The square of opposition in orthodmodular logic

HECTOR FREYTES; CHRISTIAN DE RONDE; GRACIELA DOMENECH

Corte, Corsica

Congreso; 2nd World Congress on the Square of Opposition.; 2010

University of Corsica

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 famous square of opposition. The square expresses the essential properties of monadic first order quantification. In an algebraic approach these properties can be represented taking into account monadic Boolean algebras [3]. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra; the square of opposition is then 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 [4] giving rise to new possible interpretations of the square. In this work, we consider the orthomodular logic enriched with a monadic quantifier and we provide interpretations of the square of opposition in several models of this logic as Boolean saturated orthomodular lattices [1], Baer*-semigroups and C*-algebras [2]. [1] G. DOMENECH, H. FREYTES and C. DE RONDE, “Scopes and limits of modality in quantum mechanics”, Annalen der Physik 15 (2006) 853–860. [2] G. DOMENECH, H. FREYTES and C. DE RONDE, “Modal type othomodular logic”, Mathematical Logic Quarterly 55 (2009) 287–299. [3] P. HALMOS, “Algebraic logic I, monadic Boolean algebras”, Compositio Mathematica 12 (1955) 217–249. [4] P. HÁJEK, Metamathematics of fuzzy logic, Dordrecht: Kluwer, 1998.