Class 14 : *** Proof system for logic
 
** Desirable properties of a proof system 
** Proof systems has a set of axioms (AXIOMS) and rules (R). 
** Set of formulas provable in the system = I(AXIOMS, R). 
   Alternatively, provable formulas have legal construction sequences over AXIOMS and R. 
 
**  Hilbert's proof system for Propositional Logic:  
*   Axioms : 
* 	Ax1 : (\alpha --> (\beta --> \alpha))
* 	Ax2 : ( (\alpha --> (\beta --> \gamma)) --> ((\alpha --> \beta) --> (\alpha --> \gamma)) )
* 	Ax3 : ( (\neg \beta --> \neg \alpha) --> (\alpha --> \beta) )
*   Rules : Modus Ponens 
	If we have proofs of \alpha, \(alpha --> \beta) 
        then we can derive \beta
 
**  Example : proved (\alpha --> \alpha) in Hilbert's proof system.