On Gödel Algebras of Concepts

Abstract

Beside algebraic and proof-theoretical studies, a number of different approaches have been pursued in order to provide a complete intuitive semantics for many-valued logics. Our intention is to use the powerful tools offered by formal concept analysis FCA to obtain further intuition about the intended semantics of a prominent many-valued logic, namely Godel, or Godel-Dummett, logic. In this work, we take a first step in this direction. Godel logic seems particularly suited to the approach we aim to follow, thanks to the properties of its corresponding algebraic variety, the class of Godel algebras. Furthermore, Godel algebras are prelinear Heyting algebras. This makes Godel logic an ideal contact-point between intuitionistic and many-valued logics. In the literature one can find several studies on relations between FCA and fuzzy logics. These approaches often amount to equipping both intent and extent of concepts with connectives taken by some many-valued logic. Our approach is different. Since Godel algebras are residuated lattices, we want to understand which type of concepts are expressed by these lattices. To this end, we investigate the concept lattice of the standard context obtained from the lattice reduct of a Godel algebra. We provide a characterization of Godel implication between concepts, and of the Godel negation of a concept. Further, we characterize a Godel algebra of concepts. Some concluding remarks will show how to associate equivalence classes of formulae of Godel logic with their corresponding formal concepts.