Quantum Logic

The article presents the chess composition as a logical art, with concrete examples. It began with Arabic mansuba, and later evolved to new-strategy designed by Italian Alberto Mari. The redefinition of mate (e.g. mate with a free field) or a theme to quasi-pseudo theme, opens the new space for combinations, and enables to connect it with other fields like computer science. The article was exhibited in Holland Park, W8 6LU, The Ice House between 18. Oct - 3. Nov. 2013.

The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...) subsets so there is a dual concept of logical entropy which is the normalized counting measure on distinctions of partitions. Thus the logical notion of information is a measure of distinctions. Classical logical entropy naturally extends to the notion of quantum logical entropy which provides a more natural and informative alternative to the usual Von Neumann entropy in quantum information theory. The quantum logical entropy of a post-measurement density matrix has the simple interpretation as the probability that two independent measurements of the same state using the same observable will have different results. The main result of the paper is that the increase in quantum logical entropy due to a projective measurement of a pure state is the sum of the absolute squares of the off-diagonal entries ("coherences") of the pure state density matrix that are zeroed ("decohered") by the measurement, i.e., the measure of the distinctions ("decoherences") created by the measurement. (shrink)

Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...) logical finite probability theory. After outlining these dual concepts in mathematical terms, we turn to a more metaphysical speculation about two dual notions of reality, a fully definite notion using Boolean logic and appropriate for classical physics, and the other objectively indefinite notion using partition logic which turns out to be appropriate for quantum mechanics. The existence-information duality is used to intuitively illustrate these two dual notions of reality. The elucidation of the objectively indefinite notion of reality leads to the "killer application" of the existence-information duality, namely the interpretation of quantum mechanics. (shrink)

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...) to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in the classical probability (...) resolution of Hardy’s paradox [1] is supported with the present derivation of a commutator for sets. (shrink)

In recent work, we introduced a new semantics for conditionals, covering a large class of what we call preconditionals. In this paper, we undertake an axiomatic study of preconditionals and subclasses of preconditionals. We then prove that any bounded lattice equipped with a preconditional can be represented by a relational structure, suitably topologized, yielding a single relational semantics for conditional logics normally treated by different semantics, as well as generalizing beyond those semantics.

Challenges to classical logic have emerged from several sources. According to recent work, the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's (...) recent approach to vagueness. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the non-classicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic. (shrink)

Geoffrey Hellman has argued that non-truth-functionality entails a change of meaning between classical and quantum logical connectives. This paper criticizes and significantly improves the argument.

The number of independent messages a physical system can carry is limited by the number of its adjustable properties. In particular, systems that have only one adjustable property cannot carry more than a single message at a time. We demonstrate this is the case for the single photons in the double-slit experiment, and the root of the fundamental limit on measuring the complementary aspect of the photons. Next, we analyze the other ‘quantal’ behavior of the systems with a single adjustable (...) property, such as noncommutativity and no-cloning. Finally, we formulate a mathematical theory to describe the dynamics of such systems and derive the standard Hilbert-space formalism of quantum mechanics. Our derivation demonstrates the physical foundation of the quantum theory. (shrink)

The mathematical formalism of quantum theory has been established for nearly a century, yet its physical foundations remain elusive. In recent decades, connections between quantum theory and information theory have garnered increasing attention. This study presents a physical derivation of the mathematical formalism quantum theory based on information-theoretic considerations in physical systems. We postulate that quantum systems are characterized by single independent adjustable variables. Utilizing this physical postulate along with the conservation of total probability, we derive the standard Hilbert space (...) formalism of quantum theory, including the Born probability rule. This comprehensive derivation of quantum theory offers a clear and concise physical foundation for the mathematical formalism of quantum mechanics. (shrink)

This book concerns the metasemantics of quantum mechanics (QM). Roughly, it pursues an investigation at an intersection of the philosophy of physics and the philosophy of language, and it offers a critical analysis of rival explanations of the semantic facts of standard QM. Two problems for such explanations are discussed: categoricity and permanence. New results include 1) a reconstruction of Einstein's incompleteness argument, which concludes that a local, separable, and categorical QM cannot exist, 2) a reinterpretation of Bohr's principle of (...) correspondence, grounded in the principle of permanence, 3) a meaning-variance argument for quantum logic, which follows a line of critical reflections initiated by Weyl, and 4) an argument for semantic indeterminacy leveled against inferentialism about QM, inspired by Carnap's work in the philosophy of classical logic. (shrink)

