Around and Beyond the Square of Opposition pp 147-173
Oppositions and Opposites
A formal theory of oppositions and opposites is proposed on the basis of a non-Fregean semantics, where opposites are negation-forming operators that shed some new light on the connection between opposition and negation. The paper proceeds as follows.
After recalling the historical background, oppositions and opposites are compared from a mathematical perspective: the first occurs as a relation, the second as a function. Then the main point of the paper appears with a calculus of oppositions, by means of a non-Fregean semantics that redefines the logical values of various sorts of sentences. A number of topics are then addressed in the light of this algebraic semantics, namely: how to construct value-functional operators for any logical opposition, beyond the classical case of contradiction; Blanché’s “closure problem”, i.e. how to find a complete structure connecting the sixteen binary sentences with one another.
All of this is meant to devise an abstract theory of opposition: it encompasses the relation of consequence as subalternation, while relying upon the use of a primary “proto-negation” that turns any relatum into an opposite. This results in sentential negations that proceed as intensional operators, while negation is broadly viewed as a difference-forming operator without special constraints on it.
KeywordsINRC Group Negation Opposite-forming operators Opposition Proto-negation
Mathematics Subject Classification03B35 03B05 03B65
- 1.Aristotle: Metaphysics Google Scholar
- 2.Aristotle: On Interpretation Google Scholar
- 3.Aristotle: Prior Analytics Google Scholar
- 4.Béziau, J.-Y.: New light on the square of opposition and its nameless corner. Log. Investig. 10, 218–233 (2003) Google Scholar
- 7.Blanché, R.: Structures intellectuelles (Essai sur l’organisation systématique des concepts). Vrin, Paris (1966) Google Scholar
- 8.Chatti, S., Schang, F.: Import, or not import? How to handle negation inside the square. Submitted paper Google Scholar
- 11.Moretti, A.: The Geometry of Logical Opposition. PhD thesis, University of Neuchâtel (2009) Google Scholar
- 13.Piaget, J.: Essai sur les transformations des opérations logiques (Les 256 opérations ternaires de la logique bivalente des propositions). Presses Universitaires de France, Paris (1952) Google Scholar
- 14.Piaget, J.: Traité de logique (Essai de logistique opératoire). Armand Colin (1972) (1st edition, 1949) Google Scholar
- 15.Schang, F.: Relative charity. Rev. Bras. Filos. 233, 159–172 (2009) Google Scholar
- 16.Schang, F.: Questions and answers about oppositions. In: Béziau, J.-Y., Payette, G. (eds.) New Perspectives on the Square of Opposition. Peter Lang, Bern (2011) Google Scholar
- 17.Schang, F.: Two Indian dialectical logics: saptabhaṅgī and catuṣkoṭi. J. Indian Counc. Philos. Res. 27, 45–75 (2011) Google Scholar
- 18.Sesmat, A.: Logique—II. Les raisonnements, la logistique. Hermann, Paris (1951) Google Scholar