Abstract
Peirce’s 1880 work on the algebra of logic resulted in a successful calculus (\(\mathbf {PC}\)) for Boolean algebra. Its leading principle (Peirce’s Rule) is that of residuation. We show how the law of distributivity, which Peirce states but does not prove in 1880, can be proved using Peirce’s Rule in \(\mathbf {PC}\). The system \(\mathbf {PC}\) is here presented as a sequent calculus, which was also Peirce’s preferred method. We then give a shorter proof in his 1896 graphical alpha system, and remark on the main findings also of historical importance.
M. Ma—The work is supported by the National Foundation for Social Sciences and Humanities (grant no. 16CZX049).
A. Pietarinen—The work is supported by the Academy of Finland (project 1270335) and the Estonian Research Council (project PUT 1305) (Principle Investigator A.-V. Pietarinen).
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Notes
- 1.
This definition of negation, that “from x anything you please necessarily follows” was, from the “formal point of view”, perfectly acceptable to Peirce. But he also thought that it does not “really define denial in terms of consequence” (Peirce to Huntington, February 14, 1904; see also [2]).
- 2.
Peirce later added a note in the margin of his copy of the paper: “But it was not an error!!! See my original demonstration in marginal note.” This marginal note has not been recovered.
- 3.
In that 1883 publication of the “Note B” in his Studies in Logic Peirce stated that two relatives are “undistributed” in a relative product and in a relative sum.
- 4.
Keynes distinguishes five different formulations of dilemmatic arguments: those given by (i) Mansel, Whately and Jevons, (ii) by Fowler, (iii) Keynes’s own formulation, (iv) by Hamilton, and (v) by Thomson. Peirce appears to mean none of theirs as “the most useful definition”. For example, he proposes the rule of dilemma to be “If \((a\overline{b} + c)(\overline{a} + c)\) then c” (see R 736, NEM IV, p. 115). This is proved using the distributivity principle thus: \(c + (a\overline{b}\overline{a})\) implies c, and by the law of contradiction, \(c + 0\) implies c. This direction of the derivation of the dilemmatic rule depends on the second distribution principle as given here in Peirce’s footnote, namely one that is not derivable from the lattice rules alone. In the other direction, the distribution principle applied is strictly syllogistic.
- 5.
Later in 1893 Peirce takes a dilemmatic argument to be “any argument whose validity depends upon the principle of excluded middle” (CP 2.474). Dilemmatic arguments would thus not be intuitionistically valid.
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Ma, M., Pietarinen, AV. (2017). Peirce’s Sequent Proofs of Distributivity. In: Ghosh, S., Prasad, S. (eds) Logic and Its Applications. ICLA 2017. Lecture Notes in Computer Science(), vol 10119. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54069-5_13
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