S T E V E N E. BOER

C L U S T E R - C O N C E P T S A N D S U F F I C I E N C Y

D E F I N I T I O N S

(Received 4 June, 1973)

In his recent paper, "Definitions and 'Clusters', ''x David Cooper intro- duces the notion of a 'sufficiency definition', i.e., a definition of the form

(1) Q = def Something having suffcient of the properties P1 .. . . . P~,

and proceeds to argue for two related claims. The first claim is that such definitions are admissible on a par with normal definitions (i.e., defini- tions of the form 'Q=aef Something having the properties P1, ..., P~'). And the second claim is that if the first claim is true, then the traditional theses

(2) Proper names have meaning just like other nouns,

and

(3) All truths are analytic or synthetic, and no truth can be both,

can successfully be defended against certain fashionable objections based on the peculiarities of terms expressing cluster-concepts. It is my conten- tion in this note that Cooper completely fails to establish his second claim, in consequence of which his first claim, even if true, loses much of its philosophical interest.

The fashionable objection to (2) is supposed to run as follows.

(4.1) The meaning of ordinary nouns is such as to specify a list of properties P1 ... . . P~ possession of which by X is both neces- sary and sufficient for application of the noun to X.

(4.2) Therefore, any property P~ which X could lack and still be- long to the extension of the noun cannot be part of the meaning of the noun.

(4.3) Proper names are cluster-terms, and the associated cluster of properties P1 ... . . Pn is such that various subsets are sufficient for application of the name, but no Pi is necessary.

Philosophical Studies 26 (1974) 119-125. All Rights Reserved Copyright �9 1974 by D. Reidel Publishing Company, Dordrecht-Holland

120 STEVEN E. BOER

(4.4)

(4.5)

Therefore, no P~ in the associated cluster can be part of the meaning of a proper name. Therefore, it is false that proper names have meaning just like other nouns.

Cooper's alleged refutation of (4) is quite simple:

(5.1) If sufficiency definitions are admissible, then we can define a proper name 'N ' as follows: N=dof The Xhaving sufficient of P~ ... . . P,.

(5.2) Such a definition endows 'N' with meaning and shows how something could lack any P, while still belonging to the ex- tension of 'N' .

(5.3) Therefore, proper names do have meaning just like other nouns at least in the sense that proper names are definable.

My criticism of (5) is equally simple: (5) misses the point altogether. For it is simply assumed in (5.2) that sufficiency definitions for proper names have the function of establishing meaning-equivalences. But this is to overlook a crucial distinction (propounded and defended by Kripke 9') between two different functions of sufficiency definitions: viz., synonym- fixing and reference-fixing. What the critic is (or should be) getting at in (4) is not affected by the admission of sufficiency definitions, for the critic can simply reply that such definitions serve in our language (or for a given speaker) to fix the reference with which the proper name is henceforth to be used, but do not thereby make the name a synonym of some definite description like 'the X having sufficient of PI,- . . , P, ' .

Indeed, there are familiar and compelling reasons for rejecting the view that definitions of form (1) give the meaning of proper names. For example, if such definitions are regarded as synonym-fixers, then corre- sponding sentences of the form

(6) PI(N) vP2(U) v... vP,(N)

must be regarded as analytic ! But this means that we must deny the possi- bility of a world in which N failed to satisfy our preconceptions (e.g., philosophers would have to deny that Aristotle could have died at birth, since they could not call anyone 'Aristotle' unless he had at least some of the major characteristics they associate with the name). The reference-

C L U S T E R - C O N C E P T S A N D S U F F I C I E N C Y D E F I N I T I O N S 121

fixing view of sufficiency definitions for proper names has, by contrast, no such bizarre consequence. For the referent of 'N' in the actual world is just that individual (if there is one) which uniquely satisfies the predi- cate 'has sufficient of P1,..., P, ' , and that individual might well have lacked some, most, or even all of Pa .... , P, (unless one has an independent doctrine of essential properties which gives a special status to some P~). Since the critic has an option which allows him to deny (5.2), and good reasons for pursuing this option, I think we must conclude that (4) has not been refuted.

The putative objection to (3) may be formulated as follows.

(7.1)

(7.2) (7.3) (7.4)

(7.5)

For some cluster-terms'Q' and associated properties P1, ..., P,, the sentence ' If X has all of P1 .. . . . 1',, then X is (a) Q' is necessarily true. Being necessarily true, the sentence cannot be synthetic. Therefore, by (3), it must be analytic. But it cannot be analytic because analytic truths are true in virtue of definitions, and cluster-terms like 'Q' do not have definitions. Therefore, some truths are neither analytic nor synthetic.

