Frege’s Puzzle and the New Theory of Reference – Quine, Marcus, and Kripke

Most of the metaphysical problems of identity, and especially recent claims about the necessity of identity, can be traced back to the great rationalist philosopher Gottfried Leibniz, who argued for the replacement of ordinary language with a lingua characterica, an ambiguity-free set of terms that would eliminate philosophical puzzles and paradoxes. Bertrand Russell, Ludwig Wittgenstein, and Rudolf Carnap all believed in this Leibnizian dream of ambiguity-free, logically true, facts about the world that may be true in all possible worlds. Unfortunately, fundamental limits on logic and language such as the Gödel and Russell paradoxes have prevented Leibniz’s ideal ambiguity-free language, but many modern paradoxes, including questions about identity and necessity, are resolvable in terms of information, as we shall see

More than any other philosopher, Leibniz enunciated clear principles about identity, including his Identity of Indiscernibles. If we can see no differences between things, they may be identical. This is an empirical fact, and must be tested empirically, as Leibniz knew. But once again, whenever we are talking about two things, that there is a difference between them, a discernible difference, is transparently obvious. Two things are numerically distinct even if they have identical internal information. Leibniz also described a corollary or converse, the Indiscernibility of Identicals.  But this idea is necessarily true, if such things as numerically distinct identical objects exist. We shall show that such things have only a relative identity, identity in some respects

Gottlob Frege implemented Leibniz’s program of a purely logical language in which statements or sentences with subjects and predicates are replaced with propositional functions, in which a term can be replaced by a variable. In modern terminology, the sentence Socrates is mortal can be replaced, setting the subject Socrates = x, and the predicate “is mortal” with F. “x is F” is replaced by the propositional function Fx, which is read “x is F,” or “x F’s.”

Names and Reference

Although Frege was very clear, generations of philosophers have obscured his clarity by puzzling over different names and/or descriptions referring to the same thing that may lead to logical contradictions – starting with Frege’s original example of the Morning Star and Evening Star as names that refer to the planet Venus. Do these names have differing cognitive value? Yes. Can they be defined qua references to uniquely pick out Venus. Yes. Is identity a relation? No. But the names are relations, words that are references to the objects. And words put us back into the ambiguous realm of language.

Over a hundred years of confusion in logic and language have consisted of finding two expressions that can be claimed in some sense to be identical, but upon substitution in another statement, they do not preserve the truth value of the statement. Besides Frege, and a few examples from Bertrand Russell (“Scott” and “the author of Waverly.” “bachelor” and “unmarried man”), Willard Van Orman Quine was the most prolific generator of substitution paradoxes (“9” and “the number of planets,” “Giorgione” and “Barbarelli,” “Cicero” and “Tully,” and others).

Just as information philosophy shows how to pick out information in an object or concept that constitutes the “peculiar qualifications” that individuate it, so we can pick out the information in two designating references that provide what Quine called “purely designative references.” Where Quine picks out information that leads to contradictions and paradoxes (he calls this “referential opacity”), we can “qualify” the information, the “sense” or meaning needed to make them referentially transparent when treated “intensionally.”

Ruth Barcan Marcus

In 1947, Ruth C. Barcan (later Ruth Barcan Marcus) wrote an article on “The Identity of Individuals.” It was the first assertion of the so-called “necessity of identity.” Five years later, Marcus’s thesis adviser, Frederic B. Fitch, published his book, Symbolic Logic, which contained the simplest proof ever of the necessity of identity, by the simple mathematical substitution of b for a in the necessity of self-identity statement:

(1) a = b, (2) ☐[a = a], then (3) ☐[a = b], by identity elimination.

Saul Kripke

The indiscernibility of identicals claims that if x = y, then x and y must share all their properties, otherwise there would be a discernible difference. Saul Kripke argues that one of the properties of x is that x = x, so if y shares the property of ‘= x,” we can say that y = x. Then, necessarily, x = y. But this is nonsense for distinct objects. Two distinct things, x and y, cannot be identical, because there is some difference in extrinsic external information between them. Instead of claiming that y has x’s property of being identical to x (“= x”), we can say only that y has x’s property of being self-identical, thus y = y. Wiggins called this result “impotent.” Then x and y remain distinct in at least this intrinsic property as well as in extrinsic properties like their distinct positions in space.

Information philosophy shows that the indiscernibles x and y have only relative identity.