putnam. more about `about
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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 OFPHILOSOPHY
MORE ABOUT 'ABOUT'
JNan earlier note, published in Mind,' a question raised by Good-
man2 was answered; it was shown that if a statement S is
absolutely about k the negation of S must be also. The solution
giveni 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 is
felt 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 it
was shown that if S implies a schema T without implying (y) T
there 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 taken
as -S itself. Further, it appeared to be a consequence of the
analysis offered that S could not be absolutely about k without
mentioning k-in direct conflict with Goodman's announced
intentions.Goodman's example of a statement absolutely about something
that it does not mention is 'Cows are animals', taken to be absolutely
about noncows. Now CG, with variables alone counted as desig-
nating expressions, took the predicate 'E' to be governed by no
special 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 be
directed 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 PHILOSOPHY
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 'Cows are 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 infer-
ence) 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 ap-
propriately governed. In showing this we will be upholding thepromise of CG's last paragraph.4
To 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 Boolean
laws. For example, where 'r', 's', and 't' stand for terms, one
might adopt (a) Er -- Ers, (b) Er(s ' t) (Ers V Ert), and
(c) Er(svt) (Ers-Ert). Now 'Cows are animals' may be
rendered '(x) (Exy D Exz)', and with the aid of two instances
of (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 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 y, ... , y. and S implies (y,)T forno i from 1 to n, then there is a schema R containing t such that -S
implies R while S implies (yi)R for no i from 1 to n. But this
is established by taking R as T D -S. Then R contains t, since
T 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 would
imply (yi)T, violating our hypothesis.
In 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 argumentcan be extendedto cases in which logical truth (and hence
differential mplication) is construedin terms broaderthan those of quantifica-
tional validity.
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MORE ABO UT 'ABO UT' 307
engaged in designation, variables of individuals and predicates of
their extensions. We now gain terms designating Boolean com-
pounds by counting truth functions of predicates as predicates. If
we wish to avail ourselves of a larger supply of designating ex-pressions 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) D 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 doesnot imply its own generalization with respect to any of its free
variables or predicate letters. Then: Every quantificational schema
is 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 Interpolation
Theorem asserts that if A D B is valid, then there exists a schema
C such that A D C is valid, C D B is valid, and C contains only
predicate 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 for
predicate 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' 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 letters.5 If this schema implies its own generalization
with respect to some of its free variables, then universal general-
ization 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 is absolutely about k if and only if the simple equivalents
of S contain free occurrencesof all the free variables and occurrences
of 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 such
cases are of no importancehere in any event, since neither valid nor inconsistent
schematacan representstatements that are absolutely about anything. It is to
be noted that there can be no effective method of discoveringa simple equivalentfor an arbitraryquantificationalschema.
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308 THE JOURNAL OF PHILOSOPHY
uses only predicate letters and free variables from T, and R a
schema containing an occurrence of t. Then the schema (R D R) *T
is equivalent to both T and S, so is implied by S, but cannot imply
its own 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 con-sequence 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 It 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 'W'. 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 is a simple equivalentof S, then
-T is a simple 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 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 in
favor 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 to
be 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 schemacontaining
free '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. Our
theoremis unaffected,since only its applicationturns on what are taken to be the
terms available.
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MORE ABOUT 'ABOUT' 309
POSTSCRIPT
In each of the developments above we construe the parts of terms
susceptible to generalization to be either lone variables or lonepredicate letters. In the first development, where no predicate
letter figures in a term, it is only the variables; in the second it 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-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 rendered
either
(1) (3x)Exw* (3x)Exy
or
(1') (3x)Bx. (3x)Cx
Is 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 we
could cite (l)'s further consequence
(z)((3x)Exw (3x)Exy' (x)(Exz D Exz))
as evidence that (1) does not imply (2) differentially with respect
to 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 respect
to 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 PHILOSOPHY
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 to establish this in order to 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) D (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'. Forthat generalization implies the generalization of R with respect to
'BxCx', which is to say that it implies '(3 x) (BxPxCx) D (3 r)Qx'.
Since both 'P' and 'Q' are foreign to T and T is consistent with
'(3 x) (BxCx)', there is an interpretation making T true and
'(3 x) (BxPxCx) D (] x)Qx' false. Since T is equivalent to S, it
follows that S implies TR differentially with respect to the inter-
section of the extensions of 'B' and 'C', even on our broadened
construal of 'generalizable parts'.Similarly, taking R' as '( 3x)Px *(x) (Bx D -Px) .D ( 3x) Bx',
S implies either '(3x)-Bx' or TR' differentially with respect to
the complement of the extension of 'B'. The argument may easily
be extended to apply to all compound predicates built by truth-
functional 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 respectto lone predicate letters and free variables.
HILARY PUTNAM
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
J. S. ULLIANUNIVERSITY OF CALIFORNIA, SANTA BARBARA