- #1

agro

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**truth value of "for all x in {}" and "there exist x in {}"**

Suppose T is a true statement. Now, given a nonempty set A, both the statement

for all x in A, T

and

there exist x in A, T

are true. However, let E be the empty set. What is the truth value of

for all x in E, T

and

there exist x in E, T

?

In the second chapter of Paul R. Halmos' book "Naive Set Theory", he stated that if the variable x doesn't appear in sentence S, then the statements

for all x, S

and

there exist x, S

both reduce to S.

Is that something that is just agreed upon? In that case, the statement

for all x in E, T

reduces to T which is true (eventhough there's nothing in E) and so does the statement

there exist x in E, T (eventhough there exist nothing in E).

I find that counterintuive although if it is indeed the agreed upon rule, I think I just have to get used to it (but any justification would greatly help . Your comments?

Thanks,

Agro