Propositional logic meets diplomacy at the workplace This is just a funny little incident that made me think. Please don't take it too seriously or the wrong way. I would still like to hear your opinion on it, though.

I wrote an email to a colleague the other day, concerning some other colleagues who have certain data but are not very keen to share them. I wrote to her: 


  If they don't give us those data, we can't calculate, etc.


She forwarded my email to one of them, but, first, very cleverly, she changed my sentence to 


  If they give us those data, we can calculate, etc.


to make it sound more positive. At first, I thought "this means the same but it's nicer', but then I wondered if, strictly logically, the two sentences were really the same.

Let:


$a$ = "they give us the data"
$b$ = "we can calculate"


I first checked if the original statement implied the modified one.

$$(\bar a \to \bar b) \to (a \to b)$$

This worked out to:

$$\bar a \lor b$$

i.e.

$$a \to b$$

So it looks like the original statement implies the modified one only when the modified one is true. This is already very puzzling to me.

Then I tried this:

$$(\bar a \to \bar b) \leftrightarrow (a \to b)$$

which worked out to:

$$(\bar a \land \bar b) \lor (a \land b)$$

i.e., the original and modified statement are equivalent if $a$ and $b$ are both true or both false. And, apparently, this is the same as:

$$(a \lor \bar b) \land (\bar a \lor b)$$

i.e.:

$$(\bar a \to \bar b) \land (a \to b)$$

Does any of the above make sense?

And if so, is there any way to explain with an example that the original and modified statement don't mean the same?

Thanks!
"If they give us the data ... and we are still alive, and the data is legible, and ....  then we can calculate"

The point is:  your original statement made it clear that having the data is a necessary condition for being able to calculate ... but that is not the same as saying that having the data will be sufficient to be able to calculate the data, which is what your colleague's statement says.
$\neg A\implies \neg B$ is equivalent to $B\implies A$, not $A\implies B$ so you are correct in saying they are not logically equivalent.

In your case they are equivalent because you really have $A\iff B$, i.e. We can do the calculation if and only if we get the data.