## Constructive canonicity in non-classical logics.(English)Zbl 0949.03019

From the text: We deal with the completeness aspect of the Sahlqvist theorem, i.e. we use a new technique which is able to prove canonicity without any previous or simultaneous reduction of the second-order one. We also extend the result in such a way that it applies to intermediate logics, to logics with the difference connective and to intuitionistic modalities. Another aspect of the proof we present here is its constructive character. We get a constructive analogue of Stone’s representation theorem for a large class of varieties of distributive lattices with further operators.

### MSC:

 03B45 Modal logic (including the logic of norms) 03B55 Intermediate logics 06D20 Heyting algebras (lattice-theoretic aspects)
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### References:

 [1] van Benthem, J., Modal logic and classical logic (1983), Bibliopolis: Bibliopolis Napoli · Zbl 0639.03014 [2] Fine, K., An incomplete logic containing $$S4$$, Theoria, 40, 23-29 (1974) · Zbl 0287.02011 [3] Ghilardi, S.; Meloni, G., Sulle relazioni di copertura nel modello dei filtri di una teoria proposizionale intuizionista, Rend. Ist. Lomb. Accad. Sci. Lett. A, 117, 223-235 (1983) [4] Ghilardi, S.; Meloni, G., Completezza per logiche intermedie, Atti degli Incontri di Logica Matematica, Vol. 11, 613-620 (1985), Siena [5] Goldblatt, R., The McKinsey axiom is not canonical, J. Symbolic Logic, 56, 554-562 (1991) · Zbl 0744.03019 [6] Joyal, A., Les thréorèmes de Chevalley-Tarski et remarques sur l’Algèbre constructive, Cahiers Top Grém. Diff., XVI, 256-258 (1975) · Zbl 0354.02038 [7] Sahlqvist, H., Completeness and correspondence in the first and second order semantics for modal logic, (Proc. 3rd Scandinavian Logic Symp. (1975), North Holland: North Holland Amsterdam), 110-143 [8] Sambin, G.; Vaccaro, V., A new proof of Sahlqvist’s theorem on modal definability and completeness, J. Symbolic Logic, 54, 992-999 (1989) · Zbl 0682.03009 [9] Shehtman, V. B., On incomplete propositional logics, Dokl. Akad. Nauk SSSR, 235, 542-545 (1977) [10] Thomason, S. K., An incompleteness theorem in modal logic, Theoria, 40, 30-34 (1974) · Zbl 0287.02012
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