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Quantum logical calculi and lattice structures

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Abstract

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus T eff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the ‘value-definiteness’ of propositions is not postulated, the calculus T eff represents a calculus of effective (intuitionistic) quantum logic.

Beginning with the tableaux-calculus the equivalence of T eff to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to T eff. The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space.

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Bibliography

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  1. E. -W. Stachow

    Present address: Institut für Theoretische Physik der Universität zu Köln, Zülpicherstr. 77, 5000, Köln 41, Germany

Authors and Affiliations

  1. University of Western Ontario, London, Canada

    E. -W. Stachow

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  1. E. -W. Stachow
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Stachow, E.W. Quantum logical calculi and lattice structures. J Philos Logic 7, 347–386 (1978). https://doi.org/10.1007/BF00245934

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