Inductive reasoning
Inductive reasoning is inference from observed cases to a conclusion that goes beyond them, so the conclusion is supported but never guaranteed. It is the reasoning that produces every general claim about the world: that a bus route takes about forty minutes, that a stamping machine jams roughly once a shift, that a crate of apples is sound. Inductive arguments are judged strong or weak, on a sliding scale, never valid or invalid, and that single change of vocabulary is the whole difference between induction and the other family of inference.
The trade is exact. An inductive conclusion tells you something the premises did not contain, which is what makes it worth having, and the price is that a future observation can always take it back. Deductive vs inductive reasoning lays the two families out point by point; this page stays inside induction.
Strength, not validity: what an inductive argument claims
Strength is the property an inductive argument has instead of validity. A strong inductive argument is one where the premises make the conclusion likely; a weak one is where they barely move it. Strength comes in degrees, it goes up when you add supporting evidence, and it goes down when you add a case that does not fit. None of that is true of deduction, where a valid argument is already finished and further evidence changes nothing.
Two consequences follow, and both are practical:
- Counterexamples do not destroy an inductive argument, they reprice it. One jammed apple in a crate of a hundred does not refute the claim that the crate is sound. It revises the estimate.
- The conclusion should be qualified to match the evidence. Eleven observations support usually. They do not support always, and writing always converts a defensible inductive claim into an indefensible universal one.
Five kinds of inductive inference
Inductive inference comes in five recognizable kinds, and naming which one you are using tells you immediately what would weaken it.
| Kind | Shape | What weakens it |
|---|---|---|
| Generalization | These sampled cases are F, so the population is F | A small or unrepresentative sample |
| Statistical syllogism | Most F are G, this is an F, so this is probably G | Knowing something specific about this case |
| Analogy | A resembles B in several ways, so A shares B's other property | A relevant difference between A and B |
| Causal inference | X is followed by Y reliably, so X causes Y | A third factor producing both, or reversed order |
| Prediction | The pattern held for the last n cases, so it holds next time | A reason the conditions have changed |
Analogy deserves its own warning, because it is the kind people use without noticing. Comparing a delivery route to a water pipe is an inductive argument, and it is only as good as the resemblance in the respect that matters. Two things can be alike in nine ways and unalike in the tenth, and the tenth is always the one carrying the conclusion.
What makes an inductive argument strong
Four properties make an inductive argument strong, and they are worth checking in this order.
- Count the cases. Nine observations support a weaker claim than ninety, and one observation supports almost nothing, however vivid it was.
- Check how the cases were chosen. A sample drawn only from the top of the crate is not a sample of the crate. Selection method matters more than size: a large badly chosen sample is worse than a small fair one, because it delivers a confident wrong answer.
- Look for variety. Nine bus trips all made on a Tuesday morning support a claim about Tuesday mornings. Spread the observations across the conditions the conclusion is supposed to cover.
- Narrow the conclusion to fit. Most weak inductive arguments have adequate evidence and an overreaching conclusion. The repair is usually to shrink the claim rather than to gather more data.
Since induction is only as good as the observations feeding it, the reliability of the record matters as much as the inference drawn from it. An observation you made and an observation somebody reported to you are not the same evidence.
How inductive reasoning is used to solve problems
To solve a problem inductively, treat the general rule as a hypothesis and go looking for the case that would break it. The approach has four moves, and the fourth is the one people skip.
- Gather the cases you actually have, including the ones that were inconvenient.
- State the pattern as an explicit sentence with a quantifier: most, about half, every observed.
- Derive a prediction from the pattern that would come out false if the pattern were wrong.
- Test the prediction on a case chosen to fail, not on another case of the kind that produced the pattern.
Geometry teaches this loop cleanly. In school geometry, inductive reasoning is the stage where you measure several triangles, notice that the angle sum keeps coming to the same figure and write a conjecture. The conjecture is inductive and it is not yet a theorem. It becomes one only when it is proved deductively from the axioms, and the reason schools separate the two stages is to show that a pattern that holds in every case you checked is still a guess.
The failure induction is prone to
Induction fails most reliably through the search for confirming cases. Peter Wason demonstrated this in 1960 with a task that is still the cleanest illustration in the literature. Participants were shown the number triple 2, 4, 6, told it obeyed a rule, and asked to discover the rule by proposing further triples, being told each time only whether the triple fitted. Most proposed triples like 8, 10, 12 and 20, 22, 24: cases that would fit the rule they had already guessed. The actual rule was far broader, any three ascending numbers, and the only way to find that out was to propose a triple you expected to be rejected.
The lesson transfers directly. Evidence that fits your generalization is nearly free, and it is nearly worthless, because a bad rule and a good rule produce the same confirming cases. Only a case that could have gone the other way carries information. Confirmation bias is the page on the general habit; here it is enough to note that it does its worst damage in exactly the reasoning that is supposed to discover new rules.
The same trap sits inside sequence puzzles and the online sequence calculators that solve them. Given 2, 4, 8, 16, a calculator will return 32 and a formula, and it is right that the formula fits. It is not evidence that the formula is the rule, because infinitely many rules fit any finite list. A tool that produces the simplest fitting pattern has told you about the pattern, not about the process that generated the numbers.
The test to run on any generalization
Before you rely on a rule you arrived at by induction, ask:
What would I expect to see if this rule were wrong, and have I gone looking for it?
If you cannot name an observation that would count against your rule, the rule is not doing any work: it is compatible with everything, so it predicts nothing. If you can name one and have never checked, that is the next thing to do, and it will teach you more than another ten cases of the kind you already have. Logical reasoning sets out the wider argument anatomy this test sits inside, and Critical thinking exercises has sampling and pattern items worked through with the answers.