A multi-labelled sequent calculus for Topo-Logic

Abstract

Abstract We present a labelled sequent calculus for a trimodal epistemic logic exhibitied in Baltag et al. (2017, Logic, Rationality, and Interaction, pp. 330–346), an extension of the so called ‘Topo-Logic’. To the best of our knowledge, our calculus is the first proof-calculus for this logic. This calculus is obtained via an adaptation of the label technique by internalizing a semantics over topological spaces. This internalization leads to the generation of two kinds of labels in our calculus and the labelling of formulae by pairs of labels. These novelties give tools to provide a simple calculus that is intuitively connected to the semantics. We prove that this calculus enjoys many structural properties such as admissibility of cut, admissibility of contraction and invertibility of its rules. Finally, we exhibit a proof search strategy for our calculus that allows us to prove completeness in a direct way by the extraction of a countermodel from a failure of proof. To define this strategy, we design a tool for controlling the generation of labels in the construction of a search tree, although the termination of this strategy is still open.