Difference between "only if" and "if" I am just being introduced to logic in mathematics. Let $P$ be "He is an honest person." and $Q$ be "He can be the President.". My lecturer was saying that "only if" means $P\rightarrow Q$, while "if" means $Q\rightarrow P$. I simply cannot wrap my head around this.
So let us consider the following two sentences.


*

*One can be the president only if one is honest.

*One can be the president if one is honest.


How are the two statements any different in meaning?
However, according to the definitions, the first statement is of the form $P\rightarrow Q$, while the second is of the form $Q\rightarrow P$ and I know that $P\rightarrow Q$ and $Q\rightarrow P$ clearly have different meanings.
Any explanations regarding the difference between both terms and how/when to use which will be greatly appreciated!
 A: I'll use a different example, just because I find it easier to explain:


*

*"I wear a jacket if it is cold"

*"I wear a jacket only if it is cold" 
In the first case, there is nothing stopping you from wearing a jacket when it is not cold; the "if" does not preclude your wearing a jacket under different circumstances. However, "only if" implies that you will only wear a jacket when it is cold, and never any other time. 
The "if" case can be though of as "if... then,", i.e. "If it is cold, then I will wear a jacket." Again, we can see how this does not stop you from wearing a jacket during other times. 
The second statement also allows the following: "If it is not cold, then I will not wear a jacket." We've said that the first statement does not preclude such cases, but the contrapositive of the second does. 
Hopefully that helps clear things up. 
A: Let me rephrase the examples you gave:


*

*One can be the president only if one is honest. $P\implies Q$
Is the same as:
If one can be the president then one is honest. (And therefore: If one is president one has to be honest.)

*One can be the president if one is honest. $Q\implies P$
Is the same as:
If one is honest then one can be the president. (And therefore: If one is honest one has not to be the president necessarily, though he could be)
A: I think you are wrong to find no difference in meaning between your statements 1 and 2. Part of the difficulty is that "can be" is a rather slippery phrase.
Suppose $X$ was born in England and $Y$ is born in the US.
"$X$ is eligible to be president if $X$ is honest" is false, because $X$ fails the born in the US requirement.
"$Y$ is eligible to be president only if $Y$ is honest" is true, because honesty is a requirement (we hope and assume).
