## On the Algebra of Logic in American Journal of Mathematics, Vol. III, 1880, pp. 15-58 WITH On the Algebra of Logic: A Contribution to the Philosophy of Notation in American Journal of Mathematics, Vol. VII, 1885, pp. 180-202

Cambridge: Cambridge University Press. 1st Edition. TWO VOLUME, HANDSOMELY BOUND FIRST EDITIONS OF PEIRCE’S SEMINAL WORK IN THE STUDY OF LOGIC, HIS 1885 LOGIC OF QUANTIFIERS, AS WELL AS AN IMPORTANT PRECURSOR, HIS 1880 PAPER “OFTEN REGARDED AS THE FIRST PRESENTATION OF BOOLEAN ALGEBRA IN ITS MODERN FORM” (Grattan-Guinnes, ed. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Volume 1, p. 608).

Charles Sanders Peirce was an accomplished scientist, philosopher, and mathematician, but he considered himself above all a logician. “His contributions to the development of modern logic at the turn of the 20th century were colossal, original and influential” (Encyclopedia of Philosophy).

The development of “a formal language in which mathematical statements could be made, and rules for correctly inferring statements from statements” was central to modern proof theory (Byrne, Peirce’s First-Order Logic of 1885, 949). From at least the time of Leibniz, their exists a history of formalizing language in order to formalize steps of inference. The introduction of quantification to the formalization of mathematical statements was a critical step in the development of such languages. Peirce’s seminal 1885 paper on the mathematics of logic, presents his theory of quantification — a complete presentation of logic in an algebraic format.

Peirce began his research into the algebra of logic late in the 1860s and began investigating laws on operations of relation algebras by at least 1870. Writing in 1884 (for the 1885 paper), Peirce was blunt about his intent: “In this paper I purpose to develop an algebra adequate to the treatment of all problems of deductive logic, showing as I proceed what kind of signs have necessarily to be employed at each stage of the development. I shall thus attain three objects. The first is the extension of the power of logical algebra over the whole of its proper realm. The second is the illustration of principles which underlie all algebraic notation. The third is the enumeration of the essentially different kinds of necessary inference; for when the notation which suffices for exhibiting one inference is found inadequate for explaining another, it is clear that the latter involves an inferential element not present in the former” (Peirce, 1885).

In “his important 1880 paper”, Peirce “broke with the Aristotelian semantics of classes and introduced modern semantics, allowing a class symbol to be empty (as well as to be the universe), and stated the truth values of the categorical propositions that we use today” (Stanford Encyclopedia of Philosophy); The Essential Peirce: 1867-1893, v. 1, 1992). Specifically, he explicitly eliminated “the Aristotelian derivation of particular statements from universal statements by giving the modern meaning for “All A is B”. In addition, he extended the algebra of logic for classes to the algebra of logic for binary relations and introduced general sums and products to handle quantification” (Stanford). Many, including the noted logician and philosopher Arthur Prior, have argued that Peirce’s 1880 paper alone “provides a complete basis for propositional logic” (Encyclopedia of Philosophy). Further still, the 1880 paper “holds a place of some importance in the history of formal logic and mathematics [in part because in it, Peirce] discusses that relationship between thinking and cerebration (or logic and physiology)" (ibid).

The “fruitfulness” of Peirce’s thinking in the 1880 paper is most evident in the watershed paper of 1885 wherein he introduces an entire “system for propositional logic based on five axioms for implication (represented by the signs ‘-<‘)” (Stanford). Peirce further presents “the quantification theory, also the germs of various forms of necessary inference were developed: the theory of truth functions, the axiomatic method, the tableaux method and the system of natural deduction” (Pietarinen, Peirce’s Development of the Quantification Theory). Today, each of these methods is so standard as to be studied in isolation from each other. Item #847

CONDITION & DETAILS: Two volumes. Large 4to. (12 X 9.5 inches; 300 x 237mm). The volumes are tightly and handsomely bound in three-quarter red calf over marbled paper boards. Five gilt-ruled raised bands at the spine (also gilt-lettered). Small spot on the first page of the second paper (see photo); small tear at the foot of the first paper, in no way impacting text; interior bright and clean. Near fine.

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Price:
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