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Many-valued logics of extended Gentzen style II

Published online by Cambridge University Press:  12 March 2014

Moto-o Takahashi
Moto-o Takahashi*
Affiliation:
St. Paul's University, Tokyo, Japan
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Extract

In the monograph [1] of Chang and Keisler, a considerable extent of model theory of the first order continuous logic (that is, roughly speaking, many-valued logic with truth values from a topological space) is ingeniously developed without using any notion of provability.

In this paper we shall define the notion of provability in continuous logic as well as the notion of matrix, which is a natural extension of one in finite-valued logic in [2], and develop the syntax and semantics of it mostly along the line in the preceding paper [2]. Fundamental theorems of model theory in continuous logic, which have been proved with purely model-theoretic proofs (i.e. those proofs which do not use any proof-theoretic notions) in [1], will be proved with proofs which are natural extensions of those in two-valued logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 4 , December 1970 , pp. 493 - 528
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Chang, C. C. and Keisler, J., Continuous model theory, Princeton University Press, Princeton, N.J., 1966.CrossRefGoogle Scholar
[2]Takahashi, M., Many-valued logics of extended Gentzen style I, Science Reports of the Tokyo Kyoiku Daigaku, Section A, vol. 9 (1967), (231), pp. 95–116.Google Scholar
[3]Takahashi, M., Continuous λ-ε logics. The Annals of the Japan Association for Philosophy of Science, vol. 3 (1970), pp. 205–215.Google Scholar