"Another true statement about real numbers is the following: If $\displaystyle x^2<0$, then $\displaystyle x=23$."
How is this a true statement?
Let p= x^2<0 ,and q= (x=23) and by the definition of a conditional statement if a true p, implies a false q, then the statement p----->q is false. IN all the other cases p----->q is true.
Since in our case p is false irrespectively of what q is ( x could be 23 or not) p---->q is true.
The above is a semantical definition of the conditional statement:
...................p------>q................................................ .........................................
The statement says that for any real x such that $\displaystyle x^2$ is negative, x is 23. That is false if and only if there is some x so that $\displaystyle x^2<0$ but x is not 23. There is no such x so the statement is true (vacuously).
There is a free download of a book here, which walks through additional similar examples in detail (see the first part on logic and sets):
Bobo Strategy - Topology