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
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