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.
"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/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)
- 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.
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)").