# Interpreting p implies q

SithsNGiggles
Interpreting "p implies q"

My Linear Algebra professor had my class work on some proofs, then introduced "truth tables," along with some notation and symbols.

I've taken a class on proofs before, but for some reason it didn't provide any background in pure logic, so I'm a bit lost with one thing my LinAlg prof wrote on the board.

He listed a few ways to interpret
$p \Rightarrow q$:
• p implies q,
• if p then q,
• q is necessary for p,
• p is sufficient for q,
• p only if q

I understand the first four items, but the last one doesn't make sense to me. Can someone please explain how it works? Thanks in advance.

SteveL27

My Linear Algebra professor had my class work on some proofs, then introduced "truth tables," along with some notation and symbols.

I've taken a class on proofs before, but for some reason it didn't provide any background in pure logic, so I'm a bit lost with one thing my LinAlg prof wrote on the board.

He listed a few ways to interpret
$p \Rightarrow q$:
• p implies q,
• if p then q,
• q is necessary for p,
• p is sufficient for q,
• p only if q

I understand the first four items, but the last one doesn't make sense to me. Can someone please explain how it works? Thanks in advance.

I agree that "only if" is the most confusing of the group. I think of it this way.

Say p => q. The only way that can be false is if either p is false, or q is true.

Say p is true. If q is false that makes the implication false. So if p is true then q must be true.

So if p => q is true, then p can be true only if q is true.

Remember, if 2 + 2 = 5 then I am the Pope. That's true.

So 2 + 2 = 5 only if I am the Pope. Can't be any other way.

SithsNGiggles

Thanks, SteveL27. The last three lines were very helpful.

ImaLooser

My Linear Algebra professor had my class work on some proofs, then introduced "truth tables," along with some notation and symbols.

I've taken a class on proofs before, but for some reason it didn't provide any background in pure logic, so I'm a bit lost with one thing my LinAlg prof wrote on the board.

He listed a few ways to interpret
$p \Rightarrow q$:
• p implies q,
• if p then q,
• q is necessary for p,
• p is sufficient for q,
• p only if q

I understand the first four items, but the last one doesn't make sense to me. Can someone please explain how it works? Thanks in advance.

The statement is true unless p is true and q is false.

Examples:
"if the moon is green cheese then 2+2=4"

That is true. It seems weird at first, but basically it is saying that 2+2=4 regardless so it doesn't matter what the moon is made of.

"if the moon is green cheese then 2+2=5" Sure. You will never be able to provide a counterexample, so it is a true statement. Vacuous, useless, but true.

"if 1+1=2 then 2+2=4" True. The second statement doesn't follow from the first so it is of no value, but it is indeed true.

'If 1+1=2 then 2+2=5" False!

As you can see, if there is no connection between p and q then any statement relating them is rather vacuous. But there is no harm in that.

SithsNGiggles