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26.07.2020 | Foundations | Ausgabe 19/2020 Open Access

Soft Computing 19/2020

The logic induced by effect algebras

Soft Computing > Ausgabe 19/2020
Ivan Chajda, Radomír Halaš, Helmut Länger
Wichtige Hinweise
Communicated by A. Di Nola.
Support of the research by ÖAD, Project CZ 02/2019, entitled “Function algebras and ordered structures related to logic and data fusion,” support of the research of the first and second author by IGA, Project PřF 2020 014, and support of the research of the first and third author by the Austrian Science Fund (FWF), Project I 4579-N, and the Czech Science Foundation (GAČR), Project 20-09869L, entitled “The many facets of orthomodularity,” are gratefully acknowledged.

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Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras \({\mathbf {E}}\), we investigate a natural implication and prove that the implication reduct of \({\mathbf {E}}\) is term equivalent to \({\mathbf {E}}\). Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.
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