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Logic in the twenties: the nature of the quantifier1

Published online by Cambridge University Press:  12 March 2014

Warren D. Goldfarb
Warren D. Goldfarb*
Affiliation:
Harvard University, Cambridge, Massachusetts 02138
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Extract

We are often told, correctly, that modern logic originated with Frege. For Frege clearly depicted polyadic predication, negation, the conditional, and the quantifier as the bases of logic; moreover, he introduced the idea of a formal system, and argued that mathematical demonstrations, to be fully precise, must be carried out within a formal language by means of explicitly formulated syntactic rules.

Consequently Frege has often been read as providing all the central notions that constitute our current understanding of quantification. For example, in his recent book on Frege [1973], Michael Dummett speaks of ”the semantics which [Frege] introduced for formulas of the language of predicate logic.” That is, “An interpretation of such a formula … is obtained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula … [T]his procedure is exactly the same as the modern semantic treatment of predicate logic” (pp. 89–90). Indeed, “Frege would therefore have had within his grasp the concepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness … but he did not do so” (p. 82).

This common appraisal of Frege's work is, I think, quite misleading. Even given Frege's tremendous achievements, the road to an understanding of quantification theory was an arduous one. Obtaining such understanding and formulating those notions which are now common coin in the discussion of logical systems were the tasks of much of the work in logic during the nineteen-twenties.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 3 , September 1979 , pp. 351 - 368
Copyright
Copyright © Association for Symbolic Logic 1979

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Footnotes

1

This paper, an interpretive essay on the history of logic, is a revised version of an invited address delivered to a joint session of the Association for Symbolic Logic and the American Philosophical Association, Western Division, in Chicago on April 29, 1977. I am grateful to Burton Dreben and Michael Friedman for helpful comments.

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