2

Peter asked:

If $a + \epsilon > b$ for each $\epsilon > 0$, can we conclude that $a > b$?

Micah commented

Be careful: [$\Large{1.}$] "for every $ϵ>0$, if $a+ϵ>b$, then $a>b$" is a different statement than
[$\Large{2.}$] "if, for every $ϵ>0, a+ϵ>b$, then $a>b$."
One of them is a lot closer to being true than the other.

Brian M. Scott expounded on why conditional statement #1 is false. But my difficulty is grislier! As English isn't my first language, I can't even distinguish between the two conditionals, which mean the same to me! What's the simplest way to distinguish between them?

As both share the same apodosis ("then $a> b$"), I only pay attention to the protasis. I spot only one difference: the placement of $\color{limegreen}{\text{"if"}}$. The protasis of 1 is

for every $ϵ>0, \color{limegreen}{\text{if }} a+ϵ>b$

The protasis of 2 is

$\color{limegreen}{\text{if}}$, for every $ϵ>0, a+ϵ>b$

2 Answers2

1

In statement 2's protasis, there is actually an if-then statement. It is "if 'if $\epsilon > 0$, then $a+\epsilon > b$'".

This is clunky to say in plain English, so people use the "for every" quantifier instead.

So statement 2 actually says:

"if

'if $\epsilon > 0$, then $a+\epsilon > b$'

,

then $a>b$".

On the other hand, statement 1 is:

"if $\epsilon > 0$, then

'if $a+\epsilon > b$, then $a>b$'

."

So which one is true?

Looking at it this way, it is clear that statement 2 is not true (because try letting $a=b=1$), but in some sense "close to being true", because changing the conclusion to $a\ge b$ makes it true.

However, statement 1 is "nowhere near being true", because try letting $\epsilon = 10$ and $a=1,b=2$.

Benjamin Wang
  • 4,667
1

Since your issue is a linguistic one, I'll write down the two statements in logical notation. Pay close attention to the placement of the brackets.

Statement 1: $$ (\forall a \in \mathbb R)\bigg[(\forall b \in \mathbb R)\bigg[(\forall \epsilon \in \mathbb R^+) \bigg[ (a + \epsilon > b) \implies (a > b) \bigg]\bigg]\bigg]$$

Statement 2: $$ (\forall a \in \mathbb R)\bigg[(\forall b \in \mathbb R)\bigg[(\forall \epsilon \in \mathbb R^+) \bigg[ (a + \epsilon > b) \bigg] \implies (a > b)\bigg]\bigg]$$

Kenny Wong
  • 33,403
  • Thanks. I see merely one difference: the placement of ]. Statement 1 ends with 3 ]'s. Statement 2 ends with 2 ]'s, because you placed one ] before $\Longrightarrow$. But how does this answer my question? I still befuddled! –  May 19 '23 at 08:03
  • @user1147844 I thought your question is about how to translate the English sentences into logical statements? – Kenny Wong May 19 '23 at 08:04
  • Yes, that's the first part. But I want to distinguish, and intuit, these logical statements too! –  May 19 '23 at 08:07