Logics of involutive Stone algebras
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An involutive Stone algebra (IS-algebra) is a structure that is simultaneously a De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributive lattice satisfying the well-known Stone identity xv x=1). IS-algebras have been studied algebraically and topologically since the 1980's, but a corresponding logic (here denoted IS≤) has been introduced only very recently. The logic IS≤ is the departing point for the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that IS≤ is a conservative expansion of the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic so as to obtain an extension of IS≤. We show that every logic thus defined can be axiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS≤ that cannot be obtained in the above-described way, but can nevertheless be axiomatized finitely by other methods. Most of our axiomatization results are obtained in two steps: through a multiple-conclusion calculus first, which we then reduce to a traditional one. The multiple-conclusion axiomatizations introduced in this process, being analytic, are of independent interest from a proof-theoretic standpoint. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of IS≤. Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS≤ are already uncountably many.
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