š¬ What Is Inductive Reasoning?
Inductive reasoning is the opposite direction of deductive reasoning. You start from specific observations and work up to a general conclusion.
- ā¬ļø Specific observations ā General rule
- ā¬ļø Pattern ā Conjecture (educated guess)
- ā¬ļø Used constantly in science, medicine, and everyday life
Important: Unlike deductive conclusions, inductive conclusions are probable but never certain. Even a very strong inductive argument can be wrong.
āļø Deductive vs. Inductive ā Side by Side
šļø Deductive
Conclusion is guaranteed if premises are true.
š Inductive
Conclusion is probable but not certain ā black swans exist in Australia!
š¢ Patterns and Conjectures
A conjecture is a conclusion reached through inductive reasoning ā an educated guess based on patterns. Conjectures can later be proven or disproven.
Number Pattern Example
Observations: 2, 4, 6, 8, 10, ā¦
Conjecture: "The next number is 12." (Pattern: add 2 each time)
Observations: 1, 4, 9, 16, 25, ā¦
Conjecture: "These are perfect squares: 1Ā², 2Ā², 3Ā², 4Ā², 5Ā²,ā¦ Next is 36 (6Ā²)."
Real-World Example
Observation 1: Every time I eat shrimp, I get a rash.
Observation 2: Last Tuesday I had shrimp and got a rash.
Observation 3: This happened 5 more times.
Conjecture: "I am probably allergic to shrimp."
š” How scientists use induction
Every scientific theory ā gravity, evolution, germ theory ā began as an inductive conjecture based on repeated observations. More evidence makes a conjecture stronger, but it is never absolutely proven (only repeatedly supported).
šŖ Strong vs. Weak Inductive Arguments
Since inductive conclusions are never guaranteed, we evaluate them on a spectrum of strength.
Strong Strong
"Every crow I have ever seen or read about in 500 ornithology studies is black. Therefore, crows are probably black."
Large, varied, reliable sample. High probability the conclusion is true.
Weak Weak
"I saw two crows by my house and both were black. Therefore, all crows everywhere are black."
Tiny sample size. Jumping to a sweeping conclusion.
What makes an inductive argument stronger?
- āļø More observations / larger sample
- āļø More varied observations (different places, times, conditions)
- āļø No contradicting evidence
- āļø Observations done carefully and reliably
š« Counterexamples: Disproving a Conjecture
A counterexample is a single example that proves a conjecture is false. You only need one counterexample to disprove a universal statement ("all," "every," "never").
Examples
Conjecture: "All prime numbers are odd."
Counterexample: 2 is prime and even. ā Conjecture disproved!
Conjecture: "If nĀ² is even, then n is even."
Test: n = 4 ā 16 is even, 4 is even. n = 6 ā 36 even, 6 even. Many cases support it.
In fact this is true ā but you'd need a formal proof to be certain, not just examples!
Conjecture: "nĀ² + n + 41 is always prime."
Counterexample: n = 41 ā 41Ā² + 41 + 41 = 41(41 + 1 + 1) ā divisible by 41. ā
š” Key Asymmetry
You need infinitely many examples to prove a universal claim true (which is why deductive proof is needed). But you need only one counterexample to prove it false.
ā ļø Watch Out: Hasty Generalization
The most common inductive error is drawing a sweeping conclusion from too few observations. This is called a hasty generalization and is covered in detail in the Logical Fallacies lesson.
Examples of Hasty Generalizations
"I had two bad experiences with dentists. All dentists must be terrible." ā
"My grandfather smoked his whole life and lived to 95. Smoking must not be that bad." ā
āļø Practice ā Click to Reveal Answers
1. What is the next number in this pattern? 3, 6, 12, 24, 48, ā¦
96. Each term is multiplied by 2. Conjecture: the pattern is doubling. 48 Ć 2 = 96.
2. "Every time I flip this coin it lands on heads. Therefore, this coin always lands on heads." Strong or weak?
Very weak. Even 10 heads in a row doesn't mean the coin is unfair. The sample is too small and random variation could explain it. Need far more flips.
3. Find a counterexample: "All rectangles are squares."
A rectangle that is 4 cm Ć 2 cm has four right angles but unequal side lengths ā it is a rectangle but not a square. ā Conjecture disproved.
4. Is this deductive or inductive? "The sun has risen every day for the last 4.5 billion years; therefore it will rise tomorrow."
Inductive. It is based on past observations, not a logical guarantee. (The deductive version would require a premise like "The sun always rises every day without exception" ā which is itself only known through observation.)
5. What is the difference between a conjecture and a proof?
A conjecture is an educated guess based on observed patterns (inductive). A proof uses deductive reasoning to show a statement must be true for all cases ā leaving no room for exceptions. Conjectures can be disproved with one counterexample; proofs cannot.
šÆ Key Takeaways
- Inductive reasoning moves from specific observations to a general conclusion.
- Inductive conclusions are probable, never certain ā one counterexample disproves them.
- A conjecture is an educated guess formed from a pattern.
- More observations + more variety = stronger inductive argument.
- A single counterexample is enough to disprove a universal conjecture.
- Drawing a sweeping conclusion from too few cases is a hasty generalization.