Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T18:52:11.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Elementary completeness properties of intuitionistic logic with a note on negations of prenex formulae

Published online by Cambridge University Press:  12 March 2014

G. Kreisel
G. Kreisel*
Affiliation:
The University, Reading, England
Get access Rights & Permissions [Opens in a new window]

Extract

The purpose of the present article is to formulate in an intuitionistically meaningful manner the completeness problems for the intuitionistic predicate calculus, and to establish the completeness of certain fragments of it. In these proofs certain translations of classical logic into intuitionistic logic are used, in particular those discovered by Kolmogorov [11] and Gödel [3], and a new one in which negations of prenex formulae are central. Since the last one is of interest also independently of the completeness problems, the details are presented separately at the end of this paper.

All arguments are intended to be intuitionistically valid unless the opposite is stated; in particular, ‘proof’ means intuitionistic proof, and ‘formal proof’ means proof in the relevant system of Heyting [4], [5].

Open problems are mentioned in Remarks 3.1 and 8.1.

German capitals denote formulae of predicate logic, Latin (italic) capitals denote predicate symbols or propositional letters.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 3 , September 1958 , pp. 317 - 330
Copyright
Copyright © Association for Symbolic Logic 1958

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Glivenko, V., Sur quelques points de la logique de M. Brouwer, Académie Royale de Belgique, Bulletin de la classe des sciences, vol. 15 (1929), pp. 183–188.Google Scholar
[2]Gödel, K., Über formal unenischeidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198.CrossRefGoogle Scholar
[3]Gödel, K., Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1932), pp. 34–38.Google Scholar
[4]Heyting, A., Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (1930), pp. 42–56.Google Scholar
[5]Heyting, A., Die formalen Regeln der intuitionistischen Mathematik, Sitzungsberichte der preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (1930), pp. 57–71, 158–169.Google Scholar
[6]Heyting, A., Logique et intuitionisme, Actes du 2me colloque international de logique mathématique, Paris 1952, 1954, pp. 75–82.Google Scholar
[7]Heyting, A., Intuitionism, An introduction, Amsterdam (North Holland), 1956.Google Scholar
[8]Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. 2, Berlin (Springer), 1939.Google Scholar
[9]Kleene, S. C., Introduction to metamathematics, New York and Toronto (Van Nostrand), Amsterdam (North Holland) and Groningen (Noordhoff), 1952.Google Scholar
[10]Kleene, S. C., Realizability, mimeographed Summaries of talks presented at the Summer Institute of Symbolic Logic in 19S7 at Cornell University, 1957, pp. 100–105; also in Constructivity in mathematics, Amsterdam (North Holland).Google Scholar
[11]Kolmogorov, A., Sur le principe de tertium non datur, Recueil mathématique de la Société mathématique de Moscou, vol. 32 (1925), pp. 646–667.Google Scholar
[12]Kreisel, G., On the interpretation of non-finitist proofs, Part 1, this Journal, vol. 16 (1951), pp. 241–267.Google Scholar
[13]Kreisel, G., Remark on free choice sequences and the topological completeness proofs, forthcoming in this Journal.Google Scholar
[14]Mostowski, A., Proofs of non-deducibility in intuitionistic functional calculus, this Journal, vol. 13 (1948), pp. 204–207.Google Scholar
[15]Mostowski, A., A formula with no recursively enumerable model, Fundamenta mathematicae, vol. 42 (1955), pp. 125–140.CrossRefGoogle Scholar
[16]Rasiowa, H. and Sikorski, R., On existential theorems in non-classical functional calculi, Fundamenta mathematicae, vol. 41 (1954), pp. 87–91.Google Scholar
[17]Rosser, J. B., Extensions of some theorems of Gödel and Church, this Journal, vol. 1 (1936), pp. 87–91.Google Scholar
[18]Scott, Dana, Completeness proofs for the intuitionistic sentential calculus, mimeographed Summaries of talks presented at the Summer Institute of Symbolic Logic in 19S7 at Cornell University, 1957, pp. 231–241.Google Scholar
[19]Tarski, A., Der Aussagenkalkül und die Topologie, Fundamenta mathematicae, vol. 31 (1938), pp. 103–134.CrossRefGoogle Scholar