Using Qualifiers to Describe Multiples

Question

I was recently solving some questions about quantifiers when I came across an example I didn't quite understand. The universe of discourse is the set of natural numbers $\{1,2,3,\dots\}$ (i.e. all variables represent natural numbers). The statement being converted to logic format was "Every multiple of $4$ is a multiple of $2$", which was then represented with the following symbolization: $$∀n((∀m\space n≠4m) ∨ (∃r\space n = 2r)).$$ When I read the symbolization, I translate it as follows: "For all numbers $n$, $n$ is either not a multiple of $4$ * all values of $m$, or there exists an $r$ such that $n$ is a multiple of $2r$". Is this an accurate translation and if not, what's a better way to word it? Also, is the symbolization truly the best way to represent the concept of "Every multiple of 4 is a multiple of 2"?

Answer 1

The most natural translation of the original statement into logical symbols is

$$\forall n\big(\exists m\,(n=4m)\to\exists r\,(n=2r)\big)\,.\tag{1}$$

What’s inside the scope of $\forall n$ has the form $p\to q$, which is equivalent to $\neg p\lor q$, so $(1)$ is equivalent to

$$\forall n\big(\forall m\,(n\ne 4m)\lor\exists r\,(n=2r)\big)\,.$$

I would translate this back into English as ‘for all $n$, $n$ is not a multiple of $4$, or $n$ is a multiple of $2$’ — not a multiple of $2r$. One could also stay reasonably close to the form of the symbolic expression with ‘for all $n$, $n$ is not a multiple of $4$, or $n$ is even’.