Chapter

Around and Beyond the Square of Opposition

Part of the series Studies in Universal Logic pp 193-200

The Square of Opposition in Orthomodular Logic

  • H. FreytesAffiliated withUniversita degli Studi di CagliariInstituto Argentino de Matemática Email author 
  • , C. de RondeAffiliated withCenter Leo Apostel and Foundations of the Exact Sciences, Vrije Universiteit BrusselDepartamento de Filosofía “Dr. A Korn”, Universidad de Buenos Aires-CONICET
  • , G. DomenechAffiliated withInstituto de Astronomía y Física del Espacio

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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