论可能世界的名字

“可能世界”是模态逻辑语义学的核心概念,一般也可以称为“世界”、“点”、“状态”、“时间”以及“情境”等。模态语言本质上是研究通常一阶模型论意义上的关系结构的简单、且具有丰富表达力的形式语言,但是,传统的模态语言没有一种机制来指称关系结构中的个体并对其进行推理。以传统模态逻辑为基础,在句法中引入“可能世界的名字”作为第二类原子命题、引入相应的算子和约束词而得到的语言称为混合语言,这一语言既发扬了模态语言积极的一面,同时又克服了模态语言前述的重大缺陷。本文首先介绍“可能世界的名字”引入的逻辑背景和主要思…     

局部语言的代数结构——关于局部语义的一些一般性观察(1)

在模态逻辑中,出于构造有穷模型这一特殊动机,用于构造过滤的公式集被设计成对所有子公式封闭的。事实上,仅就构造过滤模型而言,一个布尔代数可以经由非独立的生成子生成,因此公式集对子公式封闭是不必要的。运用子代数域的封闭算子,我们可以在命题逻辑的意义上定义一种贯彻了D.M.Armstrong的Combinatorialism本体论的可能世界模型。我们姑且称相应的理论为局部语义。不难发现,局部语义可以被应用于大多数涉及到信念状态,上下文语境的哲学逻辑领域以及人工智能领域。     

以谓词表达模态

以谓词表达模态,在表达力上比算子具有更明显的优势,但蒙塔古悖论和语义的不完善曾使模态谓词遭遇技术困境。采用"模态"与"真"相结合的研究策略,不仅可以在语形上利用真谓词限制模态谓词的表达力实现模态谓词的相容性,亦能在语义上借助语义真理论定义模态谓词的可能世界语义学。因此,以谓词表达模态可以促进模态逻辑的研究,实现各种模态概念与真概念的结合,体现真概念在逻辑研究中的基础性地位。     

模态逻辑的集合论语义与互模拟不变性

含有命题变元的非良基集合能够被看作解释模态语言的模型。任给非良基集合a,一个命题变元p在a上真当且仅当p属于a。命题联结词的解释与古典命题逻辑相同。一个公式3A在a上真当且仅当存在集合b属于a,使得A在b上是真的。在一个集合中,属于关系被看作可及关系。在这种思想下,我们可以定义从模态语言到一阶集合论语言的标准翻译。对任意模态公式A和集合变元x,可以递归定义一阶集合论语言的公式ST(A,x)。在关系语义学下,van Benthem刻画定理是说,在带有唯一的二元关系符号R的一阶语言中,任何一阶公式等价于某个模态公式的标准翻译当且仅当这个一阶公式在互模拟下保持不变。因此,模态语言是该一阶关系语言的互模拟不变片段。同样,我们可以在集合上定义互模拟关系,证明van Benthem刻画定理对于集合论语义和集合上的互模拟不变片段成立,即模态语言是一阶集合论语言的集合互模拟不变片段。     

强可能性与弱必然性的逻辑

本文考察了强可能性和弱必然性这两种真势模态的逻辑性质。称一个命题是强可能的,当且仅当它在某个可及的但非现实的可能世界中成立;称一个命题是弱必然的,当且仅当它在所有可及的但非现实的可能世界中都成立。强可能性与弱必然性互为对偶。在表达力上,强可能性算子不同于可能性算子。尽管如此,刻画强可能性和弱必然性的逻辑(简称为强可弱必逻辑)仍然是正规模态逻辑,而且它的极小逻辑在形态上类似于极小正规模态逻辑K。本文主要在形式技术上对强可弱必逻辑作了初步研究:比较了它和模态逻辑以及一阶逻辑在语言表达力上的区别,给出了它的极小和扩张的公理刻画以及必然和弱必然的双模态逻辑的公理刻画。     

论模态逻辑的集合论语义

引入非良基集合可以为模态逻辑提供一种新的语义学。这种语义是在集合上解释模态语言,使用集合中作为元素的集合之间的属于关系解释模态词,并在集合中采用命题变元作为本元,从而解释原子命题的真假。在这种新的语义下,从模型构造的角度看可以引入几种非标准的集合运算：不交并、生成子集合、p-态射、树展开等等,证明模态公式在这些运算下的保持或不变结果。利用这些结果还可以证明一些集合类不是模态可定义的。     

