The Origins of the Use of the Argument of Trivialization in the Twentieth Century

History and Philosophy of Logic 31 (2):111-121 (2011)
Abstract
The origin of paraconsistent logic is closely related with the argument, 'from the assertion of two mutually contradictory statements any other statement can be deduced'; this can be referred to as ex contradictione sequitur quodlibet (ECSQ). Despite its medieval origin, only by the 1930s did it become the main reason for the unfeasibility of having contradictions in a deductive system. The purpose of this article is to study what happened earlier: from Principia Mathematica to that time, when it became well established. The two main historical claims that I am going to advance are the following: (1) the first explicit use of ECSQ as the main argument for supporting the necessity of excluding any contradiction from deductive systems is to be found in the first edition of the book Grundz ge der Theoretischen Logik (Hilbert, D. and Ackermann, W. 1928. Grundz ge der Theoretischen Logik . Berlin: Julius Springer Verlag); (2) ukasiewicz's position regarding the logical constraints against contradictions varies considerably from his studies on the principle of (non-) contradiction in Aristotle, published in 1910 and what is stated in his 'authorized lectured notes' on mathematical logic that appeared in 1929. The two texts are: 1) a paper in German ( ukasiewicz, J. 1910. ' ber den Satz des Widerspruchs bei Aristotles'. Bulletin International de l'Acad mie des sciences de Cracovie, Classe d'Histoire et de Philosophie, pp. 15-38) [English translation: ukasiewicz, J. 1971. 'On the principle of contradiction in Aristotle', Review of Metaphysics , XXIV , 485-509]; and 2) a book in Polish. ukasiewicz, J. 1910. O zasadzie sprzecznosci u Aristotelesa Studium krytyczne , Warsaw: Panstwowe Wydawnictwo Naukowe [German translation: ukasiewicz, J. 1993. ber den Satz des Widerspruchs bei Aristotles . Hildesheim: Georg Olms Verlag]. The lecture notes were then published as a book ( ukasiewicz, J. 1958. Elementy Logiki Matematycznej . Warszawa: Panstwowe Wydawnictwo Naukowe [PWN] and then translated into English ( ukasiewicz, J. 1963. Elements of Mathematical Logic. Oxford, New York: Pergamon Press/The Macmillan Company) . The second half of this article will concentrate on ukasiewicz's position on ECSQ. This will lead me to propose that to regard him as a forerunner of paraconsistent logic by virtue of those early writings is accurate only if his book published in Polish is considered but not if the analysis is restricted to the paper originally published in German (as has been the case for the principal reconstructions of the history of paraconsistent logic). Furthermore, I will stress that in the 1929 book he presented one formalization of ECSQ as an axiom for sentential calculus and, also, he used ECSQ to defend the necessity of consistency, apparently independently of Hilbert and Ackermann's book. At the end, I will suggest that the aim of twentieth century usage of ECSQ was to change from the centuries-long philosophical discussion about contradictions to a more 'technical' one. But with paraconsistent logic viewed as a technical solution to this restriction, then, the philosophical problem revives but having now at one's disposal an improved understanding of it. Finally, ukasiewicz's two different positions about ECSQ open an interesting question about the history of paraconsistent logic: do we have to attempt a consistent reconstruction of it, or are we prepared to admit inconsistencies within it?
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Jean Van Heijenoort (ed.) (1967). From Frege to Gödel. Cambridge, Harvard University Press.

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