Combined causal logics of minkowski spacetime.


V.L. Vasyukov


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.






Birkhoff G. Lattice Theory, Providence, Rhode Island, 1967.

Cegla W. Causal Logic of Minkowski Space // Current Issues in Quantum Logic / S. Beltrametti and B. van Fraassen (eds.), N.Y.: Plenum, 1981, pp. 419-424.

Dalla Chiara M .-L. Quantum Logic // Handbook of Philosophical Logic / D.Gabbay and F.Guenthner (eds.), vol III, Reidel, 1986, pp. 427-469.

Goldblatt R. Diodorean Modality in Minkowski Spacetime // Studia Logica, v.39, 1980, pp. 219-236.

Jaskowski S. Rachunek zdan dla systemow dedukcyjnych sprzecznych // Studia Soc. Sci. Torunensis, Sectio A, Vol. I, No. 5, 1948. (English translation in [6]).

Jaskowski S Propositional Calculus for Contradictory Deductive Systems // Studia Logica, v.24, 1969, pp. 143-157.

Lemmon E. Algebraic Semantics for Modal Logics I. // Joum. Symb. Log., 1966, vol. 31, No 1, p.46-65.

Perzanowski J. Towards Post-Tractatus Ontology // Wittgenstein - Towards Re-Evaluation / Eds. R.Haller & J.Brandi, Verlag Holder-Pichler-Tempsky, Wien, 1990, pp. 185-199.

Smirnov V.A. Internal and External Logics 11 Bull. Sect. Logic, v.17, №3/4, 1988, pp. 170-181.

Vasyukov V.L. Jaskowski-Vasiliev Combined Discursive Logic // Forthcoming in: Logic and Logical Philosophy.


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