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.
A combinatorial proof of is shown below.