Note on Supervenience and Definability

The idea of a property’s being supervenient on a class of properties is familiar from much philosophical literature. We give this idea a linguistic turn by converting it into the idea of a predicate symbol’s being supervenient on a set of predicate symbols relative to a (first order) theory. What this means is that according to the theory, any individuals differing in respect to whether the given predicate applies to them also differ in respect to the application of at least one of the predicates in the set. The latter relationship we show turns out to coincide with something antecedently familiar from work on definability: with what is called the piecewise (or modelwise) definability, in the theory in question, of the given predicate in terms of those in the set. 1 The idea of the supervenience of a property P on a set S of properties has become increasingly familiar in the philosophical literature of the past twenty-five years.1 At its simplest, this relation holds between P and S when it is impossible for there to be two individuals alike in respect to each property in S which are not alike in respect to P. (Objects “are alike” or “agree” in respect to a property when both have the property or both lack the property.) In possible worlds terms, we can take this as saying that two individuals in any arbitrarily selected world which agree, in that world, on all properties in S, also agree on P in that world. This is what is sometimes called weak or intraworld supervenience, and it contrasts, in particular, with strong or interworld supervenience, by which is meant that any two individuals in an arbitrarily selected pair of worlds which agree, in the respective worlds, on the properties in S, also agree on P.2 The purpose of the present note is to consider a version of supervenience applied to monadic predicates in the language of a first-order theory and to relate it to a well-known concept from definability theory. The reasoning will be entirely elementary. 2 Philosophical discussions of supervenience often involve the question of whether this or that notion of supervenience amounts to “reducibility” in some sense. Our interest will be in how what we shall be calling supervenience relative to (or in) a theory Received November 5, 1997; revised October 21, 1998 244 LLOYD HUMBERSTONE is related to various theory-relative notions of definability. We therefore review the latter concepts here. (A full discussion with references to the original sources may be found in the useful survey in Rantala [7]; the reason we define explicit definability as a special case of piecewise definability is that we shall need the latter, more general, concept below.) An n-ary predicate symbol F in the language of a theory T is piecewise definable in T if for some formulas φ1, . . . , φm in that language, none of which contains occurrences of F or has variables other than x1, x2, . . . , xn free in it, we have T ∀x1, . . . ,∀xn(Fx1, . . . , xn ←→ φ1)∨ · · ·∨ ∀x1, . . . ,∀xn(Fx1, . . . ,xn ←→ φm). We add “in terms of such-and-such items of nonlogical vocabulary” (presumed not to include F) when the only nonlogical expressions to appear in φ1, . . . , φm are drawn from the listed items. For present purposes, the items in question will always be predicate symbols. F is explicitly definable in T (“in terms of a given set of predicates”) if something of the above form is provable in T where m = 1 (and only predicates in the given set occur in φ1). Finally, F is implicitly definable in T in terms of predicates G1, . . . , Gk when any two models of T with the same domain and the same extensions for G1, . . . , G2 also assign the same extension to F. (“Model of T” means, of course: structure for the language of T in which all sentences of T are true.) The way implicit definability was just characterized renders it recognizably a supervenience relation: “no difference here (on interpretation of F) without a difference there (on interpretation of G1, . . . , Gk).” Accordingly, Beth’s Theorem, which states that whenever F is implicitly definable in T in terms of G1, . . . , Gk, F is explicitly definable in T in terms of G1, . . . , Gk, may be seen as claiming that some sort of supervenience implies some sort of reducibility.3 Note, however, that the objects among whom agreement in certain respects implies agreement in another are certain first-order structures, namely, models of the theory T . (It is these which, if they interpret each of G1, . . . , Gk alike, must also treat F alike.) Accordingly, this is the wrong sort of supervenience to connect with the supervenience notions (weak, strong, . . . ) we began with. Those concerned agreement among the objects having or lacking the properties—which we can think of here simply as sets—represented by such predicates, rather than among models assigning various sets as the predicates’ extensions. Let us, then, consider a theory-relative notion of supervenience in which the objects among which agreement counts are the individuals in the domains of the theory’s models, and what constitutes agreement is agreement as to whether or not these predicates are true of the individuals concerned. This would be the most direct analogue of the notion of supervenience of properties on sets of properties introduced in our initial paragraph, except that now properties give way to predicates, and the whole thing is done relative to a first-order theory. The most straightforward adaptation of those initial ideas will be to monadic predicates, since it is these that are true or false of the individuals in the domain (of a model of a theory). It may be of interest to consider what would become of the developments to follow if the definition to be given below were stated more generally for n-adic predicates (n ≥ 1), but we shall not consider this here. (Certainly, some of the points made apropos of Corollaries 3.3 and 3.4 at the end of our discussion would not hold in the more general setting.) Suppose, then, that T is a first-order theory and among the primitive monadic SUPERVENIENCE AND DEFINABILITY 245 predicates of the language of T are F, G1, . . . , Gk. (We reserve these letters to stand for distinct monadic predicates until further notice.) Then we say that F is supervenient on G1, . . . , Gk in T just in case T ∀x∀y(((G1x ←→ G1 y) ∧ · · · ∧ (Gkx ←→ Gk y)) → (Fx ←→ Fy)). (Note that the ‘←→’ in the consequent can be replaced by ‘→’ without loss of logical strength.) We adapt the above talk of agreement and likeness to the present setting and say that two objects in the domain of a first-order structure agree on (or are alike with respect to) a predicate if both lie in the extension of that predicate in the structure or if both lie outside that extension. Then the present notion is akin to weak, rather than strong, supervenience in the sense of our opening paragraph because the relation just defined holds when in any model of T (compare “in an arbitrarily selected world”), agreement on each of G1, . . . , Gk implies agreement on F. Clearly, if F is explicitly definable in T in terms of G1, . . . , Gk , then F is supervenient on G1, . . . , Gk in T . For any individuals agreeing on the latter predicates in a model of T will agree on the open formula constructed from them and serving as the φ in a definition ∀x(Fx ←→ φ(x)),4 and hence, since we are only considering structures which are models of T , will agree on F. We can see that the converse does not hold—supervenience does not imply explicit definability (relative to an arbitrarily given theory)—because the argument just provided for the implication from explicit definability to supervenience works just as well to show that piecewise definability implies supervenience. This is because of the way models of T are considered one at a time. (The point here is that, where Th(M)—the theory of M—is the set of closed first-order formulas in the language of a (first-order) structure M which are true in M, then piecewise definability in T amounts to explicit definability in Th(M) for every model M of T . Indeed, piecewise definability is sometimes called modelwise definability.) Thus we have piecewise definability =⇒ supervenience, so we could not also have supervenience =⇒ explicit definability; otherwise we should have—something that is certainly not the case—piecewise definability =⇒ explicit definability. Readers for whom it is clear that piecewise definability is strictly weaker than explicit definability should skip the following paragraph. (A referee for this journal has suggested that an example along the following lines be included for the benefit of readers who do not find it obvious that piecewise definability does not imply explicit definability. Let T be the theory comprising the consequences of ∀x(Fx ←→ G1x) ∨ ∀x(Fx ←→ G2x), in the language whose nonlogical vocabulary consists of the predicate symbols figuring in this axiom. By the choice of axiom, F is evidently piecewise definable in terms of G1 and G2 in T . Let M be a structure with the three-element domain {a, b, c} with {a, b} as the extension of G1 and {a, c} as the extension of G2. We can expand M to a model of T in each of two distinct ways: (i) by taking {a, b} as the extension of F, and (ii) by taking {a, c} as the extension of F. Since the extension of F is thus not fixed by those of G1 and G2 in this model of T , T does not implicitly define F in terms of G1 and G2 , and so F is not explicitly definable in T in terms of them, its piecewise definability notwithstanding.) Though supervenience does not coincide with explicit definability, the possibility remains open of a coincidence with the weaker property of piecewise definability. 246 LLOYD HUMBERSTONE Let us introduce this possibility (which Proposition 3.2 (below) shows is indeed realized) by looking at a special—and especially manageable—case, in which k = 1. Suppose, then, that F is supervenient on G in some theory T . T ∀x∀y((Gx ←→ Gy) → (Fx ←→ Fy)). (1) Thus we have (2) and (3). T ∀x∀y((Gx ∧ Gy) → (Fx → Fy)). (2) T ∀x∀y((¬Gx ∧ ¬Gy) → (Fx → Fy)). (3) We can manipulate (2) and (3) so that atomic subformulas in the same variable are grouped together, getting (4) and (5), respectively. T ∀x∀y((Gx ∧ Fx) → (Gy → Fy)). (4) T ∀x∀y((¬Gx ∧ Fx) → (¬Gy → Fy)). (5) Having separated the variables, we can now massage (4) and (5) into the disjunctive forms (6) and (7). T ∀x(Gx → ¬Fx) ∨ ∀x(Gx → Fx). (6) T ∀x(Fx → Gx) ∨ ∀x(¬Fx → Gx). (7) Calling the first and second disjunct of the formula in (6), (6i), and (6ii), respectively, and similarly in the case of (7), we note that (6i), (7i) ∀x(¬Fx) (6i), (7ii) ∀x(Fx ←→ ¬Gx) (6ii), (7i) ∀x(Fx ←→ Gx) (6ii), (7ii) ∀x(Fx) where is (classical) first-order logical consequence. Thus from (6), (7), and therefore from (1), it follows that T ∀x(Fx ←→ Gx) ∨ ∀x(Fx ←→ ¬Gx) ∨ ∀x(Fx) ∨ ∀x(¬Fx). (8) It is easy to see that each of the four disjuncts here, and hence the disjunction, similarly implies the formula in (1), which is therefore equivalent to that in (8). The latter is not quite in the form required for piecewise definability, but we can adjust the last two disjuncts so that the letter of the condition is satisfied; we abbreviate Gx ∨ ¬Gx to x and its negation to ⊥x. T ∀x(Fx ←→ Gx) ∨ ∀x(Fx ←→ ¬Gx) ∨ ∀x(Fx ←→ x) ∨ ∀x(Fx ←→ ⊥x). (9) Since (9) follows from (1), for the special case of supervenience of F on a single predicate G, we have shown that such supervenience implies piecewise definability. Noting that the four disjuncts involve, on the right, each of the four nonequivalent Boolean compounds that can be constructed from Gx, we would expect something similar in the general case, in which the hypothesis is that F is supervenient in T on G1, . . . , Gk. For the proof in the general case, however, it would be cumbersome in the extreme to retrace the analogue of the passage from (1) to (9), and we use a different style of argument—one for which no particular originality is claimed: see note 2. SUPERVENIENCE AND DEFINABILITY 247 3 Formulas of the form ±G1x ∧ ±G2x ∧ · · · ∧ ±Gkx, where ±Gix is either the formula Gix or else the formula ¬Gix, will be called elementary G-conjunctions; we reserve ‘ψ’ as a variable ranging over these conjunctions. By ψM we mean the set of elements satisfying such a formula in the structure M. (See note 4.) Similarly FM (alias (Fx)M) is the extension assigned to F by M. We call ψ F-favorable just in case ψM ∩ FM = ∅. (We should strictly say ‘F-favorable relative to M’ but the relevant structure will be clear from the context.) Lemma 3.1 If M is a model for a theory in which F is supervenient on G1, . . . , Gk then FM = ∪ {ψM|ψ is an F-favorable elementary G-conjunction}.