New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, 2007, Volume 1: Invited Lectures and Special Sessions

Fuzzy formal logics were introduced in or-der to handle graded truth values insteadof only ‘true’ and ‘false’. A wide range ofsuch logics were introduced successfully, likeMonoidal T-norm based Logic, Basic Logic,Gödel Logic, Lukasiewicz Logic etc. How-ever, in general, fuzzy set theory is not onlyconcerned with vagueness, but also with un-certainty. A possible solution is to use inter-vals instead of real numbers as membershipvalues. In this paper, we present an approachwith triangle algebras, which are algebraiccharacterizations of interval-valued residu-ated lattices. The variety of these structurescorresponds in a sound and complete way to alogic that we introduce, called Triangle Logic(in the same way as, e.g., BL-algebras andBasic Logic). We will show that this truth-functional approach, along with the residu-ation principle, has some consequences thatseem to obstruct an easy and proper interpre-tation for the semantics of Triangle Logic.