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Density elimination and rational completeness for first-order logics. (English) Zbl 1133.03028
Artemov, Sergei N. (ed.) et al., Logical foundations of computer science. International symposium, LFCS 2007, New York, NY, USA, June 4–7, 2007. Proceedings. Berlin: Springer (ISBN 978-3-540-72732-3/pbk). Lecture Notes in Computer Science 4514, 132-146 (2007).
Summary: Density elimination by substitutions is introduced as a uniform method for removing applications of the Takeuti-Titani density rule from proofs in first-order hypersequent calculi. For a large class of calculi, density elimination by this method is guaranteed by known sufficient conditions for cut-elimination. Moreover, adding the density rule to any axiomatic extension of a simple first-order logic gives a logic that is rational complete; i.e., complete with respect to linearly and densely ordered algebras: a precursor to showing that it is a fuzzy logic (complete for algebras with a real unit interval lattice reduct). Hence the sufficient conditions for cut-elimination guarantee rational completeness for a large class of first-order substructural logics.
For the entire collection see [Zbl 1121.03005].

MSC:
03F05 Cut-elimination and normal-form theorems
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B52 Fuzzy logic; logic of vagueness
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