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to (...) the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over ℤ₂ where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2. (shrink)

Logical monism is the view that there is ‘One True Logic’. This is the default position, against which pluralists react. If there were not ‘One True Logic’, it is hard to see how there could be one true theory of anything. A theory is closed under a logic! But what is logical monism? In this article, I consider semantic, logical, modal, scientific, and metaphysical proposals. I argue that, on no ‘factualist’ analysis (according to which ‘there is One True Logic’ expresses (...) a factual claim, rather than an attitude like approval), does the doctrine have both metaphysical and methodological import. Metaphysically, logics abound. Methodologically, what to infer from what is not settled by the facts, even the normative ones. I conclude that the only interesting sense in which there could be One True Logic is noncognitive. The same may be true of monism about normative areas, like moral, epistemic, and prudential ones, generally. (shrink)

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

Ian Rumfitt has argued that rational adjudication against classical logic in quantum mechanics is not only unnecessary, but impossible as well. This paper explains why his argument fails.

This paper provides an overview of Janusz Czelakowski’s contributions to the theory of partial Boolean (σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-)algebras, and, more in general, to the foundation of Quantum Mechanics. Particular attention is paid to the logic of partial Boolean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-algebras, to characterizations of PBAs embeddable into Boolean (σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-)algebras, and their representation as self-adjoint idempotent (...) elements of partial commutative algebras with involution. Also, applications to the theory of orthomodular posets as well as Czelakowski’s theory of partial Boolean algebras in a broader sense will be discussed. Finally, further representation theorems and their importance for quantum logic will be outlined. (shrink)

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation ◻, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations (...) ⊲, R, and Q, satisfying some first-order conditions, used to represent (L,¬), ◻, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative ◻, and an additive ◊ embeds into the lattice of propositions of a frame (X,⊲,R,Q). Building on our recent study of "fundamental logic", we focus on the case where ¬ is dually self-adjoint (a≤¬b implies b≤¬a) and ◊¬a≤¬◻a. In this case, the representations can be constrained so that R=Q, i.e., we need only add a single relation to (X,⊲) to represent both ◻ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X,⊲,R). (shrink)

Timothy Williamson has maintained that the applicability of classical mathematics in science raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We show that this is false.

The rapid advancements in healthcare technologies have resulted in an enormous increase in biomedical data, creating the need for innovative approaches to harness this information for early disease detection. Big Data Analytics (BDA) combined with Artificial Intelligence (AI) offers unprecedented opportunities to analyze complex biomedical signal patterns and predict the onset of diseases at an early stage. The application of AI techniques like machine learning and deep learning in conjunction with BDA allows for the detection of subtle patterns in large (...) datasets that might be missed by conventional diagnostic methods. This paper explores the integration of Big Data Analytics and AI for early disease detection through the analysis of biomedical signal patterns, such as electrocardiograms (ECG), electroencephalograms (EEG), and wearable sensor data. AI algorithms, trained on vast datasets, can recognize early warning signals of various diseases such as cardiovascular disorders, neurodegenerative diseases, and respiratory conditions. The ability to analyze these signals in real time enables early intervention, potentially improving patient outcomes and reducing healthcare costs. A comprehensive literature review highlights the use of BDA and AI in biomedical applications and the challenges associated with data volume, quality, and heterogeneity. We propose a system methodology that combines AI models with real-time Big Data Analytics to process biomedical signals efficiently and accurately. The proposed system leverages AI for feature extraction and classification while utilizing Big Data tools to handle the massive influx of data from biomedical devices and sensors. This approach not only enhances diagnostic precision but also paves the way for personalized healthcare solutions. (shrink)

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers (...) a calculus for classical logic. (shrink)

