2

I am studying the book 'How to Prove It: A Structured Approach by Daniel J. Velleman'

Now according to the textbook:

In mathematics, or always means inclusive or, unless specified otherwise, so we will interpret ∨ as inclusive or.

That means that we have a general interpretation of 'or'. My question is do we have a general interpretation of 'either...or' as well? If yes, then what is it?

In the book 'either...or' is interpreted as inclusive or. For example the statement

Either John went to the store, or we’re out of eggs.

is interpreted as

If we let $P$ stand for the statement “John went to the store” and $Q$ stand for “We’re out of eggs,” then this statement could be represented symbolically as $P \lor Q$.

I found this on Wikipedia (go to 'Exclusive "or" in natural language'), which may be useful.

Navneet
  • 517

1 Answers1

6

In mathematical writing, "either...or" is used with the same meaning as "or", namely it is inclusive. The word "either" is just used to make the English flow naturally, and does not have any mathematical content.

Eric Wofsey
  • 342,377
  • One professor taught me that "or" was inclusive while "either…or" was exclusive. At that time that felt like a really frustrating constraint on the prose of my proofs, but I ended up internalizing it. Do you have any examples of the inclusive "either…or" in established mathematical literature? An inclusive "either…or" certainly feels more natural, but I want to be sure. – Foobie Bletch Sep 20 '21 at 15:28
  • 2
    For instance, on page 8 of https://pi.math.cornell.edu/~hatcher/AT/AT.pdf (a graduate-level textbook) where it says "either $X$ or $Y$ has finitely many cells" that is clearly inclusive. (I found that by just grabbing the first mathematical text I had on hand in electronic format and searching it for "either".) – Eric Wofsey Sep 20 '21 at 15:40