The critics of paraconsistency and of many-valuedness and the geometry of oppositions

Logic and Logical Philosophy 19 (1-2):63-94 (2010)
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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



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Alessio Moretti
Université de Neuchâtel

Citations of this work

Why the Logical Hexagon?Alessio Moretti - 2012 - Logica Universalis 6 (1-2):69-107.

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References found in this work

A Natural History of Negation.Laurence R. Horn - 1989 - University of Chicago Press.
The logic of paradox.Graham Priest - 1979 - Journal of Philosophical Logic 8 (1):219 - 241.
A Natural History of Negation.Laurence R. Horn - 1989 - Philosophy and Rhetoric 24 (2):164-168.
An Introduction to Non-Classical Logic.Graham Priest - 2001 - Bulletin of Symbolic Logic 12 (2):294-295.
Sur l'opposition des concepts.Robert Blanche - 1953 - Theoria 19 (3):89-130.

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