Semantics for Gödel Logics

01.08.2007 - 31.07.2008
Forschungsförderungsprojekt
Logic (and mathematics) can be considered as the study of the interplay between syntax ¿ strings of symbols which are transformed into other strings of symbols ¿ and semantics ¿ the (intended) interpretation of these strings of symbols. This was the important step: to separate these two parts allowing logicians at the beginning of the last century to advance logics and mathematics and thus initiating a golden century of mathematics. But why do we deal with semantics at all, couldn¿t we restrict ourselves to purely syntactic methods, i.e. to methods accessible by computational means? This was the idea of Hilbert¿s program at the beginning of the last century, to create a solid foundation of mathematics which can be treated with only syntactical methods ¿ the dream of every student, the universal computer where you feed a mathematical question and get the correct answer. Kurt Gödel destroyed this hope by exhibiting that with purely syntactical methods we can never cover all mathematical truths. So the study of semantics, especially in the case of different semantics for the same syntactical system, cannot be replaced by purely syntactical methods. The study of different semantics is actually the study of the very objects of mathematics that logic is related to. What we today call ¿Gödel logics¿ was developed from and for the analyzation of very different mathematical objects: The real numbers, Kripke frames, and Heyting algebras. The semantics based on subsets of the real interval [0,1] are concerned with the topological and order-theoretic properties of the reals, and the expressiveness of first order language with respect to these properties. These semantics have been mainly developed by Baaz et al. with substantial contributions by the applicant. Kripke frames form the prime semantics for Intuitionistic Logic and Modal Logics, and the study of logics of specific Kripke frames (linear with constant domains) has been carried out mainly by Japanese scientists. These Kripke frames can also be considered as a semantic for Gödel logics as was recently shown by the applicant in the course of the MC fellowship. Besides their function as semantics for Intuitionistic Logic, Kripke frames have been used in the development of modal and temporal logics. Algebraic semantics based on special Heyting algebras were introduced by Hàjek, whereas Heyting algebras are special instances of lattices. In his work Hàjek considered t-norm based logics as the foundation of fuzzy logics. In the wide class of t-norm based logics three fundamental logics form a basis of all the other logics: Łukasiewicz logic, Product logic, and Gödel logic. By combining these logics we get all possible continuous t-norm based logics. If all these semantics were the same, it would be useless to deal with all of them. But while semantics can coincide in the base case (e.g. the propositional logic), their extensions often exhibit interesting properties and differences of the semantics. A typical example is the class of finite Kripke frames versus the class of all Kripke frames: The propositional logics for both classes coincide, but the quantified propositional logic of the former class is decidable, while the one of the later is not even recursively enumerable. While the semantics do exhibit different properties for different extensions, they are still linked together via the syntax of these logics. This syntax¿semantic relation is one of the most important ones in logic. In fact modern logic can be seen as a history of the interplay between syntactical and semantical studies, and their relations. We aim at a unified presentation of these semantics, a transfer of results and techniques between these semantics, and the development of criteria for discussing and comparing semantics for first order many valued logics.

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Grant funds

  • European Commission (EU) 6.FP: MOBILITY - Human resources and mobility Marie-Curie Grants 6.Rahmenprogramm für Forschung European Commission - Framework Programme European Commission Call identifier MERG-CT-2007 Application number 046422

Forschungsschwerpunkte

  • Computational Intelligence: 100%

Schlagwörter

DeutschEnglisch
Gödel LogikGödel logics
AlgebraAlgebra
Intermediäre LogikenIntermediate logics
SemantikSemantics
AnalysisAnalysis

Publikationen