More about 'About'

More about `About' Hilary Putnam; J. S. Ullian The Journal of Philosophy, Vol. 62, No. 12. (Jun. 10, 1965), pp. 305-310. Stable URL: http://links.jstor.org/sici?sici=0022-362X%2819650610%2962%3A12%3C305%3AMA%60%3E2.0.CO%3B2-U The Journal of Philosophy is currently published by Journal of Philosophy, Inc..

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http://www.jstor.org Sun Jun 17 04:48:49 2007

JVKE 10, 1963

VOLUMELXII, KO. 1 2

NORE ABOVT 'ABOUT'

IN

an eariier note, published in ,Vt?~d,l a question raised by Goodman2 was answered; it was shown that if a statement S is absolutely about k the negation of S must be also. The solution give11 applied directly to only a special class of cases, and so was open to misunderstanding. I n the present note I\-e extend the r.:irlier solution and then provide an alternative treatment which is felt to h v e independent interest. The relevant portion of Goodman's explication runs as follows: S zs absolutely about h if and only if some statement 2' follons from S differentially nith respect to I: . . . [where] a statenlent T follo~vs,from S differentialiy n i t h ~espectto k if 1' contains an expression designating k and fol1ov;s logically from S, nhile no generalization of T mith respect to any part of that expression also follows logically from S (p. 7).

I n CG it was noted that Scan always be expressed set-theoretically, mith membership signified by a two-plsce predicate Then it was shown that if S implies a schema 1' without implying ( y ) T thcre is a schema R such that --Simplies R without implying (y)R. In the proof given it was apparent that R could always be taken as --S itself. Further, it appeared t o be a coilsequence of the analysis of;"ered that S could not be absolutely about k without mentioning /;-in direct conflict with Goodman's announced intentions. Goodman's example of a statement absolutely about something are animals', taken to he absolutely that it does not mention is 'COTT~ about noneo\\-s. Now CG, ~ i t vhariables alone counted as designating expressions, took the predicate 'E' to be governed by no special axioms, xvl~iehamounted to the exclusion of all set-theoretic J. S. Ullian, "Corollary to Goodman's Explication of 'About'," l f i n d , 71, 284 (October, 1962) : 545.

In what follo\vs this paper w~llbe referred to as CG. Nehon Goodman, "About," Afznd, 70, 277 (January, 1961): 1-24. a Clearly, precise treatment of the problem a t hand requires that attention be directed to formal representations of the sentences in question, e.g., their "translations" into predicate calculus. The analysis (both Goodman's and ours) applies prec~selyto such formalizations, and so lrith as much success to sentences of natural language as there is success in achieving their formal representation.

305 @ Copyright 1965 by Journal of Philosophy, Inc

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principles-even those involving the Boolean operations-from the logical apparatus considered to be at hand. Given a free variable whose designation was taken to be the class of cows, there was no ready means for referring to the class of noncows; so the requisite inference from 'COWSare animals' was not forthcoming. If the analysis of CG is to lend itself to this more general case (more general in its broader construal of what is to count as logical inference) it must be shown that the argument of CG can be extended to hold when the Boolean laws are counted as part of the logical apparatus-more precisely, when Boolean operations upon what are counted as terms yield what are again counted as terms appropriately governed. I n showing this we will be upholding the promise of CG's last ~ a r a g r a p h . ~ To this end we first supplement the analysis of CG as follows: Again take variables as terms, but now allow Boolean combinations of terms to count as terms as well. Adopt axiom schemata (or definition schemata) to make 'E' conform t o the Boolean laws. For example, where 'r', 's', and 't' stand for terms, one might adopt (a) Erg --= -- Ers, (b) Er(s u t ) --= (Ers V Ert), and (c) E r (s u t) = (Ers-Ert). Now 'Cows are animals7 may be rendered '(x)(Exy > Exz)', and with the aid of two instances of (a) we may derive by quantificational logic '(x)(ExZ 3 Exfj)', containing the term 'Q' which designates the class of noncows. '(y) (x) (Ex2 > Exg)' can clearly not be so derived; so 'Cows are animals' turns out, as desired, to be absolutely about noncows. To extend the argument of CG to cover the present case--or any parallel case where the terms are built by operations upon variables -it will suffice to show that if S implies a schema T that contains a term t built from the variables yl, . . . , y, and S implies (yi)T for no i from 1 to n, then there is a schema R containing t such that --S implies R while NS implies (yd)R for no i from 1 to n. But this is established by taking R as T > --S. Then R contains t, since T does, and R is clearly implied by -8. I n fact, since S implies 1', R is equivalent t o --S. SO, if 4 3 implied (yJR for some i, then --S would imply (yg)--S, and, by the argument of CG, S would imply (yi)T, violating our hypothesis. I n the development just outlined it is expressions built from variables that are taken as designating ;in CG variables alone serve in this capacity. Now an alternative treatment is forthcoming if we vary in another direction the stock of expressions taken as designating. Let us think of variables and predicates alike as 4 "The argument can be extended to cases in which logical truth (and hence differential implication) is construed in terms broader than t,hose of quantificational validity.

