On a derivation of the necessity of identity

Saul Kripke's lecture " Identity and Necessity " [1971] begins with a short discussion of a formal derivation of a law of necessity of identity. The derivation has become rather well known, but perhaps not so well understood. The present commentary attempts to improve understanding of it by enlarging on four remarks of Kripke's: 1 (A) This is an argument which has been stated many times in recent philosophy. (B) Let us interpret necessity here weakly. We can count statements as necessary if whenever the objects mentioned therein exist, the statement would be true. (C) If … we can talk about modal properties of an object at all, that is, in the usual parlance, we can speak of modality de re… then [the conclusion of the derivation] has to hold. (D) [B]y itself [the conclusion of the derivation] does not… say anything about identity statements at all. It says that for every object x and object y, if x and y are the same object, then it is necessary that x and y are the same object. The derivation in " Identity and Necessity " proceeds in three steps: 2 (1) ∀xx = x (2) ∀x∀y(x = y → (x = x → x = y)) (3) ∀x∀y(x = y → x = y)