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

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 G 1 Ł∀ and a noncumulative hypersequent calculus G 2 Ł∀ without structural inference rules. We compare th...

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Veröffentlicht in:Siberian advances in mathematics 2018-04, Vol.28 (2), p.79-100
1. Verfasser: Gerasimov, A. S.
Format: Artikel
Sprache:eng
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Zusammenfassung: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 G 1 Ł∀ and a noncumulative hypersequent calculus G 2 Ł∀ 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 G 1 Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G 2 Ł∀-provable. For a tableau version of the calculus G 2 Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G 2 Ł∀-provable sentence in the prenex form.
ISSN:1055-1344
1934-8126
DOI:10.3103/S1055134418020013