Two Instances of Peirce’s Reduction Thesis

Abstract

A main goal of Formal Concept Analysis (FCA) from its very beginning has been the support of rational communication by formalizing and visualizing concepts. In the last years, this approach has been extended to traditional logic based on the doctrines of concepts, judgements and conclusions, leading to a framework called Contextual Logic. Much of the work on Contextual Logic has been inspired by the Existential Graphs invented by Charles S. Peirce at the end of the 19th century. While his graphical logic system is generally believed to be equivalent to first order logic, a proof in the strict mathematical sense cannot be given, as Peirce’s description of Existential Graphs is vague and does not suit the requirements of contemporary mathematics.

In his book ’A Peircean Reduction Thesis: The Foundations of topological Logic’, Robert Burch presents the results of his project to reconstruct in an algebraic precise manner Peirce’s logic system. The resulting system is called Peircean Algebraic Logic (PAL). He also provides a proof of the Peircean Reduction Thesis which states that all relations can be constructed from ternary relations in PAL, but not from unary and binary relations alone.

Burch’s proof relies on a major restriction on the allowed construction of graphs. Removing this restriction renders the proof much more complicated. In this paper, a new approach to represent an arbitrary graph by a relational normal form is introduced. This representation is then used to prove the thesis for infinite and two-element domains.