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An interpolation theorem in many-valued logic

Published online by Cambridge University Press:  12 March 2014

Masazumi Hanazawa  and
Mitio Takano
Masazumi Hanazawa
Affiliation:
Department of Mathematics, Saitama University, Urawa 338, Japan
Mitio Takano
Affiliation:
Department of Mathematics, Niigata University, Niigata 950-21, Japan
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Extract

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ 1,…, Γ M ) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):

  • (I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that

  • (i) every predicate or propositional variable occurring in D occurs in A and B, and

  • (ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.

  • What shall be proved in this paper is the following (in representative form):

  • (II) If a sequent ({A 1}, {A 2}, …, {AM }) is valid, then there is a formula D such that

  • (i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A 1,…, AM , and

  • (ii) the following M sequents are valid:

  • ({A 1},{D},…,{D}),({D},{A 2},…,{D}),…,({D},{D},…,{AM }).

  • Clearly the former can be obtained as a corollary of the latter.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 2 , June 1986 , pp. 448 - 452
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

[1] Gill, R. R. R., The Craig-Lyndon interpolation theorem in 3-valued logic, this Journal, vol. 35 (1970), pp. 230–238.Google Scholar
[2] Miyama, T., The interpolation theorem and Beth's theorem in many-valued logics, Mathematica Japonica, vol. 19 (1974), pp. 341–355.Google Scholar
[3] Rousseau, G., Sequents in many-valued logic. I, Fundamenta Mathematicae, vol. 60 (1967), pp. 23–33.CrossRefGoogle Scholar
[4] Takahashi, M., Many-valued logics of extended Gentzen style. I, Science Reports of the Tokyo Kyoiku Daigaku, vol. 9 (1967), pp. 271–292.Google Scholar