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A contractionless semilattice semantics

Published online by Cambridge University Press:  12 March 2014

Steve Giambrone ,
Robert K. Meyer  and
Alasdair Urquhart
Steve Giambrone
Affiliation:
Department of Philosophy, University of Southwestern Louisiana, Lafayette, Louisiana 70504
Robert K. Meyer
Affiliation:
Department of Philosophy, Australian National University, Canberra, Act 2601, Australia
Alasdair Urquhart
Affiliation:
Department of Philosophy, University of Toronto, Toronto, Ontario M5S 1A1, Canada
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Extract

Semilattice semantics for relevant logics were discovered independently by Routley and Urquhart over 10 years ago. A semilattice semantics was first published in [10], where the weak theory of implication of [8] and [3] (i.e., R →, the pure implication fragment of the system R of relevant implication) is shown to be consistent and complete with respect to it. That result was extended in [11], But the semantics is explored in greatest detail in [12]. As reported in [4], Fine outfitted the positive semilattice semantics for R+ with a suitable Hilbert-style axiomatisation. (We refer to the system as ◡R+.) In 1980 Charlwood supplied a subscripted system of natural deduction. (See [1] and [2].) A subscripted Gentzen system was devised in [5] and [6].

Obviously, the central idea of the semilattice semantics is to impose relevant-style valuations on a semilattice (with an identity) used as the underlying model structure. However, in [12] the contractionless semantics are obtained (quite reasonably) by dropping the idempotence postulate and thus changing the relatively simple semilattice structure into a commutative monoid. Here we show that the semilattice structure can be regained for positive, contractionless relevant implication. Although we have no proofs as yet, we think that this semantics will pave the way for showing completeness for the corresponding subscripted Gentzen and natural deduction systems, as well as the Hilbert-style axiomatization, ◡RW+.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 2 , June 1987 , pp. 526 - 529
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1]Charlwood, G., Representations of semilattice relevance logic, Ph.D. Thesis, University of Toronto, Toronto, 1978.Google Scholar
[2]Charlwood, G., An axiomatic version of positive semilattice relevance logic, this Journal, vol. 46 (1981), pp. 231–239.Google Scholar
[3]Church, Alonzo, The weak theory of implication, Kontrolliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften (Menne, A.et al., editors), Kommissions-Verlag Karl Alber, Munich, 1951, pp. 22–37.Google Scholar
[4]Fine, Kit, Completeness for the semilattice semantics, this Journal, vol. 41 (1976), p. 560 (abstract).Google Scholar
[5]Giambrone, Steve, Gentzen systems and decision procedures for relevant logics, Ph.D. Thesis, Australian National University, Canberra, 1983.Google Scholar
[6]Giambrone, Steve and Kron, Alexander, Four relevant Gentzen systems, Studia Logica, vol. 46 (1986) (to appear).Google Scholar
[7]Meyer, Robert K., Improved decision procedures for pure relevant logics, typescript, 1973.Google Scholar
[8]Shaw-Kwei, Moh, The deduction theorems and two new logical systems, Methodos, vol. 2 (1950), pp. 56–73.Google Scholar
[9]Routley, Richard and Meyer, Robert K., The semantics of entailment, Truth, syntax and modality (Leblanc, H., editor), North-Holland, Amsterdam, 1973, pp. 199–243.CrossRefGoogle Scholar
[10]Urquhart, Alasdair, Completeness of weak implication, Theoria, vol. 37 (1971), pp. 274–282.CrossRefGoogle Scholar
[11]Urquhart, Alasdair, Semantics for relevant logics, this Journal, vol. 37 (1972), pp. 159–169.Google Scholar
[12]Urquhart, Alasdair, The semantics of entailment, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, Pennsylvania (available through University Microfilms, Ann Arbor, Michigan), 1973.Google Scholar
[13]Grätzer, George, Lattice theory, Freeman, San Francisco, California, 1971.Google Scholar
[14]Giambrone, Steve and Urquhart, Alasdair, Proof theories for semilattice logics, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987) (to appear).CrossRefGoogle Scholar