A Fuzzy Formal Logic for Interval-valued Residuated Lattices

Fuzzy formal logics were introduced in order to handle graded truth values instead of only ‘true’ and ‘false’. A wide range of such logics were introduced successfully, like Monoidal T-norm based Logic, Basic Logic, Gödel Logic, Lukasiewicz Logic etc. However, in general, fuzzy set theory is not only concerned with vagueness, but also with uncertainty. A possible solution is to use intervals instead of real numbers as membership values. In this paper, we present an approach with triangle algebras, which are algebraic characterizations of interval-valued residuated lattices. The variety of these structures corresponds in a sound and complete way to a logic that we introduce, called Triangle Logic (in the same way as, e.g., BL-algebras and Basic Logic). We will show that this truthfunctional approach, along with the residuation principle, has some consequences that seem to obstruct an easy and proper interpretation for the semantics of Triangle Logic.