Logic as a vehicle for sound reason has a long and lustrous history and while most developments follow the traditional notions of binary truth and crisp sentences, great efforts have been placed into the problem of reasoning with uncertainties. To this end the field of "fuzzy logic" is now of great importance both theoretically and practically. The present monograph seeks to extend and clarify the treatment of non-classical notions of logic and, more broadly, information representation in general. This is done using two theoretical developments presented with additional discussions concerning possible applications.
The first theoretical development takes the form of a novel and strictly categorical term monad that readily allows for a multitude of non-classical situations and extensions of the classical term concept. For example, using monad composition we may represent and perform substitutions over many-valued sets of terms and thereby represent uncertainty of information. As a complementary example, we may extend this term monad to incorporate uncertainty on the level of variables and indeed the operators themselves. These two notions are embodied inthe catchphrases "computing with fuzzy" and "fuzzy computing".
The second theoretical development is a direct generalization of the notion of general logics, a successful categorical framework to describe and interrelate the various concepts included under theumbrella term of 'logic'. The initial leap towards general logics was the introduction of institutions by Goguen and Burstall. This construction cover the semantic aspects of logics and in particular, axiomatizes the crucial satisfaction relation. Adding structures for, e.g., syntactic entailment and proof calculi, Meseguer established general logics as a framework capable of describing a wide range of logics. We will in our generalization further extend general logics to more readily encompass non-classical notions of truth such as in fuzzy logics and logics operating over non-classical notions of sets of sentences.