Drago, Antonino
(2020)
ALETHEIA, DOUBLE NEGATION AND NEGATION.
[Preprint]
Abstract
Abstract. The main result of present paper is that 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, negation’s definition 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”. But, after Plato and Romans the affirmative meaning of the idea of truth prevailed. 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’s 1932 paper is summarized. It shows that he reasoned according to arguments of non-classical logic about the foundations of the intuitionist logic. Through a comparative analysis of all scientific theories which are based, like the previous one, on a general problem my previous paper suggested an alternative model of organization of a theory to the deductive-axiomatic one. Then general informal logic is founded according to this model. 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. Hence, negation takes two specific and mutually incompatible meanings; in classical logic it leads to the non-contradiction principle, whereas in intuitionist logic it does not opposes to the corresponding affirmation. This essentially ambiguous nature of the negation gives reason of the insufficiency of Natural Deduction.
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