The question I wanted to ask is exactly as discussed on Math SE [here](https://math.stackexchange.com/questions/958708/difficulty-understanding-why-p-implies-q-is-equivalent-to-p-only-if-q). What's bugging me is why the first conversion from $p \to q$ to $p$ only if $q$ doesn't make sense in everyday speech.

Here's the accepted answer on that page:

> 
> "I win the lottery only if I give you \$1 billion" is equivalent to "if
>  I don't give you \$1 billion, then I won't win the lottery", and the
>  latter statement is the contrapositive to "if I win the lottery, then
>  I will give you \$1 billion."
> 
> 
> If you have doubts about why the contrapositive of a statement is an
>  equivalent statement, have you tried making a truth table?
> 
> 
> 

My problem is, sure the converted statement makes sense when I consider its equivalent and then contrapositive, but why doesn't it make sense in the original form?

Does this disconnect point to a fundamental difference between Boolean Logic and Human Language that human beings are not able to keep apart (like in my case) without adequate practice? Perhaps it's a question more related to philosophy or psychology, in which case I apologize for posting it here. 

Another explanation I came across is [this one](http://www.wwnorton.com/college/phil/logic3/ch10/ponlyifq.htm). Here the statements "I'll stay home tomorrow only if I'm sick" and "If I stay home tomorrow then I'm sick" make perfect sense. I think that might be because in this example, the cause comes after the effect, whereas in the earlier example, the cause comes before the effect.

Honestly, I'm hopelessly confused at this juncture, and any guidance is much appreciated.