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Interpreting p implies q

  1. Sep 11, 2012 #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.
  2. jcsd
  3. Sep 11, 2012 #2
    Re: Interpreting "p implies q"

    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.
  4. Sep 11, 2012 #3
    Re: Interpreting "p implies q"

    Thanks, SteveL27. The last three lines were very helpful.
  5. Sep 12, 2012 #4
    Re: Interpreting "p implies q"

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

    "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.
  6. Sep 12, 2012 #5
    Re: Interpreting "p implies q"

    Thanks ImaLooser. Your examples were pretty helpful too.
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