Combined causal logics of minkowski spacetime.

Abstract

Getting started from ideas o f N.A.Vasiliev and exploiting some conceptions o f G.Frege, V.A.Smirnov introduced several combined calculi o f sentences and events consisting o f two parts, the abstract (external) logic depending on epistemological assumptions and the empirical (internal) logic depending on ontological ones Early the author proposed to approach algebra o f events as the discursive system o f S. Jaskowski (c f [5]). One more interesting possibility would be an exploitation o f an S4.2-modal algebra instead o f an S 5-modal algebra for discursive logic as an algebra o f events. As it was shown by R.Goldblatt [4] in the Diodorean interpretation o f modality where the operator o f necessity _ is read as “it is now and always will be the case that ” time would be modelled by the four-dimensional Minkowskian geometry that forms the basis o f Einstein 's special theory o f relativity. In this case ”event" у coming after event x just in case a signal can be seen from x to у at a speed at most that o f the speed o f light (i.e. у is in the causal future o f x). Passing to the S4.2-modal algebra o f the “histories” (subsets o f events or causal paths) we thus obtain a combined calculus o f sentences and histories. The same would be done in a more abstract way if we consider an algebra o f the histories as a complete orthomodular lattice following to W.Cegla's approach (cf [2]). For both formulations o f combined causal logic o f Minkowski spacetime a semantic o f (event) bundles and semantic o f possible worlds is built and some metamathematical results are obtained.