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In How to Prove it by Velleman, for defining disjunctions, he gives the difference between exclusive "OR", and inclusive "OR."

Given two events $P$ and $Q$, the disjunction is defined for them as:

  • Inclusive: One of $P$ and $Q$, or both.

  • Exclusive: One of $P$ and $Q$, but not both.

Quoting from his book:

"In mathematics, or always means inclusive or, unless specified otherwise, ..." (Velleman, 2006, p.15)

My question is -

Why did the inclusive definition of disjunction become the convention?

Was it coincidental, or is there some aspect to the inclusive definition that makes it more convenient?

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It is such "or" one includes, and "xor" is the exception. Hence the convention. –  Pedro Tamaroff Feb 23 at 6:19

4 Answers 4

up vote 7 down vote accepted

Velleman could be read as saying that (i) in English, "or" has two meanings when used as a propositional connective, one corresponding the familiar inclusive disjunction of formal logic, the other expressing exclusive disjunction, but (ii) there's a special convention that in mathematical English, "or" is used only in the first way. Hence the OP's question -- why the special convention in maths? Two points about this.

(I) Arguably (i) is just wrong. As background recall that a standard approach to explicating how we manage to interpret what we read or hear is that the overall message conveyed is a result of the interaction of two things, first the semantic content (the "literal meaning") of the sentence used, and second contextual and pragmatic clues.

It is not infrequently said that "or" is semantically ambiguous, i.e. it has two different literal meanings. But arguably a much smoother theory seems to be that "or" is in fact semantically unambiguous, meaning inclusive disjunction. And on those occasions where we hear/read an utterance of "$A$ or $B$" as also conveying "but not both", the additional implicature is either deduced from background knowledge that $A$ excludes $B$ or has some other contextual, pragmatic, source. (Sometimes the context is unclear or complicated and we don't know whether the speaker does or does not mean to rule out the case where both disjuncts are true: but that doesn't mean that the meaning of the sentence used is not determinate.)

Note for example that in the sort of cases typically invoked to supposedly illustrate the uses of exclusive "or", it would -- on the semantic story -- be a contradiction to add "or both" whereas it normally seems like a coherent cancelling of a pragmatic (typically Gricean) implicature. Note again that 'either ... or' in English seems to have a uniform semantic negation, 'neither ... nor ...' (which couldn't be the negation if 'or' is exclusive). And so it goes.

There is a large literature on this, unsurprisingly. For a recent review, see Lloyd Humberstone's bible, The Connectives (MIT, 2011) -- which is 1492 pages mostly on 'and', 'or', 'if' and 'not'! §6.12 is titled "Exclusive/inclusive" and gives considerations against the semantic ambiguity thesis.

(II) On a charitable reading, Velleman is probably not actually asserting (i), i.e. he is not actually defending the disputed theory that ordinary 'or' is semantically ambiguous. He indeed notes that logicians distinguish an inclusive from an exclusive formal disjunction. But he is sensibly not pausing to fuss about whether we are right to sometimes interpret bare "or" in English as semantically meaning exclusive disjunction. In maths at any rate, he is saying, we do take "or" to be by default the usual Boolean inclusive disjunction which has very nice properties like being nicely dual with conjunction, satisfying De Morgan's Laws etc., being nicely related to existential quantification, etc. etc. Those neat features of logical incisive disjunction are reason enough to concentrate on it (and we can always add a "but not both" clause if it is important to formalize an exclusive message). But note Velleman could, consistently with what he says, add the whispered aside that I would add here: "Pssst! Just between you and me, I think that's what "or" always semantically means, but it would be far too distracting to argue the case here."

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One may be able to make a case that examples of natural-language exclusive or are usually pragmatically determined, but there are many examples in which it would not be a contradiction to add or both, though it would subvert the intended sense (Eat some vegetables, or you’ll be doing the dishes for a week). And natural language either-or typically is exclusive, just as entweder-oder is in German; neither-nor (weder-noch) simply isn’t its negation. But all of this, it seems to me, is beside the point of the original question. –  Brian M. Scott Apr 19 '13 at 8:20
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Leaving aside the claim about German, I disagree on every count! For a start, cancelling a pragmatic implicature may subvert the overall intended message -- but it is tendentious to say this is subverting the sense (semantic content, Fregean Sinn). But of course, this just goes to show how wise Velleman was not to get into this sort of debate! –  Peter Smith Apr 19 '13 at 8:30
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I don’t think that wisdom has much to do with it: I really think that it has little to do with teaching people to read and write mathematics. –  Brian M. Scott Apr 19 '13 at 8:34
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I'm afraid those last two comments are rather point-missing. The standard theory has it that speaker and hearer (writer and reader) have to exploit the interaction between the literal meaning of a sentence and the context in which it used in order to arrive at the message conveyed. So we can't simply read back from the possibility of different messages conveyed in different contexts by a given sentence that the blame is to be pinned on semantic ambiguity (the sentence having different literal meanings). ... –  Peter Smith Feb 23 at 12:54
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... Sure that sometimes happens, but not always. So the fact that "or" sentences can be used in different contexts to express different messages (sometimes the overall message is exclusive) doesn't by itself show that this is because the sentence is semantically ambiguous. As far as the superficial phenomena go, it could be that the multiple possible readings of the message conveyed are a function of context. It's a theoretic issue to weigh up what the best account of the empirical phenomena is: and (as I said) the non-ambiguity thesis has arguable support. For more, see Humberstone. –  Peter Smith Feb 23 at 12:56

The operations logical AND $(\wedge)$, inclusive OR $(\vee)$ are dual, in the sense that the following hold.

  1. $\neg (A \wedge B) \leftrightarrow \neg A \vee \neg B$
  2. $\neg(A \vee B) \leftrightarrow \neg A \wedge \neg B.$

This means they have essentially the same properties. They're both associative, commutative, and idempotent; and they distribute over one another. So in conclusion, inclusive OR has nice properties, and it interacts nicely with logical AND.

It also seems to show up a lot more often.

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Also, inclusive OR works well when chained: "A or B or C" means that at least one of the three is true. "A xor B xor C" does not mean that exactly one is true; it also holds when all three are true (regardless of parentheses). –  Carl Mummert Apr 19 '13 at 13:48

George Boole, when he originally developed his Laws of Thought to apply mathematics to reasoning, used the exclusive or. However, the system was quite cumbersome in comparison to modern methods.
As others took up his ideas, they found that the inclusive or was far better behaved and easier to work with. For instance, suppose we want to say "It is not the case that P or Q but not both". We get a "Either it is not the case that P and not the case that Q, or it is the case that both P and Q". Contrast this with "It is not the case that P or Q or both". To negate this, we have "It is not the case that P and it is not the case that Q".

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The most common case in mathematics is probably when "or both" is obviously impossible, in which case it doesn't matter if you use inclusive or exclusive or. For example, if we say n = 2 or 3, we know it can't be both.

In cases where it does matter, the inclusive disjunction is radically more likely to be case. I think it's because any kind of proof by cases will lead to inclusive or. Suppose you were proving a theorem of the form that "A or B implies C". If you prove that "A implies C" and "B implies C", then the inclusive or version immediately holds. If only the exclusive or version holds, then the proof strategy would probably be something like "A and not B implies C", and "B and not A implies C", which is a more complex kind of result.

Off the top of my head, I can't think of that many results where an exclusive or arises. The only one that comes to mind is the Fredholm alternative.

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