Modal-type orthomodular logic

In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripke-style semantic founded on Baer *-semigroups as in [20].Comment: submitted to the Mathematical Logic Quarterl

A Topological Study of Contextuality and Modality in Quantum Mechanics

Kochen-Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer consistently to actual and possible properties of the system. By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen-Specker theorem over this new structure. This allows us to prove that contextuality remains a central feature even in the enriched propositional system.Comment: 10 pages, no figures, submitted to I. J. Th. Phy

Scopes and Limits of Modality in Quantum Mechanics

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.Comment: 9 pages, no figure

Semilattices global valuations in the topos approach to quantum mechanics

In the framework of the topos approach to quantum mechanics a kind of global valuation is introduced and studied. It allows us to represent certain features related to the logical consequences of properties about quantum systems when its phase space is endowed with an intuitionistic structureFil: Freytes Solari, Hector Carlos. Università di Cagliari; Italia. Universidad Nacional de Rosario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: de Ronde, Christian. Universidad de Buenos Aires. Facultad de Filosofía y Letras. Instituto de Filosofía "Dr. Alejandro Korn"; Argentina. Center Leo Apostel; Bélgica. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Domenech, Graciela. Center Leo Apostel; Bélgica. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Many worlds and modality in the interpretation of quantum mechanics: an algebraic approach

Many worlds interpretations (MWI) of quantum mechanics avoid the measurement problem by considering every term in the quantum superposition as actual. A seemingly opposed solution is proposed by modal interpretations (MI) which state that quantum mechanics does not provide an account of what `actually is the case', but rather deals with what `might be the case', i.e. with possibilities. In this paper we provide an algebraic framework which allows us to analyze in depth the modal aspects of MWI. Within our general formal scheme we also provide a formal comparison between MWI and MI, in particular, we provide a formal understanding of why --even though both interpretations share the same formal structure-- MI fall pray of Kochen-Specker (KS) type contradictions while MWI escape them.Comment: submitted to the Journal of Mathematical Physic

The Contextual Character of Modal Interpretations of Quantum Mechanics

In this article we discuss the contextual character of quantum mechanics in the framework of modal interpretations. We investigate its historical origin and relate contemporary modal interpretations to those proposed by M. Born and W. Heisenberg. We present then a general characterization of what we consider to be a modal interpretation. Following previous papers in which we have introduced modalities in the Kochen-Specker theorem, we investigate the consequences of these theorems in relation to the modal interpretations of quantum mechanics.Comment: 21 pages, no figures, preprint submitted to SHPM

Canonical extensions and ultraproducts of polarities

J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames

An Intrisic Topology for Orthomodular Lattices

We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices but also a way to distinguish classical and quantum logics.Comment: Under submission to the International Journal of Theoretical Physic