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Correction to a definition of negation

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University, New Haven, Connecticut 06520
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Extract

In [3] a definition of negation was presented for the system K′ of extended basic logic [1], but it has since been shown by Peter Päppinghaus (personal communication) that this definition fails to give rise to the law of double negation as I claimed it did. The purpose of this note is to revise this defective definition in such a way that it clearly does give rise to the law of double negation, as well as to the other negation rules of K′.

Although Päppinghaus's original letter to me was dated September 19, 1972, the matter has remained unresolved all this time. Only recently have I seen that there is a simple way to correct the definition. I am of course very grateful to Päppinghaus for pointing out my error in claiming to be able to derive the rule of double negation from the original form of the definition.

The corrected definition will, as before, use fixed-point operators to give the effect of the required kind of transfinite induction, but this time a double transfinite induction will be used, somewhat like the double transfinite induction used in [5] to define simultaneously the theorems and antitheorems of system CΓ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 47 - 50
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[1]Fitch, Frederic B., An extension of basic logic, this Journal, vol. 13 (1948), pp. 95–106.Google Scholar
[2]Fitch, Frederic B., A simplification of basic logic, this Journal, vol. 18 (1953), pp. 317–325.Google Scholar
[3]Fitch, Frederic B., A definition of negation in extended basic logic, this Journal, vol. 19 (1954), pp. 29–36.Google Scholar
[4]Fitch, Frederic B., Recursive functions in basic logic, this Journal, vol. 21 (1956), pp. 337–346.Google Scholar
[5]Fitch, Frederic B., A consistent combinatory logic with an inverse to equality, this Journal, vol. 45 (1980), pp. 529–543.Google Scholar