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n-Contractive BL-logics

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Abstract

In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in Hájek (Metamathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom \({\phi^n\rightarrow\phi^{n+1}}\), for some \({n\in\mathbb{N}^+}\)) axiomatic extensions of BL and their corresponding varieties: BLn, SBLn, BL n and SBL n . Concerning BLn we have that every BLn-chain is isomorphic to an ordinal sum of MV-chains of at most n + 1 elements, whilst every BL n -chain is isomorphic to an ordinal sum of MV n -chains (for SBLn and SBL n a similar property holds, with the difference that the first component must be the two elements boolean algebra); all these varieties are locally finite. Moving to the content of the paper, after a preliminary section, we will study generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we will analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.

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Authors and Affiliations

  1. Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, via Saldini 50, 20133, Milano, Italy

    Matteo Bianchi

  2. Dipartimento di Scienze Matematiche e Informatiche “Roberto Magari”, Università degli Studi di Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Franco Montagna

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  1. Matteo Bianchi
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  2. Franco Montagna
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Correspondence to Matteo Bianchi.

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Bianchi, M., Montagna, F. n-Contractive BL-logics. Arch. Math. Logic 50, 257–285 (2011). https://doi.org/10.1007/s00153-010-0213-8

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