模糊对象和模糊性的逻辑

本文最初写于2001年,旨在探究语言上的、概念上的与本体论中的模糊性问题。文章始于这样一个问题,即模糊性仅仅是一个词的特征,抑或它也是我们的概念的特征,甚或是实在的特征？其中主要的关注点在于模糊对象的问题和相关的模糊同一性的问题。本文首先论证,即便模糊概念总可以得到进一步澄清,每个概念工具也必定是模糊的。其次,本文认为,如果不仅仅是词语,而且这些词语所表达的概念也是模糊的,那么可以论证,对象也必定是模糊的。第三,那种认为模糊对象必定具有模糊的同一性的思路是一种误解。最后一点尤为重要,因为对于模糊对象的抵制部分就来自于下述观念,即这种对象是不可能的,因为它们不遵守逻辑的同一律。本文也论述了,通过放弃二值原则以使得模糊陈述具有真和假之间的某个值或根本就没有真值的办法,也无助于解决模糊性问题,而且恰恰会造成相反的情况,即妨碍我们找到问题的解决之道。本文表明,在为真和既不真又不假之间,难以清晰地划分界限,正如在模糊的情形下要在真和假之间划界一样困难。本文的方法是利用量化模态逻辑的框架和语义学来处理这些问题。基于哥德尔模式的模态逻辑系统T,本文提出了一种关于清晰性和模糊性的量化逻辑,其中利用清晰性算子来代替必然性算子。文章还进一步讨论了模糊性和概念工具的发展之间的关联。     

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.     

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.     

Willem Blok and Modal Logic

We present our personal view on W.J. Blok's contribution to modal logic.     

A Modal Sortal Logic

An intensional semantic system for languages containing, in their logical syntax, sortal quantifiers, sortal identities, (second-order) quantifiers over sortals and the necessity operator is constructed. This semantics provides non-standard assignments to predicate expressions, which diverge in kind from the entities assigned to sortal terms by the same semantic system. The nature of the entities assigned to predicate expressions shows, at the same time, that there is an internal semantic connection between those expressions and sortal terms. A formal logical system is formulated that is proved to be absolutely consistent, sound and complete with respect to the intensional semantic system.     

Positive Monotone Modal Logic

Studia Logica - Positive monotone modal logic is the negation- and implication-free fragment of monotone modal logic, i.e., the fragment with connectives and . We axiomatise positive monotone modal...     

First-Order Classical Modal Logic

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.     

The Modal Logic of Discrepancy

Discrepancies between an agents goals and beliefs play an important, if implicit, role in determining what a rational agent is motivated to do. This is most obvious in cases where an agent achieves a complex goal incrementally and must deliberate anew as each milestone is reached. In such cases the concept of goal/belief discrepancy defines an appropriate space to which a degree-of-achievement yardstick can be applied. This paper presents soundness and completeness results concerning a logic for reasoning about goal/belief discrepancy, and it is suggested that a certain species of goal/belief discrepancy captures the concept of desire.     

A Modal Logic of Metaphor

The purpose of this paper is to suggest a formal modelling of metaphors as a lingustic tool capable of conveying meanings from one conceptual space to another. This modelling is done within DDL (dynamic doxastic logic).     

IF Modal Logic and Classical Negation

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML _s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML _s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar from ML; the resulting asymmetric simulation concept is used to prove that IFML _s is not closed under complementation. In fact we obtain a much stronger result: the only IFML _s formulas admitting their classical negation to be expressed in IFML _s itself are those whose truth-condition is in fact expressible in ML.     

On an Intuitionistic Modal Logic

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.     

Proof Analysis in Modal Logic

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel–Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.     

Three-valued Logics in Modal Logic

Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.     

Actuality in Propositional Modal Logic

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5.     

Euclidean Hierarchy in Modal Logic

For a Euclidean space , let L _n denote the modal logic of chequered subsets of . For every n 1, we characterize L _n using the more familiar Kripke semantics, thus implying that each L _n is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics L _n form a decreasing chain converging to the logic L of chequered subsets of . As a result, we obtain that L is also a logic over Grz, and that L has the finite model property. We conclude the paper by extending our results to the modal language enriched with the universal modality.