A Three-Valued Approach to Overcome Paradoxes in Syntactic Theory.pdf

A Three-Valued Approach to OvercomeParadoxes in Syntactic TheoryManfred KerberSchool of Computer ScienceThe University of BirminghamBirmingham B15 2TT, Englande-mail: M.Kerber@cs.bham.ac.ukURL: http://www.cs.bham.ac.uk/~mmkTel.: (+44)-121-414-4787, Fax.: (+44)-121-414-4281Abstract. A powerful syntactic theory as well as expressive modal log-ics have to deal with self-referentiality. Self-referentiality and paradoxesseem to be close neighbours and depending on the logical system, theyhave devastating consequences, since they introduce contradictions andtrivialise the logical system. There is a large amount of dierent at-tempts to tackle these problems. Some of them are compared in thispaper, futhermore a simple approach based on a three-valued logic is ad-vocated. In this approach paradoxes may occur and are treated formally.However, it is necessary to be very careful, otherwise a system built onsuch an attempt trivialises as well. In order to be able to formally dealwith such a system, the reason for self-referential paradoxes is studied inmore detail and a semantical condition on the connectives is given suchthat paradoxes are excluded.1 IntroductionThe symbolic representation of knowledge and belief plays a crucial ro?le in arti-cial intelligence, in particular in multi-agent systems. For their representationtwo principally dierent formal systems, modal logic and meta-systems havebeen developed. Both of them come with a detailed ramication of realisationsdepending on what is actually adequate and what is not.The predominant formalism for representing knowledge and belief in agentsystems is based on modal logics (there is a vast amount of literature on modallogic. I just point to Fittings general introduction, where further pointers can befound, see [4, p.365{448]). However, meta-systems have some crucial advantagesover modal logic, although some problems as well. As Davis points out in [2, p.77]the dierence between the two approaches is pretty much the same a