Elimination of Extra-Logical Postulates

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Elimination of Extra-Logical Postulates W. V. Quine; Nelson Goodman The Journal of Symbolic Logic, Vol. 5, No. 3. (Sep., 1940), pp. 104-109. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28194009%295%3A3%3C104%3AEOEP%3E2.0.CO%3B2-Q The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.

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'I'm J O ~ A or L STUSOLIO LOOIC Volume 6, Numbco.8, Septembe 1040

ELIMINATION OF EXTRA-LOGICAL POSTULATES W. V. QUINE and NELSON GOODMAN

1. Simple cases of eliminability. Under certain circumstances a postulate can be eliminated in favor of a mere definition, or convention of notational abbreviation. Suppose e.g. that a given system presupposes the machinery of ordinary logic and contains in addition a single extra-logical primitive: the relation Pt, "is a spatial part of." Suppose this relation is governed by a single extra-logical postulate, to the effect that Pt is transitive:

Now instead of Pt, the relation 0 of spatial overlapping might be taken aa primitive; P t could then be defined in terms of 0 as follows: The transitivity of Pt then follows by purely logical principles from the definition. The statement (1) with its 'Pt' clauses expanded according to (2) is a purely logical theorem, demonstrable independently of any stipulations or aesumptions concerning the properties of 0. Thus, through a change involving neither increase nor decrease of primitive ideas, the need of adopting (I) as a postulate is removed. Or again, consider a system comprising ordinary logic and the sole extralogical primitive S, the relation of simultaneity, governed by a sole postulate to the effect that S is symmetrical:

-

(z)(v)(z S Y 3 Y S 4. (3) Here instead of S the relation N, "is no later than," might be taken as primitive; S could then be defined in terms of N as follows: Postulation of (3) then becomes unnecessary, for (3) is an abbreviation, according to (4), of a theorem which is demonstrable within pure logic. A system having a single extra-logical postulate of transitivity, or of symmetry, is of course very trivial. In practice one would be interested in a good many properties of P t besides transitivity and its consequences, and hence would have adopted further postulates. Here, contrary to what we observed in the case of a single postulate, the shift to 0 will in general net no real saving; for, the desired properties of P t other than transitivity remain to be supplied in an indirect and probably more complex way by appropriate postulation regarding 0. But suppose we write together in a single conjunction all the desired postulates regarding Pt, thus getting a single postulate adequate to all the properties of Pt Received July 9, 1940. 104

ELIMINATION

OF EXTRA-LOGICAL

POSTULATES

105

which we may ever care to prove. What if we were able to eliminate this entire conjunctive postulate by shifting our primitive and appropriately defining Pt, just as we eliminated the transitivity postulate above? This would be surprising enough to challenge the whole notion of postulational economy. We shall show that a wholesale elimination of this sort is in fact possible in a wide range of cases. Indeed, it will turn out that in order to be wholly e l i i nable in the described fashion a postulate set need fulfill little more than the requirement of consistency.

2. Conditions on the definiens. What sort of definition, in general, will serve thus to eliminate a set of postulates? We may limit our attention to extra-logical postulates, in interpreted systems presupposing logic. Only extra-logical postulates are amenable to the kind of elimination suggested; for this consists in reducing the postulates via definitions to statements which are validated automatically by the logic presupposed. For the present let us limit our attention further to the case of a system involving no extra-logical primitives beyond a single constant term, say 'dl. This term is to be thought of as carrying some specific extra-logical meaning. Now if a definition of 'd' in terms of a new primitive is to eliminate the postulates of the system, the definiens must be such that its substitution for 'd' turns the postulates into logical truths. Let us use 'P(d)' as an abbreviation for the conjunction of the postulates; and let us represent the definiens in question as 'j(a)', where 'a' is the new primitive and no other extra-logical expression is involved. The requirement is, then, that 'P(f(a))' be logically true; in other words, that '(s)P(j(z))'--which contains only logical signs-be true. But the definiens must also meet a second requirement. No matter what meaning may have been chosen for 'd', under the restrictions imposed by the postulates, we must be able so to interpret 'a' that the proposed definiens will reproduce that meaning of 'd'. For example, in the case of the system which had (1) as its only postulate, it would have been unsatisfactory to eliminate the postulate by defining Pt in terms of a new primitive relation K as

.

iy(3z)(3w)(z K z y K w ) , because we could not find any meaning for 'K' according to which this dehiens would have the intended meaning "part of." Indeed, we can be sure that there is no relation K satisfying this condition; for if there were, the part-whole relation would be symmetrical. Our second requirement, then, is that there be some interpretation of 'a' such that d = j(a); in short, it is required that ( 3 ~(d) = j(x)). Thus definition of 'd' as 'j(a)' will reduce the postulates to logical truths, and still yield the intended meaning of 'dl (under some interpretation of 'a'), if and only if (x)P(f(s))

.

= I(x>>.

(5)

3. Existence of logical models. The contemplated elimination of 'P(d)' in favor of a logical truth cannot of course be effected unless 'P(d)' is in fact true;

106

w.

