question about vacuous truth and function I'm confusing about vacuous truth. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n)=2n$. we can calculate function values if $n$ belongs to domain. but what if it does not? The value of $f(1.5)$ does not belong to $f$ since it is not true that $(1.5, y) \in f=\\{(n,2n) | n\in \mathbb{N} \\}$. then how about the truth value of $f(1.5)=3$? 1. $(1.5,3) \notin f \leftrightarrow f(1.5) \neq3$. thus $f(1.5) \neq 3$ 2. since $1.5$ cannot match with any element in range of $f$, it cannot be false $f(1.5)=3$. so vacuously $f(1.5)=3$.

It is false that $f(1.5)=3$. As you have pointed out, $(\forall y)(1.5,y)\
otin f$, so $(\forall y)y\
eq f(1.5)$. In particular $3\
eq f(1.5)$. The fact that $1.5$ is not in the domain of $f$ does not stop the proposition "$f(1.5)=3$" from being false.