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An interpolation lemma for the pure implicational calculus

Published online by Cambridge University Press:  12 March 2014

Roy Edelstein
Roy Edelstein*
Affiliation:
Ohio State University, Columbus, Ohio 43210
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Extract

Although a variety of proofs are available for the Craig-Lyndon Interpolation Lemma, they all seem to make essential use of negation, with the result that they are inapplicable to deductive theories in which negation cannot be expressed. It is here shown that this use of negation can be avoided by extending the Interpolation Lemma to the pure implicational calculus PCI.

Determinate Parameter Lemma. For each formula A, ⊨PA ⊃ A for some sentence parameter PA occurring in A.

Proof. By induction on the construction of A.

Interpolation Lemma For PCI. Let Γ be the set of sentence parameters common to A and C.If ⊨A ⊃ and C is not valid, then there is a formula B containing no sentence parameter not in Y such that ⊨A ⊃ B and ⊨B ⊃ C.

Proof. Since ⊨PA ⊃ A by the Determinate Parameter Lemma and ⊨A ⊃ C, ⊨PA ⊃ C. But C is by hypothesis an invalid formula, so PA must occur in C as well as in A. Thus PA Є Γ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 3 , September 1975 , pp. 443 - 444
Copyright
Copyright © Association for Symbolic Logic 1975

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