涵义语义与关于概称句推理的词项逻辑

Sense Semantics and Term Logics for Generic Reasoning

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λ-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。

Generic reasoning makes a feature of having terms as its units, and senses of terms play an important role in such reasoning. The standard approach represents senses by λ-expressions, which are nevertheless not simple and natural enough. The Aristotelean syllogism is a term logic, but it is extensional and monotonic. It is, therefore, necessary to consider some new kind of term logic for generic reasoning, which is built on terms on the one hand, but is intensional and non-monotonic on the other. Toward this end, the present paper discusses sense semantics and generic syllogism. The basic ideas behind sense semantics are as follows： words express senses in the first place, and then, with the help of senses, they denote and express concepts. Generic syllogism, which is much more commonly used than the ordinary syllogism, has two basic forms： GAG and Gaa, and two systems GAG and Gaa for them are established based on sense semantics. There are three parts in this paper. The first part （section 1） is about similarity and difference between the Aristotelean syllogism and the generic syllogism. The Aristotelean syllogism Barbara （AAA-1）, which can be written as （A（M, P） ∧ A（S, M）） → A（S, P） （AAA）, is not the form of the kind of reasoning such as ＂Birds fly, penguins are birds, so penguins fly＂, because there are generic sentences in it. The correct form for this reasoning should be （G（M, P） A A（S, M）） 〉 G（S, P） （GAG）. There is another kind of syllogism, namely, ＂Birds fly, Tweety is a bird, so Tweety flys＂. For the same reason, the form of this kind of reasoning should be （G（M, P） ∧ Ms ） 〉 Ps （Gaa）, but not （A（M, P） ∧ Ms） → Ps （Aaa）. Based on the view that a term has four meanings, that is, sense, reference （extension）, concept and intension, in the second part of this paper （section 2 and 3）, frames and models for GAG and Gaa are defined. This is the sense semantics, which can explain the validity of generic syllogism. In the third part （section 4）, two systems GAG and Gaa are given, and the formulae GAG and Gaa are theorems of these two systems respectively. GAG is a system based on MQ with one extra axiom A（S, M） → （G（M, P）〉G（S, P））. However, to obtain the system Gaa based on MQ, we need three extra axioms, two of which are about generic reasoning. This shows that Gaa is more complicated than GAG.