Logical Reasoning

This chapter treats sentences as mathematical objects. You learn what makes a statement true or false, how to negate it, how to combine statements with “and”, “or” and “if… then”, and how to tell whether an argument is valid.

It is precise, careful work rather than heavy computation.

Questions are wordy and reward students who read slowly. Paper 2 often asks you to complete a truth table, negate a compound statement, or judge an argument’s validity with a reason.

The maths is light; the marks are in precision and clear justification.

KSSM adds two quantifier words to statements: 'all' (every case) and 'some' (at least one case). The trap is negating them.

The negation of 'all' is not 'no' it is 'some ... are not'.

So the negation of 'All squares are rectangles' is 'Some squares are not rectangles', which only needs one exception to be true. The negation of 'some' is 'no' or 'none': the negation of 'Some numbers are negative' is 'No numbers are negative'.

Read the quantifier first, picture what a single counter-example would look like, and you will negate correctly every time.

From one implication 'if p, then q' you can build three related statements. The converse swaps them: 'if q, then p'.

The inverse negates both: 'if not p, then not q'. The contrapositive swaps and negates: 'if not q, then not p'.

The single fact worth memorising is that only the contrapositive always shares the same truth value as the original 'if it rains, the ground is wet' and 'if the ground is not wet, it did not rain' stand or fall together. The converse and inverse can easily be false even when the original is true, which is exactly the slip the page warns about.

KSSM separates two kinds of reasoning, and questions often name one. Deductive reasoning goes from a general rule to a specific case: 'All multiples of 4 are even; 20 is a multiple of 4; therefore 20 is even' if the premises are true, the conclusion must be true.

Inductive reasoning goes the other way, spotting a pattern from specific cases and stating a general conclusion: 2, 4, 6, 8 are even, so 'the nth even number is 2n'. For argument questions, identify the premises and the conclusion first, check that each premise is accepted, and only then judge whether the conclusion is forced.

An argument can be valid in form yet rest on a false premise, so keep the two checks separate.

This example blends a quantified statement, its negation, and the truth values of compound statements, a common Paper 2 combination.