Fuzzy quantum logics and infinite-valued Łukasiewicz logic

A new, physically more plausible definition of a fuzzy quantum logic is proposed. It is shown that this definition coincides with the previously studied definition of a fuzzy quantum logic; therefore it defines objects which are traditional quantum logics with ordering sets of states. The new definition is expressed exclusively in terms of fuzzy set operations which are generated by connectives of multiple-valued logic studied by ukasiewicz at the beginning of the 20th century. Therefore, the logic of quantum mechanics is recognized as a version of infinite-valued ukasiewicz logic.     

Typed Quantum Logic

The aim of this paper was to lift traditional quantum logic to its higher order version with the help of a type-theoretic method. A higher order axiomatic system is defined explicitly and then a sound and complete class of models is given. This is an attempt to provide a quantum counterpart of classical set theory or intuitionistic topos.     

Quantum logic of sequential events and their objectivistic probabilities

A propositional calculus for quantum mechanical systems is presented which formalizes sequential connectives and then and or then for yes-no experiments in the framework of complex Hilbert space. Properties of these connectives are compared with some well-known lattice-theoretical results in quantum logic. Probabilities and objectivization versus the Copenhagen interpretation are discussed in connection with Young's two-slit experiment.     

Three-valued derived logics for classical phase spaces

Reichenbach proposed a three-valued logic to describe quantum mechanics. In his development, Reichenbach presented three different negation operators without providing any criteria for choosing among them. In this paper we develop two three-valuedderived logics for classical systems. These logics are derived in that they are based on a theory of physical measurement. In this regard they have some of the characteristics of the quantum logic developed by Birkhoff and von Neumann. The theory of measurement used in the present development is the one used previously in developingbivalent derived logics for classical systems. As these systems are derived logics, many of the ambiguities possessed by systems such as Reichenbach's are avoided.     

Weak density of states

Let L be a quantum logic, here an orthoalgebra, and let be a convex set of states on L. Then generates a base-normed space, and the dual-order unit-normed space contains a canonically constructed homomorphic copy of L, denoted by e(L). A convex set of states on L is said to be ample provided that every state on L is obtained by restricting an element of the base of the bi-dual order unit-normed space to e(L). For a quantum logic L we show that a convex set of states is ample if and only if is weakly dense in the convex set of all states on L. The notion of ampleness is then discussed in the context of Gleason-type theorems for W^* algebras and JBW algebras and also in the context of classical logics.Dedicated to Prof. Peter Mittelstaedt on the occasion of his sixtieth birthday. Research supported by the Swiss National Science Foundation.     

Propositions and orthocomplementation in quantum logic

We briefly analyze two partial order relations that are usually introduced in quantum logic by making use of the concepts of yes-no experiment and of preparation as fundamental. We show that two distinct posetsE andL can be defined, the latter being identifiable with the lattice of quantum logic. We consider the posetE and find that it contains a subsetE ₀ which can easily be orthocomplemented. These results are used, together with suitable assumptions, in order to show that an Orthocomplementation inL can be deduced by the Orthocomplementation defined inE ₀, and also to give a rule to find the orthocomplement of any element ofL.Research sponsored by C.N.R. (Italy).     

Modalities and quantum mechanics

Three approaches concerning the usage of modalities in the language of quantum mechanics were considered; Mittelstaedt and I built up a dialog semantics for modalities on a metalinguistic level, and a calculus of quantum modal logic is known that is complete and sound with respect to this dialogic semantics. Van Fraassen replaced the usual interpretation of quantum mechanics (with the projection postulate) by his modal interpretation based on a modal object language. Dalla Chiara translated a nonmodal object language for quantum mechanics and the appropriate quantum logic into a modal language. Specifically we are interested in the similarities and the differences of these three approaches.     

Correspondence principle for the quantum net

Finkelstein's suggestion for a flexible logic is taken up in the context of his causal net theory. We interpret on the net certain concepts that are first expressed in terms of the canonical flexible logic of the macroscopic world, namely, the logic of sheaves over the manifold model, here taken to be flat. From this we infer a correspondence principle in the form of a simple (model-dependent) semantics which translates certain concepts between the purely quantum world of the net and the familiar classical-quantum hybridized world of the macroscopic model. As an application, we derive and solve the reticular version of the massless Dirac equation by analyzing the Dirac operator on the net, where its behavior is easily apprehended.     

Quantum logics and hilbert space

Starting with a quantum logic (a -orthomodular poset) L. a set of probabilistically motivated axioms is suggested to identify L with a standard quantum logic L(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space.     

