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CONSERVATIVITY OF HEYTING IMPLICATION OVER RELEVANT QUANTIFICATION

Published online by Cambridge University Press:  09 July 2009

ROBERT GOLDBLATT
ROBERT GOLDBLATT*
Affiliation:
Centre for Logic, Language and Computation, Victoria University of Wellington
*
*CENTRE FOR LOGIC, LANGUAGE AND COMPUTATION, VICTORIA UNIVERSITY OF WELLINGTON, P.O. BOX 600 WELLINGTON, NEW ZEALAND. E-mail:[email protected]
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Abstract

It is known that propositional relevant logics can be conservatively extended by the addition of a Heyting (intuitionistic) implication connective. We show that this same conservativity holds for a range of first-order relevant logics with strong identity axioms, using an adaptation of Fine’s stratified model theory. For systems without identity, the question of conservatively adding Heyting implication is thereby reduced to the question of conservatively adding the axioms for identity. Some results in this direction are also obtained. The conservative presence of Heyting implication allows the development of an alternative model theory for quantified relevant logics.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 2 , Issue 2 , June 2009 , pp. 310 - 341
Copyright
Copyright © Association for Symbolic Logic 2009

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References

BIBLIOGRAPHY

Anderson, A. R., Belnap, N. D. Jr, & Dunn, J. M. (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton University Press.Google Scholar
Birkhoff, G. (1967). Lattice Theory (third edition). American Mathematical Society, Providence, Rhode Island.Google Scholar
Dunn, J. M. (1993). Star and perp: Two treatments of negation. Philosophical Perspectives, 7, 331–357.CrossRefGoogle Scholar
Fine, K. (1974). Models for entailment. Journal of Philosophical Logic, 3, 347–372. Reprinted in Anderson et al. (1992, §51).CrossRefGoogle Scholar
Fine, K. (1988). Semantics for quantified relevance logic. Journal of Philosophical Logic, 17, 22–59. Reprinted in Anderson et al. (1992, §53).CrossRefGoogle Scholar
Goldblatt, R. I. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 19–35.CrossRefGoogle Scholar
Mares, E. D. (1992). Semantics for relevance logic with identity. Studia Logica, 51(1), 1–20.CrossRefGoogle Scholar
Restall, G. (1998). Displaying and deciding substructural logics 1: Logics with contraction. Journal of Philosophical Logic, 27, 179–216.CrossRefGoogle Scholar
Restall, G. (2000). An Introduction to Substructural Logics. Routledge, London.CrossRefGoogle Scholar
Routley, R., & Meyer, R. K. (1973). The semantics of entailment. In Leblanc, H., editor. Truth, Syntax and Modality. North-Holland, Amsterdam, pp. 199–243.CrossRefGoogle Scholar