Here is a problem from Kenneth Rosen's Discrete Mathematics and its Applications, Section 1.4
Let P (x) be the statement “x can speak Russian” and let Q(x) be the
statement “x knows the computer language C++.” Express each of these
sentences in terms of P (x), Q(x), quantifiers, and logical
connectives. The domain for quantifiers consists of all students at
your school. This is a homework problem and I am not expecting a
complete answer.Every student at your school either can speak Russian or knows C++.
The answer to this problem was ∀x(P(x) V Q(x))
But I think it should be ∀x(P(x) ⊕ Q(x))
by the exclusive or XOR not inclusive or as the statement is either or
where is the correct answer? and why?
Best Answer
Does "either P or Q" really rule out the case where both "P" and "Q" holds?
Surely not always. You ask, why has the virus spread faster than predicted? I say "Either many people are not self-isolating or the virus is even more easily transmitted that we thought." In saying that, do I definitely rule out both being true? If I add "maybe both" am I thereby taking back part of what I earlier said? Arguably not.
Note too it seems natural to deny something of the form "either P or Q" by "no, neither P nor Q" -- but that's not the negation of exclusive or.
OK, there's more to be said. But it is certainly debatable how far implications of exclusivity on particular occasions of use of "either ... or ..." are generated by the literal meaning (fixing the truth-conditions) of what is said, and how far they are generated pragmatically by context. And, as far as logic is concerned, it isn't worth getting into a fight about that; the whole point of adopting formal languages is we sidestep such murky issues. But it does mean that Rosen's suggested inclusive rendering is not definitely wrong!