Validity & Counterexamples

An argument is a set of premises offered in support of a conclusion. The central notion of logic is validity: an argument is valid when there is no assignment that makes every premise true and the conclusion false. Equivalently, if the premises are all true, the conclusion must be true — validity is about the form, not about whether the premises actually hold.

Enter premises and a conclusion below. The tool checks every row; a row that makes all premises true but the conclusion false is a counterexample, highlighted in red.

Reading the verdict

• Valid: no counterexample row exists. We write $P_1, \dots, P_n \models C$ (the premises semantically entail $C$).
• Invalid: at least one counterexample row exists — that row is the proof of invalidity.

Two arguments worth comparing in the tool:

• Modus ponens — premises $P \rightarrow Q$ and $P$, conclusion $Q$ — is valid.
• Affirming the consequent — premises $P \rightarrow Q$ and $Q$, conclusion $P$ — is invalid: the row $P$ false, $Q$ true makes both premises true and the conclusion false.

Validity vs. soundness

Validity says nothing about whether the premises are actually true. An argument is sound when it is valid and all its premises are in fact true — only a sound argument’s conclusion is guaranteed true in the real world. Logic certifies validity; checking the premises is someone else’s job.

The link to tautologies

There is a tight connection between validity and the conditional: the argument with premises $P_1, \dots, P_n$ and conclusion $C$ is valid iff the single formula $(P_1 \land \dots \land P_n) \rightarrow C$ is a tautology. So validity testing and tautology testing are two views of the same thing — both are settled by the truth table.

Takeaways

• Valid = no assignment makes all premises true and the conclusion false ($\models$).
• A single counterexample row proves invalidity.
• Sound = valid and premises actually true.
• An argument is valid iff (conjunction of premises) $\rightarrow$ conclusion is a tautology.

References

• Paul Teller, A Modern Formal Logic Primer, Prentice Hall (1989) — free at tellerprimer.ucdavis.edu. Curriculum follows the primer; explanations and examples here are original.