Disjunctive multiple-conclusion consequence relations.(English)Zbl 1446.06007

Summary: The concept of multiple-conclusion consequence relation from [D. J. Shoesmith and T. J. Smiley, Multiple-conclusion logic. Cambridge etc.: Cambridge University Press (1978; Zbl 0381.03001)] and [D. Scott, Proc. Symp. Pure Math. 25, 411–435 (1974; Zbl 0318.02021)] is considered. The closure operation $$C$$ assigning to any binary relation $$r$$ (defined on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing $$r$$, is defined on the grounds of a natural Galois connection. It is shown that the very closure $$C$$ is an isomorphism from the power set algebra of a simple binary relation to the Boolean algebra of all multiple-conclusion consequence relations.

MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 03B50 Many-valued logic

Citations:

Zbl 0381.03001; Zbl 0318.02021
Full Text:

References:

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