M O R E ABOUT 'ABOUT'

307

engaged in designation, variables of individuals and predicates of their extensions. We now gain terms designating Boolean compounds by counting truth functions of predicates as predicates. If we wish to avail ourselves of a larger supply of designating expressions we may construe 'predicate' yet more broadly and so provide ourselves with further terms. Our development will be neutral with regard to the question of what is to count as a term. Now let us say that a quantificational schema S ( P ) containing the predicate letter 'P' implies its own generalization with respect to 'P' if S ( P ) > S(Q) is valid, where 'Q' is a predicate letter not occurring in S ( P ) and S(Q) is the result of putting 'Q' for all occurrences of 'P' in S ( P ) . Let us call a schema simple if it does not imply its own generalization with respect to any of its free variables or predicate letters. Then : Ez~eryquantificational schema i s equivalent to a simple schema. For let S ( P ) be a schema that implies its own generalization with respect to 'PI. Then S ( P ) > S(Q) is valid, where S ( P ) does not contain 'Q'. The Craig Interpolation Theorem asserts that if A 3 B is valid, then there esists a schema C such that A > C is valid, C > B is valid, and C contains only predicate letters common to A and B. Thus there exists an Sf containing only letters in S ( P ) other than 'P' such that S ( P ) > S' and S' > S(Q) are both valid. By the Rule of Substitution for predicate letters, S' > S ( P ) is also valid (substituting 'P' for 'Q' in S' > S (Q)). Thus S' is equivalent to S ( P ) . If S' implies its own generalization with respect to one of the remaining predicate letters, iteration of the argument guarantees existence of S", S"', . . . , and eventually a schema which is equivalent to S ( P ) and which does not imply its own generalization with respect to any of its predicate lettem6 If this schema implies its own generalization with respect to some of its free variables, then universal generalization with respect to those variables yields the desired simple equivalent of S(P). Clearly, any two simple equivalents of a schema S contain exactly the same predicate letters and free variables. Now S i s absolutely about k i f and only i f the simple equivalents of S contain free occurrences of all the free variables and occurrences of all the predicate letters that occur in sonze term designating k . For let T be a simple equivalent of S, t a term designating k which Of course a simple equivalent of S will be devoid of predicate letters entirely if S is either valid or inconsistent. Presumably such sentences as '(u)(y = y)' and its denial will be available as simple equivalents in these cases. But such cases are of no importance here in any event, since neither valid nor inconsistent schemata can represent statements that are absolutely about anything. I t is to be noted that there can be no effective method of discovering a simple equivalent for an arbitrary quantificational schema.

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uses only predicate letters and free variables from T, and R a schema containing an occurrence of t. Then the schema (R > R) . T is equivalent to both T and S, so is implied by S, but cannot imply it~sown generalization with respect to a free variable or predicate letter of t lest T fail to be simple. Conversely, if S is absolutely about k then S implies some schema T* differentially with respect to k with some term t the relevant term designating k. Also any simple equivalent T of S implies T*. If t contains a free variable or predicate letter foreign to T, then T implies the generalization of T* with respect to that free variable or predicate letter, so does S, and the assumed differential implication is contradicted. Given only minimally much in the way of terms, it is a consequence of this result that any statement that is absolutely about anything at all is absolutely about both the universal class and the null class.6 I t will be recalled that this is precisely one of the consequences 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 '8'. Similarly it can be established that S implies its own generalization with respect to a predicate letter 'P'if and only if -S does. This tells us that if T i s a simple equivalent of 8, then --T i s a si?nple equivalent of --S, and it follows from the result above that S and --S must always be absolutely about the same things. The development just given does ask that we construe predicates as designating expressions. But, to its credit, it requires no quantification over classes and so keeps us in the full sense within first-order logic. And it demands no special axioms or definitions, since first-order logic itself provides the strength of such schemata as (a)-(c), to which appeal mas necessary in the earlier development. T h e case in which a schema's simple equivalents contain no predicate letters (say ( y = z') falls into step here if we invoke (i) elimination of singular terms in favor of predicates, (ii) use of the additional apparatus of our earlier development, or (iii) construal of 'y=' and ' = z ' as terms. Without such an expedient, ' y = z' fails to be absolutely about anything but y and z. Under (i) we take 'I"' and 'G' as true of only y and only z respectively, then transform 'y = z' into '(3w)(x)(Fx = . x = w).(3w)(x)(Gx=.x = w ) . ( x ) ( w ) ( F x . G w 3 . x = w)'; under (ii) we have the term ' y v ?j' available; under (iii) we have the alternation of ' y = ' with its denial. Further, note that without similar expedient 'Fy' fails to be absolutely about the class of objects distinct from y. Adoption of (i) or (ii) remedies this (if it be thought to need remedy), as does adoption of a principle (iii') under which 'y =' is to be regarded as a predicate of any schema containing free 'y,' and similarly for other variables. One is fully free to adopt such expedients in general if one wishes to enlarge the stock of terms on hand. Our theorem is unaffected, since only its application turns on what are taken to be the t e r n s available.

