Journal of Symbolic Logic

On the Role of Implication in Formal Logic

Jonathan P. Seldin

Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.


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 $\lambda$-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.

Article information

J. Symbolic Logic, Volume 65, Issue 3 (2000), 1076-1114.

First available in Project Euclid: 6 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03B40: Combinatory logic and lambda-calculus [See also 68N18]
Secondary: 03F05: Cut-elimination and normal-form theorems 03B20: Subsystems of classical logic (including intuitionistic logic)

Implication Negation Combinatory Logic Lambda Calculus Comprehension Principle Normalization Cut-Elimination BCK Logic


Seldin, Jonathan P. On the Role of Implication in Formal Logic. J. Symbolic Logic 65 (2000), no. 3, 1076--1114.

Export citation


  • See Correction: Erratum: On the Role of Implication in Formal Logic. J. Symbolic Logic, Volume 66, Issue 4 (2001), 1975--1975.