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

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.