Tonio, How much do you know about implications?

I know quite a bit, thanx for asking.

It is well known that “P implies Q” is logically equivalent to “P only if Q”

I may be misunderstanding the language, but in my book , and until showed and convinced otherwise, the phrase "P only if Q" is the same as Q --> P or, in ordinary language, it means " P happens only if Q happens", and that's why I wrote that imo the word "only" does not change the direction of the implication.

The only way I can logically understand the word "only" wrt impication is in the notorious iff = if and only if, so "A iff and only if B" means "If B then A, and if A then B".

What I know is that P --> Q is logically eq. with ~ Q --> ~ P, with ~ meaning negation.

So unless you can convince me that in english language "P only if Q"

is the same as P --> Q I can't see how from the plain meaning of words we can reach your conclusion.

If you have some source where this is explained I'd thank you very much since, of course, I could be wrong and I could be misunderstanding some well-founded common understanding in logical mathematic spoken english which, btw, is my

3rd. langauge.

Tonio

These two statements are logically equivalent.

The series

converges only if

.

If the series

converges then