El presente trabajo pretende explicitar que los operadores modales, como construcciones lógicas, insertos en las interpretaciones modales (MI) de la mecánica cuántica son usados de manera informal sin una semántica modal adecuada. Primero se estudiarán en detalle los motivos por los que ninguna lógica cuántica puede ofrecer una base apropiada para formalizar estos operadores en contextos mecánico-cuánticos. A continuación, se presentará el enfoque de las historias cuánticas como una nueva lógica cuántica (NQL) intrínsecamente booleana como posible herramienta para formalizar operadores (...) modales en el contexto de las MI. Tras comentar brevemente las principales MI, se propondrá abordar un estudio de las modalidades presentes en ellas desde esta NQL prestando siempre especial atención al problema de la medida como hilo conductor. Finalmente, se pretende señalar cuáles han de ser los primeros pasos necesarios para ofrecer una exposición lógicamente consistente de las modalidades en contextos cuánticos relevantes con NQL como herramienta, así como señalar las ventajas que esta perspectiva podría llegar a ofrecer. (shrink)

Este artículo presenta una aproximación lógico-filosófica al problema de las medidas sin interacción (ifm, por sus siglas en inglés) presentes en ciertos experimentos físicos mecánico-cuánticos. Se explicitarán tanto las posibles vías para abordar el estudio de las IFM desde una perspectiva formal, como algunos de los principales retos a la hora de llevar a cabo dicha aproximación.

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; (...) if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

Hardy and Unruh constructed a family of non-maximally entangled states of pairs of particles giving rise to correlations that cannot be accounted for with a local hidden-variable theory. Rather than pointing to violations of some Bell inequality, however, they pointed to apparent clashes with the basic rules of logic. Specifically, they constructed these states and the associated measurement settings in such a way that the outcomes satisfy some conditionals but not an additional one entailed by them. Quantum mechanics avoids the (...) broken ‘if …then …’ arrows in such Hardy–Unruh chains, as we call them, because it cannot simultaneously assign truth values to all conditionals involved. Measurements to determine the truth value of some preclude measurements to determine the truth value of others. Hardy–Unruh chains thus nicely illustrate quantum contextuality: which variables do and do not obtain definite values depends on what measurements we decide to perform. Using a framework inspired by Bub and Pitowsky and developed in our book Understanding Quantum Raffles (co-authored with Michael E. Cuffaro), we construct and analyze Hardy–Unruh chains in terms of fictitious bananas mimicking the behavior of spin-1/2 particles. (shrink)

Dynamic quantum logic (DQL) is studied to represent the proposition of dynamism in quantum information theory as traditional quantum logic cannot deal with it. Although DQL includes many important notions for quantum physics and quantum information theory, there are still missing elements. Some concepts of measuring a specific physical quantity cannot be represented in the exiting DQL. In this study, we add a new concept of measurement to DQL, and discuss the property of this new logic.

Under the influence of Quine’s famous manifesto, many philosophers have thought that logical theories are scientific theories that can be ‘adopted’ and tested as scientific theories. Here we argue that this idea is untenable. We discuss it with special reference to Putnam’s proposal to ‘adopt’ a particular non-classical logic to solve the foundational problems of quantum mechanics in his famous paper ‘Is Logic Empirical?’ (1968), which we argue was not really coherent.

Why you don’t need quantum physics to understand the Universe (and everything in it).

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...) extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics. (shrink)

In this paper we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics. We give a characterization of the degree of non-functionality which is compatible with the propositional structure of quantum theory, showing that having truth-functional connectives, together with some assumptions regarding the relation of logical consequence, commits us to the adequacy of the interpretation sets of these connectives. An advantage of our proof is that it is independent of (...) the number of truth values involved, generalizing previous works. We also show the failure of the adequacy of every Nmatrix that is a model of the non-distributive lattice of quantum propositions. En este trabajo discutimos las nociones de adecuación y veritativo-funcionalidad en la lógica cuántica, desde el punto de vista de una semántica no determinista. Damos una caracterización del grado de no-funcionalidad compatible con la estructura proposicional de la teoría cuántica, mostrando que tener conectivos veritativo-funcionales, junto con algunos presupuestos relativos a la relación de consecuencia lógica, nos compromete con la adecuación de los conjuntos de interpretación de estos conectivos. Un punto ventajoso de nuestra prueba es que se independiza de la cantidad de valores de verdad involucrados, generalizando trabajos previos. Probamos también que falla la adecuación de cualquier Nmatriz que sirva como modelo para el retículo no distributivo de proposiciones cuánticas. (shrink)

