Hypersequent calculi can formalize various non-classical logics. In [9] we presented a non-commutative variant of HC for the weakest temporal logic of linear frames Kt4.3 and some its extensions for dense and serial flow of time. The system was proved to be cut-free HC formalization of respective temporal logics by means of Schütte/Hintikka-style semantical argument using models built from saturated hypersequents. In this paper we present a variant of this calculus for Kt4.3 with a constructive syntactical proof of cut elimination.

This chapter contains sections titled: The use of epistemic modalities is intended as a development of the earlier account of discriminability, not an alternative to it; they should therefore be understood in terms of knowledge rather than certainty. The non‐transitivity of intentional indiscriminability is a robust phenomenon in epistemic modal logic. For KT5 is one of the strongest modal systems of philosophical interest; on the current reading, it presents facts about cognition as themselves entirely open to cognition. In KT5, we (...) cannot distinguish between non‐intentional indiscriminability and identity. At least two responses to this fact are possible. We can retreat from KT5 to a weaker system such as KT4, or we can treat indiscriminability as intentional within KT5. The failure of intentional indiscriminability to be transitive in KT5 can be traced to the subject's inability to tell which object some of the presentations present. (shrink)

This paper shows that the following common assumption is false: that in modal-logical representations of higher-order vagueness, for there to be borderline cases to borderline cases ad infinitum, the number of possible distinct modalities in a modal system must be infinite. (Open access journal).