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It is widely accepted that the logic of quantum mechanics is based on orthomodular posets. However, such a logic is not dynamic in the sense that it does not incorporate time dimension. To fill this gap, we introduce certain tense operators on such a logic in an inexact way, but still satisfying requirements asked on tense operators in the classical logic based on Boolean algebras or in various non-classical logics. Our construction of tense operators works perfectly when the orthomodular poset in question is finite. We investigate the behaviour of these tense operators, e.g. we show that some of them form a dynamic pair. Moreover, we prove that if the tense operators preserve one of the inexact connectives conjunction or implication as defined by the authors recently in another paper, then they also preserve the other one. Finally, we show how to construct the binary relation of time preference on a given time set provided the tense operators are given, up to equivalence induced by natural quasi-orders. [ABSTRACT FROM AUTHOR]

Russell's theory of descriptions depends essentially on the concept of identity, in particular to eliminate definite descriptions by contextual definitions. But quantum objects, according to a possible metaphysical interpretation I shall assume without discussion here, seem not to satisfy the standard theory of identity, and do not have individuality in a sense. Even so, physicists refer informally to both 'a quantum so-and-so' and, worst, to 'the quantum so-and-so', as if these expressions were something like descriptions. In this article, we sketch a possible theory of 'quantum (definite) descriptions' using a weaker concept of indiscernibility (or indistinguishability), and advance some consequences such a theory may have for the discussions on the logical foundations of quantum theory. [ABSTRACT FROM AUTHOR]

In this paper we discuss some questions proposed by Prof. Newton da Costa on the foundations of quasi-set theory. His main doubts concern the possibility of a reasonable semantical understanding of the theory, mainly due to the fact that identity and difference do not apply to some entities of the theory’s intended domain of discourse. According to him, the quantifiers employed in the theory, when understood in the usual way, rely on the assumption that identity applies to all entities in the domain of discourse. Inspired by his provocation, we suggest that, using some ideas presented by da Costa himself in his seminars at UFSC (the Federal University of Santa Catarina) and by one of us (DK) in some papers, these difficulties can be overcome both on a formal level and on an informal level, showing how quantification over items for which identity does not make sense can be understood without presupposing a semantics based on a ‘classical’ set theory.

Orthomodular lattices were introduced to get an algebraic description of the propositional logic of quantum mechanics. In this paper, we set up axiomatization of this logic as a Hilbert style implicational logical system L, i.e., we present a set of axioms and derivation rules formulated in the signature {→,0}. The other logical operations ⋁, ⋀, ¬ are expressed in terms of implication (which is the so-called Dishkant implication) and falsum. We further show that the system L is algebraizable in the sense of Blok and Pigozzi, and that orthomodular lattices provide an equivalent algebraic semantics for it. [ABSTRACT FROM AUTHOR]

Quasi-set theory is a theory for dealing with collections of indistinguishable objects. In this paper we discuss some logical and philosophical questions involved with such a theory. The analysis of these questions enable us to provide the first grounds of a possible new view of physical reality, founded on an ontology of non-individuals, to which quasi-set theory may constitute the logical basis. [ABSTRACT FROM PUBLISHER]