Speakable in quantum mechanics

At the 1927 Como conference Bohr spoke the famous words “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.” However, if the Copenhagen interpretation really adheres to this motto, why then is there this nagging feeling of conflict when comparing it with realist interpretations? Surely what one can say about nature should in a certain sense be interpretation independent. In this paper I take Bohr’s motto seriously and develop a quantum logic that avoids assuming any form of realism as much as possible. To illustrate the non-triviality of this motto, a similar result is first derived for classical mechanics. It turns out that the logic for classical mechanics is a special case of the quantum logic thus derived. Some hints are provided as to how these logics are to be used in practical situations and finally, I discuss how some realist interpretations relate to these logics.     

三值量子逻辑进路探析

量子测量实验显示部分经典逻辑规则在量子世界中失效。标准量子逻辑进路通过特有的希尔伯特空间的格运算揭示出一种内在于微观物理学理论的概念框架结构,也即量子力学测量命题的正交补模或弱模格,解释了经典分配律的失效,它在形式化方面十分完美,但在解释方面产生了一些概念混乱。在标准量子逻辑进路之外,赖欣巴赫通过引入"不确定"的第三真值独立地提出一种不同的量子逻辑模型来解释量子实在的特征,不是分配律而是排中律失效,但是他的三值量子逻辑由于缺乏标准量子逻辑的上述优点而被认为与量子力学的概率空间所要求的潜在逻辑有很少联系。本文尝试引入一种新的三值逻辑模型来说明量子实在,它有以下优点:(1)满足卢卡西维茨创立三值逻辑的最初语义学假定;(2)克服赖欣巴赫三值量子逻辑的缺陷;(3)澄清标准量子逻辑遭遇的概念混乱;(4)充分地保留经典逻辑规则,特别是标准量子逻辑主张放弃的分配律。     

Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.     

Toward a More Natural Expression of Quantum Logic with Boolean Fractions

This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, ‘a if b’ or ‘a given b’, ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be “inapplicable” (or “undefined”) in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b=d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the ‘influence-at-a-distance’, quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that “hidden variables” are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. Partial support for this work is gratefully acknowledged from the In-House Independent Research Program and from Code 2737 at the Space & Naval Warfare Systems Center (SSC-SD), San Diego, CA 92152-5001. Presently this work is supported by Data Synthesis, 2919 Luna Avenue, San Diego, CA 92117.     

Fuzzy Topology and Łukasiewicz Logics from the Viewpoint of Duality Theory

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of ?ukasiewicz n-valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n-valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal ?ukasiewicz n-valued logic with truth constants, which generalizes Jónsson-Tarski duality for modal algebras to the n-valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics.     

Logical Cornestones of Judaic Argumentation Theory

In this paper, the four Judaic inference rules: qal wa- ? omer, gezerah ? awah, heqe ?, binyan ’av are considered from the logical point of view and the pragmatic limits of applying these rules are symbolic-logically explicated. According to the Talmudic sages, on the one hand, after applying some inference rules we cannot apply other inference rules. These rules are weak. On the other hand, there are rules after which we can apply any other. These rules are strong. This means that Judaic inference rules have different pragmatic meanings and this fact differs Judaic logic from other ones. The Judaic argumentation theory built up on Judaic logic also contains pragmatic limits for proofs as competitive communication when different Rabbis claim different opinions in respect to the same subject. In order to define these limits we build up a special kind of syllogistics, the so-called Judaic pragmatic-syllogistics, where it is defined whose opinion should be choosen in a dispute.     

Denotation of contextual modal type theory (CMTT): Syntax and meta-programming

The modal logic S4 can be used via a Curry–Howard style correspondence to obtain a λ-calculus. Modal (boxed) types are intuitively interpreted as ‘closed syntax of the calculus’. This λ-calculus is called modal type theory—this is the basic case of a more general contextual modal type theory, or CMTT.CMTT has never been given a denotational semantics in which modal types are given denotation as closed syntax. We show how this can indeed be done, with a twist. We also use the denotation to prove some properties of the system.     

A Dynamic-Logical Perspective on Quantum Behavior

In this paper we show how recent concepts from Dynamic Logic, and in particular from Dynamic Epistemic logic, can be used to model and interpret quantum behavior. Our main thesis is that all the non-classical properties of quantum systems are explainable in terms of the non-classical flow of quantum information. We give a logical analysis of quantum measurements (formalized using modal operators) as triggers for quantum information flow, and we compare them with other logical operators previously used to model various forms of classical information flow: the “test” operator from Dynamic Logic, the “announcement” operator from Dynamic Epistemic Logic and the “revision” operator from Belief Revision theory. The main points stressed in our investigation are the following: (1) The perspective and the techniques of “logical dynamics” are useful for understanding quantum information flow. (2) Quantum mechanics does not require any modification of the classical laws of “static” propositional logic, but only a non-classical dynamics of information. (3) The main such non-classical feature is that, in a quantum world, all information-gathering actions have some ontic side-effects. (4) This ontic impact can affect in its turn the flow of information, leading to non-classical epistemic side-effects (e.g. a type of non-monotonicity) and to states of “objectively imperfect information”. (5) Moreover, the ontic impact is non-local: an information-gathering action on one part of a quantum system can have ontic side-effects on other, far-away parts of the system.     

