Identity in Cocchiarella's T*

Abstract

No topic, except perhaps The Counterfactual Conditional and The Ontological Argument (singly or together), has been of more interest to philosophers of a generally analytic stripe than Leibniz's principle of the identity of indiscernibles. We are content for present purposes to take that principle roughly as follows. (1) If two objects have all their properties in common, they are identically the same. In a logic rich enough to admit quantification over whatever properties there are of whatever things there are, (1) may be pleasantly put as follows. (2) (P)(Pa Pb) = a =b Hardly anyone wants to deny the converse of (1), and those who do must be suspected of invincible ignorance on the topic of usemention. I. e., it is obvious that (3) Every object has exactly the properties that it has. In symbols, (4) a = b (P)(Pa Pb) All of this has suggested to logicians since Russell and Whitehead, at least, that identity may be profitably thought of as a defined notion. I.e., (5) a_ b df (P)(Pa Pb) Formal justification for (5) can be found in that fact that, for all kinds of logics, type theories, set theories, and so forth, identity as construed by (5) turns out to have just the right properties-