Induction by Simple Enumeration

Why it is called the weakest form

Because a single counter-instance destroys it completely, and because nothing in the argument protects against one.

The classical illustration is the swan. Europeans observed swans for centuries, every one of them white, and concluded that all swans are white. The conclusion was overturned in 1697, when black swans were found in Western Australia. The number of confirming instances had been enormous and it counted for nothing.

No number of confirming instances makes the conclusion certain, and one contrary instance makes it false. That asymmetry is the whole case against relying on simple enumeration where anything better is available.

Two further weaknesses are examinable.

It gives no reason. A causal induction tells you why the connection holds and can be tested by intervening. Simple enumeration says only that it has held so far, so it cannot distinguish a real connection from a coincidence.

The instances are usually not varied. People observe what is nearest, so the sample is drawn from one place and one period. The swans were all European.

Why it survives anyway

If it is so weak, why is it the commonest form of reasoning in ordinary life and in practice?

Because it is the starting point. All the better forms of induction begin with somebody noticing a regularity, and noticing a regularity is simple enumeration. A causal hypothesis has to be suggested before it can be tested, and the suggestion comes from the enumeration.

Because it is often all there is. Where the subject matter does not permit experiment, and human affairs mostly do not, counting instances may be the only method available.

Because it is strong enough for many purposes. Nobody demands a causal explanation before accepting that a particular registry office is always closed on the second Saturday. For practical action, a well supported enumeration is frequently enough, and the pragmatists of chapter 150 would say that is all anyone should ask.

Improving it

An enumeration is strengthened by exactly the tests given at sequence 90, and the improvements are worth stating in the form of instructions.

Increase the number of instances, remembering that the returns diminish quickly.

Vary the circumstances. Twenty instances gathered in twenty different conditions are worth far more than two hundred gathered in one. This is the single most effective improvement available.

Look for the exception rather than waiting for it. An enumeration that has survived a genuine search for counter-instances is worth much more than one that has merely not tripped over any.

Weaken the conclusion. "All crows are black" is destroyed by one white crow. "Crows are generally black" and "the next crow will probably be black" survive it. Since the conclusion is what the argument has to support, claiming less is the cheapest way of claiming something defensible.

Look for a connection. The moment a reason can be given why the property should attach to the class, the argument stops being simple enumeration and becomes something better.

Where the law uses it, and where it distrusts it

Custom. A custom is proved by showing a practice observed uniformly and continuously over a long period, which is an enumeration of instances, and the law then treats the practice as a rule. The requirements the law imposes, antiquity, continuity, uniformity, are precisely the tests for strengthening an enumeration.

Course of dealing. Where parties have dealt on the same terms many times, a court will infer that the current transaction was on those terms too. Again an enumeration, and again the law asks about the number and the consistency of the earlier dealings.

Business practice and market usage are established the same way.

And where the law distrusts it. Evidence that a person has behaved in a certain way on other occasions is generally not admissible to show that he behaved that way on this occasion. The reasoning would be an enumeration, and the law rejects it, not because the inference is worthless but because it is weak and highly prejudicial. That is a legal judgment about the strength of a form of induction, made by a rule of evidence, and it is exactly the kind of thing this syllabus exists to let a student notice.

A worked example

A tenant claims that rent has always been payable on the tenth of each month, though the lease is silent. He proves that on each of the previous thirty six months the landlord accepted rent on the tenth without objection.

The argument.

1. On each of the last thirty six occasions rent was accepted on the tenth without objection.

Therefore rent is payable on the tenth.

Assess it. Number: thirty six, which is substantial. Variety: poor, because the instances are all of the same kind and all between the same parties, though that limitation matters less here, since the conclusion is only about these parties. Modesty: the conclusion asserts a term of the tenancy, which is a good deal more than the premises establish; a modest version, that the parties treated the tenth as acceptable, follows much more comfortably. Search for exceptions: unknown, and the landlord is the person who can supply them.

