Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-12T18:51:55.044Z Has data issue: false hasContentIssue false
  • English
  • Français

Three Logical Theories

Published online by Cambridge University Press:  14 March 2022

John Corcoran
John Corcoran*
Affiliation:
State University of New York at Buffalo
Get accessRights & Permissions [Opens in a new window]

Abstract

This study concerns logical systems considered as theories. By searching for the problems which the traditionally given systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as well as with logical truth and logical consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, “strong soundness” and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of proofs. The first deals with the concept of rigor or “gaplessness” in proofs. The second is a completeness condition for a system of proofs. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual historical development of this area of logic.

Information

Type
Research Article
Information
Philosophy of Science , Volume 36 , Issue 2 , June 1969, pp. 153 - 177
Copyright
Copyright © 1969 by The Philosophy of Science Association

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.)

Article purchase

Temporarily unavailable

References

[1] Anderson, J. M., and Johnstone, H. W., Natural Deduction, Belmont, California, 1962.Google Scholar
[2] Bochenski, I. M., A History of Formal Logic (tr. Thomas, Ivo), Notre Dame, Indiana, 1961.Google Scholar
[3] Church, A., Introduction to Mathematical Logic, Princeton, 1956.Google Scholar
[4] Copi, I. M., and Gould, J. A., Readings on Logic, New York, 1964.Google Scholar
[5] Gödel, K., “Die Vollstandigkeit der Axiome das logischen Functionenkalkuls,” Monatshefte für Mathematik und Physik, vol. xxxvii, 1930, p. 349.CrossRefGoogle Scholar
[6] Henkin, L., “The completeness of the first order functional calculus,” Journal of Symbolic Logic, vol. 14, 1949, p. 159.CrossRefGoogle Scholar
[7] Hilbert, D., and Ackermann, W., Principles of Mathematical Logic (tr. Hammond, Leckie, and Steinhardt), New York, 1950.Google Scholar
[8] Hiż, H., “A warning about translating axioms,” American Mathematical Monthly, vol. LXV, 1958, p. 613.CrossRefGoogle Scholar
[9] Kalish, D., and Montague, R., Logic: Techniques of Formal Reasoning, New York, 1964.Google Scholar
[10] Kneale, W., and Kneale, M., The Development of Logic, Oxford, 1962.Google Scholar
[11] Lewis, C. I., and Langford, C. H., Symbolic Logic, 2nd ed., New York, 1959.Google Scholar
[12] Lightstone, A. H., The Axiomatic Method, Prentice-Hall, Englewood Cliffs, N.J., 1964.Google Scholar
[13] Mates, B., Elementary Logic, New York, 1965.Google Scholar
[14] Mendelson, Elliot, Introduction to Mathematical Logic, Princeton, 1964.Google Scholar
[15] Parry, W. T., “Comments on a variant form of natural deduction,” Journal of Symbolic Logic, vol. 30, 1965, p. 119.CrossRefGoogle Scholar
[16] Post, E. L., “Introduction to general theory of elementary propositions,” American Journal of Mathematics, vol. 43, 1921, p. 163.CrossRefGoogle Scholar
[17] Quine, W. V. O., Methods of Logic (revised edition), New York, 1959.Google Scholar
[18] Robinson, A., On the Metamathematics of Algebra, Amsterdam, 1951.Google Scholar
[19] Tarski, A., Logic, Semantics and Metamathematics (tr. Woodger, J. H.), Oxford, 1956.Google Scholar
[20] Whitehead, A. N., and Russell, B., Principia: Mathematica to 56, Cambridge, 1962.Google Scholar