Paper 2 logic, made precise.

What Paper 2 tests

Paper 2 uses the mathematical content of Paper 1 and adds mathematical reasoning and simple ideas from elementary logic: statements, implications, proof and purported arguments. It is not a formal university logic course. The challenge is reading ordinary mathematical language with complete precision.

The direction of a conditional

Start with the statement “if P, then Q”. It says that whenever P holds, Q must follow. It does not say that Q can happen only because of P.

The arrows are used here as a compact explanation. The official Notes on Logic and Proof state that candidates are not expected to recognise or use this symbolic notation.

Necessary and sufficient

If P implies Q, then P is sufficient for Q, while Q is necessary for P. These are two descriptions of the same arrow, read from opposite ends.

A quick example

Being divisible by 6 is sufficient for being divisible by 3. Being divisible by 3 is necessary for being divisible by 6. The number 9 shows why divisibility by 3 is not sufficient for divisibility by 6.

Negating quantified statements

Negation changes both the quantifier and the claim. To deny that every object has a property, you need at least one object without it. To deny that some object has a property, you must say that none do.

A counterexample must obey the hypothesis

To disprove “if P then Q”, find a permitted case where P is true and Q is false. An example where P is already false proves nothing about the implication.

Try small integers, zero, negative values, fractions and boundary cases before constructing something elaborate. The purpose is not to find a strange object. It is to break exactly the conclusion while preserving exactly the assumption.

Proof and disproof methods in the specification

• Direct deductive proof: begin with the assumptions and deduce the result.
• Proof by cases: divide all permitted possibilities into exhaustive cases.
• Proof by contradiction: assume the desired conclusion is false and derive an impossibility.
• Disproof by counterexample: disprove a universal statement with one valid exception.

Proof by contrapositive applies the equivalence between a statement and its contrapositive: instead of proving P ⇒ Q directly, prove ¬Q ⇒ ¬P.

How to practise this material

First translate statements without solving the surrounding algebra. Then practise deciding whether an implication is true. Finally, place the same ideas back inside full TMUA questions, where the logical relationship is rarely announced as the main topic.