Two rules of replacement deal with conditionals. Transposition (Trans) turns a conditional into its contrapositive, which you know is equivalent from Converse, Inverse and Contrapositive. Material implication (Impl) turns a conditional into a disjunction. They extend Commutation, Association and Distribution (Com, Assoc, Dist).
| Rule | Form | Form |
|---|---|---|
| Trans | $p \to q$ | $\neg q \to \neg p$ |
| Impl | $p \to q$ | $\neg p \lor q$ |
Key idea
Why it works
Worked examples
Worked example: Transposition
- Swap the parts and negate both: $\neg \neg s \to \neg r$.
- DN may then simplify $\neg \neg s$ to $s$ in another step.
Answer: $\neg \neg s \to \neg r$
Worked example: Material implication
- Read Impl right to left: $\neg p \lor q$ :: $p \to q$.
Answer: $p \to q$
Worked example: Impl in a proof
- Line 3: $q \to r$ by Impl 2 (since $\neg q \lor r$ :: $q \to r$).
- Line 4: $p \to r$ by HS 1, 3.
Answer: Two lines.
Common mistake
Worked example: Trans in a proof
- Careful: HS needs the consequent of one conditional to be the antecedent of the other.
- Trans turns $p \to q$ into $\neg q \to \neg p$, which chains with nothing here.
- Instead, Trans on $\neg q \to r$ gives $\neg r \to \neg \neg q$; that does not help either.
Answer: This argument is actually invalid (try $p$ F, $q$ T, $r$ F). Not every pair of conditionals chains.
Worked example: Trans that works
- Line 3: $\neg q \to \neg p$ by Trans 1.
- Line 4: $\neg q \to r$ by HS 3, 2.
Answer: Trans made the consequent $\neg p$ match the antecedent of premise 2.
The lesson of that example: before reshaping, check that the argument is valid. A short truth table takes a minute and can save a long search for a proof that does not exist.
Impl also explains why $p \lor q$ is equivalent to $\neg p \to q$: replace $p$ by $\neg p$ in the rule, giving $\neg p \to q$ :: $\neg \neg p \lor q$, and DN finishes the job. Translating "unless" as $\lor$ in Translating Conditionals rested on exactly this equivalence.