Logics of involutive Stone algebras

An involutive Stone algebra (IS-algebra) is simultaneously a De Morgan and (i.e., pseudo-complemented distributive lattice satisfying the identity $$\mathop {\sim }x \vee \mathop }\mathop \approx 1$$ ). IS-algebras have been studied algebraically topologically since 1980s, but corresponding logic (here denoted $$\mathcal {IS}_{\le }$$ ) has introduced only very recently. This departing point of present study, which we then extend to wide family previously unknown logics defined from IS-algebras. We show that conservative expansion Belnap-Dunn four-valued order-preserving variety algebras), give finite Hilbert-style axiomatization for it. More generally, introduce method expanding conservatively every super-Belnap strengthening logic) so as obtain an extension . thus can be axiomatized by adding fixed set multiple-conclusion rule schemata base logic. Our results entail (which known uncountable) embeds into extensions In fact, in case, establish finitary are already uncountably many. When possesses disjunction, reduce calculus traditional one; some axiomatizations analytic independent interest proof-theoretic standpoint. also consider few cannot obtained above-described way, nevertheless finitely other methods.