Techniques of Proof and Inference Rules
Natural deduction
A proof derives a conclusion from premises using inference rules. Common implicational rules include modus ponens (P → Q, P, therefore Q), modus tollens (P → Q, ¬Q, therefore ¬P), hypothetical syllogism, disjunctive syllogism, simplification, conjunction, addition, and constructive dilemma. Replacement rules include De Morgan’s laws, contraposition, distribution, and double negation.
Conditional proof and reductio
Conditional proof assumes the antecedent to derive the consequent, then discharges the assumption as a conditional. Reductio ad absurdum assumes the negation of the desired conclusion and derives a contradiction.
Worked example
Premises: P → Q; Q → R; P. Prove R.
- P → Q Premise
- Q → R Premise
- P Premise
- Q 1, 3 modus ponens
- R 2, 4 modus ponens Each line is justified by a rule and cited premises.
Traps that cost marks
- Using modus tollens incorrectly as Q → P.
- Forgetting to discharge a conditional-proof assumption.
- Applying a replacement rule to only part of a formula without proper scope.
- Treating addition as reversible; from P ∨ Q one cannot infer P.
- Failing to cite the exact lines used.