Logics which allow Degrees of Truth and Degrees of Validity
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Date
2005-02-07
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Universität Dortmund
Abstract
In dieser Dissertation werden Semantiken logischer Systeme, die sowohl Vagheit (im Sinne gradueller Wahrheitsbewertung logischer Formeln) als auch Unsicherheit (im Sinne gradueller Vertrauensbewertung logischer Formeln) auszudrücken erlauben, vom Standpunkt der mathematischen Logik aus betrachtet. Üblicherweise werden zur Repräsentation von Vagheit mehrwertige Logiken verwendet, wobei zur konkreten Wahrheitsbewertung häufig Formeln mit konkreten Wahrheitswerten verknüpft werden. Zur Repräsentation von Unsicherheit (im Sinne unvollständigen Wissens oder Vertrauens) werden Formeln der zweiwertigen Logik mit Vertrauensgraden bewertet. Es wurden eine Vielzahl logischer Systeme mit bewerteten Formeln in der Literatur beschrieben, mit zum Teil stark abweichenden Interpretationen der Struktur und Semantik von Markierungen. Zum Teil wird jedoch die Bedeutung von Markierungen nicht präzise definiert, was die Interpretation von Inferenzergebnissen erschwert bis unmöglich macht. Solange nicht eine präzise quantitative Theorie wie etwa die Wahrscheinlichkeitstheorie zur Erklärung von Markierungswerten verwendet wird, gibt es keine kanonische Erklärung für die Bedeutung einer Markierung. Führt dies dann zu einer Vielzahl möglicher Erklärungen, ohne dass diese anhand präzise dargelegter Kriterien verglichen werden können, so wird der Nutzen solcher gradueller Bewertungen insgesamt fraglich. Ein Weg zur Verbesserung dieser Situation liegt darin, Bewertungssysteme anhand grundlegender Bedeutungsunterschiede der Bewertungen in Klassen einzuteilen. Hier werden besonders Bewertungen des Wahrheitsgehalts sowie des Vertrauens in die Gültigkeit logischer Formeln betrachtet. Es gibt gut ausgearbeitete Theorien bewerteter Logiken, die zu der einen oder der anderen Klasse gehören. In dieser Dissertation wird ein sehr allgemeines System zur Definition von Markierungen zur Bewertung von Wahrheit bzw. Vertrauen beschrieben, zusammen mit den sich daraus ergebenden kanonischen Definitionen des Modellbegriffs sowie der semantischen Folgerung für markierte Formeln. Die resultierenden markierten Logiken sind sehr ausdrucksstark und erlauben sowohl Vagheit als auch Unsicherheit als auch Kombinationen beider gradueller Konzepte zu repräsentieren. Semantiken solcher Logiken werden im Allgemeinen und für interessante Spezialfälle studiert.
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.
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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Keywords
Fuzzy-Logik, Mehrwertige Logik, Possibilistische Logik, Verbandslogik, Modelltheorie, Widerlegung, Wahrheitswerte, Vertrauensgrade, Fuzzy logic, Many-valued logic, Possibilistic logic, Lattice logic, Model theory, Refutation, Degrees of truth, Degrees of trust