An argument is a list of premises followed by a conclusion. Logic cannot tell you whether the premises are true, but it can tell you whether the conclusion follows from them. That test uses nothing more than the truth tables and equivalences you have already built.
Key idea
An argument is valid when, in every row of the truth table where all the premises are true, the conclusion is true too. One row with true premises and a false conclusion (a counterexample row) makes it invalid. Valid forms include modus ponens ($P \to Q$, $P$, so $Q$) and modus tollens ($P \to Q$, $\neg Q$, so $\neg P$).
Why it works
Validity is about form, not content. "If it snows, school is closed; it snows; so school is closed" and "If $n$ is divisible by $6$, it is even; $n$ is divisible by $6$; so $n$ is even" have the same shape, $P \to Q$, $P$, therefore $Q$. Checking the shape once with a truth table settles every argument of that shape. Modus tollens works because $P \to Q$ is equivalent to its contrapositive $\neg Q \to \neg P$, so it is modus ponens in disguise.
| Name | Form | Valid? |
|---|---|---|
| Modus ponens | $P \to Q$, $P$, therefore $Q$ | yes |
| Modus tollens | $P \to Q$, $\neg Q$, therefore $\neg P$ | yes |
| Hypothetical syllogism | $P \to Q$, $Q \to R$, therefore $P \to R$ | yes |
| Disjunctive syllogism | $P \vee Q$, $\neg P$, therefore $Q$ | yes |
| Affirming the consequent | $P \to Q$, $Q$, therefore $P$ | no |
| Denying the antecedent | $P \to Q$, $\neg P$, therefore $\neg Q$ | no |
Worked examples
Worked example: Spotting the form
"If the battery is dead, the car will not start. The car will not start. Therefore the battery is dead." Valid?
- Let $P$: "the battery is dead," $Q$: "the car will not start." The argument is $P \to Q$, $Q$, therefore $P$.
- That is affirming the consequent. In the row $P$ false, $Q$ true, both premises are true and the conclusion is false.
- In real terms: the car might not start because it is out of gas. Invalid.
Answer: Invalid (affirming the consequent)
Worked example: Chaining rules
Premises: $P \to Q$, $Q \to R$, $\neg R$. What can you conclude about $P$?
- From $Q \to R$ and $\neg R$, modus tollens gives $\neg Q$.
- From $P \to Q$ and $\neg Q$, modus tollens again gives $\neg P$.
- So $\neg P$ follows. A truth table with $8$ rows confirms it: the only row with all three premises true has $P$ false.
Answer: $\neg P$
Common mistake
Affirming the consequent. "If $P$, then $Q$; $Q$ is true; so $P$ is true" sounds reasonable, but $Q$ can be true for other reasons. Check: look for a row with $P$ false and $Q$ true. Both premises hold there and the conclusion fails, so the form is invalid.
A valid argument can have a false conclusion if a premise is false. An argument that is valid and has true premises is called sound, and its conclusion must be true.