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A proof of the cut-elimination theorem in simple type theory

Published online by Cambridge University Press:  12 March 2014

Satoko Titani
Satoko Titani*
Affiliation:
University of Washington, Seattle, Washington 98195
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Extract

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.

The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.

Inductive definition of types.

0 and 1 are types.

If τ1, …, τn are types, then (τ1, …, τn) is a type.

Basic symbols.

a1τ, a2τ, … for free variables of type τ.

x1τ, x2τ, … for bound variables of type τ.

An arbitrary number of constants of certain types.

An arbitrary number of function symbols with certain argument places.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 2 , June 1973 , pp. 215 - 226
Copyright
Copyright © Association for Symbolic Logic 1973

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References

BIBLIOGRAPHY

[1]Schütte, K., Syntactical and semantical properties of simple type theory, this Journal, vol. 25 (1960), pp. 305–326.Google Scholar
[2]Takahashi, M., A proof of cut-elimination theorem in simple type theory, Journal of the Mathematical Society of Japan, vol. 19 (1967), pp. 399–410.CrossRefGoogle Scholar
[3]Takeuti, G., On the fundamental conjecture of GLC. I, Journal of the Mathematical Society of Japan, vol. 7 (1955), pp. 249–275.Google Scholar
[4]Titani, S., An algebraic formulation of cut-elimination theorem, Journal of the Mathematical Society of Japan, vol. 17 (1965), pp. 72–83.CrossRefGoogle Scholar