Foundations of Mathematical Logic by Haskell B. Curry

Author:Haskell B. Curry [Curry, Haskell B.]
Language: eng
Format: epub
ISBN: 9780486153056
Publisher: INscribe Digital
Published: 2012-06-30T04:00:00+00:00

(The other case of the left-hand proof is similar.)

In the case of V*, the situation is dual to that of *A.

In the case of *P the derivation of the Formulation I rule from that of Ketonen204 is

Conversely, the Ketonen form of *P is identical with that case of *P in Formulation I in which is void. If void prosequences on the right are not admitted, we take the of Ketonen’s *P to be C, 3; then the rule can be derived in Formulation I thus:

This completes the proof.

The Ketonen modification of *P has somewhat the opposite effect of that of Formulation II, since it makes the parametric constituents uniform in all the premises, and thus (r4)’ is true without exception. Of course, it is only applicable in case *P is multiple; in the singular cases we must leave *P as in Formulation I, and the exception to (r4)’ must stand.

Theorems 2 and 3 use the structural rules *K* and *W* to establish the equivalence of Formulations I and II, on the one hand, and of the Ketonen rules and those of Formulation I, on the other. In generalized situations where the *K* and *W* do not both hold, these equivalences may fail. In such a case one of the variant formulations may have definite advantages.

9. Some simple properties. We shall establish here two rather simple, but unrelated, properties.

Theorem 4. Let the prime statement scheme (p1) be replaced by the scheme

(p1)’

E ⊩ E

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