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A consequence relation for graded inference within the frame of infinite-valued Łukasiewicz logic. (English) Zbl 1271.03037
This paper presents a family of consequence relations \({}^{\eta}\!\!\blacktriangleright_{\zeta}\) for thresholds \(\eta,\zeta \in [0,1]\) inspired by the family of consequence relations of the form \({}^{\eta}\!\!\vartriangleright_{\zeta}\) first defined by J. Paris in a simplified version for \(\eta = \zeta\) in [Log. J. IGPL 12, No. 5, 345–353 (2004; Zbl 1071.03011)] and further studied and extended by J. B. Paris et al. in [Adv. Soft Comput. 46, 291–307 (2008; Zbl 1272.68395); Int. J. Approx. Reasoning 50, No. 8, 1151–1163 (2009; Zbl 1191.68697)] and by the author in his PhD thesis (Manchester, 2008). The latter formalize graded inference in terms very similar to those of the currently presented relations \({}^{\eta}\!\!\blacktriangleright_{\zeta}\), with the main difference being that the underlying semantics for the thresholds \(\eta\), \(\zeta\) of the relation \({}^{\eta}\!\!\vartriangleright_{\zeta}\) was probabilistic, whereas the current interpretation refers to truth degrees – the standard interpretation for the thresholds in the context of Łukasiewicz logic (see, e.g., [P. Hájek, Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030); J. B. Paris, The uncertain reasoner’s companion. A mathematical perspective. Cambridge: Cambridge University Press (1994; Zbl 0838.68104)]). Graded inference among Łukasiewicz sentences are formalized in the following terms: given a set of premises \(\Gamma\) and a threshold \(\eta\), a consequence \(\theta\) is a sentence entailed by Łukasiewicz logic that holds to a degree at least \(\zeta\) whenever the premises hold to a degree at least \(\eta\), denoted \( \Gamma {}^{\eta}\!\!\blacktriangleright_{\zeta} \theta\).

MSC:
03B50 Many-valued logic
68T37 Reasoning under uncertainty in the context of artificial intelligence
03B48 Probability and inductive logic
03B53 Paraconsistent logics
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