The research observed, in parallel and comparatively, a surveillance state’s use of communication & cyber networks with satellite applications for power political & realpolitik purposes, in contrast to the outer space security & legit scientific purpose driven cybernetics. The research adopted a psychoanalytic & psychosocial method of observation in the organizational behaviors of the surveillance state, and a theoretical physics, astrochemical, & cosmological feedback method in the contrast group of cybernetics. Military sociology and multilateral movements were adopted in the diagnostic (...) studies & research on cybersecurity, and cross-channeling in communications were detected during the research. The paper addresses several key points of technicalities in security & privacy breach, from personal devices to ontological networks and satellite applications - notably telecommunication service providers & carriers with differentiated spectrum. The paper discusses key moral ethical risks posed in the mal-adaptations in commercial devices that can corrupt democracy in subtle ways but in a mass scale. The research adopted an analytical linguistics approach with linguistic history in unjailing from the artificial intelligence empowered pancomputationalism approach of the heterogenous dictatorial semantic network, and the astronomical & cosmological research in information theory implies that noncomputable processes are the only defense strategy for the new technology-driven pancomputationalism developments. (shrink)

The extravagances of quantum mechanics never fail to enrich daily the debate around natural philosophy. Entanglement, non-locality, collapse, many worlds, many minds, and subjectivism have challenged generations of thinkers. Its approach can perhaps be placed in the stream of quantum logic, in which the “strangeness” of QM is “measured” through the violation of Bell’s inequalities and, from there, attempts an interpretative path that preserves realism yet ends up overturning it, restating the fundamental mechanisms of QM as a logical necessity for (...) a strong realism. (shrink)

This paper presents a natural deduction system for orthomodular quantum logic. The system is shown to be provably equivalent to Nishimura’s quantum sequent calculus. Through the Curry–Howard isomorphism, quantum $$\lambda $$ -calculus is also introduced for which strong normalization property is established.

A non-classical subsystem of orthomodular quantum logic is proposed. This system employs two basic operations: the Sasaki hook as implication and the _and-then_ operation as conjunction. These operations successfully satisfy modus ponens and the deduction theorem. In other words, they form an adjunction in terms of category theory. Two types of semantics are presented for this logic: one algebraic and one physical. The algebraic semantics deals with orthomodular lattices, as in traditional quantum logic. The physical semantics is given as a (...) procedure for deriving a final segment of a series of yes-no experiments. (shrink)

ion turns equivalence into identity, but there are two ways to do it. Given the equivalence relation of parallelness on lines, the #1 way to turn equivalence into identity by abstraction is to consider equivalence classes of parallel lines. The #2 way is to consider the abstract notion of the direction of parallel lines. This paper developments simple mathematical models of both types of abstraction and shows, for instance, how finite probability theory can be interpreted using #2 abstracts as “superposition (...) events” in addition to the ordinary events. The goal is to use the second notion of abstraction to shed some light on the notion of an indefinite superposition in quantum mechanics. (shrink)

We implement a recent characterization of metaphysical indeterminacy in the context of orthodox quantum theory, developing the syntax and semantics of two propositional logics equipped with determinacy and indeterminacy operators. These logics, which extend a novel semantics for standard quantum logic that accounts for Hilbert spaces with superselection sectors, preserve different desirable features of quantum logic and logics of indeterminacy. In addition to comparing the relative advantages of the two, we also explain how each logic answers Williamson’s challenge to any (...) substantive account of determinacy: For any proposition p, what could the difference between “p” and “it’s determinate that p” ever amount to? (shrink)

In order to provide a unified framework for studying non-commutative algebraic logic, Rump and Yang used three axioms to define quantum B-algebras, which can be seen as implicational subreducts of quantales. Based on the work of Rump and Yang, in this paper we shall continue to investigate the properties of three axioms in quantum B-algebras. First, using two axioms we introduce the concept of generalized quantum B-algebras and prove that the opposite of the category GqBAlg of generalized quantum B-algebras is (...) equivalent to the category LogPQ of logical pre-quantales, but we can not prove that pre-quantales can be used as the injective objects in GqBAlg. Next, we use one axiom to propose the concept of C-algebras and show that a C-algebra is a group if and only if each of its elements is dualizing. Further, by dualizing elements of a C-algebra X, we can define different binary operations on X such that X is a moniod. Finally, we by the Zig–Zag relation discuss some properties of quantum B-algebras. (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)

