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Algebraic semantics for a modal logic close to S1

机译：接近S1的模态逻辑的代数语义

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摘要

The modal systems S1-S3 were introduced by C. I. Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1. We extend S1 by a Substitution Principle (SP) which generalizes a reference rule of S1. In system S1 + SP, the relation of strict equivalence I center dot satisfies the identity axioms of R. Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity). This enables us to develop a framework of algebraic semantics which captures S1 + SP as well as the Lewis systems S3-S5. So from the viewpoint of algebraic semantics, S1 + SP turns out to be an interesting modal logic. We show that S1 + SP is strictly contained between S1 and S3 and differs from S2. It is the weakest modal logic containing S1 such that strict equivalence is axiomatized by propositional identity.
机译：模态系统S1-S3由C.I. Lewis引入，作为严格含义的逻辑。虽然S2和S3有Kripke语义，但S1没有已知的自然语义。我们用替代原理（SP）扩展了S1，该原理概括了S1的参考规则。在系统S1 + SP中，严格等价关系I中心点满足R. Suszko的非弗拉芒逻辑的身份公理，该公理适应于模态逻辑语言（我们称这些公理为命题身份公理）。这使我们能够开发代数语义的框架，该框架可捕获S1 + SP以及Lewis系统S3-S5。因此，从代数语义的角度来看，S1 + SP成为一种有趣的模态逻辑。我们显示S1 + SP严格包含在S1和S3之间，并且与S2不同。它是包含S1的最弱模态逻辑，因此通过命题同一性公理化了严格的等价关系。

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