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A Remark on the Intersection of Tow Logics

Published online by Cambridge University Press:  22 January 2016

Satoshi Miura
Satoshi Miura*
Affiliation:
Toyota Technical College
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The intuitionistic logic LJ and Curry’s LD (cf. [1], [2]) are logics stronger than Johansson’s minimal logic LM (cf. [3]) by the axiom schemes ⋏→x and y ∨ (y→⋏), respectively. However, LM can not be taken literally as the intersection of these two logics LJ and LD, which is stronger than LM by the axiom scheme (⋏ → x) VyV (y→⋏). In pointing out this situation, Prof. K. Ono suggested me to investigate the general feature of the intersection of any pair of logics. In this paper, I will show that the same situation occurs in general. I wish to express my thanks to Prof. K. Ono for his kind guidance.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 26 , February 1966 , pp. 167 - 171
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Curry, H. B.: The system LD, J. Symbolic Logic, vol. 17 (1952), pp. 35–42.CrossRefGoogle Scholar
[2] Curry, H. B.: Foundations of mathematical logic, (1963), New York.Google Scholar
[3] Johansson, I.: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Compositio Mathematica, vol. 4 (1936), pp. 119–136.Google Scholar
[4] Ono, K.: On a practical way of describing formal deductions, Nagoya Math. J., vol. 21 (1962), pp. 115–121.CrossRefGoogle Scholar
[5] Ono, K.: A certain kind of formal theories, Nagoya Math. J., vol. 25 (1965), pp. 59–86.Google Scholar
[6] Ono, K.: On universal character of the primitive logic, Nagoya Math. J., vol. 27 (1966), pp. 331–353.Google Scholar
[7] Peirce, C. S.: On the algebra of logic—A contribution to the philosophy of notation, Amer. J. of Math., vol 7 (1885), pp. 180–202.Google Scholar