Is "$1$ can be irrational." true? The statement

$1$ can be irrational.

means that $1$ is either rational, or irrational. So that statement should be true. The negation of that statement is

$1$ cannot be irrational.

But that should also be true, because $1$ is a natural number, hence $1$ cannot be irrational. I'm very confused.
 A: If we accept the interpretation of "$x$ can have property $P$" as "$x$ either has property $P$ or doesn't have property $P$," and the interpretation of "$x$ cannot have property $P$" as "it is not the case that $x$ can have property $P$," then $(1)$ is true and $(2)$ is false - for silly reasons.
However, this is sort of missing the real issue: the premise of the question - that "$1$ can(not) be irrational" can be faithfully translated into classical logic in a straightforward way - is incorrect.

This is a situation where the simplest logical framework - straightforward classical logic - does a bad job of implementing natural reasoning. The issue is the idea of possibility implicit in the word "can(not):" there's a sense of quantifying over "possible worlds" here, which is hard to do in standard classical logic. And we can tell that there's something wrong with ignoring this feature: if we interpret "$x$ can have property $P$" as "either $x$ has property $P$ or $x$ doesn't have property $P$," then that's always trivially true - so this is good evidence that that does a terrible job of accurately interpreting the word "can." (And it gets even worse when we think about "cannot.")
A better framework for dealing with this sort of assertion is provided by modal logic, or more generally the possible world semantices (not a typo: I mean the plural of "semantics") and their relatives. According to this approach, we roughly have the following situation: the sentence

"$1$ can be irrational"

would mean

"There is some possible world in which $1$ is irrational,"

whereas

"$1$ cannot be irrational"

would mean

"In every possible world, $1$ is rational"

or if you prefer,

"There is no possible world in which $1$ is irrational."

(I said "roughly" above because there's some nuance to the notion of possibility here: we really have a notion of relative possibility, or accessibility, at play which I'm ignoring here.)
Of course as you probably suspect the devil is in the details, but this is ultimately an approach which makes sense and does a much better job of capturing how expressions like the above are treated in natural language.
