I am going through a logic textbook and found the following truth table for a basic conditional statement: \begin{array}{|c|c|c|} \hline a& b & a→b \\ \hline T & T & T \\ \hline T & F & F \\ \hline F & T & T \\ \hline F & F & T \\ \hline \end{array}
This makes sense to me, however I am very confused by the 3rd row. Lets say you have a premise a which states $x>0$, and a premise b which states $x<0$. If we pick values where a is false (so for example, all negative real numbers), then we will get b as true. In reality of course, this relation overall is False by the 2nd row. But by the 3rd rows logic it is True. My question is why the 3rd row is not "indeterminate"; surely both a False and True conclusion are possible from just having information that premise a is False and premise b is True. I think the reasoning for having $a→b$ as True seems symmetrical to a possible reasoning saying $a→b$ is False, as a could represent any number of possible premises that never will map onto b regardless of if they are false or true. But the Truth table would only yield a "correct" evaluation of the truth of $a→b$ when a is true and $a→b$ is false.
