Carnap’s Problem for Modal Logic

Abstract

We take Carnap’s problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap’s problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to be interpreted as in relational Kripke semantics. Except when restricted to finite domains, no modal logic characterizes exactly the Kripkean interpretations of $\Box $ . Moreover, we show that, in contrast with the case of first-order logic, the obvious requirement of permutation invariance is not adequate in the modal case. After pointing out some known facts about modal logics that nevertheless force the Kripkean interpretation, we focus on another feature often taken to embody the gist of modal logic: locality. We show that invariance under point-generated subframes (properly defined) does single out the Kripkean interpretations, but only among topological interpretations, not in general. Finally, we define a notion of bisimulation invariance—another aspect of locality—that, together with a reasonable closure condition, gives the desired general result. Along the way, we propose a new perspective on normal neighborhood frames as filter frames, consisting of a set of worlds equipped with an accessibility relation, and a free filter at every world.

Carnap’s Problem for Intuitionistic Propositional Logic

“…Carnap's question can be asked for any consequence relation in any logical language, relative to any formal semantics for that language. Bonnay and Westerståhl [6] deals with Carnap's problem in modal logic and [7] discusses it for some logics with generalized quantifiers.…”

“…If A is a Heyting algebra and j is a nucleus on A, then A j is a Heyting algebra. Now recall from (6) the Heyting algebras of the form Up(F ), where F = (X, ≤). When j is a nucleus on this algebra, Bezhanishivili and Holliday call (F , j) a nuclear frame.…”

“…More generally, an arbitrary nuclear interpretation (over F ) is a map I F ,j as in Definition 2.1:3, with semantic values [[ϕ]] I F ,j v calculated as specified there. 6 Example 2.4 (Kripke semantics). Given F , let j K be the identity function on Up(F ); it is obviously a nucleus on Up(F ).…”

“…Remark 3.11. It may be worth noting that, by contrast, categoricity facts about classical modal logic (see [6]) only require validating all theorems. This is precisely because it is classical.…”