On quasi-varieties of multiple valued logic models

Multiple valued logic (abbreviated MVL and also known as ‘many-valued’ or ‘multi-valued’ logic) has a long tradition [17, 15, 7] and needs no presentation. In this paper we lift the concept of quasi-variety of first-order models from classical (two valued) logic to MVL and study some of its important properties. Our results can be seen as MVL extensions or generalizations of corresponding classical results. In the classical two valued framework quasi-varieties of models have been studied rather extensively by pioneers such as Mal’cev [16]; in their algebraic version they also play an important role in general or universal algebra [10]. Because of their close relationship with initiality and because of the great role played by the latter concept in programming language semantics [9, 13] and logic programming (there known as ‘least Herbrand models’) [14], they have also been studied in computing science motivated works such as [18] within the very general categorical abstract model theoretic framework of the so-called ‘theory of institutions’ of Goguen and Burstall [8]. A more recent upgraded study of quasi-varieties at the level of abstract institutions can be found in [4]. The motivation for our investigation of quasi-varieties ofMVLmodels and of their relationship to the existence of initial models (of theories) may be seen from two different but complementary directions.