Peirce's Development Of The Algebra Of Relations
Source: DAI, 42, no. 10A, (1981): 4473
Charles Sanders Peirce is almost universally credited with the development of the algebra of relations. However, in spite of the significance of this contribution to both logic and the history of logic, no extended study has been made of Peirce's algebraic development of the logic of relations. This is due both to the obscurity of Peirce's writing and to the inability to make sense of many of his claims when read within a contemporary logical framework.
Peirce is deceptively close to contemporary logic and yet at crucial points subtly different. The uniqueness of this historical position is reflected in the distinct notions of definability that underlie Peirce's development of the logic of relations. Failure to note this subtle difference in definability renders Peirce's claims false or at least subject to misconstrual.
The claim proposed throughout this thesis is that Peirce's development of the algebra of relations should be viewed as a series of attempts to find an adequate articulation of the theory of relations, rather than as a gradual clarification of the concept of a relation and subsequent development of the algebra of relations.
Peirce's discussion of relatives rather than relations in his 1870 paper "Description of a Notation For the Logic of Relatives, Resulting From An Amplification of the Conceptions of Boole's Calculus of Logic," has been shown to be a function of his methodology rather than a confusion of concepts. Moreover, the reconstruction provided for Peirce's notation for relatives indicates that the relatives can be seen as having an underlying relational structure which is mapped into classes. This reconstruction provides a simple theory within which Peirce's claims for relations can be construed with minimal departure from his basic notions of definability.
The influence of Matrix Theory on Peirce's development of the algebra of relations has not been generally recognized. Relative multiplication, Peirce's central mode of combination of concepts is derived from the multiplication schema for the linear associative algebras developed by Benjamin Peirce. Moreover, the articulation given to the final form of the algebra of relations in the 1882 paper is an elementary matrice development of the theory of relations.
Finally, the elements constitutive of a theory of relations given in final form in Peirce's 1883 paper, are all shown to be present, albeit in rudimentary form, in Peirce's first paper on the algebra of relations in 1870.
The above offer substantive support to the claim that development of an adequate structure within which to articulate the theory of relations, rather than a gradual clarification of the concept of relation, constitutes Peirce's development of the algebra of relations.
Accession No: AAG0536760