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。     

早期分析哲学中的信念逻辑

弗雷格的涵义/意谓理论为信念句子的句法和语义分析提供了一个框架。一个信念句子由专名、动词"相信"和从句组成。根据弗雷格的从句理论,从句有间接意谓,即它的思想。按弗雷格的分析,可以建立信念逻辑的形式语言和语义。这种语义是混合式的,引入涵义、个体和真值三类本体。还可以建立一个弗雷格式的信念逻辑系统。它是不需要可能世界语义学的信念逻辑。罗素先后提出处理信念语境中同一替换律失效问题的摹状词理论、信念关系论和逻辑原子主义。但罗素没有对信念句子的句法和语义作出明确的分析。     

面向汉语认知与功能特征的语法逻辑

本文研究范畴语法的两种扩充,一是从认知特征角度的扩充,将范畴语法扩充为认知特征范畴语法,通过具有完全性的逻辑证明解决了一些不合语言事实的句子判别问题;二是从功能特征角度的扩充,提出逻辑推理的形式和进一步将二者统一的可能性问题。     

括号表示法：一种中国式表示法

逻辑常项是各种逻辑系统研究的核心，使用合适的方式表示逻辑常项，可以为逻辑研究提供良好的技术工具和清晰的呈现方式。根据逻辑常项符号表示法的产生和发展历程，可以将其归纳为三种形态：自然语言表示法、符号表示法和形式化表示法。逻辑常项表示法的变迁，不仅决定着逻辑形态的呈现形式，而且决定着逻辑研究的持续和深入发展。中置法和前置法是国际逻辑学界通行的两种主要的逻辑常项表示法，与之不同，作者受舍弗(H. M. Sheffer)函数和张清宇先生相关工作的启发，提出了一种新的逻辑常项表示法——括号表示法。在该表示法中，表示逻辑常项的符号只有一对左右括号。作者阐明在任意给定的逻辑系统中，只要使用一对括号就可以定义出该系统的所有逻辑常项，彰显了括号表示法强大的归约功能和表达功能。作者还证明了括号表示法其形式语言表达的唯一性。在此基础上，作者阐明：比起中置法，括号表示法表达更简洁；比起波兰表示法，括号表示法表达更清晰。括号表示法是一种整体表示法，是由中国学者提出并系统构建的符号表示法，因此也可以称为中国式表示法。     

牟宗三对“逻辑是否可修正”问题的解答

牟宗三区分了逻辑和逻辑系统,认为逻辑系统是逻辑理性自己的显现,逻辑系统的形成需要遵守先验性的轨约原则和构造原则。由于轨约原则和构造原则的先验性以及逻辑概念的有定、有尽性,所以,牟宗三认为逻辑及其系统也具有绝对性和先验性乃至理性上的必然性和定然性。笔者认为这个理论对于"逻辑是否可修正"这一问题可以给出解答。     

时态数据库属性推理的类型逻辑

1994年Gabbay等论证了时态逻辑的公理化系统和证明论方法不适于时态数据库推理建模,因此目前主要使用非公理化的时态逻辑做推理。然而非公理化的时态逻辑缺乏公理化性质约束,形式晦涩,无直观性与运算性,因而一般不用于知识推理。另一方面,1983年Allen提出了13种时态关系运算,并使用区间逻辑对时态关系进行表达,但这些运算只能表示时间本身的运算关系,未能体现时间与属性之间的映射关系,不能表达时态数据库属性间的推理与运算。此外,时态数据库中仍存在着许多开问题,例如在做属性推理与运算时出现的Now语义的不确定性问题。基于此,我们提出一种基于时态数据库属性推理的类型逻辑系统,其主要思路为将属性映射为类型,将类型映射为时间向量子集,以时间向量集为逻辑语义模型,在推理中从句法逻辑系统剥离对时间的表示,减少逻辑算子,与时间相关的运算单纯由语义模型支持,从而降低复杂性,提高运算能力。并且由于该类型逻辑系统是基于构造性语义的,能直观解释Now的不确定性问题。文中给出了相对该系统的时态数据库属性推理的应用示例,从技术可操作性上介绍了根据该类型逻辑设计的时态推理中间件原型及工作流程,最后从元理论范畴证明了系统可靠性与完全性的逻辑性质及切割消除与判定性的证明论性质。     

从方法论的角度看动态认知逻辑的研究

本文主要探讨动态认知逻辑技术结果背后的思想,着重强调其在方法论方面的一般想法。文章在公开宣告逻辑的基础上展开讨论,重点考察了归约公理的意义,揭示了其本质是在基本语言中提前解析动态信息对认知产生的影响。文章以公共知识为例,说明了并非所有的逻辑算子都能找到归约公理,有时候我们需要丰富基本语言的表达力。而且,我们从如何给出一个逻辑的角度提出,动态认知逻辑实际上是在“动态化”认知逻辑,这种动态化的方法可以应用在其他静态的逻辑系统中。我们以动态偏好逻辑为例,说明了这一过程是如何实现的。     

