Turning a sentence into formal logic

Question

Before anyone marks this as duplicate, I note that a similar question has been brought up before, but I would like to point out that I am covering new ground here.

Part I

Consider the statement: "There is no largest natural number.".

In formal logic, I can think of two ways to represent this.

1. $(\forall x \in \mathbb{N})(\exists y \in \mathbb{N})[y>x]$

2. $(\neg \exists x \in \mathbb{N})(\forall y \in \mathbb{N})[y\leq x]$

Some people felt that the second representation is a better fit for the statement than the first. May I ask, those who are of this view, why? Perhaps, in formal logic writing, is there an unspoken rule for what kind of quantifiers should come before others? For example, should $\forall$ come before $\exists$, or otherwise? Does either make the logic "flow" better?

Part II

Consider the statement: "There is a largest natural number.".

3. $(\exists x \in \mathbb{N})(\forall y \in \mathbb{N})[y\leq x]$

4. $(\exists x \in \mathbb{N})(\forall y \in \mathbb{N})[y\leq x]$

Now this is where things start to get really mind-boggling for me. It is trivial to note that the statement in Part II is the negation of the statement in Part I. This would mean that we need only negate 1. and 2. to get 3. and 4. respectively, where 3. and 4. should accurately represent the statement in Part II.

I may have negated 1. and 2. incorrectly, but assuming I did not, negating 1. and 2. actually results in the same statement. Then, since they both negate to give the exact same statement (phrasing), why should either one fit better for the sentence in Part I? Or rather, how is it possible that two sentences, with different phrasings, although same meaning, when negated, can give the same exact sentence? Should they not give two negations with different phrasings but the same meaning as well?

Conclusion

I would like to state that I am new to this area of Mathematics and this topic really intrigues me, so I am trying to find out a little more from all of you bright minds on stackexchange!

I would also like to take this opportunity to seek out any other ways to represent the above statement and its negation, other than those I have mentioned, if any!

Answer 1

1 and 2 are not logically equivalent, since purely logically we are dealing with two different predicate symbols $>$ and $\leq$, and hence the negation of $y>x$ is not the same as $y \leq x$.

As such, the logical negation of $2$ is indeed $4$, but $3$ is not the logical negation of $1$. Instead, that would be:

$(\exists x \in \mathbb{N})(\forall y \in \mathbb{N})[\neg y>x]$

But, within the context of arithmetic (or, if you want, once we provide the standard axioms for $\leq$ and $>$), the statements do end up saying the same thing, and so we can say they are arithmetically equivalent, and so from that point of view, both are perfectly acceptable ways to symbolize the statement.

Still, should one be preferred over the other? Does one translate the statement more 'directly'? Here, I can see that someone might argue that 'There is no ...' translates as 'there is not a ..', and thus as $\neg \exists x...$, so I could see that fom that perspective someone might prefer $2$ over $1$

I have never seen anyone arguing that a certain order of quantifiers is to be preferred over the other though, so I don't think that is the reason why some people prefer $2$ over $1$

Answer 2

First, your expression of (2) is just a bit odd. The more usual form would be: $$\neg(\exists x\in\mathbb{N})(\forall y\in\mathbb{N})[y\leq x]$$(1) arises from (2) by pushing the negation through the quantifications according to the usual rules (plus a special property of $>$). If you want the parallel negated statements, then, you should first put a $\neg$ in front of (1), and then push it through. So we get:

1. $\neg (\forall x\in\mathbb{N})(\exists y\in\mathbb{N})[y> x]$
2. $(\exists x\in\mathbb{N})(\forall y\in\mathbb{N})[y\leq x]$

These are equivalent, but look different, just like your (1) and (2).