JCBNLA
May 13, 2022, 05:38 AM
In propositional logic, the disjunction "P or Q", symbolized as "P v Q" can be taken to be "inclusive" or "exclusive." Inclusive would mean "P or Q or both", whereas exclusive would mean "P or Q but not both."
It is the convention among logicians that a disjunction is to be taken as inclusive, and not exclusive. My understanding is that the reason inclusive was chosen over exclusive is that the exclusive disjunction makes working with logic more cumbersome.
My question is this: Besides making logic more cumbersome, is there something illogical about the exclusive disjunction? The truth table says that if both disjuncts are true, then the disjunction as a whole is false--because, by definition, only one of the disjuncts can be true in an exclusive disjunction. But consider this disjunction: the number 7 is prime or the number 7 is odd. Both disjuncts are true, but it seems absurd to say both disjuncts taken together is false.
It is the convention among logicians that a disjunction is to be taken as inclusive, and not exclusive. My understanding is that the reason inclusive was chosen over exclusive is that the exclusive disjunction makes working with logic more cumbersome.
My question is this: Besides making logic more cumbersome, is there something illogical about the exclusive disjunction? The truth table says that if both disjuncts are true, then the disjunction as a whole is false--because, by definition, only one of the disjuncts can be true in an exclusive disjunction. But consider this disjunction: the number 7 is prime or the number 7 is odd. Both disjuncts are true, but it seems absurd to say both disjuncts taken together is false.