Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Actually, the geometrical presentation of logical oppositions has been inspired by Apuleius (≅123–170 C.E.): in his Peri Hermeneias, he added the symmetric relation of subcontrariety as paralleling the line of contrariety; this didn’t still result in a square, but a crossed polygon. Then Boethius (480–525 C.E.) turned this quadrilateral into a genuine square by introducing the further relation of subalternation.
- 2.
- 3.
Tom and Jim are normally treated as individual constants, but they are taken here to be predicates following the Quinean paraphrase. This helps to compare these with the next opposite-forming operators, given that both grammatical categories proceed as one-place predicates or functors.
- 4.
Returning to Blanché’s triangle of contraries, this can be seen as either a set of three 2-place relations ({□α,□∼α}, {□α,◊α∧◊∼α} and {□∼α,◊α∧◊∼α}), or a single 3-place relation ({□α,□∼α,◊α∧◊∼α}).
- 5.
The number of the logical connectives in any logical system is \(m(^{m^{n}})\), where m is the number of truth-values and n is the number of arguments combined by n-ary connectives. In the standard logic of binary connectives, m=n=2, hence a total number of \(2(^{2^{2}}) = 2^{4} = 16\) connectives. Now some alternative cases may be entertained, by changing either the number of truth-values or the number of combined sentences. Piaget focused upon the second option in [13] by considering the \(2(^{2^{3}}) = 2^{8} = 256\) ternary connectives within the standard two-valued logic.
- 6.
- 7.
E.g. R(1000)=0001, and OP(1000,0001)=OP(α∧ψ,α↓ψ)=CT(Φ,Ψ); R(1100)=0011, and OP(1100,0011)=OP(α,∼α)=CD(Φ,Ψ); R(1110)=0111, and OP(1110,0111)=OP(α∨ψ,α↑ψ)=SCT(Φ,Ψ); R(1111)=1111, and OP(1111,1111)=OP(⊤,⊤)=SB(Φ,Ψ); R(0000)=0000, and OP(0000,0000)=OP(⊥,⊥)=SB(Φ,Ψ). See note 13 about the last two cases.
- 8.
Note the logical difference between “contraries of contradictories” and “the contrary of the contradictory”: the first plural expression includes one relation SCT and two functions N in the form CT(N(α),N(ψ)), whereas the second singular expression includes no relation and two functions R and N in the form R(N(α)).
- 9.
[7, p. 137]. Let A(α∧ψ)=1000 be a case of determinate with only one truth-case, i.e. the yes-answer a 1(α)=1; accordingly, this sentence entails the three indeterminates A(α∨ψ)=1110, A(α←ψ)=1101, and A(α→ψ)=1011. Note that this containment relation can be extended to what Blanché called as “semi-determinates”, i.e. the binary sentences with two truth-cases or yes-answers: the determinate A(α↛ψ)=0100 entails the two semi-determinates A(α)=1100 and A(α↮ψ)=0110, for instance.
- 10.
Initially, Czeżowski’s hexagon was about quantified sentences and consisted in supplementing universal {A,U,E} and particular {I,Y,O} with singular sentences {Ü, Ÿ} (Sa∧Pa, Sa∧∼Pa). But again, our purely formal approach of oppositions abstracts from the content of the sentences and uniquely takes the logical values into account.
- 11.
[7, p. 122] (the author’s translation).
- 12.
The choice of such a 3D polygon is motivated by a basic construction rule in [11], namely: a geometrical structure of logical oppositions is to be constructed so that its contradictory lines must intersect each other in the center of the polygon without overlapping each other.
- 13.
OP(α,α) is not counted among the range of SB, because it is assumed there that the two relata are different sentences. But strictly speaking, the relation of self-identity can be seen as a kind of subalternation since it satisfies its definition: A(α)∩A(α)=A(α) and A(α)∪A(α)=A(α). See note 6.
- 14.
See [11], where the logical tetracosahedron is presented as a gathering of one logical cube and ten logical hexagons. Six of these hexagons are Blanché’s, Béziau’s and Smessaerts’s strong hexagons, i.e. such that A(U)=A(A)∪A(E) and A(Y)=A(∼I)∩A(∼E); and the remaining four ones are Pellissier’s weak hexagons such that A(U)≠A(A)∪A(E) and A(Y)≠A(∼I)∩A(∼E).
References
Aristotle: Metaphysics
Aristotle: On Interpretation
Aristotle: Prior Analytics
Béziau, J.-Y.: New light on the square of opposition and its nameless corner. Log. Investig. 10, 218–233 (2003)
Blanché, R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953)
Blanché, R.: Sur la structuration du tableau des connectifs interpropositionnels binaires. J. Symb. Log. 22, 178 (1957)
Blanché, R.: Structures intellectuelles (Essai sur l’organisation systématique des concepts). Vrin, Paris (1966)
Chatti, S., Schang, F.: Import, or not import? How to handle negation inside the square. Submitted paper
Czeżowski, T.: On certain peculiarities of singular propositions. Mind 64, 392–395 (1955)
Luzeaux, D., Sallantin, J., Dartnell, C.: Logical extensions of Aristotle’s square. Logica Univers. 2, 167–187 (2008)
Moretti, A.: The Geometry of Logical Opposition. PhD thesis, University of Neuchâtel (2009)
Pellissier, R.: Setting n-opposition. Logica Univers. 2, 235–262 (2008)
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)
Piaget, J.: Traité de logique (Essai de logistique opératoire). Armand Colin (1972) (1st edition, 1949)
Schang, F.: Relative charity. Rev. Bras. Filos. 233, 159–172 (2009)
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)
Schang, F.: Two Indian dialectical logics: saptabhaṅgī and catuṣkoṭi. J. Indian Counc. Philos. Res. 27, 45–75 (2011)
Sesmat, A.: Logique—II. Les raisonnements, la logistique. Hermann, Paris (1951)
Slater, H.: Paraconsistent logics? J. Philos. Log. 24, 451–454 (1995)
Smessaert, H.: On the 3D visualization of the logical relations. Logica Univers. 3, 212–231 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Schang, F. (2012). Oppositions and Opposites. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0379-3_11
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0378-6
Online ISBN: 978-3-0348-0379-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)