# Logics which allow Degrees of Truth and Degrees of Validity

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- Universität Dortmund
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## Abstract

In this dissertation, the semantics of logical systems which are able to express vagueness and graded truth assessment as well as doubt and graded trust assessment are investigated from the point of view of mathematical logic. Traditionally, logics for modelling graded truth have been many-valued logics which allow truth values between 0 (false) and 1 (true). In applications, sometimes truth values are attached to formulae to assess the truth of the formula. In logics for modelling graded trust, usually trust (or plausibility, or possibility, or belief) degrees are attached to formulae from classical two-valued logic to assess the trust in the knowledge expressed by this formula. Several logical systems using labelled formulae (i. e. formulae to which some label is attached) have been described in the literature, with varying interpretations concerning structure and semantics of labels. In many cases, however, the meaning of a label is not precisely specified, casting doubt on what, from a semantic point of view, is really formalised by labelled formulae or a corresponding inference mechanism. Without a specific background theory for the meaning of labels (as is given, for instance, by probability theory), of course no canonical paradigm for specifying the structure and processing of labels exists. Consequently, several different such paradigms have been developed. Differences between these systems combined with the lack of a precisely defined semantics for labels have led to critique of such logical systems as a whole, because it must seem suspicious if from one and the same knowledge base of labelled formulae, it is possible to infer totally different results, without a clear semantic theory which can explain the differences. There have been attempts to clarify this situation, especially by distinguishing whether a system of labelled logical formulae is used for the representation of graded truth assessment or graded trust (or possibility, necessity, plausibility, uncertainty, belief ) assessment with respect to the states of affairs being modelled. Logical systems which can accomplish one or the other task have been studied and compared. In this dissertation, a very general approach to the definition of labels for expressing graded truth and graded trust is described. This definition gives rise to a canonical definition of the concepts of model and semantic consequence for the resulting logic of labelled formulae. The expressive power of such logics is very high. A label can express uncertainty about truth or trust or any combination of both. A systematic study of the semantics of these logical systems is given here, as well as a discussion and comparison of special cases.

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