In Validity you learned that an argument is valid when it is impossible for the premises to be true and the conclusion false. Truth tables turn "impossible" into something you can check: every possible situation is a row.
Key idea
Why it works
Worked examples
Worked example: A valid form
- Rows with both premises T: $\neg p$ needs $p$ F; then $p \lor q$ needs $q$ T.
- That is the row $p$ F, $q$ T, where the conclusion $q$ is T.
- No counterexample.
Answer: Valid
Worked example: An invalid form
- Rows with both premises T: $p$ T, $q$ T; and $p$ F, $q$ T.
- In the second, the conclusion $p$ is F.
Answer: Invalid; the counterexample is $p$ F, $q$ T.
Worked example: Three letters
- A counterexample needs $p \to r$ F: $p$ T, $r$ F.
- Then $p \to q$ needs $q$ T, but $q \to r$ needs $q$ F.
- No row works.
Answer: Valid
Common mistake
Remember from Validity and Truth Together that validity is not truth. A valid argument can have a false conclusion when a premise is false, and an invalid argument can have all true statements. The truth table looks at every row, the actual one and all the others, which is exactly why it tests form and not facts.
The table also tells you more than yes or no. When an argument is invalid, the counterexample row describes a concrete situation that breaks it, which is often the most convincing way to show someone the reasoning fails.
- Make a column for each premise and the conclusion.
- Mark the rows where every premise is T.
- If the conclusion is T in all of them: valid. If not: invalid, and that row is a counterexample.