Abstract
In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus (CPC). We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Béziau, J.-Y.: The power of hexagon. Log. Univers. 6, 1–2 (2012). (Special Issue: Hexagon of Opposition. Ed. J.-Y. Béziau, pp. 1–43)
Béziau, J.-Y.: Synopsis of Robert Blanché, ‘Sur le systme des connecteurs interpropositionnels’. In: Hallward, P. (ed.) Concept and Form: The Cahiers pour l‘Analyse and Contemporary French Thought. Centre for Research in Modern European Philosophy (CRMEP) and Kingston University (2011). http://cahiers.kingston.ac.uk/synopses/syn10.7.html
Blanché R.: Sur la structuration du tableau des connectifs interpropositionnels binaires. J. Symb. Log. 22, 1, 17–18 (1957)
Blanché, R.: Structures intellectualles. Essai sur l’organisation systématique des Concepts. Vrin, Paris (1966)
Blanché R.: Sur le systeme des Connecteurs interpropositionnels. Casher pour l’Anal. 10, 131–149 (1969)
Dubois, D., Prade, H.: From Blanchés hexagonal organization of concept to formal concept analysis and possibility theory. Log. Univers. 6, 1–2 (2012). (Special Issue: Hexagon of Opposition, Ed. J.-Y. Béziau, pp. 149–169)
Gottschalk W.H.: The theory of quaternality. J. Symb. Log. 18, 193–193 (1953)
Luzeaux D., Sallantin J., Dartnell C.: Logical extensions of Aristotle’s square. Log. Univers. 2(1), 167–187 (2008)
Moretti, A.: The geometry of opposition and the opposition of logic to it. In: Savardi, U. (ed.) The Perception and Cognition of Contraries. McGraw-Hill, Milano (2009)
Moretti, A.: Way the logical hexagon? Log. Univers. 6, 1–2 (2012). (Special Issue: Hexagon of Opposition. Ed. J.-Y. Béziau, pp. 69–107)
Piaget, J.: Traité de logique. Essai de syllogistique opératoire. Armand Coin, Paris (1949)
Pogorzelski, W.A.: Elementarny Słownik Logiki Formalnej (Elemantary vocabulary of formal logic). In: Editorial Office of Warsaw University—Białystok Branch, Białystok, pp. 245–247 (1989)
Sauriol P.: Remarques sur la Théorie de l’hexagone logique de Blanché. Dialogue 7, 374–390 (1968)
Smessaert H.: On the 3D visualization of logical relations. Log. Univers. 3(2), 303–332 (2009)
Smessaert H., Demey L.: Logical geometries and information in the square of opposition. JoLLI 23(4), 527–565 (2014)
Wybraniec-Skardowska, U.: Foundations for the formalization of metamathematics and axiomatizations of consequence theories. In: Adamowicz, Z. et al. (eds.) Provinces of Logic Determined, Essays in the Memory of Alfred Tarski. Annales of Pure and Applied Logic, vol.127, Part IV, pp. 234–266. Elsevier Sciences, London (2004)
Wybraniec-Skardowska U., Waldmajer J.: On pairs of dual consequence operations. Log. Univers. 5(2), 177–203 (2011)
Wybraniec-Skardowska, U.: Handbook of the World Congress on the Square of Opposition IV. In: Bézieu, J.-Y., Gan-Krzywoszyńska, K. (eds.) Pontifical Lateran University, Vatican, pp. 149–150 (2014)
Zellweger, S.: Untapped potential in Peirce’s iconic notation for sixteen binary connectives. In: Houser, N., Roberts, D. (eds.) Studies in the Logic of Charles Peirce, pp. 334–386. Indiana University Press, Bloomington (1997)
Żarnecka Biały, E.: Historia logiki dawniejszej (History of formerly logic). In: Dialogikon Series. Jagiellonian University Press, Kraków (1995)
Żarnecka Biały, E.: Mała logika (Small Logic), 3rd edn. Jagiellonian University Press, Kraków (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wybraniec-Skardowska, U. Logical Squares for Classical Logic Sentences. Log. Univers. 10, 293–312 (2016). https://doi.org/10.1007/s11787-016-0148-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-016-0148-x