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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 $T \rightarrow F$?

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    "why are tauologies the way we conclude things in propositional calculus?" Prop calculus is *sound* that means: if we derive a formula $\varphi$ in the calculus (in symbols $\vdash \varphi$) then $\varphi$ is a *tautology* (in symbols: $\vDash \varphi$). – 2017-02-22
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    What do you mean with "valid implication" ? Bu truth table, the *conditional* $T \to F$ evaluates to $F$. – 2017-02-22
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    @MauroALLEGRANZA I think I am confusing the the conditional with how to "prove" the conditional..sorry – 2017-02-22

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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 $P$.