sentences:  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) and Joachim Hereth Correia and Reinhard Pöschel (2006), have offered proofs of the Reduction Thesis.