Abstract
The following remarks apply to many functional calculi, each of which can be variously axiomatized, but for clarity of exposition we shall confine our attention to one particular system Σ. This system is to have the usual primitive symbols and formation rules of the pure first-order functional calculus, and the following formal axiom schemata and formal rules of inference.Axiom schema 1. Any tautologous wff (well-formed formula).Axiom schema 2. (a) A ⊃ B, where A is any wff, a and b are any individual variables, and B arises from A by replacing all free occurrences of a by free occurrences of b.Axiom schema 3. (a)(A ⊃ B)⊃(A⊃ (a)B). where A and B are any wffs, and a is any individual variable not free in A.Rule of Modus Ponens: applies to wffs A and A ⊃ B, and yields B.Rule of Generalization: applies to a wff A and yields (a)A, where a is any individual variable.A formal proof in Σ is a finite column of wffs each of whose lines is a formal axiom or arises from two preceding lines by the Rule of Modus Ponens or arises from a single preceding line by the Rule of Generalization. A formal theorem of Σ is a wff which occurs as the last line of some formal proof.
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