Propositional Logic · step 7 of 13

Natural deduction & the standard proof rules

Natural deduction proves a conclusion from premises using introduction and elimination rules — one pair per connective — instead of truth tables. Every line carries a justification naming the rule and the earlier lines it used. The distinctive feature is the discharge of assumptions: to prove A → B you assume A, derive B, then close the box and discharge A, which is exactly how mathematicians actually argue.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Work backwards from the goalIf the goal is A → B, open a box assuming A and aim for B. If it is A ∧ B, prove each conjunct separately. The goal shape dictates the rule.
Break the premises downApply elimination rules to what you have: ∧E splits a conjunction, →E (modus ponens) fires an implication, ∨E does a case split.
Use reductio when nothing else fitsTo prove ¬A, assume A and derive a contradiction ⊥, then apply ¬I. To prove A classically, assume ¬A and derive ⊥.
Close every boxAn assumption may only be used inside its own box. Citing a line from a closed box is the single most common proof error.

Worked example

Prove p → r from the premises p → q and q → r.

1. p → q (premise).
2. q → r (premise).
3. Assume p — open a box.
4. q, by →E on lines 1 and 3.
5. r, by →E on lines 2 and 4.
6. Close the box: p → r, by →I discharging the assumption on line 3.

Answer. p → r is derived in six lines, discharging the assumption p at the final step.

Where marks get dropped

These are the specific errors that cost credit on natural deduction & the standard proof rules questions — QED's rubric penalises each of them separately.

Using a line from inside a closed box. Once an assumption is discharged, everything derived under it is out of scope.
Forgetting to discharge. A proof that ends inside an open box has not proved the implication — it has only proved the consequent under an assumption.
Applying →E backwards. From p → q and q you cannot conclude p; that is affirming the consequent, not modus ponens.

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Answer in real notation

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Natural deduction & the standard proof rules — frequently asked questions

Which rules am I allowed to use?

The introduction and elimination pair for each of ∧, ∨, →, ¬ and ⊥, plus the classical rule (double-negation elimination or reductio). Derived rules like modus tollens usually need to be proved first unless your course lists them.

Is natural deduction better than a truth table?

For anything beyond three variables, yes — the table grows exponentially while a proof stays short. Natural deduction also scales to predicate logic, where truth tables do not exist.

How is a natural deduction proof marked?

Line by line. Each line needs a correct formula, a correct rule name, and correct line references. QED scores those as separate rubric criteria, so a valid proof with sloppy justifications still drops marks.

The rest of Propositional Logic

Connectives, truth tables, equivalences and normal forms. Each subtopic below has its own method, worked example and mark-losing traps.

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