Before considering Cooper's attempt to refute (7), we should notice that (7) is something of a Straw Man. In the first place, (7) rests on the contro- versial principle

(8) All synthetic sentences are contingent.

I say 'controversial' because there are more than a few philosophers who would maintain that sentences like 'Everything that has color has shape' are necessarily true but fail to qualify as analytic - hence are presumably synthetic necessary truths, s Such a philosopher would dismiss (7) as un- sound and would regard the apparatus of sufficiency definitions as a gratuitous complication. Secondly, (7.4) relies on a hopelessly narrow conception of analyticity as truth in virtue of definitions. Although popular in the heyday of Positivism, this construal of analyticity has few adherents today. Most contemporary proponents of the analytic/synthetic dichoto- my would be unmoved by an objection based on 'lack of definitions', for their explications of analyticity are grounded elsewhere, in more powerful

122 STEVEN E. BOER

concepts. 4 For the sake of argument, however, let us suppress these quibbles and treat (7) as a genuine threat to (3).

Cooper's refutation of (7) takes following shape:

(9.1) (9.2)

(9.3)

(9.4)

(9.5)

Suppose that sufficiency definitions are admissible. Let 'Q' be a cluster-term which generates a necessarily true conditional of the sort mentioned in (7.1). Then we can define 'Q' as 'Something having sufficient of P1,...,P,'. Consequently, the truth of the conditional ' If X has all of P~,..., P, then X is Q' will follow from the definition. Therefore, ' If X has all of P1 ... . , P, then X is Q' will be analytic.

But an important step has been suppressed at (9.4) - a step which, once made explicit, is open to serious doubt. Suppose we were to demand from Cooper a proof that the truth of the 'duster'-conditional

(10) If Xhas all o f P 1 , . . . , P ~ then Xis Q.

hinges merely on the sufficiency definition in (9.3). It seems clear from his remarks that he would respond with the following argument:

(11.1) (ll.Z)

(11.3) (11.4)

Suppose X has all of P~ .... . pn. Then X has sufficient of P1 ..... P, (i.e., has a combination sufficient to warrant applying 'Q' to X). Then, by the sufficiency definition in (9.3), X is Q. Therefore, if X has all of P1,..., P,, then X is Q.

Now the point of (11) is to show that, given the admissibility of sufficiency definitions, we could plausibly regard (10) as analytic because its conse- quent could be categorically derived from its antecedent using only the definition. But (7) is not so easily refuted, for the critic could grant that one might invent or discover a sufficiency definition for 'Q' and yet he could deny that (10) is true in virtue of such a definition. To surmount this objection, Cooper must at least show that we have some good reason(s) for regarding the truth of (10) as depending on the sufficiency definition. And this is something which (9)-cum-(ll) fails to show. For there can perfectly well be conditionals like (10) which are necessarily true but whose truth in no way depends on any sort of sufficiency definition.

C L U S T E R - C O N C E P T S A N D S U F F I C I E N C Y D E F I N I T I O N S 123

What Cooper has in mind when he speaks of an individual possessing all of the properties in the associated cluster, is a paradigmatic instance of the cluster-concept, i.e., a 'paradigm case' of Q-hood. But what he overlooks is that there are different sorts of paradigms corresponding to different sorts of cluster-concepts. Some cluster-concepts admit of'strong' paradigms, i.e., the associated cluster P1 ..... P, is such that a single thing couM have all of P1,.-., P,. These concepts, which I shall call 'strongly backed', are usually taught by reference to things that do have all of P1 ..... P,. On the other hand, some cluster-concepts admit only of 'weak' paradigms, i.e., the associated cluster P1 .... , P, is such that it is impossible (on logical or metaphysical grounds) that there should be a thing having all of P~ .... , P,, although there might be things having very many of P~, .... P,. Let us call these concepts 'weakly backed'. Many family-re- semblance concepts are weakly backed cluster-concepts. Take, e.g., the properties associated with 'game'. A (weakly) paradigmatic game is some- thing having many (perhaps even a maximally consistent set) of the im- portant associated properties; but there could not be a 'strongly' para- digmatic game, since some of the properties associated with game-hood are (logically or metaphysically) incompatible with one another.

