Searching for the roots of non-contradiction and excluded-middle

Abstract

This paper tries to obtain frameworks in which one can prove as theorems, and with few assumptions, the laws of non-contradiction (NC) and excluded-middle (EM), for a large class of very general systems including orthocomplemented and De Morgan lattices, as well as the theories of L -fuzzy sets. Its goal is nothing else than to begin with a search for the algebraic roots of these laws. For such a goal, and similarly to ancient Aristotelian logic, the laws are referred to self-contradiction and not to falsity as it is done in modern logics. It is additionally shown that there are some structures where it is possible to find special operations that separate those operations that verify NC from those verifying EM.