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Modal Companion
In logic, a modal companion of a superintuitionistic (intermediate) logic ''L'' is a normal modal logic that interprets ''L'' by a certain canonical translation, described below. Modal companions share various properties of the original intermediate logic, which enables to study intermediate logics using tools developed for modal logic. Gödel–McKinsey–Tarski translation Let ''A'' be a propositional intuitionistic formula. A modal formula ''T''(''A'') is defined by induction on the complexity of ''A'': :T(p)=\Box p, for any propositional variable p, :T(\bot)=\bot, :T(A\land B)=T(A)\land T(B), :T(A\lor B)=T(A)\lor T(B), :T(A\to B)=\Box(T(A)\to T(B)). As negation is in intuitionistic logic defined by A\to\bot, we also have :T(\neg A)=\Box\neg T(A). ''T'' is called the Gödel translation or Gödel–McKinsey– Tarski translation. The translation is sometimes presented in slightly different ways: for example, one may insert \Box before every subformula. All such variants are prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willem Blok
Willem Johannes "Wim" Blok (1947–2003) was a Dutch logician who made major contributions to algebraic logic, universal algebra, and modal logic. His important achievements over the course of his career include "a brilliant demonstration of the fact that various techniques and results that originated in universal algebra can be used to prove significant and deep theorems in modal logic." Blok began his career in 1973 as an algebraist investigating the varieties of interior algebras at the University of Illinois at Chicago. Following the 1976 completion of his Ph.D. on that topic, he continued on to study more general varieties of modal algebras. As an algebraist, Blok "was recognised by the modal logic community as one of the most influential modal logicians" by the end of the 1970s. He published many papers in the ''Reports on Mathematical Logic'', served as a member on their editorial board, and was one of their guest editors. Along with Don Pigozzi, Wim Blok co-authored the mono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |