The logic of historical necessity as founded on two-dimensional modal tense logic

Journal of Philosophical Logic 28 (4):329-369 (1999)
We consider a version of so called T x W logic for historical necessity in the sense of R.H. Thomason (1984), which is somewhat special in three respects: (i) it is explicitly based on two-dimensional modal logic in the sense of Segerberg (1973); (ii) for reasons of applicability to interesting fields of philosophical logic, it conceives of time as being discrete and finite in the sense of having a beginning and an end; and (iii) it utilizes the technique of systematic frame constants in order to handle the problem of irreflexivity in tense logics, well known since Gabbay (1981). Axiomatizations are given for two infinite hierarchies of two-dimensional modal tense logics, one without and one with the characteristic operators for historical necessity and possibility. Strong and weak completeness results are obtained for both hierarchies as well as a result to the effect that two approaches to their semantics are equivalent, much in the spirit of Di Maio and Zanardo (1996) and von Kutschera (1997)
Keywords historical necessity  finite two-dimensional coordinate system  frame constants
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DOI 10.1023/A:1004425728816
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Katsuhiko Sano (2010). Axiomatizing Hybrid Products. Journal of Applied Logic 8 (4):459-474.

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