## Journal of Symbolic Logic

### A First Approach to Abstract Modal Logics

#### Abstract

The object of this paper is to make a study of four systems of modal logic (S4, S5, and their intuitionistic analogues $IM4$ and $IM5$) with the techniques of the theory of abstract logics set up by Suszko, Bloom, Brown, Verdu and others. The abstract concepts corresponding to such systems are defined as generalizations of the logics naturally associated to their algebraic models (topological Boolean or Heyting algebras, general or semisimple). By considering new suitably defined connectives and by distinguishing between having the rule of necessitation only for theorems or as a full inference rule (which amounts to dealing with all filters or with open filters of the algebras) we are able to reduce the study of a modal (abstract) logic $\mathbb{L}$ to that of two nonmodal logics $\mathbb{L}^-$ and $\mathbb{L}^+$ associated with $\mathbb{L}$. We find that $\mathbb{L}$ is "of $IM4$ type" if and only if $\mathbb{L}^-$ and $\mathbb{L}^+$ are both intuitionistic and have the same theorems, and logics of type S4, $IM5$ or $S5$ are obtained from those of type $IM4$ simply by making classical $\mathbb{L}^-,\mathbb{L}^+$ or both. We compare this situation with that found in recent approaches to intuitionistic modal logic using birelational models or using higher-level sequent-systems. The treatment of modal systems with abstract logics is rather new, and in our way to it we find several general constructions and results which can also be applied to other modal systems weaker than those we study in detail.

#### Article information

Source
J. Symbolic Logic, Volume 54, Issue 3 (1989), 1042-1062.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743039

Mathematical Reviews number (MathSciNet)
MR1011191

Zentralblatt MATH identifier
0687.03008

JSTOR