Computational Logic on Fock Space

Fock space may provide an important mathematical model for quantum computation. For this reason, it may be useful to generalize previous work on computational logic to the Fock space framework. The basic construction of this computational logic is the set D(H) of density operators on a Fock space H. We first define n-sector p _n() and total probabilities p() of elements D(H). We next discuss NOT, AND, and OR operations on D(H). Natural equivalence classes and Scotian elements are described. We also discuss minimal and maximal elements and quantum numbers for the equivalence classes. We finally treat the operation and the stronger equivalence classes associated with this operation.     

Fuzzy Quantum Logics as a Basis for Quantum Probability Theory

Representation of an abstract quantum logic withan ordering set of states S in the form of a family L(S) of fuzzy subsets of S which fulfils conditionsanalogous to Kolmogorovian conditions imposed on -algebra of random events allows us toconstruct quantum probability calculus in a waycompletely parallel to the classical Kolmogorovianprobability calculus. It is shown that the quantumprobability calculus so constructed is a propergeneralization of the classical Kolmogorovian one. Someindications for building a phase-space representation ofquantum mechanics free of the problem of negativeprobabilities are given.     

Quantum Logic as a Basis for Computations

It is shown that computations can be founded on the laws of the genuine(Birkhoff—nvon Neumann) quantum logic treated as a particular version ofukasiewicz infinite-valued logic. A new way of encoding nonexact data whichencodes both the value of a number and its fuzziness is introduced. A simpleexample of a full adder that works in the proposed way is given and it is comparedwith other designs of quantum adders existing in the literature. A controversybetween the meaning of the very term quantum logic as used recently withinthe theory of quantum computations and the traditional meaning of this term isbriefly discussed.     

A Model with Quantum Logic, but Non-Quantum Probability: The Product Test Issue

We introduce a model with a set of experiments of which the probabilities of the outcomes coincide with the quantum probabilities for the spin measurements of a quantum spin- particle. Product tests are defined which allow simultaneous measurements of incompatible observables, which leads to a discussion of the validity of the meet of two propositions as the algebraic model for conjunction in quantum logic. Although the entity possesses the same structure for the logic of its experimental propositions as a genuine spin- quantum entity, the probability measure corresponding with the meet of propositions using the Hilbert space representation and quantum rules does not render the probability of the conjunction of the two propositions. Accordingly, some fundamental concepts of quantum logic, Piron-products, classical systems and the general problem of hidden variable theories for quantum theory are discussed.     

Baser*-semigroups and the logic of quantum mechanics

The theory of orthomodular ortholattices provides mathematical constructs utilized in the quantum logic approach to the mathematical foundations of quantum physics. There exists a remarkable connection between the mathematical theories of orthomodular ortholattices and Baer*-semigroups; therefore, the question arises whether there exists a phenomenologically interpretable role for Baer *-semigroups in the context of the quantum logic approach. Arguments, involving the quantum theory of measurements, yield the result that the theory of Baer *-semigroups provides the mathematical constructs for the discussion of operations and conditional probabilities.Supported in part by the United States Atomic Energy Commission.     

Interpreting Quantum Logic as a Pragmatic Structure

Many scholars maintain that the language of quantum mechanics introduces a quantum notion of truth which is formalized by (standard, sharp) quantum logic and is incompatible with the classical (Tarskian) notion of truth. We show that quantum logic can be identified (up to an equivalence relation) with a fragment of a pragmatic language \(\mathcal {L}_{G}^{P}\) of assertive formulas, that are justified or unjustified rather than trueor false. Quantum logic can then be interpreted as an algebraic structure that formalizes properties of the notion of empirical justification according to quantum mechanics rather than properties of a quantum notion of truth. This conclusion agrees with a general integrationist perspective that interprets nonstandard logics as theories of metalinguistic notions different from truth, thus avoiding incompatibility with classical notions and preserving the globality of logic.     

Peirce, Clifford, and Dirac

There is a clear line of progression from the logic of relations of Charles Sanders Peirce through the algebras of William Kingdon Clifford. Further, it has been shown how one can obtain the nonrelativistic quantum theory of spin one-half particles from Peirce logic. Continuing the hypothetical history, it is demonstrated here that the relativistic Dirac theory can also be related to Peirce logic. The most natural way to accomplish this is to represent the Dirac wave functions themselves as Clifford numbers rather than as spinors. The wave functions can thus appear as 4× 4 matrices. All quantities in this quantum theory can actually be expressed in terms of the Clifford basis, independent of a specific matrix representation.     

Relative compatibility and joint distributions of observables

The notion of relative compatibility of observables is treated and its relation to the existence of joint distributions is obtained. The case of conventional quantum mechanics is studied and a generalization to the case of the quantum logic approach to quantum mechanics is given. It is shown that relative compatibility is equivalent to the existence of so-called type 1 joint distributions.