Can we determine the truth value of the statement "if A then B" knowing only that B is definitely false whenever A is true, for example, "If you live in Paris, then you live in London."
From what I understand about truth tables, this on the face of it is indetermined because it could be true or false based on whether you live in Paris and not in London, or you don't live in Paris.
But at the same time, the fact that we know that whenever A is true B has to be false, seems to go against the purpose of implication, that whenever A is True B has to be True, and from that alone, the conditional statement can't be anything other than false?
The book I'm reading says that the statement in the above example is false.
Best Answer
Your understanding is quite correct, and the book is wrong. You can’t determine the truth value of $A \Rightarrow B$ just by knowing $A \Rightarrow \neg B$. The key here is to remember than in mathematical logic an implication is also True whenever its antecedent (i.e “$A$”) is False.
If $A$ is true, you can indeed conclude $(A \Rightarrow B)$ is false (I’ll let you work it out, you don’t even need truth tables). If $A$ is false however, both statements are “vacuously” true, as is any statement of the form $A \Rightarrow \text{Stuff}$.
If you are not satisfied just by the abstract reasoning, you can easily verify by using the truth table for implication.