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I n each of the developments above we construe the parts of terms susceptible t o generalization to be either lone variables or lone predicate letters. I n the first development, where no predicate letter figures in a term, it is only the variables; in the second i t is both. This construal is reflected in the central argument of CG, in its uses here, and in the theorem above relating simple equivalents to absolute aboutness. We show now that the construal can be liberalized-perhaps in keeping with Goodman's intentions-without affecting the extension of 'absolutely about'. Consider the sentence 'There are brown things and there are cows', which, following our respective developments, can be rendered either

Is this absolutely about brown co~vs? On our earlier account we see that it is, by regarding (1)'s consequence (2)

(3x)Exw. (3x)Exy. (x)(Ex(w ny) > EX(Wn y))

or (1')'s parallel consequence (2')

( 3 x)Bx . ( 3 x) Cx . (x) (BxCx > BxCx)

But if we allowed generalization with respect to compound terms we could cite (1)'s further consequence (2)

((3x)Exw- (3x)Exy. (x) (Exz 3 Exz))

as evidence that (1) does not imply (2) differentially with respect to w n y. Similarly we could cite (1')'s further consequence (3x)Bx. (3x)Cx. (x)(Qx > Qx) as evidence that (1') does not imply (2') differentially with respect to the intersection of the extensions of 'By and 'C'. Now with the help of an instance of schema (c) we derive from (1) the conjunction of (1) with

while (1') directly yields its own conjunction with (x) (Bx

r

Cx)

> ( 3 x) (BxCx)

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Since no generalization will defeat these as appropriate differential implications, we conclude that the cited sentence is absolutely about brown cows. Now our problem is this: to show that the liberalization of 'generalizable parts' nowhere narrows the extension of 'absolutely about'. We need t o establish this in order t o be assured that the relationship between absolute aboutness and simple equivalents is not dependent on our earlier construal, even though the proof given for that result was so dependent. For the remainder of the Postscript we confine our attention to the second of our developments; similar arguments are available for the first. Let T be a simple equivalent of a consistent schema S, let 'B' and 'C' be predicate letters occurring in T, and let 'P' and 'Q' be predicate letters foreign to T. Suppose these all to be monadic predicates. If S implies '- ( 3 x) (BxCX)', then, since S is consistent, S does so differentially with respect to the intersection of the extensions of 'B' and 'C'. So suppose it doesn't; that is, suppose S and its equivalent T consistent with ' ( 3 x) (BxCx)'. Let R be the valid schema ' ( 3 x) (BxPxCx) 3 ( 3 x) (BxCx)'. Then TR is equivalent to T, hence is implied by S. But T does not imply the "generalization" of TR with respect to 'BxCx'. For that generalization implies the generalization of R with respect to 'BxCx', which is t o say that it implies ' ( 3 x) (BxPxCx) > ( 3 7)Qr'. Since both 'P' and 'Q' are foreign to T and T is consistent with '(3x) (BxCx)', there is an interpretation making T true and ' ( 3 x) (BxPxCx) > ( 3 x)Qx' false. Since T is equivalent to S, it follows that S implies TR differentially with respect t o the intersection of the extensions of 'B' and 'C', even on our broadened construal of 'generalizable parts'. Similarly, taking R' as ' ( 3 x)Px. (x) (Bx 3 --Px) .3 ( 32)-Bx', X implies either '-- ( 3 x)--Bx' or TR' differentially with respect to the complement of the extension of 'B'. The argument may easily be extended t o apply t o all compound predicates built by truthfunctional composition, and variants of it yield like results for cases where quantifiers are allowed as parts of predicates. So the extension of 'absolutely about' is the same whether we allow generalization with respect to compound terms or only with respect to lone predicate letters and free variables.

HILARY PUTNAM MASSACHUSETTS INSTITUTE OF TECHNOLOGY

J. S. ULLIAN UNIVERSITY OF CALIFORNIA, SANTABARBARA

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[Footnotes] 1

Corollary to Goodman's Explication of 'About' J. S. Ullian Mind, New Series, Vol. 71, No. 284. (Oct., 1962), p. 545. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28196210%292%3A71%3A284%3C545%3ACTGEO%27%3E2.0.CO%3B2-J 2

About Nelson Goodman Mind, New Series, Vol. 70, No. 277. (Jan., 1961), pp. 1-24. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28196101%292%3A70%3A277%3C1%3AA%3E2.0.CO%3B2-T

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