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Some Remarks on Evaluations of the Primitive Logic

Published online by Cambridge University Press:  22 January 2016

Hiroakira Ono
Hiroakira Ono*
Affiliation:
Research institute for mathematical sciences, Kyoto University
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In [4], K. Ono introduced the notion of evaluations of the primitive logic LO and proved that any semi-evaluation E is an evaluation of LO if E satisfies the following conditions:

  • (E1) p* → 0 = 0,

  • (E2) p* → p* = 0,

  • (E3) 0 → p* = p*,

  • (E4) p* → (p* → q*) = p* → q*,

  • (E5) p* → (q* → r*) = q* → (p* → r*),

  • (E6) p* → q* = 0 implies (r* → p*) → (r* → q*) = 0,

  • (E7) (x)p*(x) → p*(t) = 0 for any t, and

  • (E8) if u* → p*(t) = 0 for any t, then u* → (x)p*(x) = 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 38 , March 1970 , pp. 63 - 69
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1970

References

[1] Curry, H., A note on the reduction of Gentzen’s calculus LJ, Bull. Amer. Math. Soc., 45 (1939) 288–293.Google Scholar
[2] Henkin, L., An algebraic characterization of quantifiers, Fund. Math., 37 (1950) 63–74.Google Scholar
[3] McKinsey, J.C.C. and Tarski, A., On closed elements in closure algebras, Ann. of Math., 47 (1946) 122–162.Google Scholar
[4] Ono, K., On a class of truth-value evaluations of the primitive logic, Nagoya Math. J., 31 (1968) 71–80.Google Scholar
[5] Ono, K., On a class of set-theoretical interpretations of the primitive logic, (to appear).Google Scholar
[6] Rasiowa, H., Algebraic treatment of the functional calculi of Heyting and Lewis, Fund. Math., 38 (1951) 99–126.CrossRefGoogle Scholar
[7] Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, Warszawa, 1963, Monografie Maternatyczne tom 41.Google Scholar