Characterizing intermediate tense logics in terms of Galois connections

: Dzik W, Jarvinen J, Kondo M

Publisher: OXFORD UNIV PRESS

: 2014

: Logic Journal of the IGPL

: LOGIC JOURNAL OF THE IGPL

: LOG J IGPL

: 22

: 6

: 992

: 1018

: 27

: 1367-0751

DOI: https://doi.org/10.1093/jigpal/jzu024

We propose a uniform way of defining for every logic L intermediate between intuitionistic and classical logics, the corresponding intermediate tense logic LKt. This is done by building the fusion of two copies of intermediate logic with a Galois connection LGC, and then interlinking their operators by two Fischer Servi axioms. The resulting system is called L2GC+FS. In the cases of intuitionistic logic Int and classical logic Cl, it is noted that Int2GC+FS is syntactically equivalent to intuitionistic tense logic IKt by W. B. Ewald and Cl2GC+FS equals classical tense logic Kt. This justifies calling L2GC+FS the L-tense logic LKt for any intermediate logic L. We define H2GC+FS-algebras as expansions of HK1-algebras, introduced by E. Orlowska and I. Rewitzky. For each intermediate logic L, we show algebraic completeness of L2GC+FS and its conservativeness over L. We prove relational completeness of Int2GC+FS with respect to the models defined on IK-frames introduced by G. Fischer Servi. We also prove a representation theorem stating that every H2GC+FS-algebra can be embedded into the complex algebra of its canonical IK-frame.