Weak and global supervenience are strong

Abstract

Kim argues that weak and global supervenience are too weak to guarantee any sort of dependency. Of the three original forms of supervenience, strong, weak, and global, each commonly wielded across all branches of philosophy, two are thus cast aside as uninteresting or useless. His arguments, however, fail to appreciate the strength of weak and global supervenience. I investigate what weak and global supervenience relations are functionally and how they relate to strong supervenience. For a large class of properties, weak and global supervenience are equivalent to strong supervenience. I then offer a series of arguments showing that it is precisely because of their strength, not their weakness, that both weak and global supervenience are useless in characterizing any dependencies of interest to philosophers.

Appendix

What follows is an informal proof that if A globally supervenes on B, then A’ strongly supervenes on B’, where X’, or what I have called ‘X relational properties’, are properties that can be defined using quantification, logical operators, X properties, and X relations. I have used quotation marks to show that the proof is based closely on the proof Stalnaker gave in “Varieties of Supervenience,” though extended to handle cases in which A and B contain relations as well as properties.¹

“First, define a complete B-description of a world w as follows: if there are n members of the domain of w, the description will begin with n existential quantifiers. If variable x corresponds to individual a, then the description will contain a conjunct Fx for each of a’s B-properties F, and a conjunct ∼Fx for each B-property that a lacks. [If variable y corresponds to individual c, then the description will contain a conjunct Rxy for each two-place B-relation R that a bears to c, and a conjunct ∼Rxy for each two-place B-relation that a doesn’t bear to c. Similarly for relations that are three-placed and greater.] The description will also include conjuncts x ≠ y for each pair of distinct variables x and y that are bound by the existential quantifiers, and a universal generalization saying that everything is one of the n things. Obviously, any two worlds that have the same complete B-description will be B-indiscernible with respect to a mapping of the domain of one onto the domain of the other [and any two worlds that are B-indiscernible with respect to a mapping of the domain of one onto the domain of the other will have the same complete B-description].

“Now suppose that A globally supervenes on B. Let w and z be any two possible worlds, and a and b any two individuals from the domains of w and z, respectively, such that a has all the same B′-properties in w that b has in z. Let ϕ be the complete B-description of w, and let x be the variable in the description that corresponds to a. Drop from ϕ the quantifier that binds x, and the result is an open sentence with one free variable that expresses the maximal B′ property that a has in w. Since b has the same B’-properties in z as a has in w, it follows that b has this property in z. But then the existential generalization of this open sentence, which is equivalent to ϕ, is true in z, and so w and z are B-indiscernible, relative to a mapping that maps a to b. So since we are assuming that A globally supervenes on B, it follows that w and z are also A-indiscernible, relative to the same mapping. [It follows from this that w and z will have the same complete A-description and, moreover, if we drop the variable in that description that corresponds to b giving us an open sentence with one free variable expressing the maximal A′ property that b has in z, a must also have that property in w. Therefore, A′] strongly supervenes on B′.”