Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T17:14:15.966Z Has data issue: false hasContentIssue false
  • English
  • Français

On the role of implication in formal logic

Published online by Cambridge University Press:  12 March 2014

Jonathan P. Seldin
Jonathan P. Seldin*
Affiliation:
Department of Mathematics, Concordia University, Montréal, Québec, Canada
*
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a “classical” version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic.

Keywords

Implicationnegationcombinatory logiclambda calculuscomprehension principlenormalizationcut-eliminationBCK logic
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1076 - 1114
Copyright
Copyright © Association for Symbolic Logic 2000

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]Bunder, M. W., Some consistency proofs and a characterization of inconsistency proofs in illative combinatory logic, this Journal, vol. 52 (1986), no. 1, pp. 89–110.Google Scholar
[2]Bunder, M. W., Tautologies that, with an unrestricted comprehension axiom, lead to inconsistency or triviality, Journal of Non-Classical Logic, vol. 3 (1986), no. 2, pp. 5–12.Google Scholar
[3]Bunder, M. W. and da Costa, N. C. A., On BCK logic and set theory, preprint 4/86, University of Wollongong, Department of Mathematics, 1986.Google Scholar
[4]Curry, Haskell Brooks, Grundlagen der kombinatorischen Logik, American Journal of Mathematics, vol. 52 (1930), pp. 509–536 and 789–834, Inauguraldissertation.CrossRefGoogle Scholar
[5]Curry, Haskell Brooks, Some properties of equality and implication in combinatory logic, Annals of Mathematics, vol. 35 (1934), pp. 849–850.CrossRefGoogle Scholar
[6]Curry, Haskell Brooks, The inconsistency of certain formal logics, this Journal, vol. 7 (1942), pp. 115–117.Google Scholar
[7]Curry, Haskell Brooks, Foundations of mathematical logic, McGraw-Hill Book Company, Inc., New York, San Francisco, Toronto, and London, 1963, reprinted by Dover, 1977 and 1984.Google Scholar
[8]Curry, Haskell Brooks and Feys, Robert, Combinatory logic, vol. 1, North-Holland Publishing Company, Amsterdam, 1958, reprinted 1968 and 1974.Google Scholar
[9]Curry, Haskell Brooks, Hindley, J. Roger, and Seldin, Jonathan P., Combinatory logic, vol. 2, North-Holland Publishing Company, Amsterdam and London, 1972.Google Scholar
[10]Hindley, J. Roger and Seldin, Jonathan P., Introduction to combinators and λ-calculus, Cambridge University Press, 1986.Google Scholar
[11]Howard, W. A., The formulae-as-types notion of construction, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism (Hindley, J. Roger and Seldin, Jonathan P., editors), Academic Press, New York, 1980, pp. 479–490. A version of this paper was privately circulated in 1969.Google Scholar
[12]Kleene, S. C. and Rosser, J. B., The inconsistency of certain formal logics, Annals of Mathematics, vol. 36 (1935), pp. 630–636.CrossRefGoogle Scholar
[13]Komori, Yuichi, Predicate logics without the structure rules, Studia Logica, vol. 45 (1986), no. 4, pp. 393–404.CrossRefGoogle Scholar
[14]Lopez-Escobar, E. G. K., A second paper “On the interpolation theorem for the logic of constant domains”, this Journal, vol. 48 (1983), pp. 595–599.Google Scholar
[15]Ono, Hiroakira, Semantical analysis of predicate logics without the contraction rule, Studia Logica, vol. 44 (1985), no. 2, pp. 187–196.CrossRefGoogle Scholar
[16]Ono, Hiroaktra and Komori, Yuichi, Logics without the contraction rule, this Journal, vol. 50 (1985), no. 1, pp. 169–201.Google Scholar
[17]Prawitz, Dag, Natural deduction, Almqvist & Wiksell, Stockholm, Göteborg, and Uppsala, 1965.Google Scholar
[18]Seldin, Jonathan P., Studies in illative combinatory logic, Ph.D. thesis, University of Amsterdam, 1968.Google Scholar
[19]Seldin, Jonathan P., Progress report on generalized functionality, Annals of Mathematical Logic, vol. 17 (1979), pp. 29–59.CrossRefGoogle Scholar
[20]Seldin, Jonathan P., Curry's program, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism (Hindley, J. Roger and Seldin, Jonathan P., editors), Academic Press, London, 1980, pp. 3–33.Google Scholar
[21]White, Richard B., A demonstrably consistent type-free extension of the logic BCK, Mathematica Japonica, vol. 32 (1987), pp. 149–169.Google Scholar