Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T03:24:52.162Z Has data issue: false hasContentIssue false
  • English
  • Français

An alternative semantics for quantified relevant logic

Published online by Cambridge University Press:  12 March 2014

Edwin D. Mares  and
Robert Goldblatt
Edwin D. Mares
Affiliation:
Centre for Logic, Language and Computation, Victoria University of Wellington, New Zealand. E-mail: Edwin.Mares@vuw.ac.nz, E-mail: Rob.Goldblatt@vuw.ac.nz
Robert Goldblatt
Affiliation:
Centre for Logic, Language and Computation, Victoria University of Wellington, New Zealand. E-mail: Rob.Goldblatt@vuw.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

The quantified relevant logic RQ is given a new semantics in which a formula ∀xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of ‘extensional confinement’: ∀x(A ⋁ B) → (A ⋁ ∀xB), with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 1 , March 2006 , pp. 163 - 187
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Anderson, Alan R., Belnap, Nuel D., and Dunn, J. M., Entailment: The logic of relevance and necessity, vol. II, Princeton University Press, Princeton, 1992.Google Scholar
[2]Armstrong, David M., A world of states of affairs. Cambridge University Press, Cambridge, 1997.CrossRefGoogle Scholar
[3]Brady, Ross (editor), Relevant logics and their rivals, vol. II, Ashgate, Aldershot, 2003.Google Scholar
[4]Chang, C. C. and Kiesler, H. Jerome, Model theory, second ed., North Holland, Amsterdam, 1977.Google Scholar
[5]Dunn, J. Michael, Algebraic completeness results for R-mingle and its extensions, this Journal, vol. 35 (1970), pp. 1–13.Google Scholar
[6]Dunn, J. Michael, Star and perp, Philosophical Perspectives, vol. 7 (1993), pp. 331–357.CrossRefGoogle Scholar
[7]Dunn, J. Michael and Restall, Greg, Relevance logic, Handbook of philosophical logic (Gabbay, G. M. and Guenthner, F., editors), vol. 6, Kluwer, Dordrecht, second ed., 2002, pp. 1–128.Google Scholar
[8]Fine, Kit, Models for entailment, Journal of Philosophical Logic, vol. 3 (1974), pp. 347–372, Reprinted in Anderson, Belnap, and Dunn (1992) § 51.CrossRefGoogle Scholar
[9]Fine, Kit, Semantics for quantified relevance logic, Journal of Philosophical Logic, vol. 17 (1988), pp. 22–59, Reprinted in Anderson, Belnap, and Dunn (1992) § 53.CrossRefGoogle Scholar
[10]Fine, Kit, Incompleteness for quantified relevance logics, Directions in relevant logic (Norman, J. and Sylvan, R., editors), Kluwer, Dordrecht, 1989, Reprinted in Anderson, Belnap, and Dunn (1992) § 52, pp. 205–225.CrossRefGoogle Scholar
[11]Halmos, Paul, Algebriac logic, Chelsea, New York, 1962.Google Scholar
[12]Meyer, Robert K. and Dunn, J. Michael, E, R, and γ, this Journal, vol. 34 (1969), pp. 460–474.Google Scholar
[13]Meyer, Robert K., Dunn, J. Michael, and Leblanc, Hughes, Completeness of relevant quantificational theories, Notre Dame Journal of Formal Logic, vol. 15 (1974), pp. 97–121.Google Scholar
[14]Restall, Greg, An introduction to substructural logics, Routledge, London, 2000.CrossRefGoogle Scholar
[15]Routley, Richard and Meyer, Robert K., The semantics of entailment (I), Truth, syntax, and modality (Leblanc, Hughes, editor), North Holland, Amsterdam, 1973, pp. 199–243.CrossRefGoogle Scholar
[16]Routley, Richard, Meyer, Robert K., Plumwood, Val, and Brady, Ross T., Relevant logics and their rivals, vol. 1, Ridgeview, Atascardero, 1982.Google Scholar
[17]Russell, Bertrand, The philosophy of logical atomism, 1918, reprinted in Russell, , The Philosophy of Logical Atomism, Open Court, LaSalle, IL, 1985.Google Scholar