Why unary and binary operations in logic: general result motivated by interval-valued logics

Abstract

Traditionally, in logic, only unary and binary operations are used as basic ones-e.g., "not", "and", "or"-while the only ternary (and higher order) operations are the operations which come from a combination of unary and binary ones. For the classical logic, with the binary set of truth values {0,1}, the possibility to express an arbitrary operation in terms of unary and binary ones is well known: it follows, e.g., from the well known possibility to express an arbitrary operation in DNF form. A similar representation result for [0,1]-based logic was proven in our previous paper. In this paper, we expand this result to finite logics (more general than classical logic) and to multi-D analogues of the fuzzy logic-both motivated by interval-valued fuzzy logics.