Cantor Derivative Logic in Topological Dynamics

Abstract

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamic topological systems (DTSs), which are pairs (X,f) consisting of a topological space X equipped with a continuous function on X. We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. We also prove a general result for the case where f is a homeomorphism, which yields soundness and completeness for the corresponding systems wK4H, K4H and GLH. Of special interest is GLC, which is the d-logic of all DTSs based on a scattered space. We use the completeness of GLC and the properties of scattered spaces to demonstrate the first sound and complete dynamic topological logic in the original trimodal language. In particular, we show that the version of DTL based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete.