Lucia:

> To my mind, these two statements are the same:
> 1. just because you can't prove something that "doesn't mean it might not
> exist"
> 2. just because you can't prove something "doesn't mean it doesn't exist"


Statement 2. means
"Given any statement S about existence,
  (that you can't prove S) doesn't imply (S is false)"

Statement 2 is obviously correct: there are many statements which are
either unprovable, or that you are not clever enough to prove, or
whose proofs would be so long that it would take millions of years to
write them down, but which are nevertheless true.

Statement 1 can be rephrased as
"Given any statement S about existence,
  (that you can't prove S) doesn't imply (S might be false)"

Mathematically, this doesn't make sense. You will never see any
theorem like "the equation cos(2x)=2cos^2(x)-1 might be correct"!
The given statement S is either true or false, but the phrase "S might
be false" is as meaningless as "the equation cos(2x)=2cos^2(x)-1 might
be correct".

Of course, in Lucia's statement 1, the word "might" in the English
language is there to express uncertainty, but really means "does";
that is, she really meant

"It might be that there are things you can't prove the existence of,
but which actually exist".

Or perhaps she meant

"just because you can't prove something that doesn't mean it cannot
exist", but she used "might" instead of "can" because when we can't
prove something, we are uncertain about its existence; I don't know, this
is really a problem about English grammar, not about logic.

--Jean-Louis Tu <[log in to unmask]>