Understanding implication in logic truth tables (excerpt from textbook)

Solution 1:

The discussion is about why the statement $\bot \to \bot$ is considered "true" rather than "false".   That is, why the truth table of the conditional connective is defined as it is.

An argument is considered valid if, it guarantees the conclusion is true when all the premises are true.

So if $\to$ is defined as it is, then the truth of both premises, $\{P\to Q, P\}$ are required to guarantee the truth of the conclusion $Q$.   (Knowing only that one is true is not enough to ensure the conclusion is true.)

$$\begin{array}{cc|cc|cc}P & Q & P\to Q & P & Q \\ \hline \bot & \bot & \top & \bot & \bot \\ \bot & \top & \top & \bot & \top \\ \top & \bot & \bot & \top & \bot \\ \top & \top & \top & \top & \top & \star \end{array}$$

However if we defined $\dot\to$ so that $\bot \dot\to \bot = \bot$ we only need the one premise $P\dot\to Q$ to guarantee the conclusion. The conclusion is always true when that premise is, whatever $P$ may be.

$$\begin{array}{cc|c|cc}P & Q & P\dot\to Q & Q \\ \hline \bot & \bot & \bot & \bot \\ \bot & \top & \top & \top & \star \\ \top & \bot & \bot & \bot \\ \top & \top & \top & \top & \star \end{array}$$