Can anyone explain why the predicate all
is true for an empty set? If the set is empty, there are no elements in it, so there is not really any elements to apply the predicate on? So it feels to me it should be false rather than true.
Why is predicate "all" as in all(SET) true if the SET is empty?
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logic
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1Roughly: if $\forall x \in S (R(x))$ is false, then there must be $x \in S$ such that $R(x)$ is false. Since there is no such $x$ for empty $S$, the statement $\forall x \in S (R(x))$ is true. – 2012-09-25
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1Okay... but you can also argue the other way around: if `∀x∈S(R(x))` is true, then all `x∈S` must have `R(x)` true. Since there is no such `x` for empty set `S`, the statement `∀x∈S(R(x))` is false. ... What happened? – 2012-09-25
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3The statement $\forall x \in S (R(x))$ does not imply that there is an $x$ for which $R(x)$ holds. It just says that whenever you give me an $x$ from $S$, I can demonstrate that $R(x)$ holds. In the end, it is just a matter of convention. However presuming vacuous universal quantification to be true results in smoother theory. I am just explaining the logic behind it. – 2012-09-25
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0I see. I don't think I understand but I'll nod. – 2012-09-25
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0http://en.wikipedia.org/wiki/Vacuous_truth – 2013-01-03