Internet-publication. — 2012. — 26 p. English. (OCR-слой).
[Preprint submitted to Fuzzy Sets and Systems. November 26, 2012].
[Simion Stoilow Institute of Mathematics of the Romanian Academy].
Abstract.We develop many-valued logic, including a generic abstract model theory, over a fully abstract syntax. We show that important many-valued logic model theories, such as traditional first-order many-valued logic and fuzzy multi-algebras, may be conservatively embedded into our abstract framework. Our development is technically based upon the so-called theory of institutions of Goguen and Burstall and may serve as a template for defining at hand many-valued logic model theories over various concrete syntaxes or, from another perspective, to combine many-valued logic with other logical systems. We also show that our generic many-valued logic abstract model theory enjoys a couple of important institutional model theory properties that support the development of deep model theory methods.
Introduction.Many-valued logic (abbreviated mvl; also known as ‘multiple-valued’ or ‘multi-valued’ logic) has a long tradition [18, 26, 30] and needs no presentation. Our paper builds on the idea that the essence of mvl is actually independent of the concrete syntactic context in which it is usually presented, that in fact it is independent of any syntactic context. We realize this idea by developing mvl over a fully abstract syntax by employing the conceptual machinery of the so-called institution theory of Goguen and Burstall [19].
This is a categorical abstract model theory that arose about three decades ago within specification theory as a response to the explosion in the population of logics in use there, its original aim being to develop as much computing science as possible in a general uniform way independently of particular logical systems.
This has now been achieved to an extent even greater than orginally thought, as institution theory became the most fundamental mathematical theory underlying algebraic specification theory (in its wider meaning) [33], also being increasingly used in other area of computer science. Moreover, institution theory constitutes a major trend in the so-called ‘universal logic’ (in the sense envisaged by Jean-Yves Beziau [2, 3]) which is considered by many a true renaissance of mathematical logic. A lot of model theory has gradually been developed at the level abstract institutions (see [10]).
The technical side of our abstract mvl development may be briefly described as follows. Given an abstract category of signatures and an abstract sentence functor (that gives the sets of sentences corresponding to the signatures) we build both a syntax and a model theory as well as a satisfaction relation between them.
Introduction.Preliminaries.Categories.
Institutions.
Examples of institutions.
Example (FOL).
Example (Many-valued logic).
Definition (Residuated lattice).
Abstract many-valued institutions.Technical preliminaries.
Definition (Quantification space).
Definition. A logic syntax is a pair (Sign, Sen): (Sign - Category, Sen - Functor).
The definition of I(L)
Definition (I(L) models).
Definition (I(L) sentences).
Proposition SenI(L) is functor.
Definition (I(L) satisfaction).
Corollary (I(L) satisfaction condition).
Remark (Relationship to [16]).
Embedding concrete many-valued institutions into I(L).MVL revisited.
Remark (The second order extension).
Remark (Adding crisp equality).
Remark (Propositional many-valued logic).
Fuzzy multi-algebras.
Example (The institution of fuzzy multi-algebras).
Institution-theoretic properties of I(L).Model amalgamation.
Definition (Amalgamation square): (A commutative square of signature morphisms is an amalgamation square...).
Definition (Semi-exactness).
The method of diagrams.
Conclusions and Future Research.Acknowledgements.References (
37 publ).