On the Structure of Proofs

The initial premise of this paper is that the structure of a proof is inherent in the definition of the proof. Side conditions to deal with the discharging of assumptions means that this does not hold for systems of natural deduction, where proofs are given by monotone inductive definitions. We discuss the idea of using higher order definitions and the notion of a functional closure as a foundation to avoid these problems. In order to focus on structural issues we introduce a more abstract perspective, where a structural proof theory becomes part of a more general theory of functional closures. A notion of proof equations is discussed as a structural classifier and we compare the Russell and Ekman paradoxes to illustrate this.