The Došen Square Under Construction: A Tale of Four Modalities

Abstract

In classical modal logic, necessity ${\mathop {\Box } A}$, possibility ${\mathop {\Diamond } A}$, impossibility ${\mathop {\Box } \lnot A}$ and non-necessity ${\mathop {\Diamond } \lnot A}$ form a Square of Oppositions (SO) whose corners are interdefinable using classical negation. The relationship between these modalities in intuitionistic modal logic is a more delicate matter since negation is weaker. Intuitionistic non-necessity ${\mathop {\boxminus }}$ and impossibility , first investigated by Došen, have received less attention and—together with their positive counterparts ${\mathop {\Box }}$ and ${\mathop {\Diamond }}$—form a square we call the Došen Square. Unfortunately, the core property of constructive logic, the Disjunction Property (DP), fails when the modalities are combined and, interpreted in birelational Kripke structures à la Došen, the Square partially collapses. We introduce the constructive logic $\mathsf {CKD}$, whose four semantically independent modalities ${\mathop {\Box }}$, ${\mathop {\Diamond }}$, ${\mathop {\boxminus }}$, prevent the Došen Square from collapsing under the effect of intuitionistic negation while preserving DP. The model theory of $\mathsf {CKD}$ involves a constructive Kripke frame interpretation of the modalities. A Hilbert deduction system and an equivalent cut-free sequent calculus are presented. Soundness, completeness and finite model property are proven, implying that $\mathsf {CKD}$ is decidable. The logics ${\mathsf {HK} {\mathop {\boxminus }}}$, ${\mathsf {HK} {\mathop {\Box }}}$, and of Došen and other known theories of intuitionistic modalities are syntactic fragments or axiomatic extensions of $\mathsf {CKD}$ .