First-order proofs without syntax
Heijltjes, Willem B., Hughes, Dominic J. D., Straßburger, Lutz · arXiv (Cornell University) · 2019
Proofs are traditionally syntactic, inductively generated objects. This paper reformulates first-order logic (predicate calculus) with proofs which are graph-theoretic rather than syntactic. It defines a combinatorial proof of a formula $ϕ$ as a lax fibration over a graph associated with $ϕ$. The main theorem is soundness and completeness: a formula is a valid if and only if it has a combinatorial proof.