Distinguish between exclusive “or” and inclusive “or” when the truth value of p and q cannot be simultaneously true

discrete mathematicslogic

I just started learning Discrete Mathematics using Kenneth H. Rosen's "Discrete Mathematics and Its Application", and I'm having trouble understanding the solution for example 9 in section 1.1.2.

Since the disjunction of p and q (or the inclusive "or") is the proposition that states that either p is true, or q is true, or both p and q are true, if the "or" in the statement is an inclusive "or", then if p and q are both true, the truth value of the statement has to be Truth. Therefore, what the explanation is trying to do (or at least what I think it's trying to do) is to show that when both p and q is true then the truth value for the statement is False which means the "or" must be an exclusive "or" . However, if p and q cannot be simultaneously true, then how can there be a truth value for the statement when p and q are both true? The truth table for the statement, whether with the inclusive "or" or the exclusive "or", will only have 2 cases: p is T and q is F, and p is F and q is T.

I guess what I want to ask is how to distinguish between exclusive "or" and inclusive "or" when the truth value of p and q cannot be simultaneously true?

Best Answer

Your observation is correct: if we know $P$ and $Q$ can't both be true, then the expressions $P\lor Q$ and $P\oplus Q$ are equivalent (in that they have the same truth value). We could write this as the following theorem:

$$\lnot(P\land Q)\implies(P\lor Q\iff P\oplus Q).$$

Your question "if p and q cannot be simultaneously true, then how can there be a truth value for the statement when p and q are both true?" is a little confusing. The truth functions corresponding to the logical operators are defined for all inputs even if some of the function values are irrelevant in a given situation.

Edit: I looked at the exercise you linked to in the comments, and the answer given in the book is questionable. They ask you to "translate" a sentence of the form "P or Q", and expect you to get $P\oplus Q$ by noticing and incorporating additional information (that P and Q can't both be true). It makes sense that one would want to do this, because saying $P\oplus Q$ is "stronger" or "more informative" than saying $P\lor Q$, but I'd argue that this is more than just a translation from English into logical symbols. If the exercise writer wanted $P\lor Q$ to be an incorrect answer, they should've said "write a logical expression that exactly describes the scenarios that are possible" or something, not just "translate".

Related Question