Disjunctive Multiple-Conclusion Consequence Relations

Marek Nowak

Bulletin of the Section of Logic (2019)

• Volume: 48, Issue: 4
• ISSN: 0138-0680

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Abstract

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The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r (dened on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing r, is dened 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.

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Marek Nowak. "Disjunctive Multiple-Conclusion Consequence Relations." Bulletin of the Section of Logic 48.4 (2019): null. <http://eudml.org/doc/295522>.

@article{MarekNowak2019,
abstract = {The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r (dened on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing r, is dened 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.},
author = {Marek Nowak},
journal = {Bulletin of the Section of Logic},
keywords = {multiple-conclusion consequence relation; closure operation; Galois connection},
language = {eng},
number = {4},
pages = {null},
title = {Disjunctive Multiple-Conclusion Consequence Relations},
url = {http://eudml.org/doc/295522},
volume = {48},
year = {2019},
}

TY - JOUR
AU - Marek Nowak
TI - Disjunctive Multiple-Conclusion Consequence Relations
JO - Bulletin of the Section of Logic
PY - 2019
VL - 48
IS - 4
SP - null
AB - The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r (dened on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing r, is dened 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.
LA - eng
KW - multiple-conclusion consequence relation; closure operation; Galois connection
UR - http://eudml.org/doc/295522
ER -

References

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1. [1] T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005. https://doi.org/10.1007/b139095 
2. [2] K. Denecke, M. Erné, S. L. Wismath (eds.), Galois Connections and Applications, Kluwer, 2004. https://doi.org/10.1007/978-1-4020-1898-5 
3. [3] F. Domenach, B. Leclerc, Biclosed binary relations and Galois connections, Order, Vol. 18 (2001), pp. 89–104. https://doi.org/10.1023/A:1010662327346 
4. [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. https://doi.org/10.1111/j.1749-6632.1993.tb52513.x 
5. [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.] https://doi.org/10.1007/BF01201353 
6. [6] G. Payette, P. K. Schotch, Remarks on the Scott-Lindenbaum Theorem, Studia Logica, Vol. 102 (2014), pp. 1003–1020. https://doi.org/10.1007/s11225-013-9519-y 
7. [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. 
8. [8] D. J. Shoesmith, T. J. Smiley, Multiple-conclusion Logic, Cambridge 1978. https://doi.org/10.1017/CBO9780511565687 
9. [9] T. Skura, A. Wiśniewski, A system for proper multiple-conclusion entailment, Logic and Logical Philosophy, Vol. 24 (2015), pp. 241–253. http://dx.doi.org/10.12775/LLP.2015.001 
10. [10] R. Wójcicki, Dual counterparts of consequence operations, Bulletin of the Section of Logic, Vol. 2 (1973), pp. 54–56. 
11. [11] J. Zygmunt, An Essay in Matrix Semantics for Consequence Relations, Wydawnictwo Uniwersytetu Wrocławskiego, 1984. 

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