Characterizing intermediate tense logics in terms of Galois connections

W. Dzik, J. Jarvinen, M. Kondo
2014 Logic Journal of the IGPL  
We propose a uniform way of defining for every logic L intermediate between intuitionistic and classical logics, the corresponding intermediate minimal tense logic LK_t. 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 here L2GC+FS. In the cases of intuitionistic logic Int and classical logic Cl, it is noted that Int2GC+FS is syntactically
more » ... ent to intuitionistic minimal tense logic IK_t by W. B. Ewald and Cl2GC+FS equals classical minimal tense logic K_t. This justifies to consider L2GC+FS as minimal L-tense logic LK_t 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.
doi:10.1093/jigpal/jzu024 fatcat:bnalmdg6ebbaleb3lmowc7mpay