TMUA Paper 2 logic & proof

Paper 2 is not a vocabulary test — but candidates without the vocabulary cannot see what questions ask. Six logical moves underlie nearly every reasoning question. Learn them once, precisely, and dozens of questions collapse into pattern-matching.

The six logical moves

Implication (P ⇒ Q). “If P then Q.” The workhorse of proof. Converse (Q ⇒ P). Not equivalent — the classic trap. Contrapositive (¬Q ⇒ ¬P). Equivalent — the engine of indirect proof. Necessity and sufficiency. Directional claims about conditions. Quantifiers (∀, ∃). “For all” versus “there exists”, and their negations. Counterexample. One case that kills a universal claim. Drill each family in /practice: arguments, necessity & sufficiency, quantifiers.

Converse vs contrapositive

Take “if it rains, the pavement is wet” (P ⇒ Q). The converse — “if the pavement is wet, it rained” — is plainly unsafe (a burst pipe wets pavements too). The contrapositive — “if the pavement is not wet, it did not rain” — is watertight. Paper 2 tests this asymmetry relentlessly: an argument that treats a converse as given commits the converse error (affirming the consequent). When options offer restatements of a claim, translate each into P/Q arrows on scrap paper and keep only the contrapositive as equivalent. Practise in the contrapositive hub and read Paper 1 vs Paper 2 for why slow analysis beats fast instinct here.

Necessary, sufficient, iff

“P is sufficient for Q” means P ⇒ Q: P guarantees Q. “P is necessary for Q” means Q ⇒ P: no Q without P. Being 18 is necessary but not sufficient for voting wisely; scoring full marks is sufficient but not necessary for a top score. “P if and only if Q” (iff) asserts both directions and must be proved twice — a favourite error-spotting setup where one direction is waved through. Drill the PRF3 hub, then test yourself with bank questions filtered to PRF topics.

Quantifiers & counterexamples

“For all x …” (∀) is destroyed by one counterexample; “there exists x …” (∃) is established by one example. Negations swap them: “not all swans are white” means “there exists a non-white swan”. Paper 2 questions hide quantifiers in ordinary words — always, guarantees, ensures (universal) versus sometimes, can, may (existential). When asked what weakens an argument, hunt the option supplying a single counterexample to a universal premise. Work the quantifier hub and the inference hub, and compare with the syllabus checklist.

Direct, contrapositive, contradiction

Direct proof chains P ⇒ … ⇒ Q (PRF1 drills). Proof by contrapositive proves ¬Q ⇒ ¬P instead — ideal when Q is a negative (“n² is even, so n is even” is easiest contrapositively). Proof by contradiction assumes P ∧ ¬Q and derives absurdity (√2 irrational is the classic). Paper 2 rarely asks you to name methods; it asks which line completes a half-written proof, or which assumption a contradiction argument really needs. Slow down, label each line P/Q/¬, and check direction. Consolidate with a timed mock once the moves feel automatic.

Spotting the invalid step

Error questions present a plausible proof and ask for the first line that does not follow. Recurring culprits: dividing by an expression that could be zero, taking square roots without ±, confusing converse with contrapositive, “proving” a universal with examples, scope slips (“for all ε …” with ε fixed too early), and hidden extra assumptions in diagrams. Method: verify each line against only the lines above it, not the conclusion you want. The moment a line needs something unstated, stop — that is the answer. Train in the ERR1 hub and ERR2 hub, and read how to prepare for the weekly Paper 2 routine that makes this reflexive.

Worked mini-examples: try three now

Theory sticks when you use it within minutes, so attempt three short exemplars before your next drill set. Start with TMUA-P2-ARG1-001 untimed: cover the options, write the conclusion in your own words in one sentence, then uncover and eliminate anything that restates a premise rather than the point being argued for. Most misses here come from choosing a true-but-side statement, so force yourself to ask “so what is the author trying to convince me of?” before you look at the options. Next try TMUA-P2-ARG1-002 with the same cover-and-paraphrase routine, but this time underline the quantifiers (all, some, must, can) and any conditional words (if, only if, unless). Then move to TMUA-P2-ARG2-001 and translate each option into P/Q arrows on scrap paper, marking which direction is claimed and whether the passage actually supports it.

After the three attempts, sort your errors honestly. If you picked the converse of the right answer, revisit the contrapositive routine above and redo the arguments hub slowly rather than drilling more questions at speed. If you imported an assumption the passage never stated, practise writing “not given” next to any option that needs extra facts, and compare with the assumptions hub plus the FAQ on negative marking and guessing. Then consolidate with five timed questions in /practice?topic=ARG1 and one half-paper under exam rules in a full mock — slow analysis first, timed pressure second, review longest of all.

Common slip & diagram reading

Diagram description: imagine a two-column logic map for this guide. The left column stacks P at the top with a downward arrow to Q, labelled original implication, while the right column stacks not-Q above not-P with a matching downward arrow labelled contrapositive, and a curved double-headed equivalence bar joins the two columns to show they stand or fall together. A second, crossed-out diagonal arrow runs from Q back to P on the left, marked converse with a red cross, and a small speech bubble beside it reads affirming the consequent. The alt-text habit to copy is to narrate arrows before truth: state which implication each option claims, keep only the contrapositive-shaped restatement as equivalent, and treat every converse-shaped paraphrase as needing fresh evidence rather than inherited truth.