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This is a very "soft" question, but regarding language in logic and proofs, should

"Either A or B"

Be interpreted as "A or B, but not both"?

I have always avoided saying "either" when my intent is a standard, inclusive or, because saying "either" to me makes it feel like an exclusive or.

Bram28
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Jonathan Hebert
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  • Those statements are all irrelevant because they are not "or" statements at all, they just use the word "either". – Jonathan Hebert Feb 05 '17 at 15:27
  • For English language usage, see this post. – Mauro ALLEGRANZA Feb 05 '17 at 15:34
  • Usually, two "alternatives" are mutually exclusive; this is not so with the $\lor$ (or) truth-functional connective. – Mauro ALLEGRANZA Feb 05 '17 at 15:36
  • I've used and heard people use "either" to emphasize that this are the only options. I never thought "either" was exclusive. " Either numbers are non-negative or they are non-positive. They can't be neither" sounds utterly fine to me. Anyway that's english; not math. – fleablood Feb 05 '17 at 16:30
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    If you want to say "but not both," say "but not both." – David K Feb 05 '17 at 16:37
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    And in trying to translate to logic, the statement "$A$ or $B$, but not both," we can use $(A\lor B) \land \lnot (A \land B)$, or, alternatively, $(A\land \lnot B) \lor (B \land \lnot A)$. So along with David K's natural language suggestion, you can also express the exclusive or by saying: Either $A$ and not $B$, or else, $B$ and not $A$. – amWhy Feb 05 '17 at 17:50
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    The purpose of my asking how this wording should be interpretted is because I want to know how to interpret it whenever I come across it, not because I wish to use this wording or need suggestions on what wording to use. – Jonathan Hebert Feb 05 '17 at 18:24
  • @JonathanHebert and where have you come across it in a math context? Do you have examples? – Sasho Nikolov Feb 05 '17 at 23:22
  • @SashoNikolov what prompted the question was proofs written by students on an assignment for a class I am TAing for. – Jonathan Hebert Feb 06 '17 at 01:35

3 Answers3

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No, you cannot depend on that. If it were that simple, we wouldn't need clunky phrases like "exclusive or" to make clear when an "or" is exclusive.

Linguistically, "either" is simply a marker that warns you in advance that an "or" is going to follow. Nothing more.

hmakholm left over Monica
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In everyday speech, "or" is usually exclusive even without "either." In mathematics or logic though "or" is inclusive unless explicitly specified otherwise, even with "either."

This is not a fundamental law of the universe, it is simply a virtually universal convention in these subjects. The reason is that inclusive "or" is vastly more common.

Matt Samuel
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Often, when people use "either ... or ..", they are trying to stress that they are expressing an exclusive disjunction.

But, not always.

Consider. I say "I want to be either rich or happy" ... I think that is a perfectly good usage of English ... but it is also clear that I mean this in an inclusive sense: it's not as if I go like "Oh, no, that's not what I wanted!" when I turn out to be both rich and happy. No, clearly I would be fine with that as well.

Probably the reason I use "either" here is because I may feel some tension between the two .. that being rich might well require hard work that prevents me from doing the things that makes me happy, or vice versa ... but clearly there is no certainty that I cannot be both ... and again, I would be just fine with both. So again, it's meant as inclusive, not exclusive.

Also consider the English use of 'neither $A$ nor $B$'. Presumably this is a linguistic contraction of 'not either $A$ or $B$'. But 'neither $A$ nor $B$' means that $A$ and $B$ are both not the case, and logically that only works if we treat 'either $A$ or $B$' in this context as an inclusive or, rather than an exclusive or.

The point is: English is super flexible, and super subtle, so there really is no hard rule here. You have to use your common sense to figure out what the speaker means in the context of when and where the speaker says it.

In fact, the ambiguity of English leads to another reason why people sometimes use ‘either … or’ without intending an exclusive or. Consider any statement of the form ‘$P$ and $Q$ or $R$’. Now, is this meant as $(P \land Q) \lor R$, or as $P \land (Q \lor R)$? From logic we know that these two statements are not equivalent, which is exactly why in logic we have handy parentheses to disambiguate. Well, we don’t have parentheses in English, but the ‘either … or’ can be used to act like them, for if what I meant was $(P \land Q) \lor R$, then I can say ‘either $P$ and $Q$ or $R$’. And if what I meant was $P \land (Q \lor R)$, then I can say ‘$P$ and either $Q$ or $R$’. Note that I can use a construction like ‘both … and’ for similar disambiguation purposes as well.

Bram28
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  • Yes we practically define "neither A nor B" as "not either A or B" (probably related to the fact that OR in mathematics is typically inclusive), but as I read your penultimate paragraph, I'm beginning to wonder whether a layperson actually understands/accepts the former as a linguistic contraction of the latter. – ryang Jan 29 '23 at 01:10