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$\leq$) has been introduced only very recently. The logic IS$\leq$ 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$\leq$ 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$\leq$. 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$\leq$ 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$\leq$. Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS$\leq$ are already uncountably many.
翻译:从1980年代开始,就从数学角度和数学角度研究了 IS-数学语言,但最近才引入了相应的逻辑(此处注解 IS$\leq$) 。 逻辑 IS\ leq$ 是当前研究的出发点, 然后我们再扩展至一个由 IS- ALGBRA 定义的未知的逻辑体系。 我们显示, IS\ leq$ 是Belnap- Duncion 四价逻辑的保守扩张( i. e. morgan algebras 多样性的秩序- 保留逻辑), 而我们给出了一个相对的逻辑( 这里注解 IS- leq leq$ ) 。 一般来说, 我们引入了一种方法, 将每个超位元的逻辑保守化扩展为由 IS- ALGER 定义的未知的逻辑体系, 也显示, 以 IS- discoental 规则为固定的逻辑的扩展 。