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Solution to the P − W problem1

Published online by Cambridge University Press:  12 March 2014

E.P. Martin  and
R.K. Meyer
E.P. Martin
Affiliation:
Australian National University, Canberra, Australia
R.K. Meyer
Affiliation:
Australian National University, Canberra, Australia
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Extract

Anderson and Belnap asked in §8.11 of their treatise Entailment [1] whether a certain pure implicational calculus, which we will call P − W, is minimal in the sense that no two distinct formulas coentail each other in this calculus. We provide a positive solution to this question, variously known as The P − W problem, or Belnap's conjecture.

We will be concerned with two systems of pure implication, formulated in a language constructed in the usual way from a set of propositional variables, with a single binary connective →. We use A, B,…, A1, B1, …, as variables ranging over formulas. Formulas are written using the bracketing conventions of Church [3].

The first system, which we call S (in honour of its evident incorporation of syllogistic principles of reasoning), has as axioms all instances of

(B) B → C →. A → B →. A → C (prefixing),

(B) A → B →. B → C →. A → C (suffixing),

and rules

(BX) from B → C infer A → B →. A → C (rule prefixing),

(B’X) from A → B infer B → C →. A → C (rule suffixing),

(BXY) from A → B and B → C infer A → C (rule transitivity).

The second system, P − W, has in addition to the axioms and rules of S the axiom scheme

(I) A → A

of identity.

We write ⊢SA (⊣SA) to mean that A is (resp. is not) a theorem of S, and similarly for P − W.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 869 - 887
Copyright
Copyright © Association for Symbolic Logic 1982

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Footnotes

1

The results of this paper formed part of the first author's doctoral thesis [7] presented to the Australian National University in 1978.

References

REFERENCES

[1]Anderson, A. R. and Belnap, N. D. Jr., Entailment: The logic of relevance and necessity, vol. 1, Princeton University Press, Princeton, N. J., 1975.Google Scholar
[2]Belnap, N. D. Jr., A formal analysis of entailment, Technical Report No. 7, Contract No. SAR/Nonr-609 (16), Office of Naval Research, New Haven, 1960.CrossRefGoogle Scholar
[3]Church, A., Introduction to mathematical logic, vol. 1, Princeton University Press, Princeton, N. J., 1956.Google Scholar
[4]Curry, H. B. and Feys, R., Combinatory logic, vol. 1, North-Holland, Amsterdam, 1958.Google Scholar
[5]Fine, K., Models for entailment, Journal of Philosophical Logic, vol. 3 (1974), pp. 347–372.CrossRefGoogle Scholar
[6]McRobbie, M. A., A proof theoretic investigation of relevant and modal logics, Ph. D. thesis, Australian National University, Canberra, 1979.Google Scholar
[7]Martin, E. P., The P − W problem, Ph.D. thesis, Australian National University, Canberra, 1978.Google Scholar
[8]Meyer, R. K., A general Gentzen system for implicational calculi, The Relevance Logic News-letter, vol. 1 (1976), pp. 189–201.Google Scholar