedifferential mplication) is construedin terms broaderthan those of quantifica-tional validity.

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MORE ABO UT 'ABO UT' 307engaged in designation, variables of individuals and predicates oftheir extensions. We now gain terms designating Boolean com-pounds by counting truth functions of predicates as predicates. Ifwe wish to avail ourselves of a larger supply of designating ex-pressions we may construe 'predicate' yet more broadly and soprovide ourselves with further terms. Our development will beneutral with regard to the question of what is to count as a term.Now let us say that a quantificational schema S(P) containing thepredicate letter 'P' implies its own generalization with respect to'P' if S(P) D S(Q) is valid, where 'Q' is a predicate letter notoccurring in S(P) and S(Q) is the result of putting 'Q' for alloccurrences of 'P' in S (P). Let us call a schema simple if it doesnot imply its own generalization with respect to any of its freevariables or predicate letters. Then: Every quantificational schemais equivalentto a simple schema. For let S(P) be a schema that im-plies its own generalization with respect to 'P'. Then S (P) D S (Q)is valid, where S(P) does not contain 'Q'. The Craig InterpolationTheorem asserts that if A D B is valid, then there exists a schemaC such that A D C is valid, C D B is valid, and C contains onlypredicate letters common to A and B. Thus there exists an S'containing only letters in S(P) other than 'P' such that S(P) D S'and S' D S(Q) are both valid. By the Rule of Substitution forpredicate letters, S' D S(P) is also valid (substituting 'P' for'Q' in S' D S(Q)). Thus S' is equivalent to S(P). If S' impliesits own generalization with respect to one of the remaining predicateletters, iteration of the argument guarantees existence of S", S"',and eventually a schema which is equivalent to S(P) andwhich does not imply its own generalization with respect to any ofits predicate letters.5 If this schema implies its own generalizationwith respect to some of its free variables, then universal general-ization with respect to those variables yields the desired simpleequivalent of S (P). Clearly, any two simple equivalents of aschema S contain exactly the same predicate letters and freevariables.Now S is absolutely about k if and only if the simple equivalentsof S contain free occurrencesof all the free variables and occurrencesof all the predicate letters that occur in some term designating k.For let T be a simple equivalent of S, t a term designating k which

6 Of coursea simple equivalentof S will be devoid of predicate etters entirelyif S is either valid or inconsistent. Presumably such sentences as '(y) (y = y)'and its denial will be available as simple equivalents in these cases. But suchcases are of no importancehere in any event, since neither valid nor inconsistentschematacan representstatements that are absolutely about anything. It is tobe noted that there can be no effective method of discoveringa simple equivalentfor an arbitraryquantificationalschema.

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308 THE JOURNAL OF PHILOSOPHYuses only predicate letters and free variables from T, and R aschema containing an occurrence of t. Then the schema (R D R) *Tis equivalent to both T and S, so is implied by S, but cannot implyits own generalization with respect to a free variable or predicateletter of t lest T fail to be simple. Conversely, if S is absolutelyabout k then S implies some schema T* differentially with respectto k with some term t the relevant term designating k. Also anysimple equivalent T of S implies T*. If t contains a free variableor predicate letter foreign to T, then T implies the generalizationof T* with respect to that free variable or predicate letter, so doesS, and the assumed differential implication is contradicted.

Given only minimally much in the way of terms, it is a con-sequence of this result that any statement that is absolutely aboutanything at all is absolutely about both the universal class and thenull class.6 It will be recalled that this is precisely one of theconsequences of Goodman's own analysis (p. 11).Now we know from CG that, if a schema S implies (y)>S,then S implies (y)S, and the converse is immediate by taking '-S'for 'W'. Similarly it can be established that S implies its owngeneralization with respect to a predicate letter 'P' if and only if,S does. This tells us that if T is a simple equivalentof S, then-T is a simple equivalent of S, and it follows from the resultabove that S and S must always be absolutely about the samethings.The development just given does ask that we construe predicates

as designating expressions. But, to its credit, it requires noquantification over classes and so keeps us in the full sense withinfirst-order logic. And it demands no special axioms or definitions,since first-order logic itself provides the strength of such schemataas (a)- (c), to which appeal was necessary in the earlier development.

6 The casein which a schema'ssimpleequivalentscontainno predicate etters(say 'y = z') falls into step here if we invoke (i) elimination of singularterms infavor of predicates, (ii) use of the additional apparatusof our earlierdevelopment,or (iii) construalof 'y=' and '= z' as terms. Withoutsuch an expedient, 'y-x'fails to be absolutely about anything but y and z. Under (i) we take 'F'and 'G' as true of only y and only z respectively, then transform 'y = z' into'( 3w) (x)(Fx =. x = w)- (3 w) (x) (Gx_ .x = w)- (x) (w)(Fx.Gw D.x = w)'; under(ii) we have the term 'y -J ' available; under (iii) we have the alternation of'y=' with its denial. Further,note that without similar expedient 'Fy' fails tobe absolutely about the class of objects distinct from y. Adoptionof (i) or (ii)remediesthis (if it be thought to need remedy), as does adoption of a principle(iii') underwhich 'y=' is to be regardedas a predicateof any schemacontainingfree 'y,' and similarly for other variables. One is fully free to adopt such ex-pedients in general if one wishes to enlarge the stock of terms on hand. Ourtheoremis unaffected,since only its applicationturns on what are taken to be theterms available.

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MORE ABOUT 'ABOUT' 309POSTSCRIPT

In each of the developments above we construe the parts of termssusceptible to generalization to be either lone variables or lonepredicate letters. In the first development, where no predicateletter figures in a term, it is only the variables; in the second it isboth. This construal is reflected in the central argument of CG,in its uses here, and in the theorem above relating simple equivalentsto absolute aboutness. We show now that the construal can beliberalized-perhaps in keeping with Goodman's intentions-with-out affecting the extension of 'absolutely about'.

Consider the sentence 'There are brown things and there arecows', which, following our respective developments, can be renderedeither(1) (3x)Exw* (3x)Exyor(1') (3x)Bx. (3x)CxIs this absolutely about brown cows? On our earlier account wesee that it is, by regarding (l)'s consequence(2) (3 x)Exw. (3 x)Exy. (x) (Ex(w ny) D Ex(w n y))or (l')'s parallel consequence(2') (3 x)Bx *( 3 x)Cx- (x) (BxCx D BxCx)But if we allowed generalization with respect to compound terms wecould cite (l)'s further consequence(z)((3x)Exw (3x)Exy' (x)(Exz D Exz))as evidence that (1) does not imply (2) differentially with respectto w n y. Similarly we could cite (l')'s further consequence

(3 x)Bx* 3 x)Cx. (x) (QxD Qx)as evidence that (1') does not imply (2') differentially with respectto the intersection of the extensions of 'B' and 'C'.Now with the help of an instance of schema (c) we derive from(I) the conjunction of (1) with

(x) (Exw -Exy) D ( 3 x) (Ex (wr y))while (1') directly yields its own conjunction with

(x) (Bx _ Cx) D ( 3x) (BxCx)

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310 THE JOURNAL OF PHILOSOPHYSince no generalization will defeat these as appropriate differentialimplications, we conclude that the cited sentence is absolutelyabout brown cows. Now our problem is this: to show that theliberalization of 'generalizable parts' nowhere narrows the extensionof 'absolutely about'. We need to establish this in order to beassured that the relationship between absolute aboutness and simpleequivalents is not dependent on our earlier construal, even thoughthe proof given for that result was so dependent.

For the remainder of the Postscript we confine our attention tothe second of our developments; similar arguments are availablefor the f...