In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition (...) theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics. (shrink)

It is argued that the theory of opposition is in position to contribute as a formal method of conceptual engineering, by means of an increasing dichotomy-making process that augments the number of elements into any structured lexical field. After recalling the roots of this theory and its logical tenets, it is shown how the processes of expansion and contraction of discourse can modify a lexical field and, with it, our collective representation of ideas. This theory can also bring some order (...) to the question of disagreement in philosophical discourse: what do philosophers disagree about; how can we clarify the distinction between verbal disagreement (focused on words) and substantive disagreement (focused on things)? The ensuing construction of conceptual systems will be exemplified through three case studies of philosophy: desire, truth judgment, and the left–right political divide. The construction rules of such systems resort to the theory of opposition, which intends to improve our understanding of what entails either agreement or disagreement about the meaning of concepts. Such a better understanding of philosophical discourse relies on its formalization in terms of closed lexical fields, thereby leading to a comparative analysis of concepts in light of logical relations between their definitions. (shrink)