### An alternative semantics for quantified relevant logic

Robert Goldblatt and Edwin D. Mares
Source: J. Symbolic Logic Volume 71, Issue 1 (2006), 163-187.

#### Abstract

The quantified relevant logic RQ is given a new semantics in which a formula ∀ x A 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.

First Page:
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1140641167
Digital Object Identifier: doi:10.2178/jsl/1140641167
Mathematical Reviews number (MathSciNet): MR2210060
Zentralblatt MATH identifier: 05038892

### References

Alan R. Anderson, Nuel D. Belnap, and J. M. Dunn Entailment: The logic of relevance and necessity, vol. II, Princeton University Press, Princeton,1992.
Mathematical Reviews (MathSciNet): MR1223997
David M. Armstrong A world of states of affairs, Cambridge University Press, Cambridge,1997.
Ross Brady (editor) Relevant logics and their rivals, vol. II, Ashgate, Aldershot,2003.
C. C. Chang and H. Jerome Kiesler Model theory, second ed., North Holland, Amsterdam,1977.
Mathematical Reviews (MathSciNet): MR491125
J. Michael Dunn Algebraic completeness results for R-mingle and its extensions, Journal of Symbolic Logic, vol. 35 (1970), pp. 1--13.
Mathematical Reviews (MathSciNet): MR288008
Digital Object Identifier: doi:10.2307/2271149
Zentralblatt MATH: 0231.02024
-------- Star and perp, Philosophical Perspectives, vol. 7 (1993), pp. 331--357.
J. Michael Dunn and Greg Restall Relevance logic, Handbook of philosophical logic (G. M. Gabbay and F. Guenthner, editors), vol. 6, Kluwer, Dordrecht, second ed.,2002, pp. 1--128.
Kit Fine Models for entailment, Journal of Philosophical Logic, vol. 3 (1974), pp. 347--372, Reprinted in Anderson, Belnap, and Dunn (1992) \S 51.
Mathematical Reviews (MathSciNet): MR437309
Digital Object Identifier: doi:10.1007/BF00257480
Zentralblatt MATH: 0296.02013
-------- Semantics for quantified relevance logic, Journal of Philosophical Logic, vol. 17 (1988), pp. 22--59, Reprinted in Anderson, Belnap, and Dunn (1992) \S 53.
Mathematical Reviews (MathSciNet): MR925613
Digital Object Identifier: doi:10.1007/BF00249674
Zentralblatt MATH: 0646.03013
-------- Incompleteness for quantified relevance logics, Directions in relevant logic (J. Norman and R. Sylvan, editors), Kluwer, Dordrecht,1989, Reprinted in Anderson, Belnap, and Dunn (1992) \S 52, pp. 205--225.
Paul Halmos Algebriac logic, Chelsea, New York,1962.
Robert K. Meyer and J. Michael Dunn E, R, and $\gamma$, Journal of Symbolic Logic, vol. 34 (1969), pp. 460--474.
Mathematical Reviews (MathSciNet): MR252207
Digital Object Identifier: doi:10.2307/2270909
Robert K. Meyer, J. Michael Dunn, and Hughes Leblanc Completeness of relevant quantificational theories, Notre Dame Journal of Formal Logic, vol. 15 (1974), pp. 97--121.
Mathematical Reviews (MathSciNet): MR337564
Digital Object Identifier: doi:10.1305/ndjfl/1093891202
Project Euclid: euclid.ndjfl/1093891202
Zentralblatt MATH: 0272.02028
Greg Restall An introduction to substructural logics, Routledge, London,2000.
Richard Routley and Robert K. Meyer The semantics of entailment (I), Truth, syntax, and modality (Hughes Leblanc, editor), North Holland, Amsterdam,1973, pp. 199--243.
Mathematical Reviews (MathSciNet): MR409114
Richard Routley, Robert K. Meyer, Val Plumwood, and Ross T. Brady Relevant logics and their rivals, vol. 1, Ridgeview, Atascardero,1982.
Zentralblatt MATH: 0579.03011
Bertrand Russell The philosophy of logical atomism,1918, reprinted in Russell, The Philosophy of Logical Atomism, Open Court, LaSalle, IL, 1985.