# Does “either A or B ” preclude “both A and B”?

In mathematics, "A or B" includes "A and B".
Does "either" mean "A or B but not (A and B)" or does it include the possibility of "A and B"?
The context might be mathematics, formal logic or ordinary language.

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just a quick comment on the answer by chaos which caught my eye... quote: "do you want to go to lunch now or later?", answer "yes". (Illustrating that the "either" part is implied by context as often as it's cancelled by context.) the answer "yes" has less to do with the exclusivity of the word "or" and more to do with the interpretation of the "do you" part of the question.. an answer of "yes" would suggest that the answerer had interpreted the question "do you" literally. furthermore, assuming that the "or" was being used in the inclusive sense the "yes" would potentially also include going – user66911 Feb 24 '14 at 8:05
Outside of mathematics (and often inside as well), the use of "or" (with or without "either") does not have the precise definition that some people appear to lust for. It is exceedingly dangerous to make assumptions about its exclusivity without further contextual clues. English is not mathematics. – Hot Licks May 13 '15 at 17:31
@HotLicks +1 for using 'exclusivity' and 'lust' in one comment, about something completely unrelated. – no comprende Mar 23 at 2:12

"Either A or B" most precisely means, in symbolic logic terms, "A `XOR` B", where `XOR` is the "exclusive or". So yes, it means "A or B but not both". It isn't always actually used with full precision, though, so, as usual, context has to be taken into account. If somebody says, "select either A or B", for example, they definitely mean that you should not select both. If they say "if either A or B is true", though, they probably mean a non-exclusive `OR`, and the condition is still true if both A and B are true. Unfortunately, if there's a generally reliable rule for telling which is meant, I'm failing to think of what it would be.

Without the "either", the presumption would be more toward "A `OR` B", where `OR` allows the case where both are true. Which is why computer geeks and propositional calculus nerds will, when asked "do you want to go to lunch now or later?", answer "yes". (Illustrating that the "either" part is implied by context as often as it's cancelled by context.)

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+1 for the lunch part – ghshtalt Feb 23 '11 at 17:20
If someone puts emphasis on the words "either" and "or" ("You may have either tea or coffee") - either with italics or intonation - that is a fairly reliable clue that the exclusive meaning is intended. – psmears Feb 23 '11 at 19:49
@chaos, Your last paragraph is incorrect: The reason "yes" is a valid response is due to a shift in precedence; it is not due to having a part cancelled by context. "Yes" is an equally valid response to `xor` questions too, e.g. "Do you want A xor B?" "Yes, I want (A xor B).", which means I want either A or B but not both. Compare that to "Yes, I want (A or B)." which means I want either A or B or both. – Pacerier Sep 24 '14 at 2:20
Short answer: whenever someone uses OR in a question, just answer Yes. Make them re-clarify their intent. – no comprende Mar 23 at 2:15

Either A or B means the same as A or B. Each can mean or used in the inclusive or exclusive sense.

Usually, the inclusive sense is used in mathematics and the exclusive sense in everyday life. In any case, further specification or context will remove any doubt.

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From wikipedia:

Either/or means "one or the other." Its usage, versus the simple or structure, is often for emphatic purposes, sometimes intending to emphasize that only one option is possible, or to emphasize that there are only two options. Its use in a sentence lets the reader/listener know in advance that a list of two or more possibilities will be given.

As you correctly recognize "or" used alone can also include the possibility of both A and B (especially important in mathematics).

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In mathematics or computing you do not need context to remove the uncertainty. You simply look for the presence or absence of an X before OR. In literacy, reading, writing, speaking and listening contextual interpretation is necessary. Numeracy is more precise in syntax that literacy.

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How to Prove It by Vellerman, a textbook on formal logics, says

In mathematics, or always means inclusive or, unless otherwise specified, ...

and the book later uses "either ... or ..." to mean ∨.

What English sentences are represented by the following expressions?

1. ...
2. ¬S ∧ (L ∨ S)
3. ...

### Solutions

1. ...
2. John isn't stupid, and either he's lazy or he's stupid. ...
3. ...

I don't know the origin of this phrasing, but looking at "neither ... nor ..." may help clarify.

"Neither A nor B" in logic unambiguously translates to ((not A) and (not B)).

By De Morgan's law, that expression is equivalent to (not (A or B)) so perhaps whoever established that convention thought that establishing "neither A nor B" as the logical inverse of "either A or B" would lead to the least surprise.

This differs from what I recall of my textbooks on logical circuit design.

Electrical engineers seem to use different notations for logic from formal systems people (+ and ⊕ instead of ∨ and + respectively), so the difference in interpreting "either ... or ..." may be a dialectal difference.

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I think you have confused yourself in some sense. "A or B" includes A or B or both. That needs to be clear.
Usage of 'either' as a conjunction has different meaning than 'Or' as it means 'A' or 'B' but not both.

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No. In everyday language, "A or B" usually does not include both, though it may sometimes. "Do you want tea or coffee?" implies not both - if you answer "both" you are stepping outside the norm. – Colin Fine Feb 23 '11 at 18:03

It means that it can't be both. I believe one would call it an "exclusive or" or XOR.

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What a great idea! You saved me lots of money! I have to pay income tax if I either live or work in this state... but now I know that (since I both live and work here) I do not need to pay taxes. Wow! – GEdgar Jul 11 '11 at 14:05
While the standard explanatory text may say "either live or work," I am sure your state's tax code is a little more specific in its wording so as to avoid such an intentional misreading by an antagonistic reader. – horatio Jul 12 '11 at 13:58
@GEdgar You have a great future in Accounting. Or Law. – no comprende Mar 23 at 2:18