How are these two statements logically equivalent?

Question

I am confused as to how the following two statements are logically equivalent:

$\forall a > 0, \,y < a $ _____(1)

and

$\forall a > 0, \,y \leq a $ _____(2)

Correct me if I'm wrong, but from what I understand,

(1) $\iff$ y $\leq 0$

And,

(2) $\iff$ y $\leq 0$ or y = a.

Could somebody help me understand how (1) $\iff$ (2)?

Thanks!

Answer 1

It wholly depends on the domain of discourse.

The statements are equivalent in the domain of real numbers (or the rationals or similar), because such a domain has no minimum positive value.

1. $\forall a{\in}\Bbb R~(a>0\to y<a)$

   • The constant $y$ is strictly less than any positive value
   • The constant $y$ is non-positive.

2. $\forall a{\in}\Bbb R~(a>0\to y\leqslant a)$

   • The constant $y$ is strictly not-greater than any positive value.
   • The constant $y$ is non-positive.

We can see that if $y$ is non-positive, both statements clearly hold true.   Now consider that if there were any positive real value equal to $y$ (that is, if $y$ were a positive real value) then there would exist a positive real values less than it; for example: $y/2$.   This would falsify both statements.

So for any real number $y$, both statements will have the same truth value.

---

However this is not true of some other domains: for example, the integers, where there is a minimum positive value.   For the integers, that is $1$.   Witness: $\forall a{\in}\Bbb Z~(a>0\to 1<a)$ is clearly not equivalent to $\forall a{\in}\Bbb Z ~(a>0\to 1\leq a)$.

Answer 2

You can't say $y=a$ is a possibility for (2) because you are quantifying over all possible $a$, so there's no specific $a$ for $y$ to equal.

The two statements aren't logically equivalent (some mathematical properties are required to show any equivalence), but they are both equivalent to $y \leq 0$ if you are quantifying over real numbers. To see why, suppose $y>0$. Clearly (1) cannot hold. For (2), take $a=\frac{y}{2}$.