Interpreting p implies q

  • #1
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
[itex]p \Rightarrow q[/itex]:
  • 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.
 

Answers and Replies

  • #2
795
7


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
[itex]p \Rightarrow q[/itex]:
  • 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.
 
  • #3


Thanks, SteveL27. The last three lines were very helpful.
 
  • #4
483
3


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
[itex]p \Rightarrow q[/itex]:
  • 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.
 
  • #5


Thanks ImaLooser. Your examples were pretty helpful too.
 

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