Introduction to "The Paradox of Identity"

0. Introduction

"A Theory of Complexes"^* presented a formal theory of the identity-relation. In this theory, variables go proxy for complexes, entities which--unlike the (logically) unstructured particulars of first-order logic--are ontologically differentiated and logically complex. The deductive apparatus of my theory is that of the formal system P.

Viewed as an abstract calculus, P says nothing about the world. Like other theoretical models, however, P can be endowed with empirical content by identifying complexes with the things for which FOL variables go proxy. The effect of this identification is to replace the background ontology of quantification theory with an ontology of complexes. I will warrant this replacement by showing that important logico-semantic characteristics of individual things and events which the usual ontology cannot explain, are referrable to features of complexes.

In Appearance and Reality, F. H. Bradley cites the features I have in mind:

If we take up anything considered real, no matter what it is, we find in it two aspects. There are always two things we can say about it; and if we cannot say both we have not got reality. There is a 'what' and a 'that', an existence and a content, and the two are inseparable. That anything should be, and should yet be nothing in particular, or that a quality should not qualify and give a character to anything is obviously impossible. (Appearance and Reality, p. 162, cited in Murphy's "Substance and Substantive," U.C. Publications in Philosophy, Vol. 9, 1927, pp. 63-87)

For complexes incorporate these marks of a thing's particularity. In contrast, the ground-level objects of the "ontology of individuals", as Van Heijenoort calls the background ontology of quantification theory, are devoid of any structure that is relevant to a thing's per se logical relations. The translation of singular assertions into the language of quantification theory thus involves replacement of the logically structured things and events such assertions are about, by ground-level objects of the ontology of individuals, "bare individuals" with no "inner structure, . . . mere pegs".¹

Logico-semantic analyses which proceed by translating such assertions into the language of quantification theory² thus obliterate the nexus of that and what constitutive of the particularity of the things these assertions are about. As a result, such analyses fail in two related respects. First, they fail to convey the complex states of affairs to which true singular identity and existence statements have reference; and second, they fail to uncover what about these states of affairs is responsible for the breakdown of the Indiscernibility of Identicals in propositional attitude and modal contexts. Nowhere is the insufficiency of classical analysis more apparent than in its failure to resolve the so-called "problem of identity".

The Problem of Identity