On the Historical Transformations of the Square of Opposition as Semiotic Object

Abstract

In this paper, we would show how the logical object “square of opposition”, viewed as semiotic object (articulated in textual or/and diagrammatic code), has been historically transformed since its appearance in Aristotle’s texts until the works of Vasiliev. These transformations were accompanied each time with a new understanding and interpretation of Aristotle’s original text and, in the last case, with a transformation of its geometric configuration. The initial textual codification of the theory of opposition in Aristotle’s works is transformed into a diagrammatic one, based on a new “reading” of the Aristotelian text by the medieval scholars that altered the semantics of the O form. Further, based on the medieval “Neo-Aristotelian” reading, the logicians of the nineteenth century suggest new diagrammatic representations, based on new interpretations of quantification of judgements within the algebraic and the functional logical traditions. In all these interpretations, the original square configuration remains invariant. However, Nikolai A. Vasiliev marks a turning point in history. He explicitly attacks the established logical tradition and suggests a new alternation of semantics of the O form, based on Aristotelian concepts that were neglected in the Aristotelian tradition of logic, notably the concept of indefinite judgement. This leads to a configurational transformation of the “square” of opposition into a “triangle”, where the points standing for the O and I forms are contracted into one point, the M(I, O) form that now stands for particular judgement with altered semantics. The new transformation goes beyond the Aristotelian logic paradigm to a new “Non-Aristotelian” logic (and associated ontology), i.e. to paraconsistent logic, although the argumentation used in support of it is phrased in (Neo-)Aristotelian style and the context of discovery is foundational (analogical to Lobachevsky’s research on the axiomatics of geometry). It establishes a bifurcation (proliferation) point in the development of logic. No unique logic is recognized, but different logics concerning different domains (ontologies, respectively). One branch of logic remains to be the “Neo-Aristotelian” one, while the new logic is “Non-Aristotelian”.