  Proof-theory is focused on formal proof systems and such formal systems have been developed for intuitionistic predicate calculus, classical predicate calculus, arithmetics, higher-order calculi, among many others. Intuitionistic and constructive logic began when people saw the possibility of reading ‘\(A \Rightarrow B\)’ as ‘if you give me an \(A\), I will give you a \(B\)’, which is a significant departure from the classical reading ‘\(A\) is false or \(B\) is true’.