Journal of Symbolic Logic

On the Role of Implication in Formal Logic

Jonathan P. Seldin

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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 $\lambda$-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

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183746170

Mathematical Reviews number (MathSciNet)
MR1791365

Zentralblatt MATH identifier
0976.03015

JSTOR
links.jstor.org

Subjects
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)

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

Citation

Seldin, Jonathan P. On the Role of Implication in Formal Logic. J. Symbolic Logic 65 (2000), no. 3, 1076--1114. https://projecteuclid.org/euclid.jsl/1183746170


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Corrections

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