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On axiomatizing fragments

Published online by Cambridge University Press:  12 March 2014

C. Smorynski
C. Smorynski*
Affiliation:
429 South Warwick Avenue, Westmont, Illinois 60559
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Extract

The main purpose of this note is to present new semi-model-theoretic proofs of some axiomatization results. Principally, we prove the result of [van Dalen and Statman] on the axiomatization of the equality fragment of the intuitionistic theory of apartness by ω-fold stability axioms. Further examples are also discussed.

The intuitionistic theories of equality and apartness are given by the following nonlogical axioms:

EQ .

AP .

In AP, one can define a notion of equality by means of the abbreviation:

.

The corresponding interpretation of E Q in AP is not faithful — since equality is a negation, we have stability of equality:

.

The equality fragment of AP is axiomatized by a suitable generalization of the stability of equality. To state it, one first defines a sequence of partial apartness relations:

.

A simple induction shows, for all n,

whence

.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 4 , December 1977 , pp. 530 - 544
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

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