A New History of Logic: The Laborious Birth of a Formal Pluralism

Abstract

The paper starts by remarking that the ancient Greek word for “truth” was “alétheia” (unveiling), which is a double negation. But, after Plato the affirmative meaning of the idea of truth has prevailed. The same meaning was reiterated by Romans’s word for truth, veritas. Not before the year 1968 the double negation law was re-evaluated, since its failure was recognized as representing, more appropriately than the failure of the excluded middle law, the borderline between classical logic and almost all non-classical kinds of logic. Moreover, its failure is easily recognized within a (scientific) text; which therefore can be analyzed in a new logical way. As an example, the analysis of Kolmogorov’s 1932 paper about the foundations of the intuitionist logic shows many interesting results, in particular his reasoning through arguments pertaining to the same intuitionist logic. In addition, previous papers have suggested—through a comparative analysis of all scientific theories which are based, like the previous one, on a general problem—that there exists a new model of organization of a theory which is alternative to the deductive-axiomatic model and it is governed by intuitionist logic. Some important logical events pertaining to non-classical logic are recognized by inspecting through both the double negations and this new model of organizing the original texts of past theories; so that the entire history of logic appears as a development along two main lines, one line representing classical logic dominated all others for a very long period of time, although at the origins the logical arguing had pertained to the alternative line. This history of logic confirms both Kuhn’s category of a paradigm and Koyré’s categories, represented by him through a couple of “characteristic propositions” in mutual conflict; analogous couples of propositions are suggested as representing the categories for adequately interpreting the entire history of logic.