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Three Logical Theories

Published online by Cambridge University Press:  14 March 2022

John Corcoran
John Corcoran*
Affiliation:
State University of New York at Buffalo
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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.

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

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