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Real-valued models with metric equality and uniformly continuous predicates

Published online by Cambridge University Press:  12 March 2014

Michael Katz
Michael Katz*
Affiliation:
Departement of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada
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Abstract

Two real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 772 - 792
Copyright
Copyright © Association for Symbolic Logic 1982

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References

REFERENCES

[1]Belluce, L.P. and Chang, C.C., A weak completeness theorem for infinite valued predicate logic, this Journal, vol. 28 (1963), pp. 43–50.Google Scholar
[2]Chang, C.C., A new proof of the completeness of the Łukasiewicz axioms, Transactions of the American Mathematical Society, vol. 93 (1959), pp. 74–80.Google Scholar
[3]Fourman, M.P., The logic of sheaves, Oxford University, Preprint, 1974.Google Scholar
[4]Giles, R., A non-classical logic for physics, Stadia Logica, vol. 33 (1974), pp. 399–417.Google Scholar
5]Giles, R., Łukasiewicz logic and fuzzy set theory, International Journal of Man-Machine Studies, vol. 8 (1976), pp. 313–327.CrossRefGoogle Scholar
[6]Goguen, J.A., The logic of inexact concepts, Synthese, vol. 19 (1969), pp. 325–373.CrossRefGoogle Scholar
[7]Hay, L.S., Axiomatization of the infinite valued predicate calculus, this Journal, vol. 28 (1963), pp. 77–86.Google Scholar
[8]Katz, M., Łukasiewicz logic and the foundations of measurement, Queen's Math. Preprint, 1979–1915. (Revised version to appear in Studio Logica.)Google Scholar
[9]Katz, M., Two systems of multi-valued logic for science, Proceedings of the 11th International Symposium on Multiple-Valued Logic, IEEE Computer Society Press, New York, 1981, pp. 175–182.Google Scholar
[10]Mansfield, R., The theory of Boolean ultrapowers, Annals of Mathematical Logic, vol. 2 (1971), pp. 279–323.CrossRefGoogle Scholar
[11]Rose, A. and Rosser, J.B., Fragments of many valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 1–53.CrossRefGoogle Scholar
[12]Scott, D.S., Lectures on Boolean-valued models of set theory, Summer Institute on Axiomatic Set Theory, UCLA, 1967.Google Scholar
[13]Scott, D.S., Background to formalization, Truth, Syntax and Modality (Leblanc, H., Editor), North-Holland, Amsterdam, 1973, pp. 244–273.CrossRefGoogle Scholar
[14]Scott, D.S., Completeness and axiomatizability in many valued logic, Proceedings of the Tarski Symposium (Henkin, L.et al., Editors), Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, R.I., 1974, pp. 188–197.Google Scholar