IMPLICATION; CONVERSE; CONTRAPOSITIVE

Confusing an implication with it's converse:

Take two sentences, like "I eat" and "I am hungry". One can say,

IF I am hungry THEN I eat.

But, there may be many other circumstances when one eats, not JUST when one is hungry, so, the converse, i.e.,

IF I eat THEN I am hungry

is not necessarily true. Perhaps it's not the best example, but ...

Perhaps a better example would be "IF it rains THEN I will find myself all wet" BUT, if you find yourself all wet, it doesn't necessarily mean that it rained -- you could have taken a shower, or gone swimming or something.

------------------------------

Another thing, it is always the case, however, that an implication and it's CONTRAPOSITIVE are equivalent. So,

IF I am hungry THEN I eat

is equivalent to

IF I don't eat THEN I am not hungry. In propositional logic, which is what i study, we symbolize these two sentences as

P -> Q is equivalent to -Q -> -P

[where '-' means NOT and '->' means 'if ... then']

Also, in logic, we say that in the implication 'IF P THEN Q', P is a SUFFICIENT (but not NECESSARY) condition for Q, while Q is a NECESSARY (but not SUFFICIENT) condition for P.

I hope I'm not flogging a dead horse here, but, my favorite way of visualizing an implication is with very simple Venn diagrams. In the implication P->Q, visualize P as a circle, and Q as a bigger circle with P inside it. That is a way to picture P->Q, because if P occurs (i.e. if you're inside circle P), then Q occurs (you are inside circle Q also), but it's dangerous to assume that if Q occurs, P occurs, because the Q circle is bigger, and can occur under different circumstances, i.e., outside the circle P.