Analyticity and Necessity in Leibniz

Analyticity and Necessity in Leibniz GREGORY W. FITCH JOHN W. NASON PRESENTS A NOW-FAMILIAR CRITICISM of Leibniz based on Leibniz's definition of truth. Nason says, "But if it were true that all true af~rmative propositions are analytic, then all such propositions are necessary and there is no contingency ."' Since Arnauld first raised this objection, philosophers have attempted to find in Leibniz a solution to this problem. Leibniz himself denied such a charge, but many commentators have found his replies inadequate. Those who defend Leibniz usually do so by introducing either Leibniz's notion of infinite analysis or his notion of possible worlds to distinguish contingent truths from necessary ones. Both solutions seem to have their difficulties, although the notion of an infinite analysis as Leibniz wanted to use it seems more obscure to the contemporary philosopher than that of possible worlds. It is my purpose here to solve the problem for Leibniz once and for all by using Leibniz's notion of possible worlds in conjunction with a more contemporary notion; namely, the notion of counterparts. In proceeding I will first consider the recent account of Leibniz and possible worlds presented by Benson Mates; 2for while there are some difficulties in Mates's view, much of what he says is true, and it is extremely helpful in understanding Leibniz. 1. Mates's System Mates's system is based on the Leibnizian notions of "complete individual concepts," "compossibility," and "possible worlds." "A complete individual concept is a set of simple properties satisfiable by exactly one thing and containing all the simple properties that would belong to that one thing if it existed. ''3 Compossibility is an equivalence relation that partitions the set of all complete individual concepts into equivalence classes, which are possible worlds, each containing infinitely many concepts, also denumerable. ' Associated with each constant of the language is a complete individual concept, and associated with each predicate (except identity) is a simple property (all the predicates are monadic except identity). An atomic sentence, ~, is true of a possible world W provided that the property associated with the predicate is a member of the concept associated with the subject (or constant) of the sentence and that the concept associated with the constant is a member of W. A sentence is necessarily true just in case it is true in all possible worlds. There are a few minor problems with Mates's account. First, as Mates defines ' "Leibniz and the Logical Argument for Individual Substances," Mind 51 (1942):213. 2 "Leibniz on Possible Worlds," in B. van Rootselaar and J. F. Stall, eds. Logic, Methodology and Philosophy of Science 1II (Amsterdam: North Holland Publishing Co., 1968), pp. 507-29; and "Individuals and Modality in the Philosophy of Leibniz," Studia Leibnitiana 2 (1972):81-118. J "Possible Worlds," p. 524. ' Ibid., pp. 524-25. [29] 30 HISTORY OF PHILOSOPHY compossibility, it is a two-place relation between concepts. "Individual concepts are said to be compossibleif they are capable of joint realization."' But this by itself will not guarantee that compossibility is an equivalence relation. While it is clearly reflexive and symmetrical, it does not appear to be transitive. Mates realizes the possibility that the relation of compossibility is not transitive but says that "it is blocked by the Leibnizian doctrine that in the actual world and in every other possible world, each concept 'mirrors' or 'expresses' all the other individual concepts in that world. Each individual of the actual world is related to all the others, and every relation is 'grounded' in simple attributes of the things related; the same is true of the other possible worlds as well. ''6 There is no question that Leibniz held some such view as Mates indicates. However, it is equally clear that Leibniz's notion of 'mirroring' is very obscure. Mates never defines the mirroring relation, and we are left uncertain whether it guarantees transitivity for the compossibility relation. Even if mirroring does the job and compossibility is an equivalence relation, it is not clear that we should identify possible worlds with these equivalence classes. If compossibility is an equivalence relation, then all the members of a given equivalence class will be pair...