V. QUINE AND NELSON GOODMAN

indeed, 'P(d)' is a logical consequence of (5). But granted that P(d), under what circumstances will there be a definiens 'f(a)' meeting the conjoint requirement (5)? We shall show that there will be such a definiens if and only if the postulate set has a logical model; i.e., if and only if there is a logical constant fulfilling the conditions which the postulates impose on the primitive. Suppose there is such a constant, abbreviated say as 'c'; so that 'P(c)' is true. Now a definiens tf(a)' conforming to (5) is

as is apparent from the following considerations. The expression corresponding to 'f(x)' is now: and clearly the statement:

is true when 'f(x)' is construed as (6). But (7) implies that which in turn implies that and hence that (x)PCf(x)) in view of our hypothesis 'P(c)'. Again, since 'P(d)' is by hypothesis true, it is obvious that and hence that (Zz)(d = where 'f(x)' is construed as before. We see therefore that there will be a definiens 'f(a)' conforming to (5) whenever the postulate set has a logical model. Now i t is easy to see conversely that the set will have a logical model whenever there is such a definiens; for the first part of (5) implies e.g. that Pdf(A)) and hence that 'f(A)' is a logical model of the postulates. The choice of 'A' here is of course arbitrary; any other logical constant, say 'V' or '32', would have served as well. Our choice is indeed subject to certain restrictions so long as logic is thought of as requiring a theory of types; but these restrictions vanish when the theory of types is abandoned in favor of one or another of the alternative theories.' 1 For one such theory and citatiom of others see Quine, Mathematical lagic Norton & Co., 1940), pp. 128-132, 155-160, 162-166.

(W.W.

ELIMINATION O F EXTRA-LOGICAL POSTULATES

107

4. The case of many primitives. Suppose now that our postulate set 'P(dl, .. , d,)' involves many primitive tenns 'dl', ... , 'd,'. Definitions of these in terms of new primitives 'al', . . . , 'a,' will be appropriate for the purpose of eliminating the postulates if and only if the respective definientia 'jl(al, . . , a,)', . .. , %(al, . . , a,)' are such that

.

.

as may be seen by reasoning analogous to that of $2. Paralleling the case of one primitive, it is easy to show that there will be such definientia if and only if the postulate set has a logical model; i.e., if and only if there are logical constants, say 'cll, .. . , 'c,', such that P(c1, . . , c,). S u p pose, first, that there are such constants. Then definientia ',fl(al, . . , a,)', %(all . .. , a,)', etc. conforming to (8) are: (iy)[P(al, . . , a,) (ly)[P(al, . .. , a,)

.y = a1 .v .-P(al,

. . , a,)

.y = CI],

.y = a .v .~ P ( a 1 , . , .y = cs], an)

etc., as is apparent from the following considerations. The expressions cor, z,)', '52(sl, .. . , z,)', etc. are now: responding to ',fl(xl, e e

(qy)[P(zll ... , 2,)

y =

XI

(ly)[P(~l,' . ' ~ n ) y =

~2

.

.v NP(XI, . , ~ n .) 9 = CI], .v . N P ( x ~ ,. .. ~ n .) = CZ], 1

l/

(9) (10)

etc.; and the statement:

is true when 'fi(xl, . . , z,)', 'fi(zl, . .. , x,)', etc. are construed as (9), (lo), etc. From (11)) by reasoning analogous to that in i3, we conclude that

The other half of (8) can likewise be established by following the analogy of $3. We see therefore that there will be dehientia conforming to (8) whenever the postulate set has a logical model; and it is easily shown conversely, again followingthe analogy of $3, that the postulate set will have a logical model whenever there are such definientia. We have confined our consideration to postulates in which the sole extralogical primitives are term; i.e., signs capable of occurring alongside '=' or 'c' or indeed wherever a free variable can occur. Our findings would not apply directly to a system which involves e.g. a primitive functor '$', admissible only in contexts of the form 'z $ y1 and not separable as a term in its own right. Actually this is no restriction, however, for extra-logical primitives other than terms are always readily avoidable. Instead of 's CB y', e.g., we can write 'd'(z;y)' where the primitive term 'd' designates the relation of z $ y to the ordered pair z;y.

108

W.

v.

QUINE AND NELSON GOODMAN

6. Consequences. We have seen that we can reduce a set of extra-logical postulates to logical truths, by defining ideas identical with the erstwhile primitives in terms of new primitives, if and only if the postulates have a logical model. The variety of extra-logical systems whose postulates can be eliminated in this way is thus exceedingly wide. This ia especially apparent when we reflect that logical constants of any degree of complexity may be used as models, and that a constant is a model for any set of postulates that ascribes to a primitive a subclass of the properties of that constant. Consider e.g. the calculus of individuals, which has the single primitive discreteness and the following three postulates?

