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Logical Systems and the Principles of Logic

Published online by Cambridge University Press:  14 March 2022

Marvin Farber
Marvin Farber*
Affiliation:
The University of Buffalo
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Extract

Doubts concerning the validity of logic are as old as the empirical criticism of science. In the last two decades the idea that truth is relative to given sets of basic assumptions has been prominent; and the controversy about the principle of excluded middle has focussed renewed attention upon the nature of logic and its fundamental principles.

Recent investigations in formal logic have contributed greatly to the understanding of the principles of logic. It is simply a misunderstanding to conclude from them that the logical principles have been suspended, despite the various formal structures which are called “logics.” The purely formal treatment of problems concerning the foundations of logic, while necessarily undertaken as far as it can go, has essential limitations.

Type
Research Article
Information
Philosophy of Science , Volume 9 , Issue 1 , January 1942 , pp. 40 - 54
Copyright
Copyright © Philosophy of Science Association 1942

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References

1 Read at the meeting of the Western Division of the American Philosophical Association, Indianapolis, April 25, 1941. The concluding section includes comments in reply to questions raised by Professor Paul Henle in his interesting discussion of the paper.

2 Cf. C. I. Lewis, “Alternative Systems of Logic,” The Monist, Oct. 1932. See also The Monist, 1933, where Lewis writes: “It was stated that these systems are not alternatives in the sense that if one is true then others must be false; but that they are alternatives in the sense that concepts and principles belonging to one cannot generally be introduced into another. Hence they mutually exclude one another in application. Inasmuch as any satisfactory logic, if capable of exact and systematic statement, would be one such statement amongst others it was further suggested that the actually applied canon of inference be determined by pragmatic considerations (over and above all questions of absolute truth).”

3 That is not to say that the revision of logic may not take the opposite direction, and argue on philosophical grounds for the limitation of its field of application.

4 Cf. Eric Temple Bell, The Search for Truth, pp. 254 f. (N. Y., 1934).

5 Monist, 1932, p. 505.

6 Lewis and Langford, Symbolic Logic, pp. 223 ff. (N. Y., 1932).

7 Monist, l. c.

8 In the case of “constructional” logic, the law does not hold for constructional propositions. But neither does it hold for such propositions from the “realistic” point of view.

9 Similar restrictions apply in the case of the principles of non-contradiction and identity.

10 Cf. E. Zermelo's relevant remarks in his paper, “Über Stufen der Quantifikation und die Logik des Unendlichen,” Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 41 (1932), pp. 85 ff. From his point of view, every “true” proposition is “provable,” just as every proposition definable in terms of a “well founded” system of propositions is also “decidable” without the need for a transition to a higher stage of quantification. In opposition to Gödel, he maintains that there are no objectively undecidable propositions. As he views it, Gödel's proof applies the “finitistic” limitation to the “provable” propositions of a system, and not to all the propositions belonging to the system. Hence the former, and not the latter, form a denumerable aggregate, and there will naturally be “undecidable” propositions in this sense. It would seem to follow, then, that the question of whether there are absolutely undecidable propositions is not touched by such “relativistic” considerations. Zermelo's unwillingness to abandon any part of past mathematical knowledge leads him to take a strong realistic stand against what he terms the “finitistic prejudice.”

11 Cf. E. Husserl, Formale und transzendentale Logik, pp. 162 ff.