In their presentation of canonical models for normal systems of modal logic, Hughes and Cresswell observe that some of these models are based on a frame which can be also thought of as a collection of two or more isolated frames; they call such frames ‘non-cohesive’. The problem of checking whether the canonical model of a given system is cohesive is still rather unexplored and no general decision procedure is available. The main contribution of this article consists in introducing a (...) method which is sufficient to show that canonical models of some relevant classes of normal monomodal and bimodal systems are always non-cohesive. (shrink)

It is observed that a consistent congruential modal logic is not guaranteed to have a consistent extension in which the Box operator becomes a truth-functional connective for one of the four one-place truth functions.

It is well known that no consistent normal modal logic contains (as theorems) both ♦A and ♦¬A (for any formula A). Here we observe that this claim can be strengthened to the following: for any formula A, either no consistent normal modal logic contains ♦A, or else no consistent normal modal logic contains ♦¬A.

We propose a logic of temporal contingency, which has operators of past and future contingency as primitive modalities. This logic is less expressive than standard temporal logic over the class of bidirectional frames, and cannot define some basic frame properties such as bidirectionality and transitivity. We present a minimal system based on two key ‘bridge axioms’ and a bimodal version of a so-called ‘almost definability’ schema in the literature. The completeness proof is highly nontrivial due to the requirement that the (...) canonical model be bidirectional. We then extend the results to the simplest dynamic case: public announcements. (shrink)

Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As (...) Humberstone notes, the issue of Post completeness in congruential modal logics is not well understood. The present article shows that in contrast to normal modal logics, the extent of the property of Post completeness among congruential modal logics depends on the background set of logics. Some basic results on the corresponding properties of Post completeness are established, in particular that although a congruential modal logic is Post complete among all modal logics if and only if its modality is truth-functional, there are continuum many modal logics Post complete among congruential modal logics. (shrink)