The Square of Opposition in Orthomodular Logic

Abstract

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square.

Keywords

Square of opposition Modal orthomodular logic Classical consequences 

Mathematics Subject Classification

03G12 06C15 03B45 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Universita degli Studi di CagliariCagliariItaly
  2. 2.Instituto Argentino de MatemáticaBuenos AiresArgentina
  3. 3.Departamento de Filosofía “Dr. A Korn”Universidad de Buenos Aires-CONICETBuenos AiresArgentina
  4. 4.Center Leo Apostel and Foundations of the Exact SciencesVrije Universiteit BrusselBrusselsBelgium
  5. 5.Instituto de Astronomía y Física del EspacioBuenos AiresArgentina

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