# I'm stuck and having trouble with ￢P ∨ Q Prove: P → Q

I am having trouble with this problem as I have just started doing logic. Is this the same as

P → Q
Prove: ￢P ∨ Q

?

• Which text book are you using? An online proof checker and text book may be helpful as supplementary material: proofs.openlogicproject.org – Frank Hubeny Apr 23 at 1:08
• Welcome to PSE. The answers hint at how to find a proof. But your question seems to be whether proving P → Q from ￢P ∨ Q is the same as proving ￢P ∨ Q from P → Q to which the answer is "No, this is not the same thing, though the proofs might look (structurally or otherwise) similar". – Jishin Noben Apr 23 at 8:13
• The two statements should have the same truth values. Would that analysis assist in showing equivalence? – Mark Andrews Apr 23 at 23:09

## 3 Answers

In a natural deduction system (if that is what you are using) to prove a conditional, such as is P → Q, you must use a Conditional Proof.

This takes the form of assuming the antecedent (that is P) aiming to derive the consequent (that is Q) through valid inferences (also using the premises; that is ￢P ∨ Q). Then discharging the assumption allow the deduction of the conditional (that is P → Q).

Now to prove Q from an assumption of P and the premise of ￢P ∨ Q, either use Disjunctive Syllogism, or a Proof by Cases.

In this particular case, the two statements are equivalent: `(￢P ∨ Q) ⊢ (P → Q)` and `(P → Q) ⊢ (￢P ∨ Q)` are both provably true statements, so `(￢P ∨ Q) ≡ (P → Q)`.

But in order to prove that equivalence, we need to prove both directions separately. To see why, consider the case where instead of `(￢P ∨ Q)` and `(P → Q)`, we have these two statements:

• `P`
• `P ∨ Q`

We can trivially prove that `(P ∨ Q)` follows from `P`; this is more or less the definition of the addition rule. But `P` does not necessarily follow from `(P ∨ Q)`, since `(￢P ∧ Q)` also satisfies that clause. We can prove it in one direction, but they are not equivalent statements.

• The two statements are logically equivalent using the material implication rule. Draw out a truth table and see for yourself that the tables are 100 percent identical. – Logikal Apr 23 at 23:45
• @Logikal I know they are equivalent. I said they are equivalent. But the OP isn't asking if they're equivalent; they are asking if, having already proven A ⊢ B, you must still prove B ⊢ A in order to establish that an equivalence exists. The answer to that is "yes". The fact that this particular equivalence has already been proven doesn't change that. – Ray Apr 24 at 1:24

If one uses a truth table generator one can show that (¬P ∨ Q) ↔ (P → Q). To see this, insert the following input into the Stanford Truth Table Tool: (~P or Q)<=>(P=>Q)

This shows that the two statements are equivalent.

However, the question asks one to prove P → Q given the premise ¬P ∨ Q. Here is how one might do this using a natural deduction proof checker: See the links below for an explanation of the rules used by this proof.

The proof would not be the same if we wanted to prove ¬P ∨ Q given the premise P → Q. That proof would look like this: Although the statements are equivalent based on a truth table generator, the proofs from one to the other may be different depending on which is the premise and which the conclusion.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/