The connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation defined in [14] can be obtained by composing two closure operators and that the resulting operator is still a closure operator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation.

In [This Zeitschrift 25 , 45-52, 119-134, 447-464], Pavelka systematically discussed propositional calculi with values in enriched residuated lattices and developed a general framework for approximate reasoning. In the first part of this paper we introduce the concept of generalized quantifiers into Pavelka's logic and establish the fundamental theorem of ultraproduct in first order Pavelka's logic with generalized quantifiers. In the second part of this paper we show that the fundamental theorem of ultraproduct in first order Pavelka's logic is preserved (...) under some direct product of lattices of truth values. (shrink)

We show that compactness is preserved by arbitrary direct products of lattices of truth values and that the Löwenheim-Skolem property is preserved by finite direct products of lattices of truth values.

ABSTRACT In this paper, we propose a formalization of fuzzy logic and obtain some results concerning the composition, exchange and compatibility with propositional connectives of fuzzy quantifiers, modifiers and qualifiers in this setting.