|Nov18-12, 03:12 AM||#1|
A propositional logic question
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?
|Nov19-12, 03:27 PM||#2|
3) For example, suppose we have the following axiom schemas (among others):
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.
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