Peirce apparently initiated an exchange of papers with De Morgan in late 1867, as a result of which De Morgan received a copy of Peirce’s “Three Papers on Logic” (the first three American Academy papers) by May 1868. In a letter dated 14 April 1868, De Morgan had promised to send Peirce a copy of his classic paper of 1860 on the logic of relations,21 but there is no direct evidence that this was ever sent. Nevertheless, Peirce had seen De Morgan’s paper by late December 1868, since he refers to it in another paper sent to the printer at that time.22 It is thus very likely that Peirce had read De Morgan’s paper before he wrote the entries in LN dated November 1868, even though those entries carry no clear references to De Morgan and use quite different examples.
Biographical issues aside, Peirce’s initial work in the logic of relations is significantly different from De Morgan’s. The most important difference is that while De Morgan was interested primarily in the composition of relations with relations, Peirce is concerned with the composition of relations with classes. Thus, while De Morgan’s paradigm is an expression such as “X is a lover of a servant of Y,” Peirce is first concerned with such expressions as “lover of a woman.” A predilection for class expressions is found even in DNLR, though this is often combined with the composition of relations, as in “lover of a servant of a woman.” This emphasis upon class expressions seems to reflect the Boolean frame of reference in which Peirce was working.
De Morgan also considered two types of “quantified relations.” The first is “X is an L of every M of Y,” which is expressed by Peirce as “involution,” or exponentiation. Even here, the LN shows him more concerned with the composition of a relation and a class, as in “lover of every woman,” than with strictly relational composition. The other form of quantified relation is “X is an L of none but M of Y,” a form which Peirce only considers in the section on “backward involution” which he added to DNLR shortly before it was printed (pp. 400–408).
These comparisons between De Morgan and Peirce make their relationship problematic. It becomes more so in view of the fact that some of De Morgan’s most dramatic results involve the contrary and the converse of a relation. While Peirce deals with contraries throughout LN and DNLR, he did not consider converses in the 1868 portions of LN, and he only deals with them in that section of the DNLR which he added at the time of printing.
We may conclude that while Peirce probably knew of De Morgan’s memoir on relations when he was working out the full notation of DNLR, his own Boolean orientation meant that he was working on these topics in his own way.
While DNLR is primarily a contribution to logic, parts of it may also be related to the developments in algebra to which his father contributed. During the years 1867–69, Benjamin Peirce presented a series of papers to the National Academy of Sciences which resulted in a book entitled Linear Associative Algebra (LAA) which was privately published in 1870, and then republished with notes by C. S. Peirce in 1881.23 In it Benjamin Peirce surveyed all the types of linear associative algebras which can be constructed with up to seven units, enormously generalizing such algebras as that of complex numbers (of the form a • 1 + bi) and Hamilton’s quaternions (a • 1 + bi + cj + dk). In the subsection on Elementary Relatives in DNLR, Peirce conjectured that all linear associative algebras could be expressed in terms of elementary relatives, which he then proved in 187524 and illustrated in his notes to his father’s book. This technique formed the foundation for the method of linear representation of matrices, which is now part of the standard treatment of the subject.
As in the case of the relationship of the DNLR to De Morgan’s paper, its relation to his father’s LAA is difficult to estimate accurately. Certainly they were working on these long papers at about the same time, so that some influence would not be surprising. In a short letter to his father that has been dated 9 January 1870, Peirce writes:
I think the following may possibly have some interest to you in connection with your algebras. I have been applying Boole’s Calculus to the Logic of Relative Terms & in doing so have got (among other operations) an associative non commutative multiplication. It is like this. Let k denote killer, w wife, m man. Then
kwm denotes the class of killers of wives of men
The letter then concludes with the colleague-and-teacher example which is found in the Elementary Relatives section of DNLR (pp. 408–11). While this letter shows that Peirce was thinking of his father’s work as he completed DNLR, it also suggests that the relationship between the two papers may not be very intimate.
DNLR was communicated to the American Academy of Arts and Sciences on 26 January 1870 and printed in the late spring. The exact time of its printing is uncertain, though it must have been printed by 17 June 1870 when Peirce left for Europe. He carried with him a letter of introduction from his father to De Morgan, to whom he apparently delivered copies of his memoir and his father’s book. Although there is no contemporary record of Peirce’s visiting De Morgan, he planned to do so and recalled such a meeting in later years. But the meeting could not have been a very happy one, since De Morgan was in very poor health by that time and incapable of sustained logical or mathematical discussion.
The Boolean substructure of DNLR consists of inclusion and the usual Boolean operations of addition (x + y), multiplication (x,y), and class complementation (1 —x), along with their standard laws. To illustrate the relational notation, let s = servant, l = lover, and w = woman. The most important notations are relative multiplication (sl, servant of a lover), relative involution (sl, servant of every lover), backward involution (sl, servant of none but a lover), and converse of a relation (
While DNLR is largely devoted to the logic of two-place relations, Peirce also includes a rather confusing discussion of “conjugative terms,” which stand for three-place relations. This is a marked advance over De Morgan’s restriction to two-place relations, but Peirce’s attempts to deal with this topic within the framework of DNLR present many problems of interpretation.
In addition to outlining a notation, DNLR contains a great many principles which may be easily interpreted in the modern logic of relations. Some significant identities are
There are also a great many inclusions, such as
along with chains of inclusions involving combinations of operations, as in
The complement of a relation is treated not only in a Boolean way, but also as an operation upon a relation, as is the operation of forming the converse of a relation. De Morgan’s principles governing these operations are given in Peirce’s notation. The universal and null relations are introduced, and their laws are stated.
While Peirce does not attempt to develop the laws of his notation in a deductive manner, he does provide demonstrations of a sort for many of his laws, especially in the section entitled “General Method of Working with this Notation” (pp. 387–417). In the first subsection on Individual Terms, many intuitively valid laws are demonstrated by reducing inclusions between classes to individual instances. In addition to its discussion of backward involution and conversion, the subsection on Infinitesimal Relatives contains the most elaborate mathematical analogies in the memoir, with very puzzling applications of such mathematical techniques as functional differentiation and the summation of series. The subsection on Elementary Relatives relates his own work to Benjamin Peirce’s linear associative