Analytical Frameworks for Non-Classical Mathematical Reasoning

Abstract

In this paper we will present to build such a formal framework for combined nonclassical reasoning as it has arisen with the attempt to bypass classical logic and traditional foundation of mathematics. Classical (binary) logic, determinism and rigid formalism are no longer adequate to deal with complex, uncertain or context dependent phenomena that arise in modern mathematics and its applications.

This analysis investigates the theoretical underpinnings, logical organization and epistemological significance of non-classical mathematics is like such paraconsistent logic, fuzzy logic and multi-valued, intuitionist logic, and multi-valued logics. Through an examination of these practices, the study pursues three goals: to reconceptualize commonalities they share and divergences among them; to demonstrate the extent to which they can be used for representations of uncertainty, inconsistency, and indeterminacy in doing mathematics.

The proposed logic covers logical, philosophical and methodological aspects of non-classical reasoning in order to deliver a systematic instrument for the evaluation of non-classical models. This framework formalizes the extent to which alternative logical systems transform mathematical inference, of proof organization, and indeed our understanding of mathematical truth. The paper emphasizes, as well its own, the role that non-classical logic plays in contributing to mathematical thought and stimulating new areas such as artificial intelligence, computational logic and complex systems.

Finally, it is considered that this work contributes to the theoretical development of mathematical logic by providing a well-structured vision which connects classical oriented views with non-classical ones, offering more analytical strength to nowadays’ mathematical reasoning.