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On Contextuality and Unsharp Quantum Logic
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Abstract
In this paper we provide a preliminary investigation of subclasses of bounded posets with antitone involution which are ``pastings'' of their maximal Kleene sub-lattices. Specifically, we introduce super-paraorthomodular lattices, namely paraothomodular lattices whose order determines, and it is fully determined by, the order of their maximal Kleene sub-algebras. It will turn out that the (spectral) paraorthomodular lattice of effects over a separable Hilbert space can be considered as a prominent example of such. Therefore, it provides an algebraic/order theoretical rendering of complementarity phenomena between \emph{unsharp} observables. A number of examples, properties and characterization theorems for structures we deal with will be outlined. For example, we prove a forbidden configuration theorem and we investigate the notion of commutativity for modular pseudo-Kleene lattices, examples of which are (spectral) paraorthomodular lattices of effects over \emph{finite-dimensional} Hilbert spaces. Finally, we show that structures introduced in this paper yield paraconsistent partial referential matrices, the latter being generalizations of J. Czelakowski's partial referential matrices. As a consequence, a link between some classes of posets with antitone involution and algebras of partial ``unsharp'' propositions is established.
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The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.
In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57–62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics.
Paraorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever is distributive.
We show that every orthomodular lattice can be considered as a left residuated l-groupoid satisfying divisibility, antitony, the double negation law and three more additional conditions expressed in the language of residuated structures. Also conversely, every left residuated l-groupoid satisfying the mentioned conditions can be organized into an orthomodular lattice.
We correct a mistake in the description of orthoalgebras as pastings of Boolean algebras. We present a corrected structural theorem.
Abstract algebraic logic is the more general and abstract side of algebraic logic, the branch of mathematics that studies the connections between logics and their algebra-based semantics. This emerging subfield of mathematical logic is increasingly becoming an indispensable tool to approach the algebraic study of any (mainly sentential) logic in a systematic way.
This book takes a bottom-up approach and guides readers, by means of successive steps of generalization and abstraction, to meet more and more complicated algebra-based semantics. An entire chapter is devoted to Blok and Pigozzi's theory of algebraizable logics, proving the main theorems and incorporating later developments by other scholars. After a chapter on the basics of the classical theory of matrices, one chapter is devoted to an in-depth exposition of the semantics of generalized matrices. Two more avanced chapters provide introductions to the two hierachies that organize the logical landscape according to the criteria of abstract algebraic logic, the Leibniz hierarchy and the Frege hierarchy. Dozens of examples of particular logics are presented and classified.
The book is addressed to mathematicians and logicians with little or no previous exposure to algebraic logic. Some acquaintance with non-classical logics is desirable. The book is written with students (or beginners in the field) in mind, and combines a textbook style in its main sections, including more than 400 carefully graded exercises, with a survey style in the discussion of newer research directions. The book includes some historical notes and numerous bibliographic references.
In this survey article we try to give an up-to-date account of certain aspects of the theory of ortholattices (abbreviated OLs), orthomodular lattices (abbreviated OMLs) and modular ortholattices (abbreviated MOLs), not hiding our own research interests. Since most of the questions we deal with have their origin in Universal Algebra, we start with a section discussing the basic concepts and results of Universal Algebra without proofs. In the next three sections we discuss, mostly with proofs, the basic results and standard techniques of the theory of OMLs. In the remaining five sections we work our way to the border of present day research, with no or only sketchy proofs. Section 5 deals with products and subdirect products, section 6 with free structures and section 7 with classes of OLs defined by equations. In section 8 we discuss embeddings of OLs into complete ones. The last section deals with questions originating in Category Theory, mainly amalgamation, epimorphisms and monomorphisms. The later sections of this paper contain an abundance of open problems. We hope that this will initiate further research.
The so-called orthogonal posets form an important tool for some investigations in the logic of quantum mechanics because they can be recognized as so-called quantum structures. The motivation for studying quantum structures is included e.g. in the monograph by Dvurečenskij and Pulmannová or in the papers by Beltrametti and Maczyński. It is shown that every space of numerical events [see Chajda and Länger (Intern J Theor Phys 50:2403, 2011b), Dorninger and Länger (Intern J Theor Phys 52:1141–1147, 2013) and references therein] forms an orthogonal poset. Hence, orthogonal posets should be axiomatized by standard algebraic machinery. However, considering supremum as a binary operation, they form only partial algebras. The aim of the paper is to involve a different way which enables us to describe orthogonal posets as total algebras and get an algebraic axiomatization as an equational theory.
Using ideas from fuzzy set theory, we generalize the notion of a quantum (probability) space introduced by Suppes to so-called fuzzy quantum spaces and we study the compatibility problem.
We study orthomodular structures formed by some sets of states of finite transition systems. These sets, called regions, can be interpreted as local states of a distributed, concurrent system, that can be modelled by a Petri net. The main result shows that such orthomodular structures have enough elements to represent meets of certain subsets of elements.
The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it an adequate concept of joint properties. A sharp or unsharp property manifests itself as an element of sharp or unsharp reality by its tendency to become actual or to actualize a specific measurement outcome. This actualization tendency - or potentiality - of a property is quantified by the associated quantum probability. The resulting single-case interpretation of probability as a degree of reality will be explained in detail and its role in addressing the tensions between quantum and classical accounts of the physical world will be elucidated. It will be shown that potentiality can be viewed as a causal agency that evolves in a well-defined way.
