Definition talk:Natural Deduction

I am looking for ways to rewrite this page in a more rigorous, yet as structured way. Am I correct in putting that the natural deduction system relies on the concept of WFFs with pooled assumptions, using only as an axiom the rule of assumption (in the form (coming down to) $p \vdash p$), treating the rest as rules of inference? That would allow me to present the stuff in a more uniform way while practically not altering the content. You have to agree that $p, (p \vdash q) \vdash q$ is hardly to be called an axiom, at best it is an impredicative (i.e., self-relying, circular) schema. --Lord_Farin 22:20, 27 June 2012 (UTC)