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On formulas of one variable in intuitionistic propositional calculus1

  • Iwao Nishimura (a1)

McKinsey and Tarski [3] described Gödel's proof that the number of Brouwerian-algebraic functions is infinite. They gave an example of a sequence of infinitely many distinct Brouwerian-algebraic functions of one argument, which means that there are infinitely many non-equivalent formulas of one variable in the intuitionistic propositional calculus LJ of Gentzen [1]. However they did not completely characterize such formulas. In § 1 of this note, we define a sequence of basic formulas P(X), P0(X), P1(X), … and prove the following theorems.

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I wish to express my hearty thanks to Prof. K. Ono and Mr. T. Umezawa for their valuable suggestions and criticisms.

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[1]Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39. (19341935), pp. 176210, 445–431.
[2] K. Gödel, Zum intuitionistischen Aussagenkalkül, Ergebnisse eines mathematischen Kolloquiums, Heft 4 (19311932, 1933), p. 40.
[3]McKinsey, J. C. C. and Tarski, A., On closed elements in closure algebra, Annals of Mathematics, vol. 47 (1946), pp. 122162.
[4]Mostowski, A., Proofs on non-deducibility in intuitionistic functional calculus, this Journal, vol. 13 (1948), pp. 204207.
[5]Umezawa, T., Über die Zwischensysteme der Aussagenlogik, Nagoya Mathematical Journal, vol. 9 (1955), pp. 181187.
[6]Umezawa, T., On intermediate prepositional logics, this Journal, vol. 24 (1959), pp. 2036.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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