Peirce's reduction thesis
Peirce's claim, since given formal proofs, that three-place relations cannot be built out of one- and two-place ones, while relations with four or more places can always be built out of three-place ones. Three really is a stopping point — for the shapes of relations.
Peirce's thesis has two halves. Downward: three-place relations cannot be built out of one- and two-place relations. Upward: any relation with four or more places can be built out of three-place ones. Three is where the ladder stops.
Formal proofs have been given, subject to conditions, by Burch and others, and the surrounding literature is careful about what those conditions cost. Taken as a claim about the shapes relations can have, it is the strongest support any argument from threeness has ever had — genuinely a proof that three is a closed inventory of relational kinds.
It is also, exactly, a claim about kinds and not about how many individuals turn up. The same author's teridentity shows why those come apart.
Where this comes up
Sources
- Robert Burch (1991); Hereth Correia & Pöschel (2006, 2011)
- Logic Journal of the IGPL (2023), doi:10.1093/jigpal/jzad010