The Deduction Theorem

Here φ and ψ are any schemata and Γ is a (possibly empty) set of schemata. The Deduction Theorem, we shall see, is a very nice feature of an axiomatic system, for it allows one to discover theorems much more readily than ordinarily is possible by building actual proofs. Its true value lies elsewhere than in efficiency, though. At the end of this note we will discuss the importance of the Deduction Theorem to meta-theory. We prove the left to right and the right to left directions of the “if and only if” statement of the theorem separately. Assume, first, that Γ `p φ ⊃ ψ. Let 〈S1, S2, . . . , φ ⊃ ψ〉 be one of the proofs of φ ⊃ ψ in Peirce from the assumptions in Γ. Then consider the sequence of schemata: 〈S1, S2, . . . , φ ⊃ ψ, φ, ψ〉 all but whose last lines are a proof of φ ⊃ ψ from the Peirce axioms and Γ, whose second to last line is the schema φ and whose last line follows