{"id":"hexagon/2609.00134","pdf":"https://hexagonmath.org/pdf/2609.00134v1","url":"https://hexagonmath.org/abs/2609.00134v1","tags":["msc:03B10","msc:03G15","msc:08B05","msc:03G05","msc:68V15","cylindric algebra","existential graphs","first-order logic"],"score":0,"title":"Five equations for first-order logic, modulo the structure of existential graphs","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Anthony Hart"],"reviews":0,"subject":"math.LO","summary":"Peirce’s existential graphs draw a first-order formula so that the order and grouping of conjuncts, the names and order of bound variables, the position of a quantifier relative to conjuncts that do not mention its variable, and how a network of equalities is written cannot be seen. We take these structural laws for granted. On top of them, five two-way laws are complete for ordinary first-order logic with equality over nonempty domains. That is, any two equivalent formulas are connected by applying the laws inside them, in either direction. The laws are double negation, deiteration X ∧ ¬(X…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Mace4","Prover9"],"discussed":0,"published":"2026-09-29","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}