The method of conversion of mathematical axioms into rules of sequent calculus reveals a perfect duality between classical and constructive basic notions, such as equality and apartness, and between the respective rules for these notions. Derivations by the mathematical rules of a constructive theory are mirror-image duals of corresponding classical derivations, the mathematical rules being obtained by shifting from the left to the right rule scheme and vice versa.
The class of geometric theories is among those convertible into rules and the duality defines the class of co-geometric theories, as in Definition 5.2. The logical rules of classical sequent calculus are invertible, which has for quantifier-free theories the effect that logical rules in derivations can be permuted to apply after the mathematical rules. In the case of mathematical rules that have variable conditions, this separation of logic does not always hold, because quantifier rules may fail to permute down. A sufficient condition for the permutability of mathematical rules is determined in this chapter and applied to give an extension of Herbrand's theorem from universal to geometric and co-geometric theories.
The use of systems of left and right rules is a matter of choice. In Section 7.2, we used a system of right rules for linear order, because it was felt easier to prove the main results. These results can be proved through a system of left rules, as well.