Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and... The rational first-order Pavelka logic is an expansion of the infinite-valued first-order Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent calculus G1Ł∀ and a noncumulative hypersequent calculus G2Ł∀ without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules. In particular, we show that every GŁ∀-provable sentence is G1Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G2Ł∀-provable. For a tableau version of the calculus G2Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G2Ł∀-provable sentence in the prenex form. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Siberian Advances in Mathematics Springer Journals

Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Allerton Press, Inc.
Subject
Mathematics; Mathematics, general
ISSN
1055-1344
eISSN
1934-8126
D.O.I.
10.3103/S1055134418020013
Publisher site
See Article on Publisher Site

Abstract

The rational first-order Pavelka logic is an expansion of the infinite-valued first-order Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent calculus G1Ł∀ and a noncumulative hypersequent calculus G2Ł∀ without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules. In particular, we show that every GŁ∀-provable sentence is G1Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G2Ł∀-provable. For a tableau version of the calculus G2Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G2Ł∀-provable sentence in the prenex form.

Journal

Siberian Advances in MathematicsSpringer Journals

Published: May 30, 2018

References

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