Why Are Axiom Schemas Important in Propositional Logic?

[Cinitiator]

Actually, I have several questions:
1) Why are axiom schemas the way they are? What do they represent? I know that infinitely many axioms can be written using the axiom schema form. However, what's the formal definition of axioms in predicate calculus? I've heard that the formal definition of axioms is any wff which has the axiom schema form. If that's the case, what's so special about some wffs which can have infinitely many forms? Do they have any distinctive properties at all?

2) Why and how are they used for proving theorems / making other inferences?

3) How is modus ponens used with such axiom schemas to prove theorems?

 

[Erland]

Actually, I have several questions:
1) Why are axiom schemas the way they are? What do they represent? I know that infinitely many axioms can be written using the axiom schema form. However, what's the formal definition of axioms in predicate calculus? I've heard that the formal definition of axioms is any wff which has the axiom schema form. If that's the case, what's so special about some wffs which can have infinitely many forms? Do they have any distinctive properties at all?

2) Why and how are they used for proving theorems / making other inferences?

3) How is modus ponens used with such axiom schemas to prove theorems?

1)-2) Axioms can be different in different formal theories. But in all theories, the axioms of predicate calculus must be chosen so that the set of theorems (which can be derived from the axioms by the rules) is the same as the set of logically valid formulas. It we restrict ourselves to propositional calculus, logically valid formula is the same as tautology, so the axioms of propositional calculus are chosen so that the theorems are exactly the tautologies.

3) For example, suppose we have the following axiom schemas (among others):

A1. P->(Q->P).
A2. (P->(Q->R))->((P->Q)->(P->R)).

Then, let us derive the theorem P->P:

1. (P->((P->P)->P))->((P->(P->P))->(P->P)). Instances of A2.
2. P->((P->P)->P). Instances of A1.
3. (P->(P->P))->(P->P). MP: 1,2.
4. P->(P->P). Instances of A1.
5. P->P. MP: 3,4.