Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-11T07:25:04.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural deduction based set theories: a new resolution of the old paradoxes

Published online by Cambridge University Press:  12 March 2014

Paul C. Gilmore
Paul C. Gilmore*
Affiliation:
Department of Computer Science, University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 2 , June 1986 , pp. 393 - 411
Copyright
Copyright © Association for Symbolic Logic 1986

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.)

Footnotes

1

The research reported on in this paper was supported by grants of the Natural Science and Engineering Research Council of Canada.

References

REFERENCES

Beth, E. W. [1955], Semantic entailment and formal derivability, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afdeeling Letterkunde, Nieuwe Reeks, vol. 18, no. 13, pp. 309–342.Google Scholar
Church, Alonzo [1941], The calculi of lambda-conversion, Princeton University Press, Princeton, New Jersey.Google Scholar
Feferman, Soloman [1984], Toward useful type-free theories. I, this Journal, vol. 49, pp. 75–111.Google Scholar
Fitch, Federick B. [1948], An extension of basic logic, this Journal, vol. 13, pp. 95–106.Google Scholar
Fitch, Federick B. [1952], Symbolic logic: an introduction, Ronald Press, New York.Google Scholar
Fitch, Federick B. [1967], A complete and consistent modal set theory, this Journal, vol. 32, 93–103.Google Scholar
Fitch, Federick B. [1974], Elements of combinatory logic, Yale University Press, New Haven, Connecticut.Google Scholar
Gentzen, Gerhard [1934/1935], Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39, pp. 176–210, 405–431.CrossRefGoogle Scholar
Gilmore, P. C. [1967], The consistency of partial set theory without extensionality, paper presented at the Summer Institute on Axiomatic Set Theory, Los Angeles, California; printed version in Axiomatic set theory , Proceedings of Symposia in Pure Mathematics, vol. 13, part 2, American Mathematical Society, Providence, Rhode Island, 1974, pp. 147–153.Google Scholar
Gilmore, P. C. [1968], A formalized naive set theory, paper presented at the Summer Conference on Intuitionism and Proof Theory, Buffalo, New York.Google Scholar
Gilmore, P. C. [1971], A consistent naive set theory: foundations for a formal theory of computation, IBM Research Report RC3413, July 22, 1971; slightly revised version, Combining unrestricted abstraction with universal quantification, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism (Seldin, J. P. and Hindley, J. R., editors), Academic Press, New York, 1980, pp. 99–124.Google Scholar
Gilmore, P. C. and Morrison, R., Foundations for self referential databases (in preparation).Google Scholar
Henkin, L. [1953], Completeness in the theory of types, this Journal, vol. 18, pp. 19–23.Google Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. I, Springer-Verlag, Berlin.Google Scholar
Jaśkowski, Stanisław [1934], On the rules of suppositions in formal logic, Studia Logica, no. 1.Google Scholar
Kowalski, R. [1974], Predicate logic as programming language, Information Processing 74 (Proceeding of the IFIP Congress, Stockholm, 1974), North-Holland, Amsterdam, pp. 569–574.Google Scholar
McCarthy, John [1963], A basis for a mathematical theory of computation, Computer programming and formal systems (Braffort, P. and Hirschberg, D., editors), North-Holland, Amsterdam, pp. 33–70.CrossRefGoogle Scholar
Prawitz, Dag [1965], Natural deduction, Almqvist & Wiksell, Stockholm.Google Scholar
van Orman Quine, Willard [1951], Mathematical logic, Harvard University Press, Cambridge, Massachusetts.CrossRefGoogle Scholar
Robinson, Abraham [1951], On the metamathematics of algebra, North-Holland, Amsterdam.Google Scholar
Russell, Bertrand [1905], On denoting, Mind (New Series), vol. 14, pp. 479–493.CrossRefGoogle Scholar
Scott, Dana [1975], Combinators and classes, λ-calculus and computer science theory, Lecture Notes in Computer Science, vol. 37, Springer-Verlag, Berlin, pp. 1–26.CrossRefGoogle Scholar
Shoenfield, Joseph R. [1967] Mathematical logic, Addison-Wesley, Reading, Massachusetts (2nd printing, 1973).Google Scholar
Smullyan, R. [1968], First-order logic, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Stoy, Joseph E. [1977], Denotational semantics: the Scott-Strachey approach to programming language theory, MIT Press, Cambridge, Massachusetts.Google Scholar
Troelstra, A. S. [1969], Principles of intuitionism, Lecture Notes in Mathematics, vol. 95, Springer-Verlag, Berlin.CrossRefGoogle Scholar