that a type hierarchy of three levels is generative of all that we need in logic. Part of the justification for Peirce's claim that three categories are both necessary and sufficient appears to arise from mathematical ideas about the reducibility of n-adic relations. According to Peirce's Reduction Thesis, (a) triads are necessary because genuinely triadic relations cannot be completely analyzed in terms of monadic and dyadic predicates, and (b) triads are sufficient because there are no genuinely tetradic or larger polyadic relations—all higher-arity n-adic relations can be analyzed in terms of triadic and lower-arity relations. Others, notably Robert Burch (1991), Joachim Hereth Correia and Reinhard Pöschel (2006), have offered proofs of the Reduction Thesis. There have been proposals by Donald Mertz, Herbert Schneider, Carl Hausman, and Carl Vaught to augment Peirce's threefolds to fourfolds; and one by Douglas Greenlee to reduce them to twofolds. Peirce introduces his Categories and their theory in "On a New List of Categories" (1867), a work which is cast as a Kantian deduction and is short but dense and difficult to summarize. The following table is compiled from that and later works. (The context for interpretants is not psychology or sociology, but instead philosophical logic. In a sense, an interpretant is whatever can be understood as a conclusion of an inference. The context for the categories as categories is phenomenology, which Peirce also called phaneroscopy and categorics.) See also Trikonic Notes Bibliography Peirce, C.S. (1867), "On a New List of Categories", Proceedings of the American Academy of Arts and Sciences 7 (1868), 287–298. Presented, 14 May 1867. Reprinted (Collected Papers, vol. 1, paragraphs 545–559), (The Essential Peirce, vol. 1, pp. 1–10), (Chronological Edition, vol. 2, pp. 49–59), Eprint. Peirce, C.S. (1885), "One, Two, Three: Fundamental Categories of Thought and of Nature", Manuscript 901; the Collected Papers, vol. 1, paragraphs 369-372 and 376-378 parts; the Chronological Edition, vol. 5, 242-247 Peirce, C.S. (1887–1888), "A Guess At the Riddle", Manuscript 909; The Essential Peirce, vol. 1, pp. 245–279; Eprint Peirce, C.S. (1888), "Trichotomic", The Essential Peirce, vol. 1, p. 180. Peirce, C.S. (1893), "The Categories", Manuscript 403 An incomplete rewrite by Peirce of his 1867 paper "On a New List of Categories." Interleaved by Joseph Ransdell (ed.) with the 1867 paper itself for purposes of comparison. Peirce, C.S., (), "The Logic of Mathematics; An Attempt to Develop My Categories from Within", the Collected Papers, vol. 1, paragraphs