A full table for four letters has 16 rows, and for five letters 32. The short truth table method skips most of them. It works backward from the one thing a counterexample must have, as defined in Validity by Truth Table.
Key idea
Assume every premise true and the conclusion false. Write down the values those assumptions force, one at a time. If you reach a letter that must be both T and F, no counterexample exists and the argument is valid. If you can finish a consistent assignment, it is a counterexample and the argument is invalid.
Why it works
A counterexample is a row with true premises and a false conclusion. Instead of searching all rows for one, we suppose one exists and see what it would have to look like. Each connective forces values: a false conditional forces T → F, a false disjunction forces both parts F, a true conjunction forces both parts T. If the forced values clash, the supposed row cannot exist.
Worked examples
Worked example: Valid by contradiction
Premises $p \to q$, $\neg q$. Conclusion $\neg p$.
- Make the conclusion F: $\neg p$ F, so $p$ is T.
- Premise 2 T: $\neg q$ T, so $q$ is F.
- Premise 1: $p \to q$ is T → F, which is F. But it must be T.
Answer: Clash, so no counterexample: valid.
Worked example: Invalid: a counterexample
Premises $p \to q$, $q \to r$. Conclusion $r \to p$.
- Conclusion F: $r$ T, $p$ F.
- Premise 1 $p \to q$ is T because $p$ is F, whatever $q$ is.
- Premise 2 $q \to r$ is T because $r$ is T.
- Pick $q$ either way, say T.
Answer: Invalid: $p$ F, $q$ T, $r$ T is a counterexample.
Worked example: When nothing is forced
Premises $p \lor q$, $p \to r$. Conclusion $r$.
- Conclusion F: $r$ F.
- Premise 2 T with $r$ F forces $p$ F.
- Premise 1 T with $p$ F forces $q$ T.
- Every premise is T and the conclusion F.
Answer: Invalid: $p$ F, $q$ T, $r$ F.
- A false conditional: antecedent T, consequent F.
- A true conjunction: both parts T.
- A false disjunction: both parts F.
- A true or false negation: its part has the opposite value.
- A biconditional forces nothing by itself until one side is known.
Common mistake
Stopping at the first forced value, or calling the argument invalid when the clash came from your own slip. Every value you write must be forced by an assumption. If at some point nothing is forced (as with $q$ in the second example), try the values one at a time: the argument is valid only if every try ends in a clash.