Is the Converse of a false statement always false?

Answer 1

Let's take an example.

If it is raining (then) the match will be cancelled.

A conditional statement is false if and only if the antecedent (it is raining) is true and the consequent (the match will be cancelled) is false. Thus the sample statement will be false if and only if it is raining but the match still goes ahead.

By convention, if the antecedent is false (if it isn't raining) then the statement as a whole is considered true regardless of whether the match takes place or not.

To recap: if told that the sample statement is false, we can deduce two things: It is raining is a true statement, and the match will be cancelled is a false statement. Also, we know a conditional statement with a false antecedent is always true.

The converse of the statement is:

If the match is cancelled (then) it is raining.

Since we know (from the fact that the original statement is false) that the match is cancelled is false, the converse statement has a false antecedent and, by convention, such statements are always true.

Thus the converse of a false conditional statement is always true. (A single example serves to show it's true in all cases since the logic is identical no matter what specific statements you apply it to.)

If you are familiar with truth tables, the explanation is much easier. Here is the truth table for A = X->Y (i.e. A is the statement if X then Y) and B = Y->X (i.e. B is the converse statement if Y then X).

X Y A B

F F T T

F F T T

T F F T

F T T F

Looking at the last two rows of the A and B columns, when either of the statements is false, its converse is true.

Answer 2

True