Inductive Reasoning

Induction by simple enumeration. From "these observed instances of A were B" to "all A are B", or to "the next A will be B". The weakest form, and MU's topic 1.8.

Analogy. From "these two things resemble each other in several respects" to "they resemble each other in a further respect". MU's topic 1.9, and the form of reasoning that precedent is built on.

Causal inference. From an observed regularity to the conclusion that one thing produces another. The methods for testing it were set out by John Stuart Mill and are the ordinary equipment of accident investigation.

Statistical inference. From the proportion in a sample to the proportion in the whole, or from a proportion in a class to a probability about an individual member of it.

Hypothesis, also called inference to the best explanation. From a set of facts to the explanation that accounts for them better than any rival. This is the form of reasoning a court uses on circumstantial evidence, and it is why that topic sits where it does.

What makes an inductive argument strong

The tests differ by kind, and the detailed ones come in the chapters on simple enumeration and analogy. Four apply generally.

Number of instances. More observations, more strength, though with rapidly diminishing returns.

Variety of instances. Ten observations under ten different conditions are worth far more than a hundred under one. A drug tested only on healthy young men supports a very narrow conclusion however many times it is tested.

Modesty of the conclusion. The less the conclusion claims, the stronger the argument for it. "The next consignment will probably be defective" is far better supported than "all consignments from this supplier are defective".

Absence of counter-instances, and honest search for them. An argument that has looked for contrary evidence and not found it is worth more than one that never looked, even if the evidence in hand is identical.

The problem of induction

Every syllabus expects this to be named, and it can be stated in four sentences.

Induction assumes that what has been observed is a guide to what has not been observed, which is to say that the future will resemble the past and the unexamined resembles the examined. What justifies that assumption? It cannot be justified deductively, because there is no contradiction in supposing that nature changes tomorrow. It cannot be justified inductively either, because that would be to argue that induction has worked before and will therefore work again, which is itself an induction and assumes the very thing in question.

This is David Hume's problem, and no answer to it commands general agreement. The practical position everybody in fact adopts, courts included, is that induction is indispensable and is used with care rather than defended. What matters for this paper is being able to state the problem, not to solve it.

A worked example

A workman sues, alleging that a defective machine caused his injury. There is no direct evidence of the defect, because the machine was repaired the same evening.

The following facts are proved: three other workmen on the same machine suffered similar injuries in the preceding six months; each injury happened when the guard was in the raised position; the machine's maintenance register shows no inspection for eleven months; and the employer repaired the machine on the evening of the accident without recording what was repaired.

The inductive argument.

1. Three previous injuries on this machine happened with the guard raised.

2. The machine had not been inspected for eleven months.

3. The machine was repaired immediately after this accident, with no record of what was done.

Therefore the machine was probably defective and the defect probably caused this injury.

Assess it by the four tests. Number: four incidents including the present one, which is a small but not negligible number. Variety: weak, because all four involve the same machine and the same guard, so the conclusion is properly limited to this machine. Modesty: the conclusion says "probably" and confines itself to this machine, which is correct. Counter-instances: unknown, and the employer is in the best position to produce them.

Now destroy it with one fact. Suppose the employer proves that the three earlier injuries were caused by workmen deliberately raising the guard to work faster, and that the repair on the evening in question was to a light fitting. Nothing already established has become false. The argument is nevertheless worthless. That is feature 4, and no deductive argument could ever be treated this way.

Distinctions that carry marks

What this does not mean

Induction is not "particular to general". That is the common pattern, not the definition. "Every consignment so far has been defective, so the next one will be" runs from particular to particular and is inductive. What makes it inductive is that the conclusion goes beyond the premises.

A weak inductive argument is not invalid. The word does not apply. It is weak, and it may be strengthened by more or better evidence, which is something no invalid deductive argument can be.

Probability here is not always a number. Courts reason inductively all day without assigning figures to anything. "Probable" means better supported than the alternatives, not calculated at some percentage.

Quick revision

Definition: an argument whose premises are claimed to give some but not conclusive support, so the conclusion may be false though the premises are true.

Six features: probable not certain; ampliative; judged by strength, which has degrees; can be destroyed by new premises; good version is cogency; never described as valid or invalid.

Kinds: simple enumeration, analogy, causal, statistical, hypothesis or inference to the best explanation.

Tests of strength: number of instances, variety of instances, modesty of the conclusion, absence of counter-instances after an honest search.

The problem of induction: the assumption that the unobserved resembles the observed cannot be justified deductively, and justifying it inductively is circular. Hume.

Not "particular to general", which is a pattern and not a definition.

Test yourself

1. Define inductive reasoning and state how it differs from deductive reasoning in the claim it makes.

Inductive reasoning is reasoning whose premises are claimed to provide some but not conclusive support for the conclusion, so that the conclusion may be false even though all the premises are true. A deductive argument claims that its conclusion follows with necessity, so that it cannot be false if the premises are true. The difference is in the strength of the claim made, and it is what makes induction ampliative and deduction safe.

2. Give the basic features of inductive reasoning.

The conclusion is probable and not certain; the reasoning is ampliative, the conclusion asserting more than the premises; it is judged by strength, which comes in degrees; further premises may destroy a strong argument without falsifying anything already asserted; its good version is cogency, meaning strength together with true premises and no relevant evidence suppressed; and the words valid and invalid have no application to it.

3. Name the kinds of inductive argument and give a legal example of one.

Simple enumeration, analogy, causal inference, statistical inference, and hypothesis or inference to the best explanation. A court convicting on circumstantial evidence uses the last of these: it takes a set of proved facts and concludes that guilt is the only explanation that accounts for all of them, which is why the Supreme Court's rule requires every other hypothesis to be excluded.

4. What tests determine the strength of an inductive argument?

The number of instances relied on; their variety, since instances gathered under differing conditions support a wider conclusion than the same number gathered under one; the modesty of the conclusion, a narrower conclusion always being better supported than a sweeping one; and whether contrary instances have been honestly looked for and not found, as against merely not encountered.

5. State the problem of induction.

Induction assumes that the unobserved resembles the observed. That assumption cannot be established deductively, because there is no contradiction in supposing that the pattern breaks. It cannot be established inductively either, since arguing that induction has succeeded before and will succeed again is itself an induction and assumes precisely what is in question. The problem is Hume's, and no solution is generally accepted.

6. Why is cogency a stricter standard than soundness?

Because it contains a requirement that soundness has no need of: that no relevant evidence has been left out. A valid deductive argument cannot be spoiled by additional information, so soundness need only demand validity and true premises. An inductive argument can be destroyed by a single further fact, so a strong argument from true premises is still not cogent if the arguer has kept back evidence pointing the other way.