邻域语义与修正真理论

古普塔和赫兹伯格在1982年各自独立地提出了修正真理论,建立了可用于分析真与相关悖论的修正序列。修正真理论根据语句在所有修正序列中的表现,对语句进行分类。然而,修正真理论在某些语句的分类上不能令人满意,如修正真理论把柯瑞悖论的逆命题断定为绝对地真,这与直觉不一致。本文将从两种路径引入邻域语义研究修正真理论。路径一是在基模型上引入邻域基模型,建立邻域基模型修正序列。这类修正序列比经典修正序列更多,增加的修正序列可使包括柯瑞悖论的逆命题在内的一些语句的病态呈现出来。路径二是通过引入邻域语义模型,使得对任意不含模态词的公式φ,模态公式□φ在后继阶段的真值可以反映φ在上一阶段的真值,并且□φ在极限阶段的真值可以反映φ在至这个极限阶前是否稳定真。从而可以通过□φ的真值来限定Tφ的真值,使得满足相应限制的模型类表示了相应的修正序列。本文最后将对两个路径进行整合,构造出能表示邻域基模型修正序列的整体修正序列模型。     

含例邻域逻辑的萨奎斯特对应理论（英文）

本文给出含例邻域逻辑的萨奎斯特对应理论。这种逻辑可以讨论一个可能世界的邻域的存在性信息,是关系语义和邻域语义的一种混合。增加的表达力和描述邻域函数的关系特征使我们可以在框架层面讨论这种语言可以定义什么性质,是否可以超越邻域语义的KW片段。我们给出对应定理的两个证明。第一个证明直接使用标准翻译和极小赋值的技术,第二个证明通过双模态翻译将含例邻域逻辑的对应问题转化为经典克里普克语义的双模态逻辑。     

样本发散型含糊类的形式刻画

含糊类是基于样本和相似性得到的类。通过样本和与样本的相似性处理含糊对象是人们在面对含糊性时常用的方法。含糊类有样本收敛和样本发散两大类型,后者应该更为普遍。样本收敛的含糊类也是有核含糊类,可以通过核来处理边界情况。但是因为样本发散含糊类同时也是无核含糊类,所以这个方法不适用于样本发散含糊类。从人们对于含糊对象的实际处理看,除了用正面的样本外,还会用到反面的样本。将这个过程加以抽象,本文引入了负样本以及提出了由正样本和负样本共同处理边界情况的方案。在形式刻画方面,主要是在一阶语言的基础上通过增加正样本谓词、负样本谓词和论题词给出了语言L*及其语义。在L*中可以进一步定义正谓词、负谓词以及中间谓词,通过这些表达式可以对于含糊对象及其性质给出相应的刻画。     

模态逻辑GL的基于广义谢弗竖的分析性模态公理系统（英文）

基于广义谢弗竖这种新算子,本文构造了模态逻辑GL的模态表列和分析性模态公理系统。广义谢弗竖是一种n元算子,为模态逻辑的表达式提供一种新记法,使分析性模态公理系统的陈述直接明了。由于谢弗竖是一种新算子,基于它的模态表列规则与通常的基于模态词和联结词的表列规则有所不同。分析性模态公理系统中的内定理证明很简单。因为分析性模态公理系统与模态表列之间存在某种对应关系,所以GL的分析性模态公理系统的完全性由GL的模态表列的完全性结果易证。GL的模态系统的完全性证明比较特殊,无法直接应用证明模态逻辑完全性的一般方法——典范模型方法,需要用一种过滤的方法挑出一些可能世界构造有穷模型。     

Quantification over Sets of Possible Worlds in Branching-Time Semantics

Temporal logic is one of the many areas in which a possible world semantics is adopted. Prior's Ockhamist and Peircean semantics for branching-time, though, depart from the genuine Kripke semantics in that they involve a quantification over histories, which is a second-order quantification over sets of possible worlds. In the paper, variants of the original Prior's semantics will be considered and it will be shown that all of them can be viewed as first-order counterparts of the original semantics.     

Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.     

Actualism,ontological commitment,and possible world semantics

Actualism is the doctrine that the only things there are, that have being in any sense, are the things that actually exist. In particular, actualism eschews possibilism, the doctrine that there are merely possible objects. It is widely held that one cannot both be an actualist and at the same time take possible world semantics seriously — that is, take it as the basis for a genuine theory of truth for modal languages, or look to it for insight into the modal structure of reality. For possible world semantics, it is supposed, commits one to possibilism. In this paper I take issue with this view. To the contrary, I argue that one can take possible world semantics seriously and yet remain in full compliance with actualist scruples.     

Merging without mystery or: Variables in dynamics semantics

In this paper we discuss the treatment of variables in dynamic semantics. Referent systems are introduced as a flexible mechanism for working with variables. In a referent system we carefully distinguish the variables themselves both from the machinery by which we manipulate them — their names — and from the information that we store in them — their values. It is shown that the referent systems provide a natural basis for dynamic semantics. The semantics with referent systems is compared with the familiar formalisms in dynamic semantics,DRT andDPL.     

