Tarski und Carnap on Logical Truth



Kann mal wer kurz sagen, was die "Quintessenz"
dieser abschließenden Zussamenfassung ist?
Aus: Tarski und Carnap on Logical Truth - or: What Is Genuine Logic?

10. 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.

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