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INFINITARY PROPOSITIONAL RELEVANT LANGUAGES WITH ABSURDITY

Published online by Cambridge University Press:  01 August 2017

GUILLERMO BADIA
GUILLERMO BADIA*
Affiliation:
Department of Knowledge-Based Mathematical Systems Johannes Kepler Universität Linz
*
*DEPARTMENT OF KNOWLEDGE-BASED MATHEMATICAL SYSTEMS JOHANNES KEPLER UNIVERSITÄT LINZ LINZ, AUSTRIA E-mail: guillebadia89@gmail.com
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Abstract

Analogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic L∞ω holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

Keywords

03B4703C75relevant logicmodel theoryinfinitary logicinterpolationRoutley–Meyer semantics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 4 , December 2017 , pp. 663 - 681
Copyright
Copyright © Association for Symbolic Logic 2017 

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References

BIBLIOGRAPHY

Anderson, A. R. & Belnap, N. D. (1992). Entailment. The Logic of Relevance and Necessity, Vol. II. Princeton, NJ: Princeton University Press.Google Scholar
Badia, G. (2016). The relevant fragment of first order logic. The Review of Symbolic Logic, 9(1), 143–166.Google Scholar
Badia, G. (2016). Bi-simulating in bi-intuitionistic logic. Studia Logica, 104(5), 1037–1050.Google Scholar
Barwise, J. (1974). Axioms for abstract model theory. Annals of Mathematical Logic, 7, 221–265.CrossRefGoogle Scholar
Barwise, J. & van Benthem, J. (1999). Interpolation, preservation, and pebble games. The Journal of Symbolic Logic, 64(2), 881–903.Google Scholar
Barwise, J. & Kunen, K. (1971). Hanf numbers for fragments of L ∞ω . Israel Journal of Mathematics, 10(3), 306–320.CrossRefGoogle Scholar
van Benthem, J. (1999). Modality, bisimulation and interpolation in infinitary logic. Annals of Pure and Applied Logic, 96(1–3), 29–41.Google Scholar
Bergstra, J. & van Benthem, J. (1994). Logic of transition systems. Journal of Logic, Language and Information, 3(4), 247–283.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Brady, R. T. (2006). Universal Logic. Stanford: CSLI.Google Scholar
Chagrov, A. & Zakharyaschev, M. (1997). Modal Logic. Oxford: Clarendon Press.CrossRefGoogle Scholar
Dickmann, M. A. (1975). Large Infinitary Languages. Oxford: North-Holland.Google Scholar
Dunn, M. & Restall, G. (2002). Relevance logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic. Dordrecht: Springer, pp. 1–128.Google Scholar
Ferguson, T. M. (2012). Notes on the model theory of De Morgan logics. Notre Dame Journal of Formal Logic, 53(1), 113–132.Google Scholar
Fine, K. (1988). Semantics for quantified relevance logic. Journal of Philosophical Logic, 17, 27–59.Google Scholar
Fine, K. (1989). Incompleteness for quantified relevance logics. In Norman, J. and Sylvan, R., editors. Directions in Relevant Logic. Dordrecht: Kluwer, pp. 205–212.Google Scholar
Hodges, W. (1993). Model Theory. Cambridge: Cambridge University Press.Google Scholar
Karp, C. (1964). Languages with Expressions of Infinite Length. Amsterdam: North-Holland.Google Scholar
Keisler, H. J. (1971). Model Theory for Infinitary Logic. Amsterdam: North-Holland.Google Scholar
Kurtonina, N. & de Rijke, M. (1997). Simulating without negation. Journal of Logic and Computation, 7(4), 501–522.Google Scholar
Mares, E. D. & Goldblatt, R. (2006). An alternative semantics for quantified relevant logic. Journal of Symbolic Logic, 71(1), 163–187.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logics. London: Routledge.CrossRefGoogle Scholar
Restall, G. (2013). Assertion, denial and non-classical theories. In Tanaka, K., Berto, F., Paoli, F., and Mares, E., editors. Paraconsistency: Logic and Applications. Dordrecht: Springer, pp. 81–89.Google Scholar
Robles, G. & Méndez, J. M. (2010). A Routley-Meyer type semantics for relevant logics including Br plus the disjunctive syllogism. Journal of Philosophical Logic, 39, 139–158.CrossRefGoogle Scholar
Routley, R. (1978). Problems and solutions in the semantics of quantified relevant logics. In Arruda, A. I., Chuaqui, R., and Da costa, N. C. A., editors. Mathematical Logic in Latin America. Proceedings of the IV Latin American Symposium on Mathematical Logic. Amsterdam: North Holland, pp. 305–340.Google Scholar
Routley, R., Meyer, R. K., Plumwood, V., & Brady, R. (1983). Relevant Logics and its Rivals, Vol. I. Atascadero, CA: Ridgeview.Google Scholar
Routley, R. & Meyer, R. K. (1973). The semantics of entailment. In Leblanc, H., editor. Truth, Syntax and Modality. Amsterdam: North Holland, pp. 199–243.Google Scholar
Routley, R. & Meyer, R. K. (1972). The semantics of entailment II. Journal of Philosophical Logic, 1, 53–73.Google Scholar
Routley, R. & Meyer, R. K. (1972). The semantics of entailment III. Journal of Philosophical Logic, 1, 192–208.Google Scholar
Thomas, M. (2015). A generalization of the Routley-Meyer semantic framework. Journal of Philosophical Logic, 44(4), 411–427.Google Scholar
Yang, E. (2013). R and relevance principle revisited. Journal of Philosophical Logic, 42(5), 767–782.Google Scholar