exclusive disjunction

In propositional logic, the **exclusive disjunction** (also called **exclusive or** or simply **xor**) of a family of statements (truth values, propositions, predicates, velc) is a statement that is true if and only if exactly one of the statements in the family is true. The corresponding operation on subsets (of a fixed set) or material sets is called *exclusive union*, and the binary operation agrees with *symmetric difference* (which is generalised to higher arities in a different way). If we think of quantifiers as infinitary operators, then exclusive disjunction becomes uniqueness quantification.

The **exclusive disjunction** of $p$ and $q$, written $p ⊻ q$ (and a host of other ways), may be defined in any of these forms:

- $\neg(p \Leftrightarrow q)$,
- $(p \vee q) \wedge (\neg p \vee \neg q)$,
- $(p \wedge \neg{q}) \vee (\neg{p} \wedge q)$.

These are all equivalent in classical logic. In intuitionistic logic, (2,3) are equivalent but (1) is weaker; (2,3) give the usual meaning in constructive mathematics.

The false statement is the identity for this operation; it is the exclusive disjunction of no statements.

In classical logic (but not in intuitionistic logic), this operation is associative, but multiple applications don't mean what you may think. Instead, $(p ⊻ q) ⊻ \equiv p ⊻ (q ⊻ r)$ comes out true if $p, q, r$ are all true; in general,

$(p_1 ⊻ (p_2 ⊻ \cdots ⊻ p_n)\cdots)$

is true if and only if an *odd* number of the statements $p_i$ is true. But when we write a multiple statement with ‘xor’, we really mean that *exactly one* of the statements is true. So by fiat, define the **exclusive disjunction**

$p_1 ⊻ p_2 ⊻ \cdots ⊻ p_n$

(without parentheses) to mean that exactly one of the $p_i$ is true; this definition is also used in intuitionistic logic.

As the indexed version of ordinary disjunction is existential quantification $\exists$, so the indexed version of exclusive disjunction is uniqueness quantification $\exists!$; this requires a primitive notion of equality to state in predicate logic.

Similarly, the **exclusive union** of two sets $A$ and $B$, written $A \uplus B$ (and a host of other ways), may be defined using exclusive disjunction:

$A \uplus B = \{ x \;|\; x \in A \;⊻\; x \in B \} .$

The **exclusive union** of a family $(A_i)_{i\colon I}$ of sets may be defined using uniqueness quantification:

$\biguplus_i A_i = \{ x \;|\; \exists!{i},\; x \in A_i \} .$

Note that the union, exclusive union, and (internal, or external up to natural isomorphism) disjoint union of a family of (pairwise) disjoint sets are all the same. For a family that is not disjoint, however, the union are exclusive union are different, the internal disjoint union does not make sense, and the external disjoint union is not isomorphic to either the union or the exclusive union (at least not naturally, and in some cases not at all).

We also have the **symmetric difference** of two sets, which is the same as the exclusive union. But we see this operation as the addition in a Boolean ring and so actually interpret it in the usual way as an associative operation. So the symmetric difference of $n$ sets indeed consists of those points that belong to an odd number of sets, and there is no infinitary symmetric difference.

It is widely said that ‘or’ in English can mean either inclusive or exclusive disjunction, while Latin has two terms, respectively ‘vel’ and ‘aut’, but this is not really correct. It's be more correct that ‘or’ can sometimes mean union and sometimes disjoint union (with the latter sometimes external and sometimes internal), but never symmetric difference. Thus, the only meaning of ‘or’ is for some kind of coproduct, which exclusive disjunction is not. (Similarly, ‘aut’ is more about disjoint union than symmetric difference.) In some cases, exclusive disjunction may be a valid *implicature* (in the sense of linguistic philosopher Paul Grice) even if the only valid *inference* (by the literal meaning of the sentence) is taken to be inclusive.

For the literature on this subject, see the Stanford Encylopedia entry and The Myth of the Exclusive ‘Or’ (*Mind*, 80 (317), 116–121).

What about the ‘or’ of parental threat? Consider the logician parent who says “Come here or I’ll smack you” to his child and smacks even after obedience as they believe in the inclusive ‘or’. -David

That's no different from ‘If you don't come here, then I'll smack you.’, which also suggests (but does not state) the converse. And in fact, no parent, logician or otherwise, is actually making the promise implied by the $\neg(p \wedge q)$ clause; if the child comes to such a parent and then kicks the parent in the shin, then the parent will still smack the child. Instead, if you *want* to make that promise, then you say ‘If you come here, then I won't smack you.’ explicitly. This has a very different tenor (unless you say it in a wink-nudge mafia kind of way), as it's a promise rather than a threat. (I know, it's *only* a promise, which is still different in tenor than a statement that is *both* promise and threat, as an exclusive disjunction would be. But I still hold that your statement is *only* a threat.) Note that a logician child who believes the parent's literal expression would still choose to come if avoiding smacking is the highest priority; but the reason is that refusal guarantees a smack, not that obedience necessarily avoids it. That is why the wise child also throws in a contrite expression and an oral apology, to improve the odds. —Toby

I see there’s a literature on the subject including “The Myth of the Exclusive ‘Or’” (Mind, 80 (317), 116–121). —David

Also: I argued above that the meaning of ‘Come here or I'll smack you’ must be weaker than exclusive disjunction, since the parent will smack the child anyway under some circumstances. However, I agree that it is stronger than inclusive disjunction, but that is because we may go beyond the literal meaning of the words and apply a Gricean implicature. To be specific, if the parent intends to smack the child regardless, then the parent should say ‘I'll smack you’ by the Maxim of Quantity, but the parent in fact said something more wordy. Thus we conclude that the parent does not intend to smack the child if the child comes, without ruling out the possibility that the parent will still smack the child for some other reason, as yet unanticipated. —Toby

Revised on October 16, 2013 21:38:55
by Anonymous Coward
(76.84.64.5)