Archive for Mathematical Logic

, Volume 50, Issue 3–4, pp 257–285 | Cite as

n-Contractive BL-logics

  • Matteo Bianchi
  • Franco Montagna
Original Article


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: BL n , SBL n , BL n and SBL n . Concerning BL n we have that every BL n -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 SBL n 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.


Many-Valued logics Basic logic n-Contractive logics Residuated lattices Varieties of lattices MV-algebras 

Mathematics Subject Classification (2000)

03B47 03B50 03C07 03G10 06B20 06D35 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanoItaly
  2. 2.Dipartimento di Scienze Matematiche e Informatiche “Roberto Magari”Università degli Studi di SienaSienaItaly

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