(I)

2 ( W ) ( ~ X ) ( YI )[Y (x)(y)[(w)(z I w . (Y)(z)((3x)[(w)(y

= (z)(z e w 2 y I41, = Y Z w) "(w)(z Z W)

(12) (13)

2 x = yl,

Iw - v - z 1W : 3 - X - L

w) " ( w ) ( x I w)I "(Y I z)). (14) The relation of class exclusion, L$(x C g ) , is a logical model of these postulates. It must be borne in mind that the elimination of the three postulates does not proceed by defining I as fg(x C 8 ) ;such a definition would give a meaning diiTerent from that originally intended. Discovery of a logical model assures us rather (by $3) that some definition can be found which will reproduce the original meaning of 'I'and still reduce the three postulates to logical truths. A definition to which the reasoning of $3 directly leads is: 7- =dt (.rz)[P(Sep) z = Sep v -P(Sep) z = &)(x C g)] where 'Sep' ('separateness') is a new primitive synonymous with the original 'I',and 'P(Sep)' is short for the conjunction of (12)-(14) with 'Sep' put for ' Since 'P(Sep)' is in fact true, this definition preserves the intended meaning of ' I ) ; but the definition makes (12)-(14) logically true independently of the truth of 'P(Sep)'. What postulate sets, now, cannot be eliminated by this procedure? Inconsistent sets, obviously; for they have no logical models. But there are also other postulate sets, presumably consistent, for which no logical model is known. One such set consists of the postulates 'd e d' and 'd e 2'; models are readily found for these respective postulates taken singly, viz. V and LA u iV, but no model is known for the pair. A still simpler set of the same kind consists of the single postulate 'd = d'. In general, of course, the question whether a postulate set has a logical model will depend on details of the underlying logic. Logic might be constructed in so strong a fashion as to endow these bizarre examples with logical models, or in so weak a fashion as to deprive even more ordinary postulate sets of their models. Particularly weak logics aside, how-

'I'

. .

.

'

.

Cf. Leonard and Goodman, A calculue of individuala, this JOURNAL, vol. 6 (1940), pp. 48-49. The postulates of that paper are here rendered in unabbreviated form, and ao recaat as not to depend on the theory of types. On abandonment of the theory of types the range of 'z', 'y', etc. ceases to be limited to individuals; hence we find ourselves called upon to decide whether 'to construe 'z y' as vacuously true or as false when z and y are not both individuals. We here arbitrarily adopt the former alternative; thus the clause '-(ZD)(Z w)' in (13) and (14) stipulates that z is an individual.

1

1

ELIMINATION OF EXTRA-LOGICAL POSTULATES

109

ever, we have yet to find a postulate set which has a plausible extra-logical interpretation but still lacks a logical model and thus resists elimination. In any case, a postulate set which can be proved consistent by the usual device of citing a logical (e.g., arithmetical) model is ipso facto eliminable in the described fashion. Such elimination cannot be dismissed as trivial on the ground that postulates and definitions are somehow essentially the same. They differ formally in the obvious circumstance that we can dispense with the need for definitions, but not the need for postulates, simply by writing all statements of the system in terms of primitives. A general proof of the possibility of eliminating postulates in favor of definitions, far from being rendered unnecessary by the claim that the two are essentially alike, is precisely the kind of evidence which would be needed t o substantiate such a claim. The discovery that extra-logical postulates are ordinarily eliminable challenges conventional notions of postulational economy as applied to extra-logical systems. We are thus moved to seek a standard according to which the economy achieved by the present method can be distinguished out as in some sense trivial -like the economy achieved by simply conjoining many postulates as one. We first observe that the present method of eliminating postulates leads to a system which is incomplete, in the sense that many statements involving the new primitive are neither demonstrable nor refutable. Hence we may be tempted to rule that postulational economy is significant only in complete systems; but this is unsatisfactory, because any extra-logical system contains logic and is therefore necessarily incomplete in view of Godel's theorem. A better solution is suggested by the concept of categoricity, or the related concept of synthetic An extra-logical system is synthetically complete if and only if ~orn~leteness.~ it is as complete as the underlying logic; i.e., if and only if every statement of the system is demonstrable or refutable or demonstrably equivalent to a statement containing only logical signs. I t would seem that economy of extra-logical postulates is significant only in systems which are synthetically complete. This constitutes a considerable modification of old standards of postulational economy. Discrimination between real and apparent economy comes to depend upon proof of the synthetic completeness of the system in question. Not only is such a proof normally very difficult, but most of the useful extra-logical calculi on record are in fact synthetically incomplete. Unless some quite W e r e n t criterion is discovered, the extensive economies here shown to be possible will in practice seldom be distinguishable from those effected in any of the usual ways. HARVARD UNIVERSITY

a The latter notion, under the name 'completeness relative to logic,' is due to Tarski. I t is easier t o formulate than the older concept of categoricity, and is related to the latter as follows: systems which are categorical (with respect to a given logic) are synthetically complete, and synthetically complete systems possessed of logical models are oategorical. These matters were set forth by Tarski a t the Harvard Logic Club in January, 1940, and will appear in a paper On completeness and categoricity of deductive theories. See also Lindenbaum and Tarski, Ober die Beschrdnktheit der Ausdrucksmittel deduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums, Heft 7 (1936), pp. 15-22, wherein 'Nichtgabelbarkeit' answers to 'synthetic completeness.'