Logical Reasoning · Form 4
Logical Reasoning: Key Terms
The key Logical Reasoning terms you need for SPM Mathematics, defined plainly in English, Malay and Chinese.
- Statement A sentence that is either true or false, but not both; a question or command is not a statement.
- Negation The opposite of a statement, which reverses its truth value.
- Implication A statement of the form "if p, then q", linking a condition to a conclusion.
- Argument A set of premises leading to a conclusion; it is valid when the conclusion follows from the premises.
- Conjunction (“and”) A compound statement joined by “and”, true only when both parts are true.
- Disjunction (“or”) A compound statement joined by “or”, true when at least one part is true.
- Converse The statement formed by swapping the antecedent and consequent of an implication.
- Deductive reasoning Reasoning from a general rule to a specific case.
- Inductive reasoning Reasoning from specific cases to a general pattern or rule.
- Quantifier A word such as “all” or “some” stating how many cases a statement covers.
- Counter-example A single case that shows a general statement is false.
Term pairs students mix up
- Statement vs open sentence, the difference is: a statement has a fixed truth value ('7 is prime' is true), while an open sentence like 'x + 2 = 5' has no truth value until x is given.
- Conjunction vs disjunction, the difference is: a conjunction ('and', ∧) is true only when both parts are true, while a disjunction ('or', ∨) is true when at least one part is true.
- Converse vs inverse, the difference is: the converse of 'if p then q' is 'if q then p' (parts swapped), while the inverse is 'if not p then not q' (both parts negated).
- Deductive vs inductive reasoning, the difference is: deductive reasoning moves from a general rule to a specific conclusion that must be true, while inductive reasoning moves from specific cases to a general conclusion that is only probable.
- Negation vs converse, the difference is: negation flips a single statement's truth value (~p), while the converse rearranges an implication's two parts and need not share the original's truth value.
How these terms are phrased in real SPM questions
- 'State whether each of the following is a statement.' answer 'statement' or 'not a statement' with a reason; a question or command has no truth value, so it is not a statement.
- 'Write the negation of the following statement.' flip the truth value; for a quantifier, 'all …' becomes 'some … not', and for an inequality the boundary stays, so '>' becomes '≤'.
- 'Complete the truth table.' fill every one of the 2ⁿ rows for ~p, p ∧ q, p ∨ q or p → q; the marks are for a complete, correct grid, not for explanation.
- 'Determine whether the argument is valid and give a reason.' this is a deductive-reasoning task; name the form or supply a counter-example, never just 'valid' or 'not valid'.
- 'Construct a compound statement using "and" / "or".' join the two given statements with the exact connective named, and do not silently switch it to the other one.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Is 'implication' the same as 'if and only if'?
No. An implication 'if p then q' promises only one direction: p leading to q.
'If and only if' is a biconditional and promises both directions at once, p leads to q and q leads back to p. Read the wording carefully, because a one-way 'if … then' and a two-way 'if and only if' have different truth tables.
What exactly counts as a counter-example?
A counter-example is one specific case that satisfies the hypothesis but makes the conclusion fail, so it proves a general claim false. To disprove 'all prime numbers are odd', the number 2 works: it is prime yet even.
One valid counter-example is enough; you do not need to explain why the claim fails in every case.
Are 'negation' and 'opposite' the same thing?
No, and the gap matters most with quantifiers. The negation of a statement is simply everything that is not it, so the negation of 'all are odd' is 'at least one is not odd'.
The 'opposite', 'none are odd', is a stronger claim and is usually wrong. Negation asks for one exception, not the reverse extreme.