Logical implication I'm stuck with a logic problem like this

I eat ice cream if I am sad.
I am not sad.
Therefore I am not eating ice cream.

Is this conclusion logical? The first sentence can be understood both like "ice cream $\implies$ sad" and vice versa. He stated that he is not sad but does that not mean that he is not eating ice cream ? I'm confused.
 A: In logic, there is a difference between implication ($a \implies b$) and equivalence ($a \iff b$). The word "if" (without "only") usually means the first one:
"I am sad" $\implies$ "I eat ice cream"
If "I am sad" is false, we cannot logically say anything about "I eat ice cream". It may be true or false. This makes sense, because being sad is not the only reason for eating ice cream.
See also: Denying the antecedent
A: $P\implies Q$ does not necessarily mean $Q \implies P$
But what does this mean exactly?
Well think of it like this.

All Carrots are vegetables

but this doesn't mean that

All vegetables are Carrots

A: Argument:

"I eat ice cream if I am sad. I am not sad.Therefore I am not eating ice cream."

Whenever you are stuck in an argument, try represent it in proper symbolism:
Glossary:
$E \equiv $ 'I eat ice cream'
$S \equiv $ 'I am sad'
Then we can represent your above argument like this:

  
*
  
*$S \to E$
  
*$\neg S$
  
  
  $\therefore \neg E$

Now is this a valid argument? (Recall the definition of a valid argument!) For instance, Let the statements (1) and (2) be true. Can $\neg E$ be false?
