Question 1206425
<pre>

This student has informed me that she is not allowed to use material
implication as I did in the above.

(P ⊃ Q) <=> (~P ∨ Q)

Why teachers don't just prove this early on is beyond me.  It makes
proofs a lot simpler.  

They wouldn't even need to prove equivalence, they could just prove
(P ⊃ Q) ⊃ (~P ∨ Q), and that would make lots of proofs easier.  It's 
easy to prove indirectly, anyway.

First we prove 

premise 
1. P ⊃ Q      conclusion ~P ∨ Q 

                 |2. ~(~P ∨ Q)      Assumption for Indirect Proof
                 |3. ~~P & ~Q       2, DeMorgan's law
                 |4. P & ~Q         3, Double negation
                 |5. P              4, Simplification
                 |6. Q            1,5, Modus Ponens
                 |7. ~Q & P         4, commutation
                 |8. ~Q             7, Simplification
                 |9. Q & ~Q       6,8, Conjunction
~P ∨ Q     lines 2-9  Indirect proof.

Therefore (P ⊃ Q) ⊃ (~P ∨ Q)

Now we reverse the conclusion and premise:

premise 
1. ~P ∨ Q      conclusion P ⊃ Q 

                 |2. P      Assumption for Conditional Proof
                 |3. ~~P            2, Double negation
                 |4. Q            1,3, Disjunctive syllogism

5. P ⊃ Q    lines 2-4      Conditional proof.

Therefore (~P ∨ Q) ⊃ (P ⊃ Q)

So we have proved

[(P ⊃ Q) ⊃ (~P ∨ Q)] & [(~P ∨ Q) ⊃ (P ⊃ Q)]

thus they are equivalent.  But all we need is the first part,

(P ⊃ Q) ⊃ (~P ∨ Q)

Edwin</pre>