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Solution to the PW problem1

Published online by Cambridge University Press:  12 March 2014

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

Extract

Anderson and Belnap asked in §8.11 of their treatise Entailment [1] whether a certain pure implicational calculus, which we will call PW, 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
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
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