Existential Graphs and Thirdness 1

Robert Burch has, in A Peircean Reduction Thesis, made a major contribution to communication between what could be called Peircean logic and the more “traditional” approach based on the work of Frege, Peano, and Russell; we may call this the FPR approach. And in so contributing, he helps bridge gaps in understanding which may stand between those focusing on the semiotic, and those more committed to that “traditional” approach. Others (than Frege, Peano, and Russell), of course, have contributed significantly to that tradition--the names I list here are intended to be guideposts to an approach to mathematical logic which differs in certain major respects from that of Peirce (see Zeman 1986). Burch provides an algebraic translation (his “PAL”) of Peirce's existential graphs which enables certain Peircean results concerning the reducibility of all relations to triadic relations to receive a formulation in terms of the FPR approach; as Burch notes, Charles S. Peirce repeatedly maintained that relations of adicity higher than three2 could be reduced to relations of adicity three or less, while relations of the first three adicities could not in general be reduced. This claim has seemed to many philosophers to be bizarre in light of various twentieth-century results in logic [rooted in the work of the FPR school!] that show that all relations can be reduced to dyadic ones (Burch 1991, vii). But in addition to laying out this link, Burch's book underlines some themes which are of considerable import in understanding Peirce's work, and, in its algebraic approach, provides us with interesting diagrammatic tools to aid us in our understanding of Peirce's work itself. A major background theme in Burch's book is the notion of Peirce's “Unitary Logical Vision” (3 ff.). According to this view (which is essentially correct), Peirce arrived at his basic view of logic quite early-perhaps by the 1860's--and, over the years following, experimented with a large variety of logical