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Bases for first-order theories and subtheories

Published online by Cambridge University Press:  12 March 2014

William Craig
William Craig*
Affiliation:
The Pennsylvania State University
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Extract

The extent to which we can grasp the content of a (non-logical) theory, expressing it economically by means of an axiom system or basis, varies greatly. In this paper we shall investigate what degree of economy, or at least regularity, can be achieved for all recursively axiomatizable first-order theories. A useful approach, also of interest in its own right, turns out to be the study of bases for subtheories, where a subtheory of a given theory consists of those theorems from which certain predicate symbols are absent. These predicate symbols might be thought of as the formal counterparts of the “purely theoretical” terms employed by a science, the theory corresponding to the science itself and the subtheory to its “observational consequences”. Roughly speaking, the types of operations involving such predicate symbols will be reduced to a minimum, so that their syntactical role in deductions will emerge more clearly.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 25 , Issue 2 , June 1960 , pp. 97 - 142
Copyright
Copyright © Association for Symbolic Logic 1960

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