69 research outputs found
Hypersequents and the Proof Theory of Intuitionistic Fuzzy Logic
Takeuti and Titani have introduced and investigated a logic they called
intuitionistic fuzzy logic. This logic is characterized as the first-order
Goedel logic based on the truth value set [0,1]. The logic is known to be
axiomatizable, but no deduction system amenable to proof-theoretic, and hence,
computational treatment, has been known. Such a system is presented here, based
on previous work on hypersequent calculi for propositional Goedel logics by
Avron. It is shown that the system is sound and complete, and allows
cut-elimination. A question by Takano regarding the eliminability of the
Takeuti-Titani density rule is answered affirmatively.Comment: v.2: 15 pages. Final version. (v.1: 15 pages. To appear in Computer
Science Logic 2000 Proceedings.
Incompleteness of a first-order Gödel logic and some temporal logics of programs
It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows
Quantified Propositional Gödel Logics
It is shown that Gqp↑, the quantified propositional Gödel logic based on the truth-value set V↑ = {1 - 1/n : n≥1}∪{1}, is decidable. This result is obtained by reduction to Büchi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of Gqp↑ as the intersection of all finite-valued quantified propositional Gödel logics
Compactness of first-order fuzzy logics
One of the nice properties of the first-order logic is the compactness of
satisfiability. It state that a finitely satisfiable theory is satisfiable.
However, different degrees of satisfiability in many-valued logics, poses
various kind of the compactness in these logics. One of this issues is the
compactness of -satisfiability. Here, after an overview on the results
around the compactness of satisfiability and compactness of -satisfiability
in many-valued logic based on continuous t-norms (basic logic), we extend the
results around this topic. To this end, we consider a reverse semantical
meaning for basic logic. Then we introduce a topology on and
that the interpretation of all logical connectives are continuous with respect
to these topologies. Finally using this fact we extend the results around the
compactness of satisfiability in basic ogic
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