# Natural Deduction: The Rules Philosophy I · Logic · https://tryals.app/learn/philosophy-i/natural-deduction-the-rules ## Proving Instead of Checking Truth tables decide everything and scale terribly. **Natural deduction** takes the other route: instead of surveying all assignments, it derives the conclusion from the premises by applying rules, one step at a time. Each step is justified by a rule and by the earlier lines it uses, so a proof is a checkable object rather than a survey. The rules come in pairs. For each connective there is an **introduction** rule saying how to derive a sentence with that connective as its main operator, and an **elimination** rule saying what may be derived from one. | Connective | Introduction | Elimination | |---|---|---| | $\wedge$ | From $P$ and $Q$, infer $P \wedge Q$ | From $P \wedge Q$, infer $P$ (or $Q$) | | $\vee$ | From $P$, infer $P \vee Q$ | From $P \vee Q$ and both cases giving $R$, infer $R$ | | $\rightarrow$ | Assume $P$, derive $Q$, conclude $P \rightarrow Q$ | From $P \rightarrow Q$ and $P$, infer $Q$ | | $\neg$ | Assume $P$, derive a contradiction, conclude $\neg P$ | From $\neg\neg P$, infer $P$ | Conditional elimination is modus ponens under a formal name. Conjunction rules are trivial. The two that carry real weight are **conditional introduction** and **negation introduction**, because both work by making an **assumption**. An assumption is a temporary supposition, opened for the sake of argument and later **discharged**. Its scope is marked by a subproof, and the crucial discipline is that nothing inside a subproof may be used after the assumption is discharged, the conclusion depends on the assumption, so it cannot escape its scope. Violating this is the commonest way to produce a proof of something false. Conditional introduction is the formal counterpart of ordinary conditional reasoning: to establish "if P then Q", suppose P and get to Q. Negation introduction is *reductio ad absurdum*: to establish "not P", suppose P and derive a contradiction. Natural deduction and truth tables agree exactly. The system is **sound**, everything provable is valid, and **complete**: everything valid is provable. Neither property is obvious, and both had to be proved. > **Common pitfall:** using a line from inside a closed subproof. Once an assumption is discharged, everything derived under it is off limits, those lines were only ever conditional on a supposition you have now withdrawn. This single restriction is what keeps the system sound. ## Practice questions 8 of this lesson's 11 practice questions, with answers. The full set is in the app. ### 1. Arrange the steps of a conditional introduction proof in order. **Answer:** 1. Assume the antecedent as a temporary supposition 2. Derive intermediate results within the subproof 3. Reach the consequent inside the subproof 4. Discharge the assumption 5. Conclude the conditional outside the subproof **Why:** To prove "if P then Q", suppose P and reach Q. The conditional is asserted only *after* discharge, and it no longer depends on the assumption, which is exactly what makes it a categorical result. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/arrange-the-steps-of-a-conditional-introduction-proof-in-order ### 2. Why may a line from a closed subproof not be used later in the proof? A. Lines inside subproofs are intermediate steps rather than valid truths B. It was derived under an assumption that has since been withdrawn C. Only the final derived line of a subproof is retained as an active rule D. Subproof steps are merely hypothetical sketches without formal numbering **Answer:** B. It was derived under an assumption that has since been withdrawn **Why:** Every line in a subproof is conditional on its assumption. Once discharged, that support is gone, so the line no longer stands on its own. This restriction is precisely what keeps the system sound. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/why-may-a-line-from-a-closed-subproof-not-be-used-later-in-the-proof ### 3. A proof of "if P then Q" by conditional introduction requires how many assumptions to be made and then discharged? **Answer:** 1 **Why:** **One**: the antecedent is assumed, the consequent derived, and the assumption discharged as the conditional is asserted. The proof that results depends only on the original premises. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/a-proof-of-if-p-then-q-by-conditional-introduction-requires-how ### 4. Which statements about natural deduction are correct? A. It can prove sentences that truth tables would judge invalid B. Assumptions must be discharged before a proof is complete C. Conditional elimination is modus ponens D. Each connective has an introduction and an elimination rule **Answer:** B. Assumptions must be discharged before a proof is complete; C. Conditional elimination is modus ponens; D. Each connective has an introduction and an elimination rule **Why:** The system is **sound** (everything provable is valid) and **complete** (everything valid is provable), so it can never disagree with the tables. Both properties had to be proved and neither is obvious. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/which-statements-about-natural-deduction-are-correct ### 5. Truth tables establish validity by surveying every valuation, whereas natural deduction constructs a step-by-step derivation from premises. Why does this fundamental difference make natural deduction preferable for complex arguments? A. It generates conclusions that hold independently of any initial premises B. It can prove valid inferences that semantic truth tables fail to capture C. It avoids the exponential explosion of checking every possible valuation D. It bypasses the requirement to confirm the formal soundness of derivations **Answer:** C. It avoids the exponential explosion of checking every possible valuation **Why:** Truth tables grow exponentially ($2^n$ rows for $n$ sentence letters), making large surveys computationally intractable. Natural deduction proves the same validities via directed syntactic steps. However, soundness and completeness guarantee neither system proves more or less than the other. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/truth-tables-establish-validity-by-surveying-every-valuation-whereas ### 6. Natural deduction is strictly weaker than truth tables: some valid arguments cannot be proved in it. **Answer:** False **Why:** False, the system is **complete**, so every valid argument has a proof, and **sound**, so nothing invalid does. The two methods agree exactly; natural deduction trades exhaustive checking for rule-following, not power. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/natural-deduction-is-strictly-weaker-than-truth-tables-some-valid ### 7. Match each proof technique to the shape of reasoning it formalises. **Answer:** - Conditional introduction → Suppose it, and see what follows - Negation introduction → Suppose it, and derive an absurdity - Disjunction elimination → Either way, the same conclusion follows - Conjunction introduction → Both hold, so their conjunction holds **Why:** Each rule formalises something people already do. Disjunction elimination is argument by cases; negation introduction is reductio. The rules were designed to look like natural reasoning, which is where the name comes from. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/match-each-proof-technique-to-the-shape-of-reasoning-it-formalises ### 8. Sort each rule by whether it introduces or eliminates a connective. **Answer:** - Introduction rule: From P and Q, infer their conjunction, From P, infer the disjunction of P with anything - Elimination rule: From a conjunction, infer one conjunct, From a conditional and its antecedent, infer the consequent **Why:** Introduction rules put a connective into the conclusion; elimination rules take one out of a premise. The symmetry is deliberate and is what makes the system easy to remember. Page: https://tryals.app/practice/philosophy-i/natural-deduction-the-rules/sort-each-rule-by-whether-it-introduces-or-eliminates-a-connective