多值逻辑语义博弈

【作者】 戴细华；

【导师】 鞠实儿；

【作者基本信息】 中山大学 ， 逻辑学， 2006， 博士

【摘要】 本文所研究的是逻辑博弈中的语义赋值博弈。即给定一个命题,在相应的模型中如何用博弈语义来定义命题的真值。根据J. Hintikka和G.. Sandu [18]的分类,至少存在4种不同的博弈方式。采用不同的方式可构造不同的博弈论语义学(Game-theoretical semantics)。本文致力于研究其中的一种:语义博弈((Semantical games)。这是一种采用维护/攻击博弈方法发展起来的博弈论语义学,它的目标是用博弈的方式来确定语句的真值。C.S. Peirce用解释者和回答者的两人博弈说明量词的语义,从而成为这一理论的先驱。在二十世纪六十年代, Hintikka将语义解释与博弈结合起来,在经典逻辑的基础上建立了一阶语言的语义赋值博弈(Semantical evaluation games),由此创立了这一理论。在此之后出现了所谓的逻辑语义学博弈化趋势[23],相对于各种不同语义系统的语义博弈理论纷纷出现。在这里所谓的博弈化是指:根据给定的语义理论,设计一种维护/攻击博弈方法,利用该方法可确定某一语句在某一赋值下的真值。本文的主要工作是在文献[18][19][24]的基础上,描述语义博弈的一般结构,建立多值逻辑语义博弈的一般理论。首先,我们综述了已有的几种主要逻辑博弈类型,在比较各种博弈类型共同点的基础上,抽象出我们所关注的语义博弈中的一般概念,从形式上给出它们的定义,为对语义博弈的一般结构进行讨论奠定基础。这些博弈虽然形式不一,但有许多共同点。第一,有两个参与者;第二,有一个命题;第三,这个命题有两种可能结果(真和假);第四,在这两种结果下有评判胜负的标准。其次,在一般概念的基础上重点讨论多值逻辑语义博弈的一般结构,将多值逻辑进行博弈化,从三值逻辑到m值逻辑。本文推广了J.V. Benthem (2000)的二值逻辑语义博弈,详细给出了三值及m值逻辑系统中的公式在模型中的语义博弈,通过博弈的结果定义命题的真值,并与Tarski真值定义进行比较。并指出:与经典二值逻辑语义博弈不同,多值逻辑语义博弈中,一方没有必胜策略并不能得出另一方一定有必胜策略。它与前面所述的二值逻辑语义博弈有许多不同之处。主要做法是:在每个公式前面增加一个标记得到标记公式,利用辩论双方对所辩公式的态度将博弈分成两种博弈类型,激进博弈和保守博弈;在激进博弈中有一个子博弈转换,它是不同于二值语义博弈的。我们给出了证伪者在保守博弈与激进博弈中必胜的不同充分必要条件。我们还给出了利用支付函数判断必胜策略的定理。在文中将表明,我们的方法可包含二值逻辑博弈情况。依据我们的模型,任意一个多值逻辑系统都可得到类似的博弈。最后,本文在一般结构的基础上给出了几个具体的语义赋值博弈模型。将Lukasiewicz,Kleene三值逻辑理论博弈化。此外我们给出了OPS逻辑系统的语义博弈。OPS是我们基于开放世界假设构造的一个新的逻辑系统,它是一个三值逻辑系统的二值化。我们首先证明它相对于Tarski语义的可靠性和完全性,然后给出其语义赋值博弈。更多还原

【Abstract】 In the thesis, we studied semantical evaluation games for many-valued logic. According to Hintikka and Sandu (1997), there are at least four different ways in playing games. Then different game-theoretical semantics can be construed with respect to different ways. We are going to study only on one of those ways of games: semantical evaluation games, briefly it is called semantical games. This is a game-theoretical semantics developed from the method of“verifying and falsifying”. Game-theoretical method is applied there in order to decide the truth value of a given sentence. Peirce is considered as one of the precursors of this method who employed interpreters and answers to explain the semantics of quantifiers in two-player games. In 1960s, Hintikka combined semantical interpretations and games; he then constructed semantical evaluation games for first order logic within the framework of classical logic. After that, a new trend called gamification of logical semantics comes into being (Johan van Benthem, 2004, Sec.1.6). Lots of semantical game theories with respect to various semantical systems emerged. Here gamification means, given a certain semantical theory, to design an approach for verifying/falsifying and to employ this approach to determine the truth value of some sentence under a valuation.The main work of this thesis is to characterize the general structure of semantical games based on the literature of van Benthem (2004), Hintikka and G.Sandu 1997, Hintikka (1973) first, and then to construct a general game-theoretical semantic.First,we introduce some types of logic games. Then we abstract general concepts of semantical games, given formal definitions. These games have some common properties though they are in different forms: first, there are two players in a game; secondly, there is a proposition to be considered as the game object; thirdly, this proposition has two possible results (true and false); fourthly, there is a criterion to decide winning or losing.Secondly, we brought the general structure for the semantical games of many-valued logic based on the general concepts. This thesis extends two-valued evaluation games proposed by Hintikka around 1960 and studied by van Benthem (2000), Hintikka and Sandu et al to many-valued games. We first put forward a theory of evaluation games for three-valued and m-valued formulas under respective models in general.,and then define truth values of a given sentence by a semantical game. We compare the truth-definition of Tarski and of ours. It is claimed in the thesis that many-valued evaluation games have something different from classical two-valued evaluation games. In this many-valued games, we cannot conclude from one player has no winning strategy that the other has a winning strategy. The main idea is that we add a mark in front of a formula, call it marked formula. According to the two participators’attitudes with respect to the debating proposition, we divide games into two types—radical games and conservative games. There is a subgame in the radical game (only a subgame?); it is different from two-valued game. And we showed that Falsifier has different winning conditions between the radical and conservative games. And we proved a theorem that assesses winning strategy by payoff function. We established a series of results in this thesis.At last we use this theory to have a gamification for Lukasiewicz and Kleene three-valued semantics as an example. And we constructed a logic system OPS based on the open-ended world assumption. It is a special kind of two-valued logic system. We proved its soundness and completeness theorems, and then its game semantics was given.更多还原