In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for (...) bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws. (shrink)

Say that metaphysical indeterminacy occurs just when there is a fact such that neither it nor its negation obtains. The aim of this work is to shed light on the issue of whether orthodox quantum mechanics provides any evidence of metaphysical indeterminacy by discussing the logical, semantic, and broadly methodological presuppositions of the debate. I argue that the dispute amounts to a verbal disagreement between classical and quantum logicians, given Eli Hirsch’s account of substantivity; but that it need not be (...) so if Ted Sider’s naturalness-based account of substantivity is adopted instead. Given the latter approach, can anything be said in order to tip the balance of the dispute either way? Some prima facie reasonable constraints on naturalness entail that the classicist is right, and the quantum world is therefore determinate. Nevertheless, there are reasons for weakening those constraints, to the effect that the dispute remains very much open. Finally, I discuss alternative accounts of metaphysical indeterminacy, and argue that they are unsuitable for framing the quantum indeterminacy debate. (shrink)

In this work, we discuss the failure of the principle of truth functionality in the quantum formalism. By exploiting this failure, we import the formalism of N-matrix theory and non-deterministic semantics to the foundations of quantum mechanics. This is done by describing quantum states as particular valuations associated with infinite non-deterministic truth tables. This allows us to introduce a natural interpretation of quantum states in terms of a non-deterministic semantics. We also provide a similar construction for arbitrary probabilistic theories based (...) in orthomodular lattices, allowing to study post-quantum models using logical techniques. (shrink)

Any logic is represented as a certain collection of well-orderings admitting or not some algebraic structure such as a generalized lattice. Then universal logic should refer to the class of all subclasses of all well-orderings. One can construct a mapping between Hilbert space and the class of all logics. Thus there exists a correspondence between universal logic and the world if the latter is considered a collection of wave functions, as which the points in Hilbert space can be interpreted. The (...) correspondence can be further extended to the foundation of mathematics by set theory and arithmetic, and thus to all mathematics. (shrink)

Noi ancora non pensiamo autenticamente. M. Heidegger (1952) Che l'umanità sia progredita non c'è alcun dubbio, ma in quale senso, e a quale prezzo?

The paper sketches an argument against modal monism, more specifically against the reduction of physical possibility to metaphysical possibility. The argument is based on semantic aspects of non-distributive quantum logic.

Desde los trabajos de von Neumann y Birkhoff hasta la actualidad, el estudio de distintas estructuras algebraicas asociadas al formalismo cuántico ha dado lugar a interesantes desarrollos. A modo de ejemplo, el teorema de Kochen-Specker ha tenido una fuerte repercusión en los fundamentos e interpretación de la teoría cuántica. En este trabajo, prestaremos especial atención al abordaje lógico-algebraico iniciado por von Neumann y Birkhoff (aunque también discutiremos otros formalismos, tales como la lógica de la superposición de Tzouvaras). Se presenta el (...) formalismo de Nmatrices como una semántica adecuada para el retículo de proyectores cuánticas, logrando una novedosa caracterización de los estados cuánticos como valuaciones de cierta Nmatriz cuántica. (shrink)

ince the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition is dual to the notion of a subset. Hence the Boolean logic of (...) subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets. (shrink)

ABSTRACT Topos quantum theory is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics.1 1 In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion of the quantum state (...) is problematic for those who view TQT as a realist reformulation of quantum theory. 1Introduction 2Topos Quantum Theory 2.1Phase space 2.2Hilbert space 2.3Beyond Hilbert space 2.4Defining realism 2.5The spectral presheaf 2.6The logic of topos quantum theory 3Interpreting States in Topos Quantum Theory 4Interpreting Truth Values and Clopen Subobjects in Topos Quantum Theory 4.1Interpreting the truth values 4.2Interpreting Subcl 5Neo-realism 5.1The covariant approach 6Conclusion. (shrink)