Logical reasoning

Thinking clearly

Logical reasoning means reaching conclusions step by step, so that each step follows from the last. It’s the backbone of maths proofs, computer programming and good arguments in everyday life.

Two kinds of reasoning

Inductive reasoning helps you discover patterns. Deductive reasoning proves them.

Counterexamples

A single counterexample proves a general statement false.

• Statement: "All prime numbers are odd." Counterexample: 2.
• Statement: "Multiplying always makes a number bigger." Counterexample: 5 × 0.5 = 2.5.

Many examples can’t prove a statement true, but one counterexample can prove it false.

Conditional statements

A conditional statement has the form "If P, then Q."

• "If a number ends in 0, then it is divisible by 10." (true)

The converse swaps the parts: "If Q, then P."

• "If a number is divisible by 10, then it ends in 0." (also true)

But converses aren’t always true:

• "If a shape is a square, then it has four sides." (true)
• Converse: "If a shape has four sides, then it is a square." (false: a rectangle is a counterexample)

Logic puzzles

Logic puzzles train deductive thinking. Use a grid to record what’s true (✓) and false (✗).

Puzzle: Ana, Ben and Cara each have one pet: a cat, a dog or a fish. Ana is allergic to fur. Ben’s pet barks. Who has which?

• Ana is allergic to fur, so she can’t have the cat or the dog → Ana has the fish.
• Ben’s pet barks → Ben has the dog.
• Only the cat is left → Cara has the cat.

So: Ana – fish, Ben – dog, Cara – cat.

Venn diagrams and "all / some / none"

• "All A are B": circle A sits inside circle B.
• "Some A are B": circles overlap.
• "No A are B": circles are separate.

Worked examples

Example 1: Find a counterexample: "The sum of two numbers is always greater than each number."

−3 + (−2) = −5, which is less than both. (Or 5 + 0 = 5.)

Example 2: Is this deductive or inductive? "The last 5 Mondays it rained, so it will rain next Monday."

Inductive (and not a reliable conclusion).

Example 3: Write the converse of "If it is a triangle, then its angles add to 180°."

"If a polygon’s angles add to 180°, then it is a triangle." (True.)

Common mistakes

• Treating examples as proof.
• Assuming the converse is true.
• Ignoring special cases like 0, 1, negatives and fractions when testing statements.

Try it yourself

1. Find a counterexample: "All multiples of 3 are odd."
2. Write the converse of "If a number is divisible by 4, then it is even." Is it true?
3. Deductive or inductive: "All rhombuses have equal sides. This shape is a rhombus. So its sides are equal."

Answers: 1) 6 2) "If a number is even, then it is divisible by 4." False (6). 3) deductive

Quick recap

• Inductive reasoning spots patterns; deductive reasoning proves conclusions.
• One counterexample disproves a statement.
• A converse isn’t automatically true.