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An algebraic approach to intuitionistic connectives

Published online by Cambridge University Press:  12 March 2014

Xavier Caicedo  and
Roberto Cignoli
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
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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
Information
The Journal of Symbolic Logic , Volume 66 , Issue 4 , December 2001 , pp. 1620 - 1636
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Balbes, R. and Dwinger, R, Distributive lattices, University of Missouri Press, Columbia, Missouri. 1974.Google Scholar
[2]Caicedo, X., Investigaciones acerca de los conectivos intuicionistas. Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales, vol. 19 (1995), pp. 705–716.Google Scholar
[3]Caicedo, X., Conectivos sobre espacios topológicos. Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales, vol. 21 (1997), pp. 521–534.Google Scholar
[4]Fitting, M. C., Intuitionistic logic, model theory, andforcing. North-Holland. Amsterdam, 1969.Google Scholar
[5]Freyd, P.. Aspects of topoi. Bulletin of the Australian Mathematical Society. vol. 7 (1972), pp. 1–76.CrossRefGoogle Scholar
[6]Gabbay, D. M.. On some new intuitionistic propositional connectives, I, Studia Lógica, vol. 36 (1977), pp. 127–139.CrossRefGoogle Scholar
[7]Gabbay, D. M.. Semantical investigations in Heyting intuitionistic logic, Reidel Publishing Company. Dordrecht, 1981.CrossRefGoogle Scholar
[8]Goldblatt, R., Topoi, the categorical analysis of logic. North Holland. Amsterdam, 1984.Google Scholar
[9]Gratzer, G., On boolean functions, (Notes on lattice theory II), Revue Roumaine de Mathématiques Purés et Appliquées, vol. 7, 1962, pp. 693–697.Google Scholar
[10]Hecht, T. and Katrinák, T.. Equational classes of relative Stone algebras. Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 248–254.CrossRefGoogle Scholar
[11]Kaarly, K. and Pixley, A. F.. Affine complete varieties. Algebra Universalis, vol. 24 (1987), pp. 74–90.CrossRefGoogle Scholar
[12]Kaarly, K. and Pixley, A. F., Polynomial completeness in algebraic systems, Chapman and Hall, Boca Ratón, 2000.CrossRefGoogle Scholar
[13]Kaminski, M., Nonstandard connectives of intuitionistic propositional logic, Notre Dame Journal of Formal Logic, vol. 29 (1988). pp. 309–331.CrossRefGoogle Scholar
[14]Monteiro, L., Algebre du calculpropositionnel trivalent de Heyting. Fundamenta Mathematicae. vol. 74(1972), pp, 99–109.CrossRefGoogle Scholar
[15]Pixley, A., Completeness in arithmetical algebras. Algebra Universalis, vol. 2 (1972), pp. 179–196.CrossRefGoogle Scholar
[16]Rasiowa, H., An algebraic approach to non-classical logics. North Holland, Amsterdam. 1974.Google Scholar
[17]Rasiowa, H. and Sikorski, R., The mathematics of metamathematics. Polish Scientific Publishers, Warsaw. 1963, Third edition. 1970.Google Scholar
[18]Reyes, G. E. and Zolfaghari, H., Bi-Heyting Algebras, topos and modalities. Journal of Philosophical Logic, vol. 25 (1996), pp. 25–43.CrossRefGoogle Scholar
[19]Thomas, I., Finite limitations on Dummett's LC. Notre Dame Journal of Formal Logic, vol. 3 (1962), pp. 170–174.CrossRefGoogle Scholar
[20]Touraille, A., The word problem for Heyting* algebras. Algebra Universalis, vol. 24 (1987). pp. 120–127.CrossRefGoogle Scholar
[21]Yashin, A. D.. New solutions to Novikov's problem for intuitionistic connectives. Journal of Logic and Computation, vol. 8 (1998), pp. 637–664.CrossRefGoogle Scholar