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An axiomatic version of positive semilattice relevance logic1

Published online by Cambridge University Press:  12 March 2014

G. Charlwood
G. Charlwood*
Affiliation:
University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
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Extract

In [8] and [9] Urquhart presented the semilattice semantics for a relevance logic with conjunction and disjunction. These semantics combined the simplest known semantics for relevant implication, with minimal modification of the truth conditions for conjunction and disjunction. A slightly different approach leads to the distinct positive relevance logic of [7] and [5]. Sound and complete axiomatic logics for the semilattice semantics were readily available, so long as disjunction was left out.

The purpose of this paper is to report the remedy of this lack. In [1] the author developed work of Fine ([3], announced in [4]) and Prawitz [6] to show soundness and completeness of an axiomatic logic and two natural deduction logics, with respect to Urquhart's semantics. Thus Prawitz' positive relevance logic is equivalent to the semilattice logic, rather than the logic of [5]. We present here the axiomatic logic and show its equivalence to the semilattice semantics.

In §1 the languages and logic are given, along with a few proof-theoretic remarks. In §2 the semantics are defined, and the soundness proof is sketched. In §3, the completeness proof is given.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 2 , June 1981 , pp. 233 - 239
Copyright
Copyright © Association for Symbolic Logic 1981

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Footnotes

1

I would like to express my thanks to my thesis supervisor, Alasdair Urquhart, and to my teachers, Bas van Fraassen, R.K. Meyer and Kit Fine, for invaluable instruction and guidance.

References

REFERENCES

[1]Charlwood, G.W., Representations of semilattice relevance logic, thesis, University of Toronto, 1978.Google Scholar
[2]Church, A., Introduction to mathematical logic, Princeton University Press, New Jersey, 1956.Google Scholar
[3]Fine, K., Completeness for the semilattice semantics with disjunction and conjunction, unpublished manuscript, c. 1975.Google Scholar
[4]Fine, K., Completeness for the semilattice semantics with disjunction and conjunction, abstract, this Journal, vol. 41 (1976), p. 560.Google Scholar
[5]Fine, K., Models for entailment, Journal of Philosophical Logic, vol. 3 (1974), pp. 347–372.CrossRefGoogle Scholar
[6]Prawitz, D., Natural deduction, Almqvist & Wiksell, Stockholm, Göteborg, Uppsala, 1965.Google Scholar
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[8]Urquhart, A., Semantics for relevant logics, this Journal, vol. 37 (1972), pp. 159–169.Google Scholar
[9]Urquhart, A., The semantics of entailment, thesis, University of Pittsburgh, 1973.Google Scholar