This talk summarizes a 2006 paper in Annals of Mathematics titled "Proofs Without Syntax":

http://arxiv.org/PS_cache/math/pdf/0408/0408282v3.pdf

The paper reformulates propositional logic as a branch of graph theory. Graph homomorphisms replace syntactic proofs. The homomorphisms are relaxed topological/categorical fibrations.

The talk should be very accessible, with many examples, and will not assume any technical background in proof theory.