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Syntactical and semantical properties of simple type theory

Published online by Cambridge University Press:  12 March 2014

Kurt Schütte
Kurt Schütte*
Affiliation:
Institute for Advanced Study, Princeton and Universität Marburg (Lahn)
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Extract

In my paper [10] I introduced the syntactical concepts “positive part” and “negative part” of logical formulas in first-order predicate calculus. These concepts make it possible to establish logical systems on inference rules similar to Gentzen's inference rules but without using the concept “sequent” and without needing Gentzen's structural inference rules. Proof-theoretical investigations of several formal systems based on positive and negative parts are published in [11]. In this paper I consider a similar formal system of simple type theory.

A syntactical concept of “strict derivability” results from the formal system in [10] by generalization of the axioms and inference rules from first to higher-order predicate calculus and by addition of inference rules for set abstraction by means of a λ-symbol which allows us to form set expressions of arbitrary types from well-formed formulas.

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Articles
Information
The Journal of Symbolic Logic , Volume 25 , Issue 4 , December 1960 , pp. 305 - 326
Copyright
Copyright © Association for Symbolic Logic 1962

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References

[1]Beth, E. W., Remarks on natural deduction, Indagationes mathematicae, vol. 17 (1955), pp. 322–325.CrossRefGoogle Scholar
[2]Beth, E. W., Semantic entailment and formal derivability, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afd. letterkunde, n.s., vol. 18 no. 13 (1955), pp. 309–342.Google Scholar
[3]Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39 (1934), pp. 176–210, 405–431.CrossRefGoogle Scholar
[4]Hasenjaeger, G., Eine Bemerkung zu Henkin's Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe, this Journal, vol. 18 (1953), pp. 42–48.Google Scholar
[5]Henkin, L., The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159–166.Google Scholar
[6]Henkin, L., Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 81–91.Google Scholar
[7]Hintikka, K. J. J., Form and content in quantification theory, Two papers on symbolic logic. Acta philosophica Fennica, no. 8, Helsinki 1955, pp. 7–55.Google Scholar
[8]Hintikka, K. J. J., Notes on quantification theory, Societas Scientiarum Fennica, Commentationes physico-mathetnaticae, vol. 17 no. 12 (1955).Google Scholar
[9]Kanger, Stig, Probability in logic, Stockholm studies in philosophy 1 (1957).Google Scholar
[10]Schütte, K., Ein System des verknüpfenden Schlieβens, Archiv für mathematische Logik und Grundlagenforschung, vol. 2 (1956), pp. 55–67.CrossRefGoogle Scholar
[11]Schütte, K., Beweistheorie, Berlin, Göttingen, Heidelberg 1960.Google Scholar
[12]Takeuti, G., On a generalized logic calculus, Japanese Journal of Mathematics, vol. 23 (1953), pp. 39–96.CrossRefGoogle Scholar