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Relational proof system for relevant logics

Published online by Cambridge University Press:  12 March 2014

Ewa Orlowska
Ewa Orlowska*
Affiliation:
Institute of Theoretical and Applied Computer Science, Polish Academy of Sciences
*
Azaliowa 29, 04-539 Warsaw, Poland, E-mail: orlowska@plearn.bitnet
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Abstract

A method is presented for constructing natural deduction-style systems for propositional relevant logics. The method consists in first translating formulas of relevant logics into ternary relations, and then defining deduction rules for a corresponding logic of ternary relations. Proof systems of that form are given for various relevant logics. A class of algebras of ternary relations is introduced that provides a relation-algebraic semantics for relevant logics.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 4 , December 1992 , pp. 1425 - 1440
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[1]Anderson, A. R. and Belnap, N. D., Entailment: The logic of relevance and necessity, Princeton University Press, Princeton, 1975.Google Scholar
[2]Belnap, N. D., Display logic, Journal of Philosophical Logic, vol. 11 (1982), pp. 375–418.CrossRefGoogle Scholar
[3]Belnap, N. D., Gupta, A., and Dunn, J. M., A consecution calculus for positive relevant implication with necessitv, Journal of Philosophical Logic, vol. 11 (1980), pp. 343–362.CrossRefGoogle Scholar
[4]Brady, R. T., Natural deduction systems for some quantified relevant logics, Logique et Analyse, vol. 27 (1984), pp. 355–377.Google Scholar
[5]Brady, R. T., The Gentzenization and decidability of RW, Journal of Philosophical Logic, vol. 19 (1990), pp. 35–73.CrossRefGoogle Scholar
[6]Brady, R. T., Gentzenization and decidability of some contraction-less relevant logics, Journal of Philosophical Logic, vol. 20 (1991), pp. 97–117.CrossRefGoogle Scholar
[7]Dunn, J. M., A ‘Gentzen system’ for positive relevant implication (abstract), this Journal, vol. 3 (1973), pp. 356–357.Google Scholar
[8]Dunn, J. M., Consecution formulation of positive R with cotenability and t, Entailment: The logic of relevance and necessity, Princeton University Press, Princeton, 1975, pp. 381–391.Google Scholar
[9]Giambrone, S. and Kron, A., Four relevant Gentzen systems, Studia Logica, vol. XLVI (1987), pp. 55–71.CrossRefGoogle Scholar
[10]Giambrone, S. and Meyer, R. K., Completeness and conservative extension results for some Boolean relevant logics, Studia Logica, vol. XLVIII (1989), pp. 1–14.CrossRefGoogle Scholar
[11]Giambrone, S. and Urquhart, A., Proof theories for semilattice logics, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 433–439.CrossRefGoogle Scholar
[12]Henkin, L., Monk, D., and Tarski, A., Cylindric algebras, North-Holland, Amsterdam, 1985.Google Scholar
[13]Johnson, F. A., A natural deduction relevance logic, Bulletin of the Section of Logic, vol. 6 (1976), pp. 164–170.Google Scholar
[14]Orlowska, E., Relational interpretation of modal logics, Algebraic Logic (Andreka, H., Monk, D., and Nemeti, I., editors), North-Holland, Amsterdam, 1991, pp. 443–471; see also Bulletin of the Section of Logic, vol. 17 (1988), no. 1, pp. 2–14.Google Scholar
[15]Orlowska, E., Interpretation of relevant logics in a logic of ternary relations, Bulletin of the Section of Logic, vol. 19 (1990), no. 2, pp. 39–48.Google Scholar
[16]Orlowska, E., Nonclassical relation and cylindric algebras (manuscript), 1990.Google Scholar
[17]Orlowska, E., Algebraic aspects of relational knowledge representation: Modal relation algebras, Lecture Notes in Artificial Intelligence, Springer, (to appear).Google Scholar
[18]McRobbie, M. A., A tableau system for positive relevant implication, Bulletin of the Section of Logic, vol. 6 (1977), pp. 131–133.Google Scholar
[19]McRobbie, M. A. and Belnap, N. D., Relevant analytic tableau, Studia Logica, vol. XXXVIII (1979), pp. 187–209.CrossRefGoogle Scholar
[20]Routley, R. and Meyer, R. K., The semantics of entailment, Truth, syntax and modality (Leblanc, H., editor), North-Holland, Amsterdam, 1973, pp. 199–243.CrossRefGoogle Scholar
[21]Routley, R., Meyer, R. K., et al., Relevant logics and their rivals. Volume 1, Ridgeview, Artascadero, 1982.Google Scholar