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Quantum Logical Calculi and Lattice Structures

E. -W. Stachow
Journal of Philosophical Logic
Vol. 7, No. 1 (Jan., 1978), pp. 347-386
Published by: Springer
Stable URL: http://www.jstor.org/stable/30226180
Page Count: 40
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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 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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