A Peircean Reduction Thesis: The Foundations of Topological Logic

A Peircean Reduction Thesis: The Foundations of Topological Logic

A Peircean Reduction Thesis: The Foundations of Topological Logic

A Peircean Reduction Thesis: The Foundations of Topological Logic

Excerpt

Charles S. Peirce repeatedly maintained that relations of adicity higher than three 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 that show that all relations can be reduced to dyadic ones. There is, for example, the well known reduction to dyadic relations of Quine (1966a). And there is the theorem of Löwenheim, which as a matter of ironic fact Löwenheim proved in terms of an algebraic logic quite similar to Peirce's and derived historically from Peirce's own algebraic logic through the volumes of Schröder. Moreover, although Peirce repeatedly stated that he had proved the claim before 1870, and although Peirce argued for the reduction thesis at some length in his 1870 paper, "Description of a Notation for the Logic of Relatives," doubt has existed over whether Peirce really did prove the claim.

From the fact that in some or other sense of "reduction" all relations can be reduced to dyadic ones, it does not, of course, follow that Peirce's claim is wrong or that his reduction thesis cannot be proved. For Peirce's understanding of "reduction" might be different from any sense of "reduction" in which wholesale reduction of all relations to the dyadic is possible. Only within the last seven or eight years has progress been made toward a partial vindication of Peirce and an understanding of his own sense of "reduction." Hans G. Herzberger proposed a "bonding algebra" as providing the basis of Peirce's thesis and partially substantiated Peirce by proving a certain version of the thesis correct for domains of sufficiently large cardinality. Kenneth L. Ketner proposed Peirce's existential graphs as basic to understanding Peirce's sense of "reduction" and partially substantiated Peirce by appealing to a notion of "valency analysis" (Ketner, 1986b). By extending both the algebraic ideas of Herzberger and the graph-theoretical ideas of Ketner, this work proposes to develop an algebraic formalism in which a reduction thesis similar to and perhaps identical to the reduction thesis Peirce had in mind can be proved for the general case. This work also proposes to show that the reduction thesis it proves is consistent with the result of Löwenheim and the result of Quine, despite the fact that these results may appear to conflict with it.

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