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Kripke semantics/HintikkaVsKripke: Kripke semantics is not a viable model for the theory of logical modalities (logical necessity and logical possibility).
Problem: the right logic cannot be axiomatized.
Solution: to interpret Kripke semantics as a non-standard semantics,...
...in the sense of Henkin's non-standard interpretation of the logic of higher levels, while the correct semantics for logical modalities would be analogous to a standard interpretation.
Kripke semantics/Hintikka: Kripke semantics is a modern model-theoretic approach that is misleadingly called Kripke semantics. E.g.
F: is a framework consisting of
SF: a set of models or possible worlds and
R: a two-digit relation, a kind of alternative relation.
Possible worlds: w1 is supposed to be an alternative, which could legitimately be realized instead of w0 (the actual world).
R: the only limitation we impose on it is reflexivity.
Truth-conditions/Modal logic/Kripke semantics/Hintikka: the truth conditions for modal sentences are then:
(TN) Given a frame F, Np is true in w0 ε SF iff. P is true in every alternative wi ∈ SF to w0.
(T.M) Given a frame F, Mp is true in w0 ε SF iff. P is true in at least one alternative wi ∈ SF to w0.
Model theory/Modal logic/Hintikka: Kanger, Guillaume and later Kripke have seen that when we add reflexivity, transitivity, and symmetry, we get a model theory for axiom systems of the Lewis type for modal propositional logic.
Kripke semantics/modal logic/logical possibility/logical necessity/HintikkaVsKripke/HintikkaVsKripke semantics: Problem: if we interpret the operators N, P as expressing logical modalities, they are inadequate: we need more than one arbitrary selection for logical possibility and necessity of possible worlds. We need truth in every logically possible world.
But in the Kripke semantics it is not necessary that all such logically possible worlds are contained in the set of alternatives. ((s). That is, there may be logically possible worlds that are not considered). (See below the logical possibility forms the largest class of possibilities).
Problem: Kripke semantics is therefore inadequate for logical modalities.
Kripke/Hintikka: Kripke has avoided epistemic logic and the logic of propositional attitudes, concentrating on pure modalities.
Therefore, it is strange that he uses non-standard logic.
But somehow it seems clear to him that this is not possible for logical modalities.
Metaphysical possibility/Kripke/HintikkaVsKripke: Kripke has never explained what these mystical possibilities actually are.
Worse: he has not shown that they are so restrictive that he can use his extremely liberal non-standard semantics.
Jaakko and Merrill B. Hintikka
The Logic of Epistemology and the Epistemology of Logic Dordrecht 1989
J. Hintikka/M. B. Hintikka
Untersuchungen zu Wittgenstein Frankfurt 1996