Ok, so it's very easy to show P v P = P (where = is logically equivalent) using a truth table as well as using a conditional proof.
- P v P Premise
- ~p Assumption
- p Disjunctive Syllogism (1, 2)
- p & ~p Conjunction (3, 4)
- ~p –> (p & ~p) CP (2–4)
- p v ~p EMI
- ~p v p Commutation (6)
- ~p v ~~p Double Negation (7)
- ~(p & ~p) De Morgan's (8)
- ~~p Modus Tollens (5, 9)
- p Double Negation
My question is, how do I show p v p = p WITHOUT using a truth table OR a conditional prove? I can only use the basic rules of inference (EMI, Disjunctive Syllogism, Addition, Conjunction, Simplification) as well as the rules of replacement (De Morgan's, Distribution, etc.)
Best Answer
This looks like a Copi exercise, so I'll use the rules in the 1998 edition of "Introduction to Logic".
1 [(p $\lor$ p)=(p $\lor$ p)] $\lor$ commutation
2 [(p $\lor$ p)=$\lnot$$\lnot$(p $\lor$ p)] 1 Double Negation
3 [(p $\lor$ p)=$\lnot$($\lnot$p $\land$ $\lnot$ p)] 2 De Morgan's ahem... Petrus Hispanus's Theorems
4 [(p $\lor$ p)=$\lnot$$\lnot$p] 3 $\land$ tautology
5 [(p $\lor$ p)=p] 4 Double Negation