## Standard completeness theorem for $$\Pi$$MTL.(English)Zbl 1071.03013

$$\Pi$$MTL is a schematic extension of the monoidal t-norm-based logic (MTL) by a new connective that is in $$[0, 1]$$ interpreted by the ordinary product of reals. It is proved that $$\Pi$$MTL satisfies the standard completeness theorem, namely, that a formula of $$\Pi$$MTL is provable iff it is true in a degree 1 in all $$\Pi$$MTL-chains in $$[0, 1]$$ with finitely many Archimedean classes. From the algebraic point of view this means that the class of $$\Pi$$MTL-algebras in $$[0, 1]$$ generates the variety of all $$\Pi$$MTL-algebras.

### MSC:

 03B52 Fuzzy logic; logic of vagueness 03B50 Many-valued logic 03G25 Other algebras related to logic
Full Text:

### References:

 [1] Birkhoff, G.: Lattice Theory, 3rd edn. Am. Math. Soc. Colloquium Publications, 1995 · Zbl 0063.00402 [2] Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn. Springer, 1996 [3] Esteva, F., Godo, L.: Monoidal t-norm Based Logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124(3), 271–288 (2001) · Zbl 0994.03017 [4] Esteva, F., Gispert, J., Godo, L., Montagna, F.: On the Standard Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic. Studia Logica 71(2), 199–226 (2002) · Zbl 1011.03015 [5] Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963 · Zbl 0137.02001 [6] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998 · Zbl 0937.03030 [7] Hájek, P.: Observations on the Monoidal T-norm Logic. Fuzzy Sets and Systems 132(1), 107–112 (2002) · Zbl 1012.03035 [8] Höhle, U.: Commutative, Residuated l-monoids. In: Non–Classical Logics and Their Applications to Fuzzy Subsets, U. Höhle, E.P. Klement (eds.), Kluwer Academic Publisher, Dordrecht, 1995, pp. 53–106 · Zbl 0838.06012 [9] Jenei, S., Montagna, F.: A Proof of Standard Completeness for Esteva and Godo’s Logic MTL. Studia Logica 70(2), 183–192 (2002) · Zbl 0997.03027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.