Degree of Negation of an Axiom

TL;DR

This paper introduces the concept of a degree of negation for axioms and propositions, extending classical negation to include partial or fuzzy negations, applicable across various fields.

Contribution

It proposes a novel concept of partial negation of axioms, inspired by fuzzy and neutrosophic logic, broadening the understanding of negation in scientific and humanistic propositions.

Findings

Introduced the degree of negation concept for axioms.

Connected the idea to fuzzy and neutrosophic logic.

Applied the concept to geometry and other fields.

Abstract

In this article we present the two classical negations of Euclid's Fifth Postulate (done by Lobachevski-Bolyai-Gauss, and respectively by Riemann), and in addition of these we propose a partial negation (or a degree of negation) of an axiom in geometry. The most important contribution of this article is the introduction of the degree of negation (or partial negation) of an axiom and, more general, of a scientific or humanistic proposition (theorem, lemma, etc.) in any field - which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood), or like in neutrosophic logic [with a degree of truth, a degree of falsehood, and a degree of neutrality (i.e. neither truth nor falsehood, but ambiguous, unknown, indeterminate)].