Modellierung vager natürlichsprachlicher Quantoren über Dialogspiele und Fuzzy Logik

Abstract

In natural language (NL), quantifiers are often used to make statements about states of affairs, like “Many people like football”, and “About half the people are female”. In particular, many and few express that some set of objects is relatively big, or small respectively. The semantics of those two quantifiers is not fixed once and for all, but rather depends on contextual information. Likewise, quantifiers like about half and almost all show a comparable behavior, as the tolerance margins that make corresponding statements acceptable can change from one situation to another. Fuzzy logic is often used to model such NL constructs, in particular contemporary t-norm based mathematical fuzzy logics (MFLs). Hintikka expressed Classical Logic (CL) game semantically, and Giles expressed Łukasiewicz logic (Ł), a MFL, game semantically. The shared underpinning is a two player zero sum game of perfect information, where the two players act strategically. Fermüller and Roschger have augmented Giless game by a third non-strategic player, thereby introducing what we call the random witness selection principle into the framework. The latter principle allows us to also express two other MFLs, Gödel logic and Product logic, game semantically. We achieve this by allowing for propositional quantification, which enables us to model the Delta operator, which is basically a projection operator, evoking discontinuous truth functions. This is needed to express Gödel implication in Giless framework. Moreover, the propositional quantifier based on the random witness selection principle, together with the Delta operator and the existential propositional quantifier, allows us to model multiplication and division of truth functions, which we need to define the connectives of Product logic. Building on this result, we show how to define all MFLs that are finitely representable in our framework. Furthermore, the gained expressibility is used to model a variety of NL quantifiers within the framework. This pursuit is conducted in a step-by-step manner, that guarantees neat interpretability of statements. First, we model semi-fuzzy quantifiers, i.e. quantifiers that can only take classical arguments, i.e. predicates that evaluate to either (definitely) true or false. Then we lift those to fully-fuzzy quantifiers in a systematic and principle guided way, by means of quantifier fuzzification mechanisms (QFMs). As a final contribution of this thesis, we define and test, by means of an implementation, a full-fledged query language, featuring quantifiers based on the random witness selection principle. The results show that probabilistic evaluations not only are suitable to model vagueness in NL, but also increase efficiency in presence of large amounts of data.