# 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.

• 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. – user126158 Mar 23 '16 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.)

• 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. – user126158 Mar 23 '16 at 2:15
• So it's either `XOR` or the other person sucks at logic. – Vlasec Aug 4 '16 at 8:51
• '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? – Edwin Ashworth Jan 14 '20 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.

• While this may be the most accurate/concise answer, the bold claim needs supporting reference/s. Linked and attributed, please. – Edwin Ashworth Jan 14 '20 at 11:15

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.

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?

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.

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)").

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

• 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. – user126158 Mar 23 '16 at 2:18

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.

• Hello, 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. – Edwin Ashworth Jan 14 '20 at 11:26
• – Hass Jan 14 '20 at 16:15
• It seems that in math it means either one or both. Is this true? – Hass Jan 14 '20 at 16:18
• Probably (see Mathwords.com, for instance), but that's off-topic on ELU. – Edwin Ashworth Jan 14 '20 at 16:47

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.

• 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