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Innocuous substitutions

Published online by Cambridge University Press:  12 March 2014

Daniel Leivant
Daniel Leivant*
Affiliation:
Ohio State University, Columbus, Ohio 43210
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Extract

In classical first-order predicate logic CL1 (without equality) only tautologies and antitautologies satisfy nontautological schemas. I.e., if F[p, Q] is a nontautological formula (fl) in the predicate letters shown, with p prepositional, then ⊬ F[K, Q] for any sentence K not containing some Q ∈ Q, unless ⊢ K or ⊢ ¬K. This is an easy consequence of the Completeness Theorem. Clearly, the analogous statement fails for intuitionistic predicate logic IL1; already when Q is empty: (i) ¬¬ K for, e.g., K ≡ p ∨ ¬p; (ii) ¬¬K → K for, e.g., K ≡ ¬p.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 2 , June 1980 , pp. 363 - 368
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[1]de Jongh, D. H. J., Formulas of one propositional variable in intuitionistic arithmetic, Report 73–03, Department of Mathematics, University of Amsterdam, 1973.Google Scholar
[2]Leivant, D., Absoluteness of intuitionistic logic, Ph. D. dissertation, Amsterdam, 1975.Google Scholar
[3]Leivant, D., Metamathematical applications of the ω-rule (to appear).Google Scholar
[4]Nishimura, , On formulas of one variable in intuitionistic propositional calculus, this Journal, vol. 25 (1960), pp. 327–331.Google Scholar