Inductive vs deductive reasoning: the difference, with ten examples to sort

Deductive reasoning: conclusions that must follow

A deductive argument aims to make its conclusion certain. In the definition used by the Internet Encyclopedia of Philosophy, an argument is valid when, if the premises are true, the conclusion must be true. Validity depends only on the argument's form. "All the reservoirs in the district are below half full; Millbrook is a reservoir in the district; so Millbrook is below half full" is valid whatever the actual water level, because the conclusion is already contained in the premises.

Two words keep the idea straight. A valid argument has a form that guarantees its conclusion if the premises hold. A sound argument is a valid argument whose premises are in fact true. A valid argument can still reach a false conclusion if one of its premises is false, and that is a fault in the premise, not in the logic.

Inductive reasoning: conclusions that go beyond the premises

An inductive argument aims to make its conclusion likely. The same encyclopedia calls an inductive argument strong when its premises render the conclusion likely, and cogent when it is strong and its premises are true. "The last forty trains on this line left on time, so tomorrow's will too" is inductive. However many on-time trains you add, the next one could still be late, because the conclusion says something the premises do not.

That gap is not a defect of a particular argument. It is what induction is. Philosophers have argued about how inductive conclusions can be justified at all since David Hume raised the question in A Treatise of Human Nature in 1739, the puzzle now called the problem of induction. Everyday life, science and reasoning tests rely on induction anyway, because learning a rule from examples is induction.

Why "specific to general" is the wrong test

A common shortcut defines induction as reasoning from the specific to the general, and deduction as the reverse. The Internet Encyclopedia of Philosophy, which cites the version in Peter Kreeft's logic textbook, calls that characterisation too crude. Some deductions move from particular to particular ("Ana is taller than Ben; Ben is taller than Cy; so Ana is taller than Cy"). Some inductions move from general to particular ("most trains on this line run on time, so this one probably will"). The direction of the argument does not decide the type.

A more common account says that a deductive conclusion is already contained in its premises, while an inductive one goes beyond them. Philosophers have not settled on a perfect dividing line, and the encyclopedia's own conclusion is that it is often more useful to ask whether an argument is valid, or how strong it is, than to sort it into a type. The three questions below do that.

How to tell which one you are looking at

Ask three questions, in this order:

1. If every premise were true, could the conclusion still be false? If not, it is a valid deduction.
2. If it could, was the argument trying to make the conclusion certain? Words such as "so it must be" signal an attempted deduction. If the form does not guarantee the conclusion, it is a deduction that fails, and often a named fallacy.
3. Otherwise, how strong is the support? Words such as "probably", "most" and "so far" signal induction. Judge it by how much evidence there is, how representative it is and whether anything counts against it.

Cue words help, but they do not decide. People say "must" about guesses, and "probably" about things that follow with certainty.

Ten examples to sort

Decide for each whether it is deductive or inductive, and, for the deductive ones, whether it is valid. The answers follow the list.

1. All the reservoirs in the district are below half full. Millbrook is a reservoir in the district. So Millbrook is below half full.
2. The last forty trains on this line left on time, so tomorrow's will too.
3. If the alarm is set, the panel light is red. The light is not red. So the alarm is not set.
4. 3, 6, 12, 24: each number is double the one before, so the next is 48.
5. If it rained overnight, the pitch is wet. The pitch is wet. So it rained overnight.
6. A random survey of 2,000 voters found 54% in favour, so most voters are probably in favour.
7. Every number divisible by 6 is divisible by 3. 84 is divisible by 6. So 84 is divisible by 3.
8. Bridges of this design have stood for a century on this kind of soil, so this one will probably stand too.
9. The key is either in the drawer or in the car. It is not in the drawer. So it is in the car.
10. In each row of a three-by-three grid the shading darkens by one step, so the missing last cell is the darkest shade.

The answers, in the same order:

1. Deductive, valid. The conclusion is contained in the premises.
2. Inductive: a prediction from a run of past cases.
3. Deductive, valid. Denying the result denies the condition, a form called modus tollens.
4. Inductive: a rule inferred from four examples. Once you accept the rule, applying it is deduction.
5. Deductive in intent, and invalid. A wet pitch has other causes, such as a sprinkler. This is the fallacy of affirming the consequent.
6. Inductive: a statistical generalisation from a sample to a population.
7. Deductive, valid.
8. Inductive: an argument by analogy.
9. Deductive, valid. With one option ruled out, the other must hold; this is a disjunctive syllogism.
10. Inductive: the rule is inferred from the rows you can see, and the last cell is filled by applying it.

Examples 4 and 10 are the point where the two kinds meet. Most reasoning puzzles ask you to induce a rule and then deduce what the rule implies for the missing piece.

Where each one appears in reasoning tests

Inductive. Number series and figure matrices are induction tests. You see a few examples of a rule and must infer the rule itself; nothing in the puzzle proves that your rule is the intended one, which is why a good puzzle has exactly one rule that fits every row and column.

Our number sequence test has 20 sequences across five kinds of rule, and names the rule behind every answer.

Our matrix reasoning test has 20 grids across five kinds of rule, with the rule named after each one.

Deductive. Syllogisms, if-then statements and ordering puzzles are deduction tests. Everything you need is in the premises, and the task is to see what must follow and to resist what merely sounds right.

Our logical reasoning questions are 20 of these: 7 syllogisms, 7 if-then statements and 6 ordering problems, with an explanation for every option.

Spotting the deductions that fail. Example 5 is a fallacy, an argument whose form looks valid and is not.

Our guide to logical fallacies names the common ones, with examples.

Then practise on the spot-the-fallacy test.

The trap in induction: looking only for confirmation

Inductive reasoning goes wrong in a characteristic way: by collecting cases that fit a rule instead of testing cases that could break it. Peter Wason showed this in 1960 with a task now known as the 2-4-6 problem. People were told that the numbers 2, 4, 6 fit a hidden rule and had to find it by proposing other triples. Wason distinguished between seeking confirming evidence alone, which he called enumerative induction, and seeking evidence that could also rule a guess out, which he called eliminative induction. Only 6 of his 29 participants reached the correct rule without first announcing a wrong one.

You can try the same kind of task in our 2-4-6 rule game. It counts how many of your test triples could only have confirmed your guess.