What does "either A or B" mean precisely, and how can I prove it? [duplicate]
"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.)
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.
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.