Does Either...Or means Exclusive Or or Inclusive Or?

Question

Let's take a compound propositions

Either it is below freezing or it is snowing.

Now if

$p$: it is below freezing 

$q$: it is snowing

Will it be $p \vee q$ or $p \oplus q$? There are some instances where semantics are required. For example

Either you are ill or appearing for exam. In this both cases can't be true, because if you are ill you can't appear for example and you must be in one state.

Answer 1

The use of "Either.....or...." in Mathematical logics is used to imply that one of the two given is true but not both at a time. Consider the statement "Ram is either Smart or Rich". Now Here the statement can be inferred as "Ram is smart or rich but not both".

However If we talk about English, English is super flexible, and super subtle, so there really is no hard rule in English, You have to use your common sense to figure out what the speaker means in the context of when and where the speaker says it.

Consider I say "Today I will study either Discrete Maths or Engineering Maths." now in English usage it is clear that I mean this in an inclusive sense, It would be perfectly fine If I manage to study both EM and DM today.

Comments

Also, Ram can be both rich and smart. There is no harm in it, in fact it can be possible as rich people are smarter and vice versa. So here inclusive or would be better.

I am using same logic as here