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Viliam_Bur comments on 17 Rules to Make a Definition that Avoids the 37 Ways of Words Being Wrong - Less Wrong
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Comments (23)
A prime number n is a number whose only factors are multiplicative units and n*a multiplicative unit (and these two sets are distinct). Typical examples include 2, 3, 5, 7 and 11. Less-typical examples include -2 and 1+i; they are often excluded from consideration in mathematics.
"whose only factors" -- that's where you are hiding the negation
("only" = "there is no other")
Well, there's a tricky thing in mathematics called "the law of excluded middle". Using the law, you can e.g. prove that a implies b is logically equivalent to (not a) or b. It also lets you do existence proofs by proving it isn't possible for there to be no examples. So in classical logic every statement is confused with its double negation.
I generally try to use intuitionistic logic though, where a->b is not logically equivalent to anything else and double negations have to be written out. You do have
, but that only goes one direction and results in a weaker statement. If you look at my other reply with an intuitionistic frame of mind, then you'll see that the "only" is an implication, with no negation in sight.
Not necessarily: see mathnerd314's comment below (or above). In fact, in “there is no other”, there is a double negation (the second being in “other”, which hides “not equal to”), which can be eliminated.