Deductive Reasoning - Certain Conclusions from Given Facts
Deductive reasoning means drawing conclusions that follow with certainty from a set of given premises. If the premises are true and the reasoning is valid, the conclusion must be true. There is no room for exceptions. It is the foundation of mathematical proof.
Premises and Conclusions
A premise is a statement assumed or known to be true.
A conclusion is what follows logically from those premises.
A valid argument is one where, if all premises are true, the conclusion must also be true.
Syllogisms
The classic form of deductive reasoning is the syllogism: two premises lead to one conclusion.
| Part | Example 1 | Example 2 |
|---|---|---|
| Premise 1 | All multiples of 4 are even. | All squares have four sides. |
| Premise 2 | 28 is a multiple of 4. | A rhombus has four sides. |
| Conclusion | Therefore 28 is even. | Therefore a rhombus is a square. — INVALID |
Example 2 shows that even when premises are true, a badly constructed argument can lead to a false conclusion. Valid form matters as much as true premises.
Deduction in Mathematics
Mathematical proof is entirely deductive. You start with axioms (statements accepted without proof) and previously proved theorems, then deduce new results step by step. Every line must follow logically from what came before.
Worked Examples
By deduction: 17 is a prime greater than 2, so 17 must be odd. 17 is odd. (Verified: 17 is indeed odd.)
Let the two even numbers be 2m and 2n, where m and n are integers (any whole numbers).
Their sum = 2m + 2n = 2(m + n).
Since (m + n) is an integer, 2(m + n) is divisible by 2.
Therefore the sum is even. ✓
This is invalid. School may be closed for other reasons (a holiday, a flood). This error is called affirming the consequent – a common logical fallacy.
Let the three consecutive integers be n, n+1, n+2.
Sum = n + (n+1) + (n+2) = 3n + 3 = 3(n+1).
3(n+1) is divisible by 3 for any integer n. Proved.
Common Logical Fallacies
| Fallacy | Structure | Example |
|---|---|---|
| Affirming the consequent | If P then Q; Q; therefore P. | If it rains the road is wet; road is wet; so it rained. (Could be a burst pipe.) |
| Denying the antecedent | If P then Q; not P; therefore not Q. | If divisible by 4 then even; not divisible by 4; so not even. (6 is even but not divisible by 4.) |
| Hasty generalisation | A few cases → universal rule. | Three students got full marks; the test must be easy. |
Key Takeaways
- Deductive reasoning: if the premises are true and the logic is valid, the conclusion must be true.
- Mathematical proof is entirely deductive – each step follows from what is already established.
- True premises alone do not guarantee a true conclusion if the reasoning structure is flawed.
- One counter-example is enough to disprove a deductive claim.
Practice Questions
- Premises: All rectangles have four right angles. A square is a rectangle. What can you deduce?
- Prove deductively that the sum of two odd numbers is always even.
- Identify the fallacy: “If a number is divisible by 6, it is divisible by 2. 10 is divisible by 2, so 10 is divisible by 6.”
- Prove that the product of any even number and any integer is always even.
- Premises: No prime number greater than 2 is even. 91 is odd. Does it follow that 91 is prime? Explain.