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Tutorial on Inconsistency-Adaptive Logics Read chapter Read first chapter

2015 | OriginalPaper | Chapter

2. Round Squares Are No Contradictions (Tutorial on Negation Contradiction and Opposition)

Author : Jean-Yves Beziau

Published in: New Directions in Paraconsistent Logic

Publisher: Springer India

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Abstract

We investigate the notion of contradiction taking as a central point the idea of a round square. After discussing the question of images of contradiction, related to the contest Picturing Contradiction, we explain why from the point of view of the theory of opposition, a round square is not a contradiction. We then draw a parallel between different kinds of oppositions and different kinds of negations. We explain why from this perspective, when we have a paraconsistent negation \(\lnot \), the formulas p and \(\lnot p\) cannot be considered as forming a contradiction. We finally introduce the notions of paranormal negation and opposition which may catch the concept of a round square.

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previous chapter Tutorial on Inconsistency-Adaptive Logics
next chapter On the Philosophy and Mathematics of the Logics of Formal Inconsistency
Footnotes
1
The logic of imagination is still a quite new and open field. A starting point was a paper by Ilkka Niiniluoto in 1985 [44]; for a critical account of this paper see [30].
 
2
The president of the jury was Kuntal Ghosh, from the Indian Statistical Institute in Kolkata where the event was taking place.
 
3
I had the opportunity at this time to meet and discuss with David Bohm in London. After that I wrote a dissertation on Plato’s cave [4] and later on I developed the paraconsistent logic Z inspired by Bohm’s ideas. About this logic, see [9], and about how it was conceived, see [10].
 
4
Larry Horn has, however, pointed out that even if we do not have a picture of the square of opposition by Aristotle, the Stagyrite suggested such a picture—see [39].
 
5
The work of Blanché has been published in [23‐25], about the hexagon see [12].
 
6
Since 2007 we are organizing a world congress on the square of opposition. The first edition happened in Montreux, the second in Corsica in 2010, the third in Beirut in 2013, the fourth in the Vatican in 2014, the next one is projected to happen in Easter Island in 2016—see http://​www.​square-of-opposition.​org. Related publications are [11, 14, 18‐22].
 
7
These are the three primitive colors. The theory of opposition can also be applied to the theory of colors, see in particular the hexagon of colors of Dany Jaspers [40].
 
8
This context is important, not only to rule out other geometries—one may claim that a point is both a straight and a curved line, so that a curved line is not a contradiction, but in standard geometry a point is not a line—but also objects out of the scope of geometry, like an abstract concept such as beauty. It is possible to say that beauty is neither a square nor a circle, but this is not necessarily a convincing example to sustain that square and circle are not contradictory.
 
9
André Breton promoted as a key feature of surrealist writing the idea of “carambolage sémantique” [29]. But this is not the same as a “dérapage sémantique.” The idea is to create a poetic effect by putting together opposed notions, leading to a sense of absurdity. Flaubert used systematically in his masterpiece Bouvard et Pécuchet [35] a process qualified as “antithetic juxtaposition” consisting of putting side by side two different opinions or theories. This was to show that human knowledge is not really coherent.
 
10
For more discussion about the variety of symbolism, see [17].
 
11
This was reported to me by Newton da Costa. He faced this phenomenon when visiting the Australopithecus in his own country in the 1970s.
 
12
As explained in [11], not satisfied with this octagon, I split it in three stars that I put together in a three-dimensional polyhedron of opposition which also perfectly reflects the duality and symmetry between these two negations. The multidimensional theory of opposition has been further developed by Moretti [42], Smessaert [49] and Pélissier [45].
 
13
For a detailed discussion about how to define a paraconsistent negation, see [5, 6].
 
14
At the metalevel, tautology and antilogy form a contrary pair, see the metalogical hexagon presented in [16].
 
15
The expression “paranormal logic” was used in the paper [32] where a paranormal logic different from De Morgan logic was introduced. De Morgan logic is derived from De Morgan algebra, for details about this, see [15].
 
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Metadata
Title
Round Squares Are No Contradictions (Tutorial on Negation Contradiction and Opposition)
Author
Jean-Yves Beziau
Copyright Year
2015
Publisher
Springer India
Book
New Directions in Paraconsistent Logic

Print ISBN: 978-81-322-2717-5

Electronic ISBN: 978-81-322-2719-9

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-81-322-2719-9_2

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