Modalities as interactions between the classical and the intuitionistic logics

Michał Walicki



We give an equivalent formulation of topological algebras, interpreting S4, as boolean algebras equipped with intuitionistic negation. The intuitionistic substructure—Heyting algebra—of such an algebra can be then seen as an “epistemic subuniverse”, and modalities arise from the interaction between the intuitionistic and classical negations or, we might perhaps say, between the epistemic and the ontological aspects: they are not relations between arbitrary alternatives but between intuitionistic substructures and one common world governed by the classical (propositional) logic. As an example of the generality of the obtained view, we apply it also to S5. We give a sound, complete and decidable sequent calculus, extending a classical system with the rules for handling the intuitionistic negation, in which one can prove all classical, intuitionistic and S4 valid sequents.


topological algebras; Boolean algebras; Heyting algebras; modal logics, intuitionistic logics

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P. Blackburn, M. de R?ke, and Yde Venema, Modal Logic. Cambridge University Press, 2001.

A. Horn, “Free s5 algebras”, NotreDame Journal of Formal Logic 29(1) (1978), 189–191.

B. Jónsson and A. Tarski, “Boolean algebras with operators I”, American J. Mathematics 73 (1951), 891–939.

H.M. MacNeille, “Partially ordered sets”, Transactions of the American Mathematical Society 42 (1937), 416–460.

J.C.C. McKinsey and A. Tarski, “The algebra of topology”, The Annals of Mathematics 45(1) (1944), 141–191.

J.C.C. McKinsey and A. Tarski, “On closed elements in closure algebras”, The Annals of Mathematics 47(1) (1946), 126–162.

H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics. PWN, Warszawa, 1963.

A.S. Troelstra and H. Schwichtenberg, Basic Proof Theory. Cambridge University Press, 2 edition, 2000.

S. Vickers, Topology via Logic. Cambridge University Press, 1989.

M. Walicki, “Modalities as interactions between the classical and the intuitionistic logics”, Technical Report 330, Department of Informatics, University of Bergen, 2006.

Print ISSN: 1425-3305
Online ISSN: 2300-9802

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