Paraorthomodular posets are bounded partially ordered sets with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from algebraic and order-theoretic perspectives. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind–MacNeille completion is paraorthomodular are provided.
The Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.
The bounded resolutions of the identity in a von Neumann algebra can be ordered by {Es(u)}≼{Et(u) ≧ET(u) u∈R. The selfadjoint operators in the algebra are partially ordered by this relation and are shown to form a conditionally complete lattice. The lattice operations are (essentially) defined by Eysa(u) =AEsa(u) for all u contained in R. This order is called spectral order and agrees with the usual order on commutative subalgebras. For positive operators, 5 is greater than or equal to T in spectral order if and only if Sn is greater than or equal to T" in the usual order for all n≧l. Kadison's well-known counterexample is shown to fail. The operator lattice defined by spectral order differs from a vector lattice in the fact that S≽T does not imply that S+C≽zT+C.
This paper discusses three types of fuzzy quantum posets, states and observables on these posets, representations of fuzzy quantum posets, representation of observables, and joint observables.
To those like myself who are mainly concerned with the methodology of the empirical sciences, the present symposium is both sobering and encouraging. It is sobering as one thinks of the scientific contrast between the majority of papers read here and the standard sources in the methodology and philosophy of science. Yet it is encouraging, because the hope is engendered that many of the methods, and perhaps above all, the intellectual standards of these papers, will extend themselves in a natural way to logical investigations of the empirical sciences. The logical and philosophical foundations of physics, for example, seem to be at about the stage where the foundations of mathematics were during most of the nineteenth century. Nearly any physicist and a large number of philosophers are prepared to deliver at a moment’s notice a lecture on the foundations of quantum mechanics. The situation is far different with respect to the foundations of mathematics. With an ever-increasing volume of deep and rigorous results, mathematicians unacquainted with the literature are not prone to deliver casually-put-together obiter dicta on logic and related topics.
For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by “≤”, and which, as a poset, is orthocomplemented by a unary operation of negation. The negation of the proposition x is denoted by NX and the least element in the logic is denoted by 0, we write NO = 1.A binary operation “→” is introduced into a logic, the operation is interpreted as material implication so that “x → y” is a proposition of the logic and is read as “x materially implies y”. If material implication has the properties11. (x → 0) = NX,
12. if x ≤ y then (z → x) ≤ (z → y),
13. if x ≤ y then x ≤ (y ≤ z)= x → z,
14. x ≤ {y → N(y → Nx)},
then the logic is an orthomodular lattice. The lattice operations of join and meet are given by
x y = Nx → N(Nx → Ny)
x y = N(X → N(x → y))
and, in terms of the lattice operations, the material implication is given by
(x → y) = (y x) NX.Moreover the logic is a Boolean algebra if, and only if, in addition to the properties above, material implication satifies
15. (x → y) = (Ny → Nx).
This little book is the outgrowth of a one semester course which I have taught for each of the past four years at M. 1. T. Although this class used to be one of the standard courses taken by essentially every first year gradu ate student of mathematics, in recent years (at least in those when I was the instructor), the clientele has shifted from first year graduate students of mathematics to more advanced graduate students in other disciplines. In fact, the majority of my students have been from departments of engi neering (especially electrical engineering) and most of the rest have been economists. Whether this state of affairs is a reflection on my teaching, the increased importance of mathematical analysis in other disciplines, the superior undergraduate preparation of students coming to M. 1. T in mathematics, or simply the lack of enthusiasm that these students have for analysis, I have preferred not to examine too closely. On the other hand, the situation did force me to do a certain amount of thinking about what constitutes an appropriate course for a group of non-mathematicians who are courageous (foolish?) enough to sign up for an introduction to in tegration theory offered by the department of mathematics. In particular, I had to figure out what to do about that vast body of material which, in standard mathematics offerings, is "assumed to have been covered in your advanced calculus course".
We present compatibility of type I, II, III fuzzy quantum posets and representations of observables on them. Our results extend results given by A. Dvurečenskij and F. Chovanec for type II in [Int. J. Theor. Phys.27 (1988) 1069–1082], by the author and A. Dvurečenskij for type I in [Acta Math. Univ. Comenianae58–59 (1991) 247–258] and by the author for type I, II in [Appl. of Math.37 (1992) 357–368]. The existence of the joint distribution and the joint observable of a sequence of observables is pointed out, too.
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.
Our first result is that the interpolation property is related to an old construction due to J. Kalman [11], which associates a Kleene algebra K(L) with each bounded distributive lattice L. More precisely, we show that K is a functor from the category of bounded distributive lattices ~ into the category Y{ of Kleene algebras, and that K has a left adjoint k : Y{--+ 5r This adjunction is an equivalence when k is restricted to the full subcategory of Y{ formed by the centered Kleene algebras which satisfy the interpolation property. It was proved independently by D. Vakarelov [25] and M. Fidel [10] that the elements of a Nelson algebra can be represented by pairs of elements of a Heyting algebra. We show that their construction is essentially obtained by restricting Kalman's functor
We show that everyD-lattice (lattice-ordered effect algebra)P is a set-theoreticunion of maximal subsets of mutually compatible elements, called blocks.Moreover, blocks are sub-D-lattices and sub-effect-algebras ofP which areMV-algebras closed with respect to all suprema and infima existing inP.
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.
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).
Unsharp Reality and the Question of Quantum Systems
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