Finite Cardinals in Quasi-set Theory

Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to them. According to some authors, this is the best way to understand quantum objects. The fact that identity is not defined for m-atoms raises a technical difficulty: it seems impossible to follow the usual procedures to define the cardinal of collections involving these items. In this paper we propose a definition of finite cardinals in quasi-set theory which works for collections involving m-atoms.     

On the Epistemic Foundation for Iterated Weak Dominance: An Analysis in a Logic of Individual and Collective attitudes

This paper proposes a logical framework for representing static and dynamic properties of different kinds of individual and collective attitudes. A complete axiomatization as well as a decidability result for the logic are given. The logic is applied to game theory by providing a formal analysis of the epistemic conditions of iterated deletion of weakly dominated strategies (IDWDS), or iterated weak dominance for short. The main difference between the analysis of the epistemic conditions of iterated weak dominance given in this paper and other analysis is that we use a semi-qualitative approach to uncertainty based on the notion of plausibility first introduced by Spohn, whereas other analysis are based on a quantitative representation of uncertainty in terms of probabilities.     

The Quasi-lattice of Indiscernible Elements

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice ( \mathfrakI{\mathfrak{I}}-lattice), which can be modeled by an algebraic structure built in quasi-set theory \mathfrakQ{\mathfrak{Q}} . This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that ‘naturally’ arises is non distributive.     

The Structure Group of a Generalized Orthomodular Lattice

Orthomodular lattices with a two-valued Jauch–Piron state split into a generalized orthomodular lattice (GOML) and its dual. GOMLs are characterized as a class of L-algebras, a quantum structure which arises in the theory of Garside groups, algebraic logic, and in connections with solutions of the quantum Yang–Baxter equation. It is proved that every GOML X embeds into a group G(X) with a lattice structure such that the right multiplications in G(X) are lattice automorphisms. Up to isomorphism, X is uniquely determined by G(X), and the embedding \(X\hookrightarrow G(X)\) is a universal group-valued measure on X.     

Categorical Abstract Algebraic Logic: Behavioral π-Institutions

Recently, Caleiro, Gon¸calves and Martins introduced the notion of behaviorally algebraizable logic. The main idea behind their work is to replace, in the traditional theory of algebraizability of Blok and Pigozzi, unsorted equational logic with multi-sorted behavioral logic. The new notion accommodates logics over many-sorted languages and with non-truth-functional connectives. Moreover, it treats logics that are not algebraizable in the traditional sense while, at the same time, shedding new light to the equivalent algebraic semantics of logics that are algebraizable according to the original theory. In this paper, the notion of an abstract multi-sorted π-institution is introduced so as to transfer elements of the theory of behavioral algebraizability to the categorical setting. Institutions formalize a wider variety of logics than deductive systems, including logics involving multiple signatures and quantifiers. The framework developed has the same relation to behavioral algebraizability as the classical categorical abstract algebraic logic framework has to the original theory of algebraizability of Blok and Pigozzi.     

Second-order Logic and the Power Set

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments.     

Unified Interpretation of Quantum and Classical Logics

Quantum logic is only applicable to microscopic phenomena while classical logic is exclusively used for everyday reasoning, including mathematics. It is shown that both logics are unified in the framework of modal interpretation. This proposed method deals with classical propositions as latently modalized propositions in the sense that they exhibit manifest modalities to form quantum logic only when interacting with other classical subsystems.     

Foundations of Everyday Practical Reasoning

“Since today is Saturday, the grocery store is open today and will be closed tomorrow; so let’s go today”. That is an example of everyday practical reasoning—reasoning directly with the propositions that one believes but may not be fully certain of. Everyday practical reasoning is one of our most familiar kinds of decisions but, unfortunately, some foundational questions about it are largely ignored in the standard decision theory: (Q1) What are the decision rules in everyday practical reasoning that connect qualitative belief and desire to preference over acts? (Q2) What sort of logic should govern qualitative beliefs in everyday practical reasoning, and to what extent is that logic necessary for the purposes of qualitative decisions? (Q3) What kinds of qualitative decisions are always representable as results of everyday practical reasoning? (Q4) Under what circumstances do the results of everyday practical reasoning agree with the Bayesian ideal of expected utility maximization? This paper proposes a rigorous decision theory for answering all of those questions, which is developed in parallel to Savage’s (1954) foundation of expected utility maximization. In light of a new representation result, everyday practical reasoning provides a sound and complete method for a very wide class of qualitative decisions; and, to that end, qualitative beliefs must be allowed to be closed under classical logic plus a well-known nonmonotonic logic—the so-called system ?.     

A Defense of Second-Order Logic

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non-isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985)], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs.     

Stability and Paradox in Algorithmic Logic

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally.     

Curry’s Paradox and ω -Inconsistency

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes¹. In this paper I show that a number of logics are susceptible to a strengthened version of Curry’s paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in ?ukasiewicz’s continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of ukasiewicz logic which individually, but not jointly, lack the problematic feature.     

The Mereological Foundation of Megethology

In Mathematics is megethology (Lewis (1993). Philosophia Mathematica, 1(1), 3–23) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos’ plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non expressed in a mereological language, a mereological foundation of set theory is achievable within first order logic. Furthermore, we show how a mereological codification of ordered pairs is achievable with a very restricted use of the notion of plurality without plural quantification.