Some argument patterns are so common that they have names. You met them in words in The Two Conditional Fallacies; now truth tables (Validity by Truth Table) prove which are valid and which are not.
| Name | Form | Valid? |
|---|---|---|
| Modus ponens | $p \to q$, $p$, so $q$ | valid |
| Modus tollens | $p \to q$, $\neg q$, so $\neg p$ | valid |
| Hypothetical syllogism | $p \to q$, $q \to r$, so $p \to r$ | valid |
| Disjunctive syllogism | $p \lor q$, $\neg p$, so $q$ | valid |
| Affirming the consequent | $p \to q$, $q$, so $p$ | invalid |
| Denying the antecedent | $p \to q$, $\neg p$, so $\neg q$ | invalid |
Key idea
Why it works
Worked examples
Worked example: Name the form
- The second premise denies the consequent.
- The conclusion denies the antecedent.
Answer: Modus tollens (valid).
Worked example: A look-alike
- The second premise affirms the consequent, not the antecedent.
- You might have passed for another reason.
Answer: Affirming the consequent (invalid).
Worked example: Bigger parts
- Let $p$ be $r \land s$ and $q$ be $\neg t$.
- The second premise $\neg \neg t$ is $\neg q$.
- The conclusion is $\neg p$.
Answer: Modus tollens, with formulas in place of the letters.
Worked example: Disjunctive syllogism in words
- The first premise is a disjunction.
- The second denies one disjunct; the conclusion is the other.
Answer: Disjunctive syllogism (valid).
Each named form covers infinitely many arguments, because the letters can stand for any statements, simple or compound. Spotting the form under bigger parts, as in the third example, is the skill that natural deduction will use on every line.
Common mistake