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。     

Proof Theory of Paraconsistent Quantum Logic

Paraconsistent quantum logic, a hybrid of minimal quantum logic and paraconsistent four-valued logic, is introduced as Gentzen-type sequent calculi, and the cut-elimination theorems for these calculi are proved. This logic is shown to be decidable through the use of these calculi. A first-order extension of this logic is also shown to be decidable. The relationship between minimal quantum logic and paraconsistent four-valued logic is clarified, and a survey of existing Gentzen-type sequent calculi for these logics and their close relatives is addressed.     

形式化量子力学不必使用量子逻辑

一般认为,标准量子力学需要使用一套它自己的逻辑系统,即量子逻辑。量子逻辑采用与一般逻辑系统不同的语义规则,因此和古典逻辑无法兼容。此篇文章将呈现一套量子力学的严格形式基础,它是对古典二值逻辑之保守扩充;保守扩充意指比原先之逻辑系统强,但较强的原因为它有较多之词汇。此套逻辑为三值逻辑。古典逻辑中为真的句子仍然为真。古典逻辑中为假的句子将被区分为强性假与中性。第三个真值一中性一考虑了非本征态情况中之观察句。本文详列了物理的公理并显示它们具有一个模型。此提案的可行性说明了量子逻辑是不必要的,并且存在一个共同的逻辑架构可提供给数学、非量子物理及量子力学使用。     

Weakly Intuitionistic Quantum Logic

In this article von Neumann’s proposal that in quantum mechanics projections can be seen as propositions is followed. However, the quantum logic derived by Birkhoff and von Neumann is rejected due to the failure of the law of distributivity. The options for constructing a distributive logic while adhering to von Neumann’s proposal are investigated. This is done by rejecting the converse of the proposal, namely, that propositions can always be seen as projections. The result is a weakly Heyting algebra for describing the language of quantum mechanics.     

Does Science Influence the Logic we Ought to Use: A Reflection on the Quantum Logic Controversy

In this article I argue that there is a sense in which logic is empirical, and hence open to influence from science. One of the roles of logic is the modelling and extending of natural language reasoning. It does so by providing a formal system which succeeds in modelling the structure of a paradigmatic set of our natural language inferences and which then permits us to extend this structure to novel cases with relative ease. In choosing the best system of those that succeed in this, we seek certain virtues of such structures such as simplicity and naturalness (which will be explained). Science can influence logic by bringing us, as in the case of quantum mechanics, to make natural language inferences about new kinds of systems and thereby extend the set of paradigmatic cases that our formal logic ought to model as simply and naturally as possible. This can alter which structures ought to be used to provide semantics for such models. I show why such a revolution could have led us to reject one logic for another through explaining why complex claims about quantum mechanical systems failed to lead us to reject classical logic for quantum logic.     

Systems of Quantum Logic

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication.     

Extended Quantum Logic

The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.     

Disjunctive Quantum Logic in Dynamic Perspective

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application towards dynamic quantum logic, namely in describing the temporal indeterministic propagation of actual properties of physical systems. This paper can as such by conceived as an addendum to Quantum Logic in Intuitionistic Perspective that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement.     

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.     

The Logic of Quantum Measurements in terms of Conditional Events

This paper shows that the non-Boolean logic of quantum measurementsis more naturally represented by a relatively new 4-operationsystem of Boolean fractions—conditional events—thanby the standard representation using Hilbert Space. After therequirements of quantum mechanics and the properties of conditionalevent algebra are introduced, the quantum concepts of orthogonality,completeness, simultaneous verifiability, logical operations,and deductions are expressed in terms of conditional eventsthereby demonstrating the adequacy and efficacy of this formulation.Since conditional event algebra is nearly Boolean and consistsmerely of ordered pairs of standard events or propositions,quantum events and the so-called "superpositions" of statesneed not be mysterious, and are here fully explicated. Conditionalevent algebra nicely explains these non-standard "superpositions"of quantum states as conjunctions or disjunctions of conditionalevents, Boolean fractions, but does not address the so-called"entanglement phenomena" of quantum mechanics, which remainphysically mysterious. Nevertheless, separating the latter phenomenafrom superposition issues adds clarity to the interpretationof quantum entanglement, the phenomenon of influence propagatedat faster than light speeds. With such treacherous possibilitiespresent in all quantum situations, an observer has every reasonto be completely explicit about the environmental–instrumentalconfiguration, the conditions present when attempting quantummeasurements. Conditional event algebra allows such explicationwithout the physical and algebraic remoteness of Hilbert space.     

Logic and Philosophy of Logic in Wittgenstein

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting thoughts that have gone widely unnoticed.     

Default Logic as Dynamic Doxastic Logic

Dynamic doxastic logic (DDL) is used in connexion with theories of belief revision. Here we try to show that languages of DDL are suitable also for discussing aspects of default logic. One ingredient of our analysis is a concept of coherence-as-ratifiability.     

Logic and Metaphor

In this work, attention is drawn to the abundant use of metaphor and analogy in works of logic. I argue that pervasiveness of figurative language is to be counted among the features that characterize logic and distinguish it from other sciences. This characteristic feature reflects the creativity that is inherent in logic and indeed has been demonstrated to be a necessary part of logic. The goal of this paper, in short, is to provide specific examples of figurative language used in logic that yield insights into the nature of the subject. I encourage the reader to take metaphors seriously, and to accept that they are not mere embellishments but key elements in our understanding of logic.     

Adaptive Logic as a Modal Logic

Modal logics have in the past been used as a unifying framework for the minimality semantics used in defeasible inference, conditional logic, and belief revision. The main aim of the present paper is to add adaptive logics, a general framework for a wide range of defeasible reasoning forms developed by Diderik Batens and his co-workers, to the growing list of formalisms that can be studied with the tools and methods of contemporary modal logic. By characterising the class of abnormality models, this aim is achieved at the level of the model-theory. By proposing formulae that express the consequence relation of adaptive logic in the object-language, the same aim is also partially achieved at the syntactical level.     

Traditional Logic,Modern Logic and Natural Language

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’.     

Logic and Ontology

A brief review of the historicalrelation between logic and ontologyand of the opposition between the viewsof logic as language and logic as calculusis given. We argue that predication is morefundamental than membership and that differenttheories of predication are based on differenttheories of universals, the three most importantbeing nominalism, conceptualism, and realism.These theories can be formulated as formalontologies, each with its own logic, andcompared with one another in terms of theirrespective explanatory powers. After a briefsurvey of such a comparison, we argue that anextended form of conceptual realism provides themost coherent formal ontology and, as such, canbe used to defend the view of logic as language.     

Quantum Logic as a Fragment of Independence-Friendly Logic

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation . Then in a Hilbert space turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann quantum logic can be interpreted by taking their disjunction to be ¬(A & B). Their logic can thus be mapped into a Boolean structure to which an additional operator has been added.     

Logic and Grammar

Grammar can be formulated as a kind of substructural propositional logic. In support of this claim, we survey bare Gentzen style deductive systems and two kinds of non-commutative linear logic: intuitionistic and compact bilinear logic. We also glance at their categorical refinements.     

Modal Logic As Dialogical Logic

The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which name worlds) I will show how to lift the dialogical conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail.