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Testing Logical Equivalence

  1. Jan 30, 2013 #1
    Hello hello, I'm not looking just for an answer per say, but am also wondering the thought process in solving problems such as the following:

    Hopefully this doesn't take up too much of someones time.

    Determine whether the following equations are logically equivalent:

    1) (∃x)( P(x) → Q(x) )

    2) (∀x)P(x) → (∃x)Q(x)
     
  2. jcsd
  3. Jan 30, 2013 #2

    micromass

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    Try to write the implication in terms of other connectives.
     
  4. Jan 31, 2013 #3
    After you have followed micromass's suggestion, the next step can make more sense to you if you contemplate an analogy (I stress that this is a way of thinking: it would not work as a formal proof)
    all quantifier like a large "and",
    existence quantifier like a large "or"
    "and" like "intersection"
    "or" like "union"
    deMorgan Laws.
    Formally, if you are not an intuitionist, you can try playing around with the equivalence between "[itex]\forall[/itex]x P" and "~[itex]\exists[/itex]x ~P", or between "[itex]\exists[/itex]x Q" and "~[itex]\forall[/itex]x ~Q"

    (by the way, it's "per se")
     
  5. Jan 31, 2013 #4

    MLP

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    If you are familiar with how to determine a formula is logically true, then you can use the fact that formulas are logically equivalent just in case their biconditional is logically true. If there is an interpretation that makes the biconditional of (1) and (2) false, then they are not logically equivalent. If there is no such interpretation, then they are logically equivalent.
     
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