Drago, Antonino
(0202)
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 reevaluated, 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 nonclassical 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 nonclassical 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 deductiveaxiomatic 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 noncontradiction 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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