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In Kripke 's Naming and Necessity, there is a footnote says that

"Lewis's elegant paper also suffers from a purely formal difficulty: on his interpretation of quantified modality, the familiar law (y) ((x)A(x) ⊃ A(y)) falls, if A(x) is allowed to contain modal operators. (For example, (∃y) ((x) ◊(x ≠y)) is satisfiable but (∃y) ◊ (y ≠ y) is not.) Since Lewis's formal model follows rather naturally from his philosophical views on counterparts, and since the failure of universal instantiation for modal properties is intuitively bizarre, it seems to me that this failure constitutes an additional argument against the plausibility of his philosophical views. "

Doesn't the quote make a failure? A(x) usually means a formula with a free variable x, however (∃y) ((x) ◊(x ≠y)) in the example is a sentence with no free variable. And (∃y) ◊ (y ≠ y) even can't be (because it will be illegal) the substituting result of (∃y) ((x) ◊(x ≠y). So even if (∃y) ((x) ◊(x ≠y)) is satisfiable but (∃y) ◊ (y ≠ y) is not, it seems not the matter of modal qualification but only because the variables in the formulae are bounded.

It seems confusing that Kripke was familiar with all these in his eighteen-year-old paper which proofs the modal logic is complete, but in NN he made this confusing footnote.

If I make some mistakes, please correct me. Thank you in advance!

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    I think it is a typo, and/or Kripke uses some peculiar shorthands. It is supposed to be (x)A(x) ⊃ A(y) (universal instantiation) and A(x) is ◊(x ≠y), y is a constant. He could put (y) on it but then he'd have to write A(x,y), not just A(x). Then (∃y) ((x) ◊(x ≠y)) probably means (by existential generalization) that one can find a constant to satisfy (x)◊(x ≠y), but not one to satisfy ◊(y ≠ y), i.e. to make (∃y)◊(y ≠ y). – Conifold Jul 28 '20 at 7:42
  • maybe that, however, in Kripke classical paper which proofs the completeness of modal logic, he begins with definition, talking about substitution. And in that paper, his use of A(x) and other things is the usual way. – AnduinWilde Jul 28 '20 at 11:15
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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

  1. (∀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

  1. (∀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

  1. (∀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.

References:

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.

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  • Thank you. Your answer helps me a lot. However, I wonder why we can take φ(x) to be ◇(x≠y) , I think it violate the rule of substitution, for after the substitution y is bounded. – AnduinWilde Jul 29 '20 at 8:19
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Long comment

I'm puzzled also, but for a different reason...

From (∃y) ((x) ◊(x ≠ y)), using a fresh term a, we have, by (∃-elim): (x) ◊(x ≠ a).

Thus, using (∀-elim) with a (legitimate) we have: ◊(a ≠ a) and finally with (∃-intro): (∃y) ◊(y ≠ y).

But how we can say that the premise is satisfiable, if we can derive from it: ◊(a ≠ a) ?

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I shall try explicate Kripke's footnote against the backdrop of Lewis's 1968 paper "Counterpart Theory and Quantified Modal Logic" and leave it to the reader to decide how felicitous Kripke's phrasing is and how rigorous Lewis's counterpart-theoretical formalism is.

Lewis defines four primitive predicates of counterpart theory:

  1. Wx: x is a possible world,
  2. Ixy: x is in possible world y,
  3. Ax: x is actual,
  4. Cxy: x is a counterpart of y.

and eight postulates:

  1. xy(IxyWy): Nothing is in anything except a world,
  2. xyz ((IxyIxz) → y = z): Nothing is in two worlds,
  3. xy(Cxy → ∃zIxz): Whatever is a counterpart is in a world,
  4. xy(Cxy → ∃zIyz): Whatever has a counterpart is in a world,
  5. xyz((IxyIzyCxz) → x = z): Nothing is a counterpart of anything else in its world,
  6. xy(IxyCxx): Anything in a world is a counterpart of itself,
  7. x(Wx ∧ ∀y(Iyx ↔ ∃Ay)): Some world contains all and only actual things (there is a unique actual world),
  8. xAx: Something is actual.

We may notice that Lewis's system does not rule out such sentences as the following:

∃x∃y(I(x, w) ∧ I(y, w) ∧ (x = y) ∧ ∃w'x'y'(I(x', w') ∧ I(y', w') ∧ C(x', x) ∧ C(y', y) ∧ (x' ≠ y'))

which says that x and y, identical to each other in a world w, may have distinct counterparts x' and y' in a world w' in which they are not identical. This may seem to be an abuse of syntax, but it is not. That is a consequence of neglecting multiple aspects of an object essentially (or, if one prefers to say, rigidly) designating the object itself. Consider 'the younger Aristotle of the Academy' and 'the elder Aristotle of the Lyceum'. In our actual world, we conceive of them as identical persons (x = y). Lewis's formalism allows that, by a twist of fate, the younger Aristotle of the Academy and the elder Aristotle of the Lyceum might keep different tracks of life and end up with distinct counterparts in another world (right, Lewis's counterpart theory could make as nice a basis for science fiction screenplays as Baudrillard's 'Simulacra and Simulation' has done for the Matrix series).

Kripke remarks this neglect by reference to the translation of Lewis's formalism into the standard modal logic, which has been known to be not overlapping. From Hilbert-Ackermann system, he cites the axiom

xA(x) → A(y), y being free for x

and applies the rule of universal generalisation to it. He points out correctly that in case A(x) is a compound wff with modality, it can turn out to be an unsatisfiable formula, whereas it would be satisfiable in Lewis's formalism.

For a thorough examination of the issue, I would recommend

Kracht, Marcus and Oliver Kutz: "Logically Possible Worlds and Counterpart Semantics for Modal Logic" in Jacquette, Dale (ed) Philosophy of Logic. Handbook of the Philosophy of Science, vol. 5, pp. 943–996. Elsevier, Amsterdam (2007).

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  • My confusion is, the example Kristen gives, (∃y) ((x) ◊(x ≠y)), does not begin with universal qualification. Even if this doesn't matter, he could not substitute x with y, since y is bounded in the formula after substitution (∃y) ◊ (y ≠ y), which is illegal. – AnduinWilde Jul 30 '20 at 5:26
  • @AnduinWilde in brief, we cannot argue when modality is involved as we do in the plain first-order predicate logic over such unstructured domains as sets. We need to appeal to structured domains such as situations and possible worlds. Notice that Kripke employs not ∀xA(x) → A(y), but ∀y(∀xA(x) → A(y)), for the former one is not valid with respect to the semantical system bearing his name, while the latter one is. – Tankut Beygu Jul 30 '20 at 17:39

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