The eight rules of inference are not enough on their own. From $\neg (p \land q)$ and $p$ you should be able to get $\neg q$, but no rule of inference fits: the first premise is a negation, not a disjunction. What you need is a way to rewrite $\neg (p \land q)$ as the equivalent $\neg p \lor \neg q$ (De Morgan's Laws). That is what rules of replacement do.
Key idea
Why it works
| Rules of inference | Rules of replacement | |
|---|---|---|
| Examples | MP, MT, HS, DS, CD, Simp, Conj, Add | DM, Com, Assoc, Dist, DN, Trans, Impl, Equiv, Exp, Taut |
| Based on | valid argument forms | logical equivalences |
| Apply to | whole lines only | any part of a line |
| Direction | one way | both ways |
Worked examples
Worked example: Inside a line
- DM is a rule of replacement.
- It replaces the antecedent with an equivalent formula.
Answer: Yes. Replacement works on parts of lines.
Worked example: Not allowed
- Simp is a rule of inference, so it needs a whole line that is a conjunction.
- Here the conjunction is only the antecedent.
- In fact the result does not follow: $(p \land q) \to r$ can be true while $p \to r$ is false (try $p$ T, $q$ F, $r$ F).
Answer: No.
Worked example: Either direction
- DN says $p$ :: $\neg \neg p$.
- Replacement may go from right to left or left to right.
Answer: Yes, both ways.
Common mistake
The next four lessons give the ten replacement rules in small groups. With all eighteen rules together, the system can prove every valid argument of statement logic, a fact you will meet again in Soundness and Completeness.