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A double deduction system for quantum logic based on natural deduction

Yannis Delmas-Rigoutsos

Journal of Philosophical Logic 26 (1):57-67 (1997)

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A Natural Deduction System for Orthomodular Logic.Andre Kornell - 2024 - Review of Symbolic Logic 17 (3):910-949.details Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate logic: (...) Download Export citation Bookmark 1 citation
The philosophy of alternative logics.Andrew Aberdein & Stephen Read - 2009 - In Leila Haaparanta (ed.), The development of modern logic. New York: Oxford University Press. pp. 613-723.details This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) Download Export citation Bookmark 13 citations
Toward a More Natural Expression of Quantum Logic with Boolean Fractions.Philip G. Calabrese - 2005 - Journal of Philosophical Logic 34 (4):363-401.details This paper uses a non-distributive system of Boolean fractions (a\|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due (...) Download Export citation Bookmark
Natural Deduction for Quantum Logic.K. Tokuo - 2022 - Logica Universalis 16 (3):469-497.details This paper presents a natural deduction system for orthomodular quantum logic. The system is shown to be provably equivalent to Nishimura’s quantum sequent calculus. Through the Curry–Howard isomorphism, quantum $$\lambda $$ -calculus is also introduced for which strong normalization property is established. Download Export citation Bookmark 1 citation

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