General Patterns of Opposition Squares and 2n-gons

Abstract

In the first part of this paper we formulate the General Pattern of Squares of Opposition (GPSO), which comes in two forms. The first form is based on trichotomies whereas the second form is based on unilateral entailments. We then apply the two forms of GPSO to construct some new squares of opposition (SOs) not known to traditional logicians. In the second part of this paper we discuss the hexagons of opposition (6Os) as an alternative representation of trichotomies. We then generalize GPSO to the General Pattern of 2n-gons of Opposition (GP2nO), which also comes in two forms. The first form is based on n-chotomies whereas the second form is based on co-antecedent unilateral entailments. We finally introduce the notion of perfection associated with 2n-gons of opposition (2nOs) and point out that the fundamental difference between a SO and a 6O is that the former is imperfect while the latter is perfect. We also discuss how imperfect 2nOs can be perfected at different fine-grainedness.

Keywords

Trichotomy Unilateral entailment General Pattern of Squares of Opposition General Pattern of 2n-gons of Opposition 

Mathematics Subject Classification

03B65 

References

  1. 1.
    Altman, A., Peterzil, Y., Winter, Y.: Scope dominance with upward monotone quantifiers. J. Log. Lang. Inf. 14(4), 445–455 (2005) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ben-Avi, G., Winter, Y.: Scope dominance with monotone quantifiers over finite domains. J. Log. Lang. Inf. 13(4), 385–402 (2004) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Blanché, R.: Sur l’opposition des concepts. Theoria 19 (1953) Google Scholar
  4. 4.
    Brown, M.: Generalized quantifiers and the square of opposition. Notre Dame J. Form. Log. 25(4), 303–322 (1984) MATHCrossRefGoogle Scholar
  5. 5.
    Gottschalk, W.H.: The theory of quaternality. J. Symb. Log. 18(3), 193–196 (1953) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Jaspers, D.: Operators in the lexicon: on the negative logic of natural language. PhD thesis, Leiden University (2005) Google Scholar
  7. 7.
    Moretti, A.: Geometry for modalities? Yes: through ‘n-opposition theory’. In: Béziau, J.-Y., Costa-Leite, A., Facchini, A. (eds.) Aspects of Universal Logic, pp. 102–145. Centre de Recherches Sémiologiques, University of Neuchâtel, Neuchâtel (2004) Google Scholar
  8. 8.
    Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Clarendon, Oxford (2006) Google Scholar
  9. 9.
    Seuren, P.A.M.: The Logic of Language. Oxford University Press, Oxford (2010) Google Scholar
  10. 10.
    Smessaert, H.: On the 3D visualisation of logical relations. Logica Univers. 3, 303–332 (2009) MathSciNetCrossRefGoogle Scholar
  11. 11.
    van Eijck, J.: Generalized quantifiers and traditional logic. In: van Benthem, J., ter Meulen, A. (eds.) Generalized Quantifiers and Natural Language, pp. 1–19. Foris, Dordrecht (1984) Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.The Hong Kong Polytechnic UniversityHong KongChina

Personalised recommendations