Since the English “or” (or “either … or”) is ambivalent, it must be defined more precisely. In English, “or” can not only be used when we talk about an alternative in a way that one alternative is true and the other false, but also when both alternatives are true as in the following case: “Either Lincoln was the 16th president or Johnson was the 17th president.” In logic, this is called an “inclusive or.” An “exclusive or,” by contrast, (“XOR” for short) is used in the sense “either … or, but not both.”
This distinction is important for the three valid argument schemes disjunctive syllogism, not-both syllogism, and XOR syllogism. If you have an alternative with regard to which you know that one possibility is true, you must use what I call the “XOR syllogism” when you argue for the falsehood of the other possibility; but if you have an alternative with regard to which you know that one possibility is false, you have to use the “disjunctive syllogism” to argue for the truth of the other one.
If you take the wrong scheme, you cannot be sure whether what you are arguing for is necessarily the case. Thus, it is important to know that the disjunctive syllogism is based on an “inclusive or,” and the XOR syllogism on an “exclusive or.” All three forms used above are defined as follows:
|
p |
q |
p v q inclusive or |
p ≠ q exclusive or |
p | q not both p and q |
|---|---|---|---|---|
|
T |
T |
T |
F |
F |
|
T |
F |
T |
T |
T |
|
F |
T |
T |
T |
T |
|
F |
F |
F |
F |
T |