What is the very meaning of negation, if any? Or, to put it in other terms: how to mean negation? Is there non-revisable properties of negation that any logical system should share? To understand negation as a logical constant leads one to a broader philosophical question, that is: don’t some minimal properties inhere to every logical constant that go beyond their definition within a closed formal system? Just as Hilbert said that geometrical notions don’t have any fix meaning outside the axiom system in which they are defined, from a formalist view any logical constant is an empty or meaningless symbol outside the closed system in which it can be defined. Beyond such a formalist view of logic, we’ll consider the notions of form and content through an overview of distinctive sorts of negations. These can be divided into two main sorts of negation for A, such that non-A can express among others:

1.      Classical negation (in symbols: ¬A), consistent and complete

2.      Non-classical negation, as:

a.       Intuitionistic negation (in symbols: ~A), consistent and paracomplete

b.      Paraconsistent negation (in symbols:*A, paraconsistent and complete

3.      Variants: relevant negation, illocutionary negation, fuzzy negation, and so on.

How are these three negations negations?

The tutorial will be developed in three steps:

A.     Negation in history (Antics, Middle Age and Modern aspects)

B.     Negation in context (logic, mathematics, natural languages, religions)

C.     Negation and Dichotomy

References:

ARISTOTLE. Categories
ARISTOTLE. Peri Hermeneias, Chapter 7
ARISTOTLE. Metaphysics, Book G7
Arnon AVRON: “On negation, completeness and consistency”, in Handbook of Philosophical Logic, D. M. Gabbay and F. Guenthner (ed.), Vol. 9, Kluwer Academic Publishers (2002), 287-320
Jean-Yves BÉZIAU: “Bivalence, excluded middle and non contradiction”, in The Logica Yearbook 2003, L. Behounek (ed.), Academy of Sciences, Prague (2003), 73-84
Jean-Yves BÉZIAU.Are paraconsistent negations negations ?” in Paraconsistency, the  logical way to the inconsistent (Proceeding of the 2^nd World Congress on Paraconsistency – Juquehy 2001), W. Carnielli et al. (eds), Marcel Dekker, New York (2002), 465-486
Jean-Yves BÉZIAU. “Paraconsistent Logic ! (A reply to Slater)”,Sorites 17(2006), pp. 17-25
Dov M. GABBAY, Heinrich WANSING (ed.): What is negation?, Kluwer, Dordrecht (1999)
Laurence R. HORN: A Natural History of Negation, CSLI Publications (2001)
Edward S. KLIMA: “The negation in English”, in The Structure of Language: Readings in the Philosophy of Language, J. A. Fodor and J. J. Katz (eds.), Englewood Cliffs, N. J. (1964), 246-323
W. V. QUINE: Philosophy of Logic, Harvard University Press (1^st ed.) (1970)
Heinrich WANSING: “Negation”, in (The Blackwell Guide to) Philosophical Logic, L. Goble (ed.), Blackwell Publishing (2001), pp. 415-436