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SPM Mathematics: How to Master Logical Reasoning
Logical reasoning is about statements, quantifiers and arguments, more like careful language than calculation. The marks go to precise wording, not clever ideas.
Statements, quantifiers and negation
A statement is a sentence that is either true or false, and the quantifiers all and some change its meaning completely. Negating correctly is the skill examiners test most: the negation of all are is some are not, not none are.
Write negations out in full rather than trusting your instinct.
Implications, converse and argument forms
An implication if p then q has a converse if q then p, and the two are not the same; mixing them up is the classic error. For arguments, learn to recognise valid deductive forms and to tell deduction (general to specific) from induction (specific to general).
These are definitions worth memorising word for word.
Distinguishing a statement from a non-statement
Before anything else in this chapter, a sentence has to qualify as a statement, something that can be judged true or false, and only one of the two. "Kuala Lumpur is the capital of Malaysia" is a statement because it's definitely true; "Is Mathematics your favourite subject?"
is not, because a question can't be marked true or false. Questions, commands and opinions ("Mathematics is hard") are the usual traps in this identification exercise, and getting this first step wrong throws off everything that follows, since quantifiers and negation only apply to genuine statements.
When a compound statement counts as true
Joining two statements with "and" creates a statement that is only true when both parts are true, if either part is false, the whole thing is false. Joining them with "or" works the opposite way: the combined statement is true as long as at least one part is true, and only false when both parts are false.
Students often assume "or" needs just one part true and stop checking there, but examiners can ask you to judge a compound statement built from a false statement and a true one, so check each part separately before deciding on the whole sentence.
Checking whether an argument is actually valid
An argument is valid only when the conclusion must be true whenever every premise is true, it isn't enough for the conclusion to sound reasonable. A fast way to test this on paper is to imagine a situation where all the premises hold and ask whether the conclusion could still fail; if you can picture even one such situation, the argument is invalid, regardless of how convincing the wording sounds.
This check catches arguments that feel right but skip a logical step, which is exactly what this sub-topic is designed to test.
Frequently asked questions
How do I know if a sentence counts as a statement?
Ask whether it can be judged definitely true or definitely false and nothing else, if it's a question, command, or opinion, it isn't a statement in this chapter's sense.
Is the converse of a true implication always false?
Not always, some converses do happen to be true, but you can't assume it; you have to check it separately, usually by trying to find a counter-example.
Why does this chapter feel different from the rest of SPM Maths?
Because the skill is closer to careful reading and precise language than to calculation, so slow, exact wording matters more here than speed.