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Some anomalies in Fitch's system QD

Published online by Cambridge University Press:  12 March 2014

M. W. Bunder  and
Jonathan P. Seldin
M. W. Bunder
Affiliation:
Oxford University, Oxfordshire, England
Jonathan P. Seldin
Affiliation:
Oxford University, Oxfordshire, England
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Extract

This note shows that the argument used in the proof of the inconsistency of Curry's system (see [1]) can also be applied to Fitch's system QD (see [3, Chapter 6]). As one vital rule of is not present in QD this argument does not lead to an actual contradiction, but it does lead to a theorem which is not a proposition if the system is consistent.

The method used below is that of [2], which is simpler than that of [1].

The axioms and rules of QD that we require are also present in [1] and [2] provided that Fitch's D is replaced by H. These axioms and rules are the following:

DD int: ⊦ D(Da),

m p: If ⊦ a ⊃ b2 and ⊦ a then ⊦ b, and

res imp int: If a ⊦ b then Da ⊦ a ⊃ b.

Let a be arbitrary and let G be [x](Dx ⊃ (x ⊃ a)). Then let X be BWBG(BWBG). We then have the following proof:

Thus, we have ⊦ X. Now suppose we also had ⊦ DX. Then by the method of the innermost subproof in the above proof, we would have ⊦ a, and since a is

arbitrary the system would be inconsistent. Hence, if QD is consistent, we do not have ⊦ DX, and so X is a theorem which is not a proposition.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 247 - 249
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Bunder, M. W., The inconsistency of , this Journal, vol. 41 (1976), pp. 467–468.Google Scholar
[2]Bunder, M. W. and Meyer, R. K., On the inconsistency of systems similar to , this Journal, vol. 43 (1978), pp. 1–2.Google Scholar
[3]Fitch, Frederic B., Elements of combinatory logic, Yale University Press, New Haven and London, 1974.Google Scholar
[4]Fitch, Frederic B., An extension of CΔ (to appear).Google Scholar
[5]Fitch, Frederic B., Excluded middle and the paradoxes, abstract, Meeting of the Association for Symbolic Logic, New York, 1975, this Journal, vol. 42 (1977), pp. 148–149.Google Scholar