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An algebraic approach to intuitionistic connectives

Published online by Cambridge University Press:  12 March 2014

Xavier Caicedo
Affiliation:
Departamento De Matemáticas, Universidad De Los Andes, Apartado Aéreo 4976, Bogotá. D.C. —Colombia, E-Mail: xcaicedo@uniandes.edu.co
Roberto Cignoli
Affiliation:
Departamento De Matemática, Facultad De Ciencias Exactas Y Naturales, Universidad De Buenos Aires — Conicet Ciudad Universitaria, 1428 Buenos Aires -Argentina, E-Mail: cignoli@dm.uba.ar
Corresponding

Abstract.

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be “completed” conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2001

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