Sense and Reference of Predicates: Comments on Frege’s Theory of Sense-Reference

Frege argued that a predicate was a functional expression and the reference of it a concept, which as a predicative function had one or more empty places and was thus incomplete. Frege’s view gives rise to what has been known as the paradox of the concept “horse.” In order to resolve this paradox, I argue for an opposite view which retains the point that a predicate is a function, i.e. that a predicative function is complete in a sense. Specifically speaking, a predicate performing the function of a predicate has at least one empty place and has no reference, while a predicate performing the function of a subject does not have any empty place but does have a reference. Frege not only regarded a concept with one or more empty places as the reference of a predicate but also took a set of objects without any empty place to be the extension of a concept with one or more empty places. Thus, it presents a complex relationship between the reference of a predicate and its corresponding extension, leading to disharmony in his theory. I argue that this is because there is a major defect in Frege’s theory of meaning, namely the neglect of common names. What he called extensions of concepts are actually extensions of common names, and the references of predicates and the extensions of common names have a substantial difference despite being closely related.     

涵义语义与关于概称句推理的词项逻辑

概称句推理具有以词项为单位的特征并且词项的涵义在其中起到了重要的作用。已有的处理用A一表达式表达涵义,不够简洁和自然。亚里斯多德三段论是一种词项逻辑,但它是外延的和单调的。这两方面的情况使得有必要考虑新的词项逻辑。涵义语义的基本观点是：语词首先表达的是涵义,通过涵义的作用,语词有了指称,表达概念。概称句三段论是更为常用的推理,有两个基本形式GAG和Gaa。在涵义语义的基础上建立的系统GAG和Gaa是关于这两种推理的公理系统。     

Completeness and Decidability Results for Some Propositional Modal Logics Containing “Actually” Operators

The addition of actually operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing actually operators has concentrated entirely upon extensions of KT5 and has employed a particular model-theoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing actually operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.     

A Semantics for Means-end Relations

There has been considerable work on practical reasoning in artificial intelligence and also in philosophy. Typically, such reasoning includes premises regarding means–end relations. A clear semantics for such relations is needed in order to evaluate proposed syllogisms. In this paper, we provide a formal semantics for means–end relations, in particular for necessary and sufficient means–end relations. Our semantics includes a non-monotonic conditional operator, so that related practical reasoning is naturally defeasible. This work is primarily an exercise in conceptual analysis, aimed at clarifying and eventually evaluating existing theories of practical reasoning (pending a similar analysis regarding desires, intentions and other relevant concepts). “They were in conversation without speaking. They didn’t need to speak. They just changed reality so that they had spoken.” Terry Pratchett, Reaper Man     

FINITE MODELS CONSTRUCTED FROM CANONICAL FORMULAS

This paper obtains the weak completeness and decidability results for standard systems of modal logic using models built from formulas themselves. This line of work began with Fine (Notre Dame J. Form. Log. 16:229–237, 1975). There are two ways in which our work advances on that paper: First, the definition of our models is mainly based on the relation Kozen and Parikh used in their proof of the completeness of PDL, see (Theor. Comp. Sci. 113–118, 1981). The point is to develop a general model-construction method based on this definition. We do this and thereby obtain the completeness of most of the standard modal systems, and in addition apply the method to some other systems of interest. None of the results use filtration, but in our final section we explore the connection.     

On Dynamic Topological and Metric Logics

We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f² (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q ⁺. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function.     

The Classification of Propositional Calculi

We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.     

中国逻辑史视野下的先秦易学与名辩学

19世纪以来的中国逻辑史研究,比照西方逻辑,仅仅将先秦的名辩学看成是中国逻辑史的主要部分。其间虽有对《周易》的涉及,但也是浅尝辄止。本文从《周易》经、传产生的年代,特别是《周易》经、传所蕴涵的逻辑内容与先秦名辩逻辑的代表人物孔子、惠施、邓析等人的思想对比,说明先秦易学逻辑对名辩逻辑产生了深刻的影响。得出《周易》的经、传发轫了中国古代逻辑中的推类理论,而名辩学尤其是墨家辩学的推类理论则是对自《周易》以来的推类逻辑传统的总结和提高的结论。