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• W2402685112 abstract "We consider GAűdel logics extended by an operator whose semantics is given by I(o(A)) = min{1, r + I(A)}. The language of propositional GAűdel logics L consists of a countably infinite set Var of propositional variables and the connectives ⊥, ⊃, ∧, ∨ with their usual arities. We will consider extensions by a unary connective o, by a unary connective 4 or by both. For any r ∈ [0, 1], a GAűdel r-interpretation I maps formulas to V such that I(⊥) = 0, I(A ∧B) = min{I(A), I(B)}, I(A ∨B) = max{I(A), I(B)}, I(A ⊃ B) = { 1 I(A) ≤ I(B), I(B) I(A) > I(B). If the language contains o resp. 4, we additionally require I(o(A)) = min{1, r + I(A)}, I(4(A)) = { 1 I(A) = 1 0 I(A) < 1. Let G be some Hilbert-Frege style proof calculus that is sound and complete for propositional GAűdel logics (without o and 4), e. g. take a proof system for intuitionistic logic, plus the schema of linearity (A ⊃ B) ∨ (B ⊃ A), see [3] or, alternatively, use one of the systems described in [4]. We prove that G enhanced by the axiom schemata (⊥ ≺ o⊥) ⊃ (A ≺ oA), (⊥ ↔ o⊥) ⊃ (A ↔ oA), and o(A ⊃ B) ↔ (oA ⊃ oB) is sound and complete w. r. t. the above semantics. Generalizing ideas from [2], we also give an algorithm that constructs a proof for any valid formula. However, this semantics fails to have a compact entailment. The above proof system can also be further combined with a proof system for 4, see [1], to yield a sound and complete calculus for the valid formulas in that language. While the propositional fragment has quite a simple structure, we will show that first order GAűdel logic enhanced by this ring operator is not recursively enumerable, using a technique by Scarpellini [5] employed for Łukasiewicz logic. This ring operator makes the borderline of similarities and contrasts between Łukasiewicz logic visible. The situation changes if one interprets o, more generally, as a function with certain monotonicity properties. ∗partially supported by Austrian Science Fund (FWF-P22416) A. Voronkov, G. Sutcliffe, M. Baaz, C. Fermuller (eds.), LPAR-17-short (EPiC Series, vol. 13), pp. 13–14 13 Godel logics with an operator shifting truth values M. Baaz, O. Fasching" @default.
• W2402685112 created "2016-06-24" @default.
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• W2402685112 date "2010-01-01" @default.
• W2402685112 modified "2023-09-27" @default.
• W2402685112 title "Gödel logics with an operator shifting truth values." @default.
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