Consider the statement,
$1.$ "If it is Tuesday, then it is raining".
In propositional logic, 1 would read as, "$p \implies q$." Now, in accordance with the rules and definitions prescribed in logic, we have a plethora of logical equivalences. We can rewrite 1 as $ \neg p \vee q$, and in English,
$2.$ "It is not Tuesday or it is raining."
By setting up a truth table, we can prove that these statements are equivalent, hence, 1 and 2 have the same meaning. However, if I were read the two statements in English, I wouldn't suspect that they have the same meaning in everyday language. My question is: does the fact that the two statements in logic have the same meaning necessarily imply that the have the same meaning in everyday language? Because I honestly don't see how they convey the same meaning.