is closed with respect to Łukasiewicz disjunction of pairwise exclusive proposi-, and satisfying conditions (a) – (d) is a quantum logic in Birkhoff - von Neumann, ] doubted whether this idea can be extended to sets of propositions about properties, , when translated from the language of fuzzy, This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter, http://creativecommons.org/licenses/by/4.0/. where quantum theories and models are used. simply two different events and statements that describe them are two different statements. Editor's statement Foreword Preface Part I. Hilbert-Space Quantum Mechanics 1. and union of sets is related to disjunction in the same way in both cases: Of course in the case of fuzzy sets and many-valued logics the specific forms of, these expressions depend on the adopted model of complementation/negation, intersec-, tion/conjunction, and union/disjunction. nski, M.J.: The orthogonality postulate in axiomatic quantum mechanics. If X is an m-dimensional submanifold of M and p m ℓ is a regular point of X m ℓ , then the image of the above morphism is the tangent space to X m ℓ-1 at p m ℓ-1 ; in this sense, p m ℓ is a frame for X m ℓ-1 at p m ℓ-1 . Nevertheless, since after experi-, mental checking any such dichotomic proposition occurs to be either true or false, quantum, logic in the BvN sense is generally treated as 2-valued logic, non-classical because of lack, Among other non-classical logics studied at that time, in general without references to, quantum physics, were various kinds of many-valued logics. Quantum-mechanical features in terms of the logic of the physical system 15. Fortunately in the case of statements, concerning results of future experiments on quantum objects the answer is unique: There, is exactly one rather specific model of infinite-valued logic endowed with globally defined, negation and partially defined conjunction and disjunction such that any BvN quantum logic, in the traditional order-theoretic sense can be isomorphically represented as such logic. ], this line of investigations ceased in the late 1950s. Institute of Mathematics, University of Gda, , allows to represent it further as a family of infinite-valued propositional, with globally defined negation and partially defined conjunction. [, There are no doubts that in Boolean algebras, which are proper models of sets of dichotomic, (‘yes-no’) propositions about properties of classical objects, conjunctions and disjunctions, of these propositions are modeled by order-theoretic operations of meet and join. The Logic Of Quantum Mechanics 1936 Item Preview remove-circle ... John von Neumann, meccanica quantistica, fisica quantistica, Quantum Mechanics, principio di non contraddizione, Law of non contradiction, non contradiction principle Collection opensource Language English. States 3. In: Hooker C.A. Let us note that this condition in the realm. logic and meaningless according to the widely accepted version of quantum mechanics. New Journal of Physics 13, 043016. In other words, quantum probabilities are ‘ontic’ not ‘epistemic’, i.e., they. Quantum Mechanics”, Revi ew of M oder n Phy sic s 20: 367–87. Although there were se, modest attempts at applying many-valued (mostly 3-valued) logic in description of quantum, Birkhoff and von Neumann’s proposal. J. Theor. The Logic of Quantum Mechanics. Massimo Morigi Superpositions of states and closure spaces 18. A related paper by von Neumann and Birkhoff, “The Logic of Quantum Mechanics”, was also published in 1936, and it is reprinted in von Neumann (1961–1963, Vol. scale into their, or any other, institutional repository. This aim was achieved in [, expense of introducing the fourth condition stating that the empty set is the only set in the, studied family that is wekly disjoint with itself. This information is retrieved by decoding the value of the corresponding classical observable. Already in 1913 in his valuable but almost for. Compound systems 8. Indeed, if Łukasiewicz intersection (, defined globally, then all elements would be, algebra. for the class of chordal claw-free graphs. partially ordered set, so it is still equipped with order-theoretic operations of meet and join. The Many-Valued Logic of Quantum Mechanics. When we agree that statements about future non-certain events should be analyzed with, the use of many-valued logic, we are still facing a problem which of the plethora of var, ious models of many-valued logics should be used. (1936). The University of Western Ontario Series in Philosophy of Science … Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls, Die logischen Grundlagen der Wahrschein-lichkeitsrechnung, Fuzzy quantum logics and infinite-valued Lukasiewicz logic, Towards many-valued/fuzzy interpretation of quantum mechanics, Abacus Logic: The Lattice of Quantum Propositions as the Poset of a Theory, Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs, The contact system on the spaces of (m,ℓ)-velocities. The philosophical debate about quantum logic between the late 1960s and the early 1980s was generated mainly by Putnam’s claims that quantum mechanics empirically motivates introducing a new form of logic, that such an empirically founded quantum logic is the ‘true’ logic, and that adopting quantum logic would resolve all the para- Transition-probability spaces and quantum systems 19. Set theory is then developed, not by taking set as a primitive concept but by assuming each set A is determined by a property P characteristic of its members: A = {x:P(x)}. form a particular model of infinitely-valued {\L}ukasiewicz logic. application to matters such as quantum information protocols. is closed with respect to the countable Łukasiewicz unions of pairwise weakly, is characterized by fuzzy set operations, in particular, ), by the second part of this theorem it is still a, be a family of fuzzy subsets of an arbitrary universe, is a lattice in which meet and join for any pair, ] that by transposition of an implication: ‘if, only partially. Contemporary physics 51, 59–83.arXiv:0908.1787BC & Ross Duncan (2011) Interacting quantum observables. Description of properties of quantum objects before they By accessing, sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of, use (“Terms”). Mathematical structures emerging from the Hilbert-Space formulation of Quantum mechanics Part II. Unable to display preview. ResearchGate has not been able to resolve any citations for this publication. Now let us note that because of associativity of, . The Logic of Quantum Mechanics - take II Bob Coecke – Oxford Univ. When this is expressed formally the result can be read in two ways according to whether the underlying logic is classical logic or Ł∞ (with the above interpretation). Year: 2002. For a good introduction to this area see [1, 3, 21, 22]. Dynamics 24. complementation is related to logical negation, intersection of sets is related to conjunction.