Tautology, Entailment and Consistency

What the Table Was For

Truth tables are a procedure; the semantic relations they compute are what logic actually cares about.

Logical truth is the base case. A tautology is true under every assignment, a contradiction under none, and a contingency under some but not all. Only contingent sentences carry information about the world, a tautology rules nothing out, which is why "it will either rain or not" is useless as a forecast.

Entailment is the central relation. A set of premises entails a conclusion when every assignment making all premises true also makes the conclusion true, written $\Gamma \models \varphi$. This is validity stated semantically, and it connects to the conditional through a result worth knowing: $\Gamma \models \varphi$ exactly when the conditional from the conjoined premises to $\varphi$ is a tautology. Argument-checking and sentence-checking are the same task in different clothes.

Consistency is the third relation. A set of sentences is consistent when some assignment makes them all true simultaneously. Inconsistency is a serious defect, because an inconsistent set entails everything: if the premises cannot all be true, no assignment makes them true with a false conclusion, so vacuously every conclusion follows. This is the principle of explosion, and it is why a contradiction in a theory is fatal rather than merely awkward.

Two relations that are easy to confuse:

Equivalence is entailment in both directions. Entailment alone is one-way: $P \wedge Q$ entails $P$, but $P$ does not entail $P \wedge Q$.

The final piece is that an argument is valid if and only if its premises are inconsistent with the negation of its conclusion. This equivalence is what licenses proof by contradiction: to establish a conclusion, assume its denial and derive an inconsistency.

Common pitfall: treating an inconsistent set of premises as simply useless. It is worse than useless, it entails every sentence whatsoever, including the negation of anything you wanted to prove. A theory that has become inconsistent has not merely lost some conclusions; it has lost the ability to rule anything out.