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.
asked Feb 23, 2011 at 17:01
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3just 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– user66911Commented Feb 24, 2014 at 8:05
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9Outside 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 LicksCommented May 13, 2015 at 17:31
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2@HotLicks +1 for using 'exclusivity' and 'lust' in one comment, about something completely unrelated.– user126158Commented Mar 23, 2016 at 2:12
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9 Answers
"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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12If 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.– psmearsCommented Feb 23, 2011 at 19:49
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11@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
xorquestions 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.– PacerierCommented Sep 24, 2014 at 2:20 -
Short answer: whenever someone uses OR in a question, just answer Yes. Make them re-clarify their intent.– user126158Commented Mar 23, 2016 at 2:15
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3'You can contact us either by phone or by email' is obviously totally idiomatic, and illustrates a grey area. Obviously the two different methods remain freely available – and a group or skilled individual could use both at once without contravening the instruction. // Have you supporting references? Commented Jan 14, 2020 at 11:38
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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1While this may be the most accurate/concise answer, the bold claim needs supporting reference/s. Linked and attributed, please. Commented Jan 14, 2020 at 11:15
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).
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?
- ...
- ¬S ∧ (L ∨ S)
- ...
Solutions
- ...
- John isn't stupid, and either he's lazy or he's stupid. ...
- ...
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.
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.
Sometimes even 'and' is used in natural language as logical OR: "You can have coffee and cake" may not mean that you can only have both and not one of them. Or "you can have chocolate spread and Gouda cheese on your sandwich".
Often, in natural language writing (especially in a formal setting, such as technical or business documentation) "and/or" is used to denote the logical OR and to prevent the confusion about the meaning of 'or'. We use 'and/or' because 'or' tends to suggest logical XOR more than logical OR.
"Either A or B" does not absolutely preclude "A and B", but the general usage and meaning tends to prefer the XOR ("not (A and B)").
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'And' and 'or' are never actually used interchangeably: "You can have coffee and cake" is short for "You can have coffee and you can have cake", even as it can be translated as "You can have either coffee or cake."– ryangCommented Jun 2 at 4:26
According to Cambridge Advanced Learner's Dictionary (CALD), it states that "either-or describes a situation in which there is a choice between two different plans of action, but both together are not possible.
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1Hello, Hassine. Your 'answer' is better than most of the above, as it gives a supporting reference. But nowadays, actual quotes, with attributed links, are just about mandatory on ELU. // You've prompted me to look for supporting references for user2683's 'answer'; I think there is never going to be agreement between the literalist and the pragmatist answers. Commented Jan 14, 2020 at 11:26
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It seems that in math it means either one or both. Is this true?– HassCommented Jan 14, 2020 at 16:18
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Probably (see Mathwords.com, for instance), but that's off-topic on ELU. Commented Jan 14, 2020 at 16:47
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1No. This definition in CALD is given for the compound adjective either-or. 'It's an either-or situation.' Commented May 25, 2023 at 15:48
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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7No. 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. Commented Feb 23, 2011 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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7What 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!– GEdgarCommented Jul 11, 2011 at 14:05
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3While 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.– horatioCommented Jul 12, 2011 at 13:58
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@GEdgar You have a great future in Accounting. Or Law.– user126158Commented Mar 23, 2016 at 2:18