Logics from Galois connections

In this paper, Information Logic of Galois Connections (ILGC) suited for approximate reasoning about knowledge is introduced. In addition to the three classical propositional logic axioms and the inference rule of modus ponens, ILGC contains only two auxiliary rules of inference mimicking the performance of Galois connections of lattice theory, and this makes ILGC comfortable to use due to the flip-flop property of the modal connectives. Kripke-style semantics based on information relations is defined for ILGC. It is also shown that ILGC is equivalent to the minimal tense logic Kt, and decidability and completeness of ILGC follow from this observation. Additionally, relationship of ILGC to the so-called classical modal logics is studied. Namely, a certain composition of Galois connection mappings forms a lattice-theoretical interior operator, and this motivates us to axiomatize a logic of these compositions. It turns out that this logic satisfies the axioms of the non-normal logic EMT4. Hence, EMT4 can be viewed to be embedded in ILGC. EMT4 is complete with respect to the neighbourhood semantics. Here, we introduce an alternative semantics for EMT4. This is done by defining the so-called interior models, and completeness of EMT4 is proved with respect to the interior semantics.