So I have to show: is complete has a unique satisfying truth assignment. So my proof has to go in both directions...

To show the " ", we can try proving its contrapositive:

"If is not complete then it has no unique truth assignment."

If there is a truth assignment then for an arbitrary A the following is one of the options that must be true:

then .

And in order for that to be true we need to be complete (because if we have , then so so ).

Suppose is not complete, then if then . So there's no truth assignment. Is this right?

To show the " ", similarly I'll try proving its contrapositive:

"If doesn't have a unique satisfying truth assignment, then it is not complete."

Suppose doesn't have a unique satisfying truth assignment. Then how could we prove that is therefore not complete? Here's where I'm stuck...

P.S: By the truth assignment I meant: