The decidability of normal K 5 logics
Michael C. Nagle · Journal of Symbolic Logic · 1981
The literature on modal logic includes a number of general completeness and decidability results. The work of Schiller Joe Scroggs [5], R.A. Bull [1], Kit Fine [2], and Krister Segerberg [6] provide examples. Scroggs showed that the proper extensions of S 5 have the finite model property and are axiomatizable. (Harrop [3] then argued that logics having these properties are decidable.) Bull extended Scroggs' result by showing that the normal extensions of S 4.3 have the finite model property. Fine subsequently provided a model-theoretic proof of Bull's result and also proved the axiomatizability of these logics. In a different direction Segerberg proved that every normal logic containing the characteristic axioms of Lewis' systems S 4 and S 5 is decidable. The present paper is in this tradition. We extend the results of Scroggs and Segerberg by showing that every normal modal logic containing the S 5 axiom has the finite model property, is axiomatizable, and thus is decidable.