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.
Best Answer
While it’s perhaps not immediately evident, the statement It is not Tuesday or it is raining does at least imply the statement If it is Tuesday, then it is raining even in everyday language: if the first statement is true, and if today really is Tuesday, then it must in fact be raining. The second probably does not imply the first in most people’s everyday language, because in everyday usage if ... then is normally taken to imply some connection $-$ perhaps not a truly causal connection, but something along those lines. Material implication ($\to$) is definitely not the same as everyday if ... then, so a formal equivalence between statements involving material implication may not translate into an equivalence between the apparent everyday counterparts.
The same problem arises to some degree with disjunction ($\lor$). In everyday usage the word or is often closer to exclusive or ($\veebar$) than to ordinary inclusive $\lor$, especially when preceded by either: It was either John or Charles; Apologize to your sister, or leave the table!
More generally, there’s a problem translating between formal and everyday language. Consider the statement Touch me, and you’ll lose some teeth! On the face of it that is $p\land q$, where $p$ is you touch me, and $q$ is you will lose some teeth, but the purely formal version closest to the real sense is $p\to q$.