Returning now to Cooper's argument, suppose that 'Q" expresses a weakly backed cluster-concept (e.g., 'Q' might be 'game'). Presumably 'Q' could be given a definition of the form (1). And the corresponding conditional (10) would indeed be a necessary truth. But (10)'s truth would in no way depend on the sufficiency definition, for the antecedent of (10) would be self-contradictory. (10) would, that is, be a logical truth (or a metaphysically necessary truth - if such there be). In terms of (11) this means that we would move directly from (11.1) to (11.3); (11.2) would be an idle partner. Thus, if (7) is couched in terms of weakly backed cluster-concepts, and if the emphasis of (7.4) is shifted to the notion of 'truth in virtue of definitions', then Cooper's argument (9) from the avail- ability of sufficiency definitions fails as a refutation. Of course, if he could produce an independent argument to the effect that all logical (or meta- physically necessary) truths are analytic, then (9) could be patched up; but no such argument is given - and finding one is not likely to be easy.

I do not, however, wish to rest my case against (9) with weakly backed cluster-concepts. For (9) does not clearly work even for strongly backed cluster-concepts. The reason for this comes out clearly when we look at

124 STEVEN E. BOER

(11). What supports (11.2)? There would seem to be only two plausible answers: either (11.2) follows from (11.1) in virtue of form, or else (11.2) follows from (11.1) in virtue of content. The former alternative is demon- strably false, and the latter is at least dubious.

Among strongly backed cluster-concepts there is an interesting species comprising what may be called 'threshold concepts'. The distinguishing feature of a threshold concept T is that, among the associated properties P~, ..., Pn, only proper subsets are sufficient for the application of 'T'. In other words, even though it is possible that something might have all of P1, ..., Pn, such a thing would not automatically be called 'T'. As a (tenta- tive) example of a threshold concept, I suggest 'good farmland'. It seems plausible to accept a sufficiency definition something like 'good farm- land=d~f land having sufficient of the substances sl , . . . , sk in the soil'. Now it might well be the case that if all of s~, ..., s k were simultaneously present in the same plot of land, then a certain chemical reaction would take place which would render the land infertile. Since we may suppose that the users of 'good farmland' know a little practical chemistry, I think we can see how they might apply the expression on the basis of the pre- sence of various subsets ofs 1 ..... Sk but definitely (or at least provisionally) withold it in the presence of all of s~, ..., s k. Such being the case, however, the inference from (11.1) to (11.2) cannot be licensed by form alone. For (11.1) can be true and (11.2) false when 'Q' expresses some threshold concept.

It appears, then, that Cooper must appeal to 'content' in order to support the move from (1 I. 1) to (11.2). And such an appeal just amounts to saying that it is 'part of the concept Q' that anything having all the associated properties ipso facto has a combination of properties which warrants the application of 'Q'. No definition of the form (1), however, can reflect this fact (if it is a fact). Yet a good definition should state the conventional connotation of the definiendum, as Cooper himself stresses (pp. 502-503). So if the move from (11.1) to (11.2) is legitimate, it must be because we have a definition of the form

(12) Q=def Something having all or sufficiently many of P1,. . . ,P, .

Unfortunately, the price of rescue via (12) is trivialization of (11). For in the presence of (12), we can move directly from (11.1) to (11.3). The fact that (i 1.2) is once again an idle partner is merely a symptom of the fact

CLUSTER-CONCEPTS AND SUFFICIENCY DEFINITIONS 125

that the second disjunct in the definiens of (12) is doing no work. In other words, the only case where (9) seems to work - viz., that of strongly backed non-threshold cluster-concepts - is a case in which the notion of 'sufficiency' plays no essential role in the argument.

Cooper could, of course, reply that (12) is a sort of sufficiency defini- tion and that, if admitted, it would at least show how some necessarily true conditionals like (10) could reasonably be regarded as analytic. That we should so regard them is, however, a conclusion warranted only by the additional premiss that definitions of the form (12) do in fact repro- duce the meanings of some cluster-terms. Cooper provides no direct ar- gument for this, and most of the visible means of supporting it appear to derive, in one way or another, from the notorious Paradigm Case Argu- ment. I conclude, then, that Cooper's arguments do not convincingly establish that the admission of definitions of form (1) provides a way of defending (2) and (3) against (4) and (7) respectively.

The Ohio State University

NOTES 1 Mind LXXXI (1972), 495-503.

In 'Naming and Necessity', in D. Davidson and G. Harman (eds.) Semantics of Natural Language, D. Reidel Publishing Co., Dordrecht, Holland, 1972, pp. 253-355. 3 Cf. R. M. Chisholm, Theory of Knowledge, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966, Chap. v. 4 Cf. D. K. Lewis, Convention, Harvard University Press, Cambridge, Mass., 1969, Chap. V.