Mittwoch, März 01, 2006

Tarski und Carnap on Logical Truth

What Is Genuine Logic? Consequences for Modal Logic
Modal logics are not genuine logics, but systems of extralogical analytical postulates or rules about the intensional sentence operator $\Box$. (It should be clear that this classification of modal logics as not genuine logics but as systems of analytical principles does in no way diminish their philosophical importance - but it avoids several confusions about the question "what is the right modal logic?". The hidden semantical parameter in modal logics is the entire Kripke frame $\langle W,R \rangle$. We have to view this frame as the variable extensional interpretation of the modal operator $\Box$: $I(\Box$ = $\langle W,R \rangle$. Modal logics are sets of modal formulas which are true for certain classes of frames from which the interpretations of $\Box$ are taken (e.g. all universal frames, which gives S5, etc.). The metalogical principles characterizing certain frame classes are extralogical meaning postulates.
How would a genuine modal logic look like? It should have a fixed frame $\langle W,R \rangle$. Naturally, $W$ should be the set of all logically possible words, and $R$ be the universal relation on $W$. Indeed - this is nothing but Carnap's original conception of modal logic (1947, pp. 173ff; and 1946, system MFL). It is an historical error to think that Carnap's modal logic was S5. Only in the propositional part of his paper (1946, system MPL) Carnap deviates from his original idea and introduces closure under substitution to arrive at a system equivalent with the Lewis system S5. But modal logic according to his original idea is much stronger than S5. In the genuine Carnapian modal logic it holds that $\Box A$ is logically true if and only if $A$ itself is logically true, for arbitrary formulas $A$ (1947, p. 174; convention 39-1). Thus, e.g., for every atomic variable $p$, $\Diamond p$ is logically true and $\Box p$ is logically false - moreover, every completely modalized
sentence
will be L-determined. Carnap's genuine modal logic has very unusual properties - for instance, it is not closed under substitution for propositional variables, and its rules are nonmonotonic.