Rule of inference
A rule of inference is a known valid argument. In logical proofs, rules of inference can be used as stepping stones for reaching the conclusion of some larger argument.
Rules of inference differ from logic laws in that you can only use a rule of inference when its hypotheses are true, which provides you with a true conclusion. On the other hand, logic laws allow propositions to be rewritten regardless of their truth values. The direction that a rule of inference can take you is therefore strictly from the hypotheses to the conclusion. Here's some rules of inference:
| Modus ponens |
\(p\) \(\underline{p \rightarrow q}\) \(\therefore q\) |
|---|---|
| Modus tollens |
\(\lnot q\) \(\underline{p \rightarrow q}\) \(\therefore \lnot p\) |
| Addition |
\(\underline{p\qquad\,\,\,\,}\) \(\therefore p \lor q\) |
| Simplification |
\(\underline{p \land q}\) \(\therefore p\) |
| Conjunction |
\(p\) \(\underline{q\qquad\,\,\,\,}\) \(\therefore p \land q\) |
| Hypothetical syllogism |
\(p \rightarrow q\) \(\underline{q \rightarrow r\quad\,}\) \(\therefore p \rightarrow r\) |
| Disjunctive syllogism |
\(p \lor q\) \(\underline{\lnot p\quad}\) \(\therefore q\) |
| Resolution |
\(p \lor q\) \(\underline{\lnot p \lor r\,\,}\) \(\therefore q \lor r\) |
⚠ In the table, \(p\) and \(q\) represent any compound
proposition, so don't assume that you must be working with individual propositions in order to use
rules of inference! Just keep an eye out for the distinct patterns of surrounding logical operations that match the hypotheses of a rule of inference,
and make sure those hypotheses are true before using it.