Bibliography on quantum logics and related structures

Pavičić, Mladen

doi:10.1007/bf00739999

Cited by 34 publications

(5 citation statements)

References 1,094 publications

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“…This does not mean that interest in applying non-classical logics to the foundations of quantum mechanics vanished completely. On the contrary, in the seventies and eighties more than thousand papers were published as well as nearly twenty books, and several international conferences on 'quantum logics and related structures' were held (see the extensive Pavičić bibliography [17] published in 1992 that contains more than 1500 entries). Although no systematic bibliographical research was performed later, it seems that later on these numbers could have even doubled.…”

Section: Many-valued Logics In Foundations Of Quantum Mechanicsmentioning

confidence: 99%

Unification of Two Approaches to Quantum Logic: Every Birkhoff – von Neumann Quantum Logic is a Partial Infinite-Valued Łukasiewicz Logic

Pykacz

2010

Stud Logica

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In the paper it is shown that every physically sound Birkhoff -von Neumann quantum logic, i.e., an orthomodular partially ordered set with an ordering set of probability measures can be treated as partial infinite-valued Lukasiewicz logic, which unifies two competing approaches: the many-valued, and the two-valued but non-distributive, which have co-existed in the quantum logic theory since its very beginning.

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Section: Many-valued Logics In Foundations Of Quantum Mechanicsmentioning

confidence: 99%

Unification of Two Approaches to Quantum Logic: Every Birkhoff – von Neumann Quantum Logic is a Partial Infinite-Valued Łukasiewicz Logic

Pykacz

2010

Stud Logica

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“…As we could see above, the implications do not play any decisive role in the definition of lattices, especially not in the definitions of OML and DL where they do not appear at all, and they also do not play a decisive role in the definition of logics. A few decades ago that was a major issue, though: "A 'logic' without an implication ... is radically incomplete, and hardly qualifies as a theory of deduction" [42] and a hunt to find a "proper implication" among the five possible ones was pursued in 1970ies and 1980ies [43][44][45]. Apart from → 1 and → 3 it turns out [25] that one can also define a →…”

Section: Proofmentioning

confidence: 99%

Classical Logic and Quantum Logic with Multiple and Common Lattice Models

Pavičić

2016

Advances in Mathematical Physics

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We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and non-distributive ortholattices as its models. In particular, we prove that quantum as well as classical logics are complete and sound with respect to these lattices. We also show that there is one common non-orthomodular lattice that is a model of both quantum and classical logics. In technical terms, that enables us to run the same classical logic on both a digital (standard, two subset, 0-1 bit) computer and on a non-digital (say, a six subset) computer (with appropriate chips and circuits). With quantum logic, the same six element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.

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“…Quantum logic was born in the seminal 1936 paper co-authored by Garret Birkhoff and John von Neumann (1936), and after about a 25-year-long dormancy, it has developed into a rich field in the past 40 years, with literally thousands of papers dealing with different aspects of it (see the bibliography of Pavicic 1992). The revival of quantum logic in the early sixties was a peculiar one however: the notion of quantum logic that came to be accepted as the standard concept differed substantially from the original idea put forward by Birkhoff and von Neumann, but the discrepancy between the standard view and the original concept was not the subject of a systematic discussion at the time quantum logic became fashionable.…”

Section: Introductionmentioning

confidence: 99%

The Birth of quantum logic

Rédei

2007

History and Philosophy of Logic

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By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff-von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that the quantum propositional system should be isomorphic to an abstract projective geometry.Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first x: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics -without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics.

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Bibliography on quantum logics and related structures

Cited by 34 publications

References 1,094 publications

Unification of Two Approaches to Quantum Logic: Every Birkhoff – von Neumann Quantum Logic is a Partial Infinite-Valued Łukasiewicz Logic

Unification of Two Approaches to Quantum Logic: Every Birkhoff – von Neumann Quantum Logic is a Partial Infinite-Valued Łukasiewicz Logic

Classical Logic and Quantum Logic with Multiple and Common Lattice Models

The Birth of quantum logic

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