Attached is a PDF document I found while looking for an article on the If-then/Implication logical operator. The truth table for If/Then is on the 8th page of the PDF file.

What if it is known that the implication compound statement is False and it is known that the antecedent is False? What can one conclude (if one can at all) about the consequent since the truth table doesn't specify the assumption that the implication compound statement is false?

Couldn't we just treat the implication compound statement as a statement and apply the implication operator and calculate it according to to the implication truth table? By this I mean: Let A,B be defined mathematical statements and C be an undefined math statement.

A is False

A=>B is False

Let A=>B be C. So C is False.

So we are essentially evaluating A=>C. Both A and C are false so according to the truth table, the statement A=>C is True.