Peirce's Algebra Of Logic And The Law Of Distribution
Source: DAI, 47, no. 05A, (1986): 1751
In 1880 Charles Peirce published "On the Algebra of Logic" in which he presented a symbolic logic that provides a complete basis for the propositional calculus. Peirce intended to base his logic exclusively on syllogistic principles, and from that basis to derive the remaining logical laws, including the two principles that constitute the law of distribution: ((a x (b + c)) =((a x b) + (a x c))) ((a + (b x c)) = ((a + b) x (a + c)))
For these he presented no proof, declaring that it was "easy" but "too tedious to give." When challenged by Schroder, who insisted that part of the law of distribution was independent, Peirce was unable to produce his proof and decided that he had been mistaken. Peirce concluded that Syllogistic was insufficient as the basis of the algebra of logic and that dilemmatic principles were required. He came to regard this requirement as the most elementary refutation of the claim that all deductive reasoning is syllogistic. Yet Peirce never accepted Schroder's suggestion that the problematic part of the law of distribution be adopted as an axiom of logic. In 1904 Peirce sent to Edward V. Huntington, for publication, a proof of the law of distribution which he claimed had been worked up for his 1880 paper.
In this dissertation I give
an account, within the above context, of Peirce's work on the law of
distribution. As part of my account I describe an unpublished Peirce manuscript
(MS 575) that contains a summary proof of the full law of distribution, and I
show that it was written before but near the publication of "On the
Algebra of Logic." I compare the MS 575 proof and the "
Accession No: AAG0558692