What does 'p only if q' mean in logic and proofs?

[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]

Thanks ImaLooser. Your examples were pretty helpful too.