universal quantification over null set

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July 16th 2007, 05:54 PM
polypus

universal quantification over null set

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

$ \forall x \in \varnothing, P(x) $

since it is equivalent to

$ \lnot \exists x \in \varnothing, \lnot P(x) $

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?
July 16th 2007, 07:36 PM
CaptainBlack

Quote:

Originally Posted by polypus

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

$ \forall x \in \varnothing, P(x) $

since it is equivalent to

$ \lnot \exists x \in \varnothing, \lnot P(x) $

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)
July 18th 2007, 01:55 PM
ray_sitf

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.
July 18th 2007, 03:07 PM
Plato

Quote:

Originally Posted by ray_sitf

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.