ABSTRACT
Proofs have been devised for the consistency and completeness of systems and the independence of axioms. This chapter utilizes examples to show how derivations or proofs are constructed. Proofs will be set out in the manner adopted for the propositional calculus. The Rule of Supposition corresponds, roughly, to the unstated rule that an axiom may be introduced at any point in a proof. The number of the suppositions on the extreme left-hand side must include the numbers of any suppositions upon which any premises for that line depend. The chapter discusses number of proofs which serve further to illustrate the methods of natural deduction. Proofs by natural deduction in the predicate calculus are set out in a similar way to proofs in the propositional calculus. The chapter presents a number of proofs as illustrations of the method of natural deduction for the predicate calculus and, especially, of the use of the newly introduced rules.
