Quantum logical calculi and lattice structures

Journal of Philosophical Logic 7 (1):347 - 386 (1978)
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Abstract

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus Teff, 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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Citations of this work

Matrix-Based Logic for Application in Physics.Paul Weingartner - 2009 - Review of Symbolic Logic 2 (1):132-163.
How Does Quantum Logic Correspond to Physical Reality?Ernst-Walther Stachow - 1977 - Journal of Philosophical Logic 6 (1):485 - 496.

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References found in this work

The Principle of Excluded Middle in Quantum Logic.P. Mittelstaedt & E. -W. Stachow - 1978 - Journal of Philosophical Logic 7 (1):181 - 208.

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