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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
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[1] T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005. · Zbl 1073.06001
[2] K. Denecke, M. Erné, S. L. Wismath (eds.), Galois Connections and Applications, Kluwer, 2004. · Zbl 1050.06001
[3] F. Domenach, B. Leclerc, Biclosed binary relations and Galois connections, Order, Vol. 18 (2001), pp. 89-104. · Zbl 0987.06005
[4] M. Erné, J. Koslowski, A. Melton, G. E. Strecker, A Primer on Galois Connections, Annals of the New York Academy of Sciences, Vol. 704 (1993), pp. 103-125. · Zbl 0809.06006
[5] G. K. E. Gentzen, Untersuchungen über das logische Schließen. I, Mathematische Zeitschrift, Vol. 39 (1934), pp. 176-210, [English translation: Investigation into Logical Deduction, [in:] M. E. Szabo, The collected Works of Gerhard Gentzen, North Holland, 1969, pp. 68-131.]
[6] G. Payette, P. K. Schotch, Remarks on the Scott-Lindenbaum Theorem, Studia Logica, Vol. 102 (2014), pp. 1003-1020. · Zbl 1329.03088
[7] D. Scott, Completeness and axiomatizability in many-valued logic, Proceedings of Symposia in Pure Mathematics, Vol. 25 (Proceedings of the Tarski Symposium), American Mathematical Society 1974, pp. 411-435. · Zbl 0318.02021
[8] D. J. Shoesmith, T. J. Smiley, Multiple-conclusion Logic, Cambridge 1978. · Zbl 0381.03001
[9] T. Skura, A. Wiśniewski, A system for proper multiple-conclusion entailment, Logic and Logical Philosophy, Vol. 24 (2015), pp. 241-253. · Zbl 1375.03016
[10] R. Wójcicki, Dual counterparts of consequence operations, Bulletin of the Section of Logic, Vol. 2 (1973), pp. 54-56.
[11] J. Zygmunt, An Essay in Matrix Semantics for Consequence Relations, Wydawnictwo Uniwersytetu Wrocławskiego, 1984. · Zbl 0559.03012
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