How the landlord attacks it. Not by disputing the thirty six instances, which are admitted. By producing a counter-instance, a single month in which he objected, or by breaking the connection: proving that the tenth was accepted as an indulgence and not as of right. A counter-instance destroys the enumeration; showing the instances have another explanation destroys the inference. Those are the two lines of attack on any argument of this form, and they are always available.

Distinctions that carry marks

What this does not mean

It is not the same as deduction from a general rule. Enumeration produces the general rule; deduction uses one that is already given.

A large number is not a substitute for variety. Two hundred observations of one kind support a narrower conclusion than twenty of twenty kinds.

Rejecting it in evidence is not calling it worthless. The rule against character evidence reflects a judgment that the inference is weak relative to its prejudicial effect, not that no inference is possible.

Quick revision

Definition: a general conclusion drawn from particular instances on the sole ground that no contrary instance has been observed. Also called induction per enumerationem simplicem.

Primary induction makes the inductive leap to instances nobody has observed, and simple enumeration is one. Secondary induction does not: it applies an established generalisation, or moves particular to particular, as analogy does.

Weakest form of induction: one counter-instance destroys it, and no number of instances makes it certain. The black swan is the standard illustration.

Three weaknesses: no reason is given; the sample is usually unvaried; nothing guards against the exception.

Why it survives: it is where every better induction starts, it is often the only method available, and it is frequently strong enough to act on.

Improvements: more instances, greater variety, an honest search for exceptions, a weaker conclusion, and a connection if one can be found.

Legal uses: custom, course of dealing, market usage. Legal distrust: the exclusion of character and similar-fact evidence.

Test yourself

1. Define induction by simple enumeration and set out its form.

It is the form of induction in which a general conclusion about a class is drawn from a number of observed instances, on the sole ground that no contrary instance has been observed. Its form is: this A is B, that A is B, every other observed A has been B, therefore all A are B. Nothing beyond the counting of instances is offered in support, which is what makes the enumeration simple.

2. Why is it called the weakest form of induction?

Because a single counter-instance falsifies its conclusion outright, while no number of confirming instances can make the conclusion certain, and nothing in the argument guards against the exception. It also gives no reason for the connection it asserts, so it cannot tell a real regularity from a coincidence, and its instances are usually drawn from a narrow range of circumstances.

3. Give the classical illustration of its failure.

The swan. For centuries every swan observed in Europe was white, and the generalisation that all swans are white was supported by an immense number of instances. Black swans were found in Western Australia in 1697 and the conclusion was destroyed at once. The number of confirming observations counted for nothing against one contrary observation.

4. How can an enumeration be strengthened?

By increasing the number of instances, though the returns diminish quickly; by varying the circumstances under which they are gathered, which is the most effective single improvement; by searching honestly for counter-instances rather than merely not encountering any; by weakening the conclusion, since a claim that crows are generally black survives a white crow while a claim that all crows are black does not; and by finding a reason for the connection, at which point the argument ceases to be simple enumeration.

5. Give two legal doctrines that rest on simple enumeration.

Custom, which is established by proving a practice observed uniformly and continuously over a long period, the law's requirements of antiquity, continuity and uniformity being the standard tests for strengthening an enumeration. And course of dealing, where a court infers that the present transaction was on the terms the parties have used many times before, the strength of the inference depending on the number and consistency of the earlier dealings.

7. Distinguish primary from secondary induction.

Primary induction infers a general conclusion from particular instances, covering instances nobody has observed, and so makes the inductive leap; simple enumeration and causal induction are of this kind. Secondary induction does not make that leap: it applies a generalisation already established, or reasons from one particular to another without asserting anything general, as analogy does. The test is whether the conclusion reaches unobserved instances.

6. How is an argument of this form attacked?

In two ways, and both are always available. First, by producing a counter-instance, which destroys the generalisation outright however many confirming instances there were. Second, by showing that the instances have some other explanation, so that the observed regularity is a coincidence or the product of a cause that will not operate in the present case. The first attacks the conclusion, the second attacks the inference.