Triangle algebras: A formal logic approach to interval-valued residuated lattices

In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of interval-valued residuated lattices (IVRLs). Furthermore, we present triangle logic (TL), a system of many-valued logic capturing the tautologies of IVRLs. Triangle algebras are used to cast the essence of using closed intervals of L as truth values into a set of appropriate logical axioms. Our results constitute a crucial first step towards solving an important research challenge: the axiomatic formalization of residuated t-norm based logics on LI, the lattice of closed intervals of [0,1], in a similar way as was done for formal fuzzy logics on the unit interval.