Propositional Calculus: Deductive Systems

Abstract

The theorems of a theory T(U) are the logical consequences of the set of axioms U. Suppose we have a formula A and we want to know if it belongs to the theory T(U). By Theorem 2.38, U ╞ A if and only if ╞ A₁ Λ ⋯ Λ A _n → A, where U = {A₁,..., A _n } is the set of axioms. Thus A ∈T(U) iff a decision procedure for validity answers ‘yes’ on the formula. However, there are several problems with this approach:

• The set of axioms may be infinite, for example, in an axiomatization of arithmetic, we may specify that all formulas of the form (x = y) → (x + 1 = y + 1) are axioms.

• Very few logics have decision procedures like the propositional calculus.

• A decision procedure may not give insight into the relationship between the axioms and the theorem. For example, in proofs of theorems about prime numbers, we would want to know exactly where primality is used (Velleman 1994, Section 3.7). This understanding can also help us propose other formulas that might be theorems.

• A decision procedure just produces a ‘yes/no’ answer, so it is difficult to recognize intermediate results, lemmas. Obviously, the millions of mathematical theorems in existence could not have been inferred directly from axioms.