On counterpossibles

The traditional Lewis–Stalnaker semantics treats all counterfactuals with an impossible antecedent as trivially or vacuously true. Many have regarded this as a serious defect of the semantics. For intuitively, it seems, counterfactuals with impossible antecedents—counterpossibles—can be non-trivially true and non-trivially false. Whereas the counterpossible "If Hobbes had squared the circle, then the mathematical community at the time would have been surprised" seems true, "If Hobbes had squared the circle, then sick children in the mountains of Afghanistan at the time would have been thrilled" seems false. Many have proposed to extend the Lewis–Stalnaker semantics with impossible worlds to make room for a non-trivial or non-vacuous treatment of counterpossibles. Roughly, on the extended Lewis–Stalnaker semantics, we evaluate a counterfactual of the form "If A had been true, then C would have been true" by going to closest world—whether possible or impossible—in which A is true and check whether C is also true in that world. If the answer is "yes", the counterfactual is true; otherwise it is false. Since there are impossible worlds in which the mathematically impossible happens, there are impossible worlds in which Hobbes manages to square the circle. And intuitively, in the closest such impossible worlds, sick children in the mountains of Afghanistan are not thrilled—they remain sick and unmoved by the mathematical developments in Europe. If so, the counterpossible "If Hobbes had squared the circle, then sick children in the mountains of Afghanistan at the time would have been thrilled" comes out false, as desired. In this paper, I will critically investigate the extended Lewis–Stalnaker semantics for counterpossibles. I will argue that the standard version of the extended semantics, in which impossible worlds correspond to maximal, logically inconsistent entities, fails to give the correct semantic verdicts for many counterpossibles. In light of the negative arguments, I will then outline a new version of the extended Lewis–Stalnaker semantics that can avoid these problems.     

Indicative and Subjunctive Conditionals

In any plausible semantics for conditionals, the semantics for indicatives and subjunctives will resemble each other closely. This means that if we are to keep the possible-worlds semantics for subjunctives suggested by Lewis, we need to find a possible-worlds semantics for indicatives. One reason for thinking that this will be impossible is the behaviour of rigid designators in indicatives. An indicative like 'If the stuff in the rivers, lakes and oceans really is H₃O, then water is H₃O' is non-vacuously true, even though its consequent is true in no possible worlds, and hence not in the nearest possible world where the antecedent is true. I solve this difficulty by providing a semantics for conditionals within the framework of two-dimensional modal logic. In doing so, I show that we can have a reasonably unified semantics for indicative and subjunctive conditionals.     

A Default-Oriented Theory of Procedural Semantics

Procedural Semantics(PS), broadly construed, is the thesis that the meaning of a symbolic expression may be identified with procedures which specify how the expression is to be used, or applied to the world. J.A. Fodor has characterized PS as a wildly implausible form of the verification theory of meaning, and has argued that PS constitutes a plausible semantic theory only for highly simplistic universes, such as “blocks worlds” and databases. Fodor argued further that insofar as PS is defensible, it is “parasitic upon” classical denotational semantics. Fodor's critique of PS provoked various rejoinders. Although these rejoinders were not always in agreement, both Wilks and Woods reasoned that some form of PS must be true if we are to render coherent certain fundamental concepts of modeloretic semantics (e.g., denotations and truth-conditions). In the present paper these arguments are reviewed and extended. It is argued that not only Tarskian semantics, but other forms of model-theoretic semantics (including possible world and Situation Semantics) may very well ontologically presuppose some form of PS. In addition, a default-oriented form of PS is presented which avoids the “decision procedure” approach of early PS. The new theory embraces aspects of Quine's pragmatism, and assumes that semantic procedures may return pragmatic (default) truth values which may be revised, if the need arises, by adjudication procedures. On the theory here described, semantic procedures do not constitute the complete meanings of symbolic expressions, but constrain these meanings. It is argued that such constraints must exist if there are to be ontological foundations for such traditional notions as denotations and extensions.     

Science, metaphysics and method

While there are many examples of metaphysical theorising being heuristically and intellectually important in the progress of scientific knowledge, many people wonder how metaphysics not closely informed and inspired by empirical science could lead to rival or even supplementary knowledge about the world. This paper assesses the merits of a popular defence of the a priori methodology of metaphysics that goes as follows. The first task of the metaphysician, like the scientist, is to construct a hypothesis that accounts for the phenomena in question. It is then argued that among the possible metaphysical theories, the empirical evidence underdetermines the right one, just as the empirical evidence underdetermines the right scientific theory. In the latter case it is widely agreed that we must break the underdetermination by appeal to theoretical virtues, and this is just what should be and largely is done in metaphysics. This is part of a more general line of argument that defends metaphysics on the basis of its alleged continuity with highly theoretical science. In what follows metaphysics and theoretical science are compared in order to see whether the above style of defence of a priori metaphysics is successful.