Aletheia, double negation and negation

TL;DR

This paper explores the concept of negation as a structural element in scientific theories, examining its historical and logical aspects, especially focusing on double negation and its role in distinguishing classical and intuitionist logic.

Contribution

It introduces a new logical analysis based on the failure of the double negation law, linking linguistic, historical, and logical perspectives to differentiate classical and non-classical logic.

Findings

Double negation law's failure marks the boundary between classical and non-classical logic.

Analysis of Kolmogorov's 1932 paper shows non-classical reasoning in foundational work.

Negation as a unary operation helps distinguish between classical and intuitionist logic.

Abstract

The definition of negation has to be referred to the totality of a theory and at last to what is defined as the organization of a scientific theory; in other words, the definition of negation is of a structural kind, rather than of an objective kind or a subjective kind. The paper starts by remarking that the ancient Greek word for truth was aletheia, which is a double negation, i.e. unveiling. Not before the 1968 the double negation law was re-evaluated, since it was recognized that its failure represents more appropriately than the failure of of excluded middle law the borderline between classical logic and almost all non-classical kinds of logic. Moreover, the failure of this law is easily recognized within a scientific text; this fact allows a new kind of logical analysis of a text. As an example, the analysis of Kolmogorov 1932 paper shows that he reasoned according to arguments of non-classical logic about the foundations of the intuitionist logic. The negation is defined as a unary operation which, under the problem of deciding whether a doubly negated proposition is equal to the corresponding affirmative proposition or not, leads to a subdivision into classical logic and intuitionist logic.