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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
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