# Why is exclusive disjunction not a standard logical operation?

This question Rules of inference for exclusive disjunction and logical biconditional shows that the inference rule for exclusive disjunction are little different from the other rules for conjunction, disjunction, implication, etc.

Then why is the exclusive disjunction so often neglected/left out? I know it can be defined a posteriori without having it as a primary connective, but even in this form it is almost never mentioned.

• See Why did the inclusive "OR" become the convention?: "The operations logical AND and inclusive OR are dual" Commented May 29 at 10:32
• @MauroALLEGRANZA Thanks for your comment, but to be clear, I'm not asking why OR is used instead of XOR, but why the two aren't usually presented together
– user1313162
Commented May 29 at 10:34
• The second reason may be historical: Boole developed his logic with $\times$ and $+$ starting from the numerical meaning, where $\times$ is an operation “the result being the class whose members are both Xs and Ys” (modern intersection). The original "logical interpretation" of $+$ was "that $u+v$ corresponded to dividing a class into two parts, evidently meaning U and V are disjoint classes." If we remove the "disjointness" requisite, this becomes the modern union of classes, that is inclusive. Commented May 29 at 10:38
• It's also nice that $\lor$ and $\land$ define a lattice structure in various situations when dealing with semantics: eg this, this. This is related to what has already been said. Commented May 29 at 12:15

Hard to say exactly ... but it's probably for a bunch of different reasons, whether historical, theoretical, or practical, and Mauro pointed to some probable reasons in the Comments.

In particular, I think inclusive or just has some nice properties that exclusive or does not have.

• $$\land$$ and $$\lor$$ are duals

• inclusive disjunction corresponds to union

• It's easy to see that $$\{ \land, \lor, \neg \}$$ forms an expressively complete set, and indeed $$\{ \neg, \lor \}$$ is complete. However, $$\{ \neg, \oplus \}$$ is not.

• It's easier to capture $$\oplus$$ using $$\lor$$ and other operators than the other way around

• Probably in terms of applications we are just dealing with inclusive or's more often than exclusive or's.

• Even if some 'or' is exclusive, treating it as inclusive doesn't change the nature and validity of something like a proof by cases. Whereas treating an inclusive or as exclusive more easily leads to invalid results.