Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics

Nobu-Yuki Suzuki

Bulletin of the Section of Logic (2017)

  • Volume: 46, Issue: 1/2
  • ISSN: 0138-0680

Abstract

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We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.

How to cite

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Nobu-Yuki Suzuki. "Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics." Bulletin of the Section of Logic 46.1/2 (2017): null. <http://eudml.org/doc/295533>.

@article{Nobu2017,
abstract = {We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.},
author = {Nobu-Yuki Suzuki},
journal = {Bulletin of the Section of Logic},
keywords = {intermediate predicate logics; existence property; disjunction property},
language = {eng},
number = {1/2},
pages = {null},
title = {Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics},
url = {http://eudml.org/doc/295533},
volume = {46},
year = {2017},
}

TY - JOUR
AU - Nobu-Yuki Suzuki
TI - Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics
JO - Bulletin of the Section of Logic
PY - 2017
VL - 46
IS - 1/2
SP - null
AB - We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.
LA - eng
KW - intermediate predicate logics; existence property; disjunction property
UR - http://eudml.org/doc/295533
ER -

References

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  1. [1] A. Church, Introduction toMathematical Logic I, Princeton University Press, Princeton (1956). 
  2. [2] L. M. Doorman, A note on the existence property for intuitionistic logic with function symbols, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36 (1990), pp. 17–21. 
  3. [3] D. M. Gabbay, V. B. Shehtman and D. P. Skvortsov, Quantification in nonclassical logic. Vol. 1, Studies in Logic and the Foundations of Mathematics, 153. Elsevier, Amsterdam (2009). 
  4. [4] H. Ishihara and T. Nemoto, A note on the independence of premiss rule, Mathematical Logic Quarterly, to appear. 
  5. [5] S. C. Kleene, Introduction to metamathematics, D. Van Nostrand Co., Inc., New York 1952. 
  6. [6] S. C. Kleene, Disjunction and existence under implication in elementary intuitionistic formalisms, Journal of Symbolic Logic 27 (1962), pp. 11–18. (An addendum, the same journal 28 (1963), pp. 154–156.) 
  7. [7] Y. Komori, Some results on the super-intuitionistic predicate logics, Reports on Mathematical Logic 15 (1983), pp. 13–31. 
  8. [8] P. Minari, Disjunction and existence properties in intermediate predicate logics, Atti del Congresso Logica e Filosofia della Scienza, oggi. San Gimignano, 7–11 dicembre 1983. Vol. I – Logica. (1986), CLUEB, Bologna (Italy). 
  9. [9] T. Nakamura, Disjunction property for some intermediate predicate logics, Reports on Mathematical Logic 15 (1983), pp. 33–39. 
  10. [10] H. Ono, A study of intermediate predicate logics, Publications of the Research Institute for Mathematical Sciences 8 (1972/73), pp. 61–649. 
  11. [11] H. Ono, Some problems in intermediate predicate logics, Reports onMathematical Logic 21 (1987), pp. 55–67. (Supplement 22 (1988), pp. 117–118.) 
  12. [12] D. Prawitz, Natural deduction. A proof-theoretical study, Acta Universitatis Stockholmiensis. Stockholm Studies in Philosophy, No. 3 Almqvist & Wiksell, Stockholm 1965. (Reprint: Dover Publications, 2006) 
  13. [13] N.-Y. Suzuki, A remark on the delta operation and the Kripke sheaf semantics in super-intuitionistic predicate logics, Bulletin of the Section of Logic, University of Łódź, 25 (1996), pp. 21–28. 
  14. [14] N.-Y. Suzuki, Algebraic Kripke sheaf semantics for non-classical predicate logics Studia Logica 63 (1999), pp. 387–416. 
  15. [15] N.-Y. Suzuki, Halldén-completeness in super-intuitionistic predicate logics, Studia Logica 73 (2003), pp. 113–130 
  16. [16] N.-Y. Suzuki, A negative solution to Ono’s problem P52: Existence and disjunction properties in intermediate predicate Logics, to appear. 
  17. [17] A. S. Troelstra and D. van Dalen, Constructivism in mathematics, Vol. I. An introduction, Studies in Logic and the Foundations of Mathematics, 121. North-Holland, Amsterdam, 1988. 
  18. [18] T. Umezawa, “Chūkan jutsugo ronri” (Intermediate predicate logics) (in Japanese), Gekkan Masematikusu, Vol. 1, No. 2 (1980), pp. 162–168. Kaiyo Shuppan Co., Ltd. 

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