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Intermediate predicate logics determined by ordinals

Published online by Cambridge University Press:  12 March 2014

Pierluigi Minari ,
Mitio Takano  and
Hiroakira Ono
Pierluigi Minari
Affiliation:
Department of Philosophy, University of Florence, 50139 Firenze, Italy
Mitio Takano
Affiliation:
Department of Mathematics, Niigata University, Niigata 950-21, Japan
Hiroakira Ono
Affiliation:
Department of Mathematics, Hiroshima University, Hiroshima 730, Japan
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Abstract

For each ordinal α > 0, L(α) is the intermediate predicate logic characterized by the class of all Kripke frames with the poset α and with constant domain. This paper will be devoted to a study of logics of the form L(α). It will be shown that for each uncountable ordinal of the form α + η with a finite or a countable η(> 0), there exists a countable ordinal of the form β + η such that L(α + η) = L(β + η). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(α) if α is an uncountable regular ordinal. Moreover, it will be proved that the mapping L is injective if it is restricted to ordinals less than ωω, i.e. α ≠ β implies L(α) ≠ L(β) for each ordinal α, β ≤ ωω.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 1099 - 1124
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

[1]van Benthem, J., Notes on modal definability, Notre Dante Journal of Formal Logic, vol. 30 (1989), pp. 20–35.Google Scholar
[2]Casari, E., Intermediate logics, Atti degli incontri di logica matematica 1982, Università di Siena (1983), pp. 243–298.Google Scholar
[3]Chagrov, A. V., On the complexity of propositional logics, Complexity problems in mathematical logic (Kanovich, M. I., editor), Kalininskiĭ Gosudarstvennyĭ Universitet, Kalinin, 1985, pp. 80–90. (Russian)Google Scholar
[4]van Dalen, D., lntuitionistic logic, Handbook of philosophical logic, Vol. III (Gabbay, D. and Guenthner, F., editors), Reidel, Dordrecht, 1986, pp. 225–339.CrossRefGoogle Scholar
[5]Hosoi, T. and Ono, H., Intermediate propositional logics (a survey), Journal of Tsuda College, vol. 5 (1973), pp. 67–82.Google Scholar
[6]Levy, A., Basic set theory, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[7]Minari, P., Kripke-definable ordinals, Atti degli incontri di logica matematica, Università di Siena, vol. 2 (1985), pp. 185–188.Google Scholar
[8]Minari, P., A strong Löwenheim-Skolem-type theorem for Kripke semantic, unpublished.Google Scholar
[9]Ono, H., A study of intermediate predicate logics, Publications of the Research Institute for Mathematical Sciences, vol. 8 (1972), pp. 619–649.CrossRefGoogle Scholar
[10]Ono, H., Some problems in intermediate predicate logics, Reports on Mathematical Logic, vol. 21 (1987), pp. 55–67.Google Scholar
[11]Ono, H., On finite linear intermediate predicate logics, Studia Logica, vol. 47 (1988), pp. 81–89.CrossRefGoogle Scholar
[12]Shehtman, V. B., Denumerable approximability of superintuitionistic and modal logics, Studies in nonclassical logics and formal systems (Mihaĭlov, A. I., editor), “Nauka”, Moscow, 1983, pp. 287–299. (Russian)Google Scholar
[13]Skvortsov, D. P., On axiomatizability of some intermediate predicate logics, Reports on Mathematical Logic, vol. 22 (1988), pp. 115–116. (summary)Google Scholar
[14]Takano, M., Ordered sets R and Q as bases of Kripke models, Studia Logica, vol. 46 (1987), pp. 137–148.CrossRefGoogle Scholar
[15]Takano, M., A negative answer to Ono's first problem: K-completeness does not imply strong K-completeness, Reports on Mathematical Logic, vol. 21 (1987), pp. 69–71.Google Scholar