Logical Connectives on Lattice Effect Algebras

Studia Logica 100 (6):1291-1315 (2012)
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Abstract

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek's BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr

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Citations of this work

L-algebras and three main non-classical logics.Wolfgang Rump - 2022 - Annals of Pure and Applied Logic 173 (7):103121.
L -effect Algebras.Wolfgang Rump & Xia Zhang - 2020 - Studia Logica 108 (4):725-750.

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References found in this work

Effect algebras and unsharp quantum logics.D. J. Foulis & M. K. Bennett - 1994 - Foundations of Physics 24 (10):1331-1352.
Implication connectives in orthomodular lattices.L. Herman, E. L. Marsden & R. Piziak - 1975 - Notre Dame Journal of Formal Logic 16 (3):305-328.
Phi-symmetric effect algebras.M. K. Bennett & D. J. Foulis - 1995 - Foundations of Physics 25 (12):1699-1722.
Modal propositional logic on an orthomodular basis. I.L. Herman & R. Piziak - 1974 - Journal of Symbolic Logic 39 (3):478-488.

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