[ Nash | da Costa | Severino | Bourdieu | Derrida | Héritier | Sylvan | Angelelli | Lawvere | Laruelle | Badiou | Zilberberg | Kripke | Lewis | Gauthier | Petitot | Pizzi | Marion | Nef | Guitart | Priest | Descola | Gärdenfors | Carnielli | Smith | Butler | Zalamea | Lordon |Béziau | Smessaert | Barot | Moretti | Berto ] – other periods
The logician, mathematician and philosopher Newton Carneiro Affonso da Costa is very important for the philosophy of opposition in so far he proposed, seemingly successfully (but this question is philosophically still open), to build mathematical systems capable of endorsing “non-trivial inconsistencies” (that is: mathematically meaningful contradictions, of the form “A and not-A”), what is by now called “paraconsistency” (the dual notion, “paracompleteness” is what is also called “mathematical intuitionism”, the rejection of the “principle of the excluded middle”, i.e. “A or not-A”). Paraconsistent logics are a central issue on the contemporary debate on the foundations of logic. Among the ancestors of this revolutionary idea one can find Meinong, Vasil’ev and Wittgenstein. Lukasiewicz is also very important here, for he has been the first logician who demonstrated in a rigorous way – also analysing the Aristotelian writings where this question is touched – that the principle of non-contradiction is an axiom which can be skipped from logical systems (although he himself did not explore directly the strange kind of logics that could arise from such a daring move). Among the many heirs of da Costa, one can mention (for they also are philosophers of opposition) Sylvan (formerly called Routley), Priest and Béziau. An important philosopher of paraconsistency is Berto, who – beside being an analytical philosopher – also is a heir of the great continental philosopher Severino (himself the most radical philosopher of non-contradiction in the world!). Paraconsistent logics have been charged by Slater (1995) of not being worth the name they bear, since what they present as a “paraconsistent negation” is in fact not a “negation” (i.e. a contradiction-forming operator) but something much weaker (a subcontrariety-forming operator), not deserving the name “negation” (this crucial distinction pertains to the so-called “logical square” or “square of opposition”). It is by some of the many answers given by several paraconsistent logicians to this challenge (among which Priest, Restall, Brown, Paoli, Béziau, …) that the new mathematical science of oppositions, “oppositional geometry” (also called “N-Opposition Theory”, “NOT”) has started, notably with the works of Moretti, Pellissier and Smessaert. The question of knowing to which extent paraconsitent logic can generate a non-standard contradiction (thus definitely defeating Slater’s argument) pertains more to oppositional geometry than to logic and it seems, so far, to be still open (a promising element of positive answer seems to be Moretti’s discovery of a technique allowing to construct mathematically sound “logical poly-simplexes of dimension m, which can provide an infinite geometrical diffraction of the notion of “contradiction”).
- Bobenrieth Miserda, A., Inconsistencias ?por qué no?, Colcultura, 1996
- da Costa, N.C.A., Ensaio sobre os Fondamentos da Lógica, Hucitec y Editora da Universidade de Sâo Paulo, Sâo Paulo, 1980
- da Costa, N.C.A., ,
- Grana, N., Sulla teoria delle valutazioni di N.C.A. da Costa, Liguori, Napoli, 1990
- Marconi, D. (ed.), La formalizzazione della dialettica. Hegel, Marx e la logica contemporanea, Rosenberg & Sellier, Torino, 1979
- Moretti A., “The Critics of Paraconsistency and of Many-Valuedness and the Geometry of Oppositions“, in: Logic and Logical Philosophy, Vol. 19, 2010
- Palau, G., Introducción filosófica a las lógicas no clásicas, Gedisa, Buenos Aires, 2002
OPPOSITIONAL GEOMETRY 