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Negation as a modal operator. (English) Zbl 0626.03006
Kripke-style models which besides the intuitionistic accessibility relation have a modal accessibility relation, and in which negation is treated as a modal impossibility operator, are given for propositional logics with negation weaker than Johansson’s negation, as well as for Johansson’s and Heyting’s propositional logics and their extensions. The weakest logic captured by these models - that one in which the modal relation is as general as possible - is properly contained in Johansson’s logic. Models of this type adequate for the Johansson propositional calculus are shown intertranslatable with the standard Kripke models for this calculus. Conditions which must be met by models of this type to capture various negation axioms, and some known extensions of the Johansson propositional calculus with these axioms, are also considered. It is shown how in models adequate for the Heyting propositional calculus the modal relation becomes definable in a certain sense in terms of the intuitionistic relation. Finally, some comments are made on models of this type for propositional calculi based on classical or intermediate negationless logics. In particular, a modelling is given for Curry’s propositional logics D and E. A shorter version of this paper, with some additional information, has appeared under the title “Negation and impossibility” [Essays on Philosophy and Logic, J. Perzanowski ed., Jagiellonian University, Cracow, 85-91 (1987)].

MSC:
03B20 Subsystems of classical logic (including intuitionistic logic)
03B45 Modal logic (including the logic of norms)
03B55 Intermediate logics
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