Multi-valued Logic Semantics Game

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.