Originally Posted by
Plato 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 $\displaystyle \sum\limits_{k = 1}^\infty {a_k } $ converges only if $\displaystyle \left( {a_n } \right) \to 0$.
If the series $\displaystyle \sum\limits_{k = 1}^\infty {a_k } $ converges then $\displaystyle \left( {a_n } \right) \to 0$