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Chapter 2 Semantics for non-standard logics

DOI: 10.1016/s0049-237x(97)80004-3
  • Linguistics
  • Logic
  • Mathematics


Publisher Summary This chapter provides a detailed description of the semantic theory for nonstandard logics. The algebraic semantics for superintuitionistic logics allows the use of algebraic semantics more effectively. It presents algebraic semantics for modal and temporal logics based on pseudo–boolean, modal, and temporal algebras. Every algebraic logic λ has a characterizing variety “Var” (λ) of the corresponding algebras. The relational second–order Kripke semantics for modal and temporal logics is also discussed. The development of Stone's theory for presentation of algebras using Kripke–style models is to develop semantic instruments for non–standard logical systems. A description of the duality between subalgebras, homomorphic images, and direct products of algebras and the corresponding operations over Kripke models are also presented. Various results concerning the finite model property––the property of approximation of logics by finite algebras and models are developed. Using algebraic semantic techniques, a fundamental connection between modal and superintuitionistic logics is determined. Some advanced tools for the finite modal property arealso considered. Examples of Kripke incomplete logics with simple axiomatic systems and theorems concerning the Kripke completeness of modal logics are also discussed.

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