For some reason, be it some bad habit or something else, I can not understand why the statement "p only if q" would translate into p implies q. For instance, I have the statement "Samir will attend the party only if Kanti will be there." The way I interpret this is, "It is true that Samir will attend the party only if it is true that Kanti will be at the party;" which, in my mind, becomes "If Kanti will be at the party, then Samir will be there."
Can someone convince me of the right way?
EDIT:
I have read them carefully, and probably have done so for over a year. I understand what sufficient conditions and necessary conditions are. I understand the conditional relationship in almost all of its forms, except the form "q only if p." What I do not understand is, why is p the necessary condition and q the sufficient condition. I am not asking, what are the sufficient and necessary conditions, rather, I am asking why.
Best Answer
Think about it: "$p$ only if $q$" means that $q$ is a necessary condition for $p$. It means that $p$ can occur only when $q$ has occurred. This means that whenever we have $p$, it must also be that we have $q$, as $p$ can happen only if we have $q$: that is to say, that $p$ cannot happen if we do not have $q$.
The critical line is whenever we have $p$, we must also have $q$: this allows us to say that $p \Rightarrow q$, or $p$ implies $q$.
To use this on your example: we have the statement "Samir will attend the party only if Kanti attends the party." So if Samir attends the party, then Kanti must be at the party, because Samir will attend the party only if Kanti attends the party.
EDIT: It is a common mistake to read only if as a stronger form of if. It is important to emphasize that $q$ if $p$ means that $p$ is a sufficient condition for $q$, and that $q$ only if $p$ means that $p$ is a necessary condition for $q$.
Furthermore, we can supply more intuition on this fact: Consider $q$ only if $p$. It means that $q$ can occur only when $p$ has occurred: so if we don't have $p$, we can't have $q$, because $p$ is necessary for $q$. We note that if we don't have $p$, then we can't have $q$ is a logical statement in itself: $\lnot p \Rightarrow \lnot q$. We know that all logical statements of this form are equivalent to their contrapositives. Take the contrapositive of $\lnot p \Rightarrow \lnot q$: it is $\lnot \lnot q \Rightarrow \lnot \lnot p$, which is equivalent to $q \Rightarrow p$.