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Subformula theorems for N-sequents

Published online by Cambridge University Press:  12 March 2014

Hugues Leblanc
Hugues Leblanc*
Affiliation:
Temple University
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Extract

With all seven of ‘∼’, ‘⊃’, ‘&’, ‘ν’, ‘≡’, ‘∀’, and ‘∃’ understood to serve as primitive signs, let an expression of the sort

A1, A2, … , An → B1, B2, … , Bm,

where A1, A2, …, Aη (η≥ 0), B1, B2, …, and Bm (m > 0 if n = 0, otherwise m ≥ 0) are wffs of the first-order quantificational calculus (QC), count as an L-sequent of QC; let an L-sequent of QC of the sort

A1, A2, … , An → B

count as an N-sequent of QC;2 and let an L-sequent S of QC count as a Gentzen L-sequent of QC if no individual variable of QC occurs both bound and free in S.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 2 , 23 July 1968 , pp. 161 - 179
Copyright
Copyright © Association for Symbolic Logic 1968

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References

[1]Bernays, P., Betrachtungen zum Sequenzen-Kalkul, Contributions to logic and methodology in honor of J. M. Bocheński, Tymieniecka, A.-T., ed., North-Holland, Amsterdam, 1965, pp. 1–44.Google Scholar
[2]Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934), pp. 176–210, 403–431.CrossRefGoogle Scholar
[3]Gentzen, G., Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathemathehe Annalen, vol. 112 (1936), pp. 493–565.CrossRefGoogle Scholar
[4]Kanger, S., Provability in logic, Almquist and Wiksell, Stockholm, 1957.Google Scholar
[5]Leblanc, H., Techniques of deductive inference, Prentice-Hall, Englewood Cliffs, N.J., 1966.Google Scholar
[6]Prawitz, D., Natural deduction, A proof-theoretical study, Almquist and Wiksell, Stockholm, 1965.Google Scholar