We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

Some algebraic structures determined by the class σ(þ) of all effects of a Hilbert space þ and by some subclasses of σ(þ) are investigated, in particular de Morgan-Brouwer-Zadeh posets [it is proved that σ(þ n )(n<∞) has such a structure], Brouwer-Zadeh * posets (a quite trivial example consisting of suitable effects is given), and Brouwer-Zadeh 3 posets which are both de Morgan and *.It is shown that a nontrivial class of effects of a Hilbert space exists which is a BZ (...) 3 poset. An ɛ-preclusivity relation on the set of all vectors of þ is introduced, and it is shown that it satisfies the regularity condition also for ε∃ [1/2, 1]. (shrink)

Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective not is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL 3) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.

On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a “state-event-probability” (SEVP) structure, using a quite standard “fuzzification” procedure, a set of unsharp events (or “effects”) is constructed and the corresponding “state-effect-probability” (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an (...) orthomodular complete lattice. As customary, by these orthocomplementations the two modal-like necessity and possibility operators are introduced, and it is shown that Ludwig's and Jauch-Piron's approaches to quantum physics are “interpreted” in complete SEFP. As a marginal result, a standard procedure to construct a lot of unsharp realizations starting from any sharp realization of a fixed observable is given, and the relationship among sharp and corresponding unsharp realizations is studied. (shrink)

In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems (...) for classical logic and Girard's proofnets for linear logic. (shrink)

Paraconsistent quantum logic, a hybrid of minimal quantum logic and paraconsistent four-valued logic, is introduced as Gentzen-type sequent calculi, and the cut-elimination theorems for these calculi are proved. This logic is shown to be decidable through the use of these calculi. A first-order extension of this logic is also shown to be decidable. The relationship between minimal quantum logic and paraconsistent four-valued logic is clarified, and a survey of existing Gentzen-type sequent calculi for these logics and their close relatives is (...) addressed. (shrink)

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent system for lattice logic. A completeness theorem with respect (...) to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics. (shrink)

Two new multilattice logics called submultilattice logic and indexed multilattice logic are introduced as a monosequent calculus and an indexed monosequent calculus, respectively. The submultilattice logic is regarded as a monosequent calculus version of Shramko’s original multilattice logic, which is also known as the logic of logical multilattices. The indexed multilattice logic is an extension of the submultilattice logic, and is regarded as the logic of multilattices. A completeness theorem with respect to a lattice-valued semantics is proved for the submultilattice (...) logic, and a completeness theorem with respect to a multilattice-valued semantics is proved for the indexed multilattice logic. (shrink)

Some algebraic structures of the set of all effects are investigated and summarized in the notion of a(weak) orthoalgebra. It is shown that these structures can be embedded in a natural way in lattices, via the so-calledMacNeille completion. These structures serve as a model ofparaconsistent quantum logic, orthologic, andorthomodular quantum logic.

This paper is concerned with a logical system, called Brouwer-Zadeh logic, arising from the BZ poset of all effects of a Hilbert space. In particular, we prove a representation theorem for Brouwer-Zadeh lattices, and we show that Brouwer-Zadeh logic is not characterized by the MacNeille completions of all BZ posets of effects.

We review the fact that an MV-algebra is the same thing as a lattice-ordered effect algebra in which disjoint elements are orthogonal. An HMV-algebra is an MV-effect algebra that is also a Heyting algebra and in which the Heyting center and the effect-algebra center coincide. We show that every effect algebra with the generalized comparability property is an HMV-algebra. We prove that, for an MV-effect algebra E, the following conditions are mutually equivalent: (i) E is HMV, (ii) E has a (...) center valued pseudocomplementation, (iii) E admits a central cover mapping γ such that, for all p, q∈E, p∧q=0⇒γ(p)∧q=0. (shrink)

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

In this paper we make some general remarks on the use of non-classical logics, in particular paraconsistent logic, in the foundational analysis of physical theories. As a case-study, we present a reconstruction of P.\ -D.\ F\'evrier's 'logic of complementarity' as a strict three-valued logic and also a paraconsistent version of it. At the end, we sketch our own approach to complementarity, which is based on a paraconsistent logic termed 'paraclassical logic'.