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.