"Bringing It All Home"

7. Bringing It All Home

It's time to give Mephisto his due. He demands to be told what all those symbols, and structured individuals, have to do with Frege's Principle and the problem of identity.

According to Frege's Principle, the meaning of a sentence remains constant when a singular term is substituted for another with the same meaning. If Frege were right, Hesperus couldn't possibly mean Hesperus and Phosphorus couldn't possibly mean Phosphorus. Granted that Hesperus = Phosphorus, if Hesperus meant Hesperus and Phosphorus meant Phosphorus, Hesperus = Phosphorus and Hesperus = Hesperus would have to mean the same thing. But they don't. Therefore, either Hesperus doesn't mean Hesperus and Phosphorus doesn't mean Phosphorus, or Frege got it wrong and the meaning of a sentence doesn't remain constant when a singular term is substituted for another with the same meaning.

Frege got it wrong. Let me say why.

Symbolize Hesperus by h and Phosphorus by p. The identity of Hesperus and Hesperus then comes to:

(3) h = h

and the identity of Hesperus and Phosphorus to:

(4) h = p

By hypothesis, Hesperus and Phosphorus are complexes--C-complexes, to be exact. As a result, (3) and (4) go over into assertions about C-complexes, thus:

(5) h.H = h.H
(6) h.H = p.P

Now, T3c says that C-complexes are identical just in case there is some complex of which their constituents are parts, so that by being parts of that compound entity these are one in substance. By hypothesis, Hesperus and Phosphorus are C-complexes. Consequently, the identity of Hesperus and Phosphorus involves the oneness in substance of the constituents of both Hesperus and Phosphorus. Not so the self-identity of Hesperus, for as T3d suggests, Hesperus's self-identity depends not upon the oneness in substance of the constituents of both Hesperus and Phosphorus, but upon the oneness in substance of the constituents of Hesperus alone. Therefore, since Hesperus = Phosphorus and Hesperus = Hesperus differ in what they assert, Hesperus = Phosphorus and Hesperus = Hesperus differ in meaning, even though Hesperus and Phosphorus are one and the same.13

Dead Meat