Why is the condition of an propsitional consequence that the implication is a tautology?

Question

Why is the condition that $Q$ is a propsitional consequence of $P$ that $P \rightarrow Q$ is a tautology reasonable?

More generally, why are tauologies the way we conclude things in propositional calculus?

Isnt a valid implication supposed to be false in the case $T \rightarrow F$?

Answer 1

In response to your first question (I'm not sure I understand the other two), assume $Q$ is a tautological consequence of $P$. If $P \to Q$ were not a tautology, then it would be false under some truth value assignment. Such an assignment must make $P$ true and $Q$ false. But the existence of such an assignment contradicts our assumption, since if $Q$ is a consequence of $P$, then $Q$ is true under any assignment that makes $P$ true. Thus, $P \to Q$ is a tautology.

Conversely, assume that $P \to Q$ is a tautology. If $P$ is true under an arbitrary truth value assignment, then so too must $Q$ be. For if $Q$ were false, then we'd have a truth value assignment that makes $P \to Q$ false, contradicting our assumption that $P \to Q$ is a tautology.

We've shown

$P \to Q$ is a tautology if and only if $Q$ is a tautological consequence of $P$.