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Math Help - universal quantification over null set

  1. #1
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    universal quantification over null set

    what is the truth value of a universal quantification over an empty set?

    <br />
\forall x \in \varnothing, P(x)<br />

    since it is equivalent to

    <br />
\lnot \exists x \in \varnothing, \lnot P(x)<br />

    it seems that it would be vacuously true, however it is intuitively rather difficult to accept. for instance if i want to prove that the null set is a subset of any arbitrary set then all i need to prove is that for any element of the null set that element is also in any given set. however by the above this is always true and it seems that there is no connection whatsoever between the predicate and the quantifier, and the statement hence seems contrived.

    any thoughts?
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  2. #2
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    Quote Originally Posted by polypus View Post
    what is the truth value of a universal quantification over an empty set?

    <br />
\forall x \in \varnothing, P(x)<br />

    since it is equivalent to

    <br />
\lnot \exists x \in \varnothing, \lnot P(x)<br />

    it seems that it would be vacuously true, however it is intuitively rather difficult to accept. for instance if i want to prove that the null set is a subset of any arbitrary set then all i need to prove is that for any element of the null set that element is also in any given set. however by the above this is always true and it seems that there is no connection whatsoever between the predicate and the quantifier, and the statement hence seems contrived.

    any thoughts?
    That all seems OK to me.

    RonL

    (sometimes we just have to get used to things that at first seem a bit peculiar)
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  3. #3
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    This is related to the issue in Logic where "p implies q" is held to be true whenever p is false. Thus if p is a contradiction, then "p implies q" is true, for any q.
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  4. #4
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    Quote Originally Posted by ray_sitf View Post
    This is related to the issue in Logic where "p implies q" is held to be true whenever p is false. Thus if p is a contradiction, then "p implies q" is true, for any q.
    We old time logic instructors have repeated the following mantra many times.
    A false statement implies any statement.
    A true statement is implied by any statement.


    Because x \in \emptyset is a false statement, then
    “if x \in \emptyset then x \in X" is a true statement.
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