The TL;DR version is that Kripke has misunderstood Lewis’ counterpart theory, and so his criticism is off-base. The longer version follows.
To understand what is going on here, it helps to have a little background. Modal logic, with the box/diamond notation, was originally conceived by C I Lewis to express modal properties of propositions, e.g. to say of some propositions that they are necessarily or possibly true. As such, it was only used with the propositional calculus, and the modalities were de dicto. It is a significant extension to combine modal operators with predicate logic to create quantified modal logic, because it implies that there are de re modalities. Some logicians reject the idea of de re modality entirely. However, Saul Kripke, David Lewis and others are fine with it. But there are many differences of opinion over exactly what logical principles should apply, both to the axioms or rules of the logic, and to the semantics of the logic, which is usually expressed using possible worlds (PW).
One of the key differences is that Kripke holds with the concept of cross-world identity, i.e. that an individual can exist in more than one PW. Lewis for his part developed a theory of counterparts, under which no individual exists in more than one PW. To illustrate the difference, when Kripke thinks of the counterfactual, “I could have won that race” he takes himself to mean that he, Kripke, won that race in some other PW. Lewis on the other hand, takes it mean that some counterpart of him, similar in many ways but not identical, won the race in some PW. For the details of the debate between them you’ll have to read their respective work.
To come back to your question, Kripke makes the accusation against Lewis that his counterpart theory violates an accepted principle of logic, namely the indiscernibility of identicals. The formula (y) ((x)A(x) ⊃ A(y)) is meant to be one way of expressing this. We could express it another way as
- (∀x)(∀y)(x=y → (φ(x) ↔ φ(y)).
(I’m using → here for the material conditional, and ↔ for the material biconditional.) Kripke’s complaint is that if we substitute a modal formula for φ then this principle fails. For example, if we take φ(x) to be ◇(x≠y) then we have
- (∀x)(∀y)(x=y → (◇(x≠y) ↔ ◇(y≠y))
Kripke claims that (2) is not acceptable as a theorem because (∃y)(∀x) ◊(x≠y) is satisfiable but (∃y) ◊(y≠y) is not, and hence (1) is violated.
Lewis’ response is that Kripke is not interpreting the diamond correctly in the context of his counterpart theory. A proper reading of (2) goes something like this: For any x and y, where x and y are identical in the actual world, there is a PW where there exists a counterpart of x and a counterpart of y and those counterparts are not identical – if and only if – there is a PW where there exists a counterpart of y that is not identical with itself.
Now there is no PW where an individual is not identical with itself, so (2) simplifies to
- (∀x)(∀y)(x=y → ¬◇(x≠y) )
This states that if x and y are identical in the actual world, then there is no PW in which there is a counterpart of x that is not identical with a counterpart of y. Or in other words there is no PW containing more than one counterpart of any individual in the actual world. Lewis says that this is indeed a logically contingent proposition: it can be false because it is possible for an individual in the actual world to have multiple distinct counterparts in a single PW. But this does not violate the indiscernibility of identicals, because (1) does hold universally.
So Lewis’ rebuttal is that he has not violated any sacrosanct principle of logic, but rather that (2) should not be interpreted as an instance of (1). Specifically, the diamond operator, within Lewis’ theory, contains a kind of implicit quantifier of its own, so that in (2) the variables x and y that lie within the scope of the ◇ are not the same as the x and y that are outside. The moral is that it is not sufficient to read a modal sentence such as (2) and say the word ‘possibly’ in your head when you see a diamond and ‘necessarily’ when you see a box. The box and diamond must be understood rigorously in terms of the logic of which they are a part.
David Lewis, “Counterpart Theory and Quantified Modal Logic”, Journal of Philosophy 65 (1968), 113-126.
Saul Kripke, Naming and Necessity (Reidel, 1972), page 45, note 13.
Allen Hazen, “Counterpart-Theoretic Semantics for Modal Logic”, Journal of Philosophy 76 (1979), 319-338.