论逻辑后承概念

塔尔斯基在其1933年的论文中基于自己开创的语义学定义了形式化语言中真这个概念。然后,他在发表于1936年的本文中,在真这个概念的定义的基础上,第一次为后承概念提出了一个实质恰当的定义,即“句子X从类K的句子逻辑地得出当且仅当K这个类的每个模型也是句子X的模型”,使得逻辑后承这个现代逻辑核心概念的定义成为标准定义。但是,这个定义也遗留了一个更为根本的问题,即逻辑词项和非逻辑词项的划分标准问题,后者将由塔尔斯基本人在1966年给出了一个划分标准,从而开辟了一个方兴未衰的逻辑哲学研究方向。     

逻辑常项与保守性：以tonk为例

根据逻辑推理论,逻辑常项的意义是由它的引入和消去规则确定的。普莱尔（Arthur Prior）提出的tonk对推理论构成了严重挑战。库克（Roy Cook）最近构造了一个禁止传递性的相干的逻辑系统,即Tonk-逻辑,并借助四值语义学重新定义了Tonk-后承概念,在这种概念之下,tonk的引入规则和消去规则都是有效的,同时系统还不是平凡的。本文探讨了保守性与常项的引入和消去规则的协调性之间的联系,并定义了两种较强的协调性概念,即HCU-协调性和HML-协调性概念。借助这两个概念,本文论证,tonk不是HCU-协调的也不是HML-协调的,因而它不是合法的逻辑常项,Tonk-逻辑也不是一种合法的逻辑系统。     

十一、与戴维森一起读《论语》：如何用语用学的概念来描述古代中国的解释实践

在对中国古代语言与逻辑的研究中,占主导地位的方法是将句子当作其主要研究对象:它所关心的是句子的语法与逻辑结构.与这种"语法的方法"相对照,还有一种"语用学的方法".这种方法将语言实践,也就是"说出一个句子(the utterance of a sentence)"这一言语行动本身,当作其主要研究对象.     

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.     

Correspondence Truth and Quantum Mechanics

The logic of a physical theory reflects the structure of the propositions referring to the behaviour of a physical system in the domain of the relevant theory. It is argued in relation to classical mechanics that the propositional structure of the theory allows truth-value assignment in conformity with the traditional conception of a correspondence theory of truth. Every proposition in classical mechanics is assigned a definite truth value, either ‘true’ or ‘false’, describing what is actually the case at a certain moment of time. Truth-value assignment in quantum mechanics, however, differs; it is known, by means of a variety of ‘no go’ theorems, that it is not possible to assign definite truth values to all propositions pertaining to a quantum system without generating a Kochen–Specker contradiction. In this respect, the Bub–Clifton ‘uniqueness theorem’ is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a particular observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The conceptual implications of the resulting account are traced down and analyzed at length. In this light, the traditional conception of correspondence truth may be viewed as a species or as a limit case of the more generic proposed scheme of contextual correspondence when the non-explicit specification of a context of discourse poses no further consequences.     

Are the Laws of Quantum Logic Laws of Nature?

The main goal of quantum logic is the bottom-up reconstruction of quantum mechanics in Hilbert space. Here we discuss the question whether quantum logic is an empirical structure or a priori valid. There are good reasons for both possibilities. First, with respect to the possibility of a rational reconstruction of quantum mechanics, quantum logic follows a priori from quantum ontology and can thus not be considered as a law of nature. Second, since quantum logic allows for a reconstruction of quantum mechanics, self-referential consistency requires that the empirical content of quantum mechanics must be compatible with the presupposed quantum ontology. Hence, quantum ontology contains empirical components that are also contained in quantum logic. Consequently, in this sense quantum logic is also a law of nature.     

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.     

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.     

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.     

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.     

Bohm's Ontological Interpretation and its Relations to Three Formulations of Quantum Mechanics

The standard mathematical formulation of quantum mechanics is specified. Bohm's ontological interpretation of quantum mechanics is then shown to be incapable of providing a suitable interpretation of that formulation. It is also shown that Bohm's interpretation may well be viable for two alternative mathematical formulations of quantum mechanics, meaning that the negative result is a significant though not a devastating criticism of Bohm's interpretation. A preliminary case is made for preferring one alternative formulation over the other. This revised version was published online in June 2006 with corrections to the Cover Date.