Abstract
Peirce’s probabilistic justification of induction has much to do with his conceptions of an inference and its validity, for he conceives induction as an inference from sample to population, and contends that it is valid in the sense in which any valid inference is valid. The following passages, taken from “On the Natural Classification of Arguments,” provides a summary description of Peirce’s conceptions of an inference and its validity.
Both this Chapter and the next have been simplified to form a paper entitled “Peirce’s Probabilistic Theory of Inductive Validity”, published in The Transactions of the Charles S. Peirce Society, Vol. II, no. 2, Fall, 1966, 86–105.
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References
The Collected Papers of Charles Sanders Peirce, 2.462.
Ibid., 2.463.
Ibid., 2.464.
Ibid., 3.160.
Ibid., 3.164.
Cf., Ibid., 2.589.
Cf., Ibid., 2.680.
Cf., Ibid., 2.623, 2.508-514, 1.559.
Cf., Ibid., 2.508-2.514.
Cf., Ibid., 2.623. Similarly, a general form of deduction and a general form of hypothesis are, according to Peirce, respectively, as follows: Cf., 2.508-2.514. All instances of M are instances of P, S1, S2, S3, etc. are instances of M, Hence, S1, S2, S3, etc. are instances of P. All instances of M are instances of P, S1, S2, S3, etc. are instances of P, Hence, S1, S2, S3, etc. are instances of M. Examples for these inferences are obviously easily constructible. Cf., 2.623.
Cf., Ibid., 2.267.
Cf., Ibid., 2.267.
Ibid., 2.696.
Ibid., 2.695
Ibid., 2.700; also Cf. 2.710. I have reformulated Peirce’s statement of this syllogism in order to make it intelligible to our modern readers.
Ibid., 2.702; also Cf. 2.718.
The precise formulation of this law and the rigorous mathematical proof of this law which I propose in Appendix II are reached by me independently of other sources. As far as I know, no standard texts in mathematical statistics or probability theory have given a proof, or sometimes, even a formulation, of this important, although seemingly obvious principle.
Cf., Ibid., 2.287; also 2.724. In spite of the fact that Peirce does not develop any systematic statistics, he has nevertheless often referred to probability formulas such as that of the probable errors. Therefore, in connection with his discussion of probable inference, it should be to our benefit to understand Peirce’s argument which uses statistical laws or laws of probability to draw conclusions of statistical deduction and induction, as we shall soon see, that a somewhat detailed presentation of the essential laws in mathematical statistics and probability theory be made in appendices.
As we shall see in the next Chapter, this same premise of a statistical deduction is also the premise of an induction.
Peirce says at one place that “The conclusion of the statistical deduction is here regarded as being ‘the proportion r of the S’s are P’s’, and the words ‘probably about’ as indicating the modality with which this conclusion is drawn and held for true.” (2.721nl).
It is clear that, if we take logic in a broad sense, so as to include the classical calculus of probability, a probable inference with a leading principle from the calculus of probability should be considered valid in the sense of having a logical leading principle.
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© 1969 Martinus Nijhoff, The Hague, Netherlands
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Cheng, CY. (1969). The Nature and Validity of Inference. In: Peirce’s and Lewis’s Theories of Induction. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-9367-2_3
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