Inductive arguments are indispensable to natural science, social science and, in general, any planning for the future.

Inductive arguments must "sacrifice" the certainty of deductive arguments, to go beyond the content of their premises.

I. Generalization or enumeration

In an inductive argument which depends on enumeration, we draw a conclusion about all the members of a class from premises which refer to some observed members of that class. Sometimes this is called sampling. Such arguments have the form:

All apples in the sample are Grade A. Therefore, all the apples in the barrel are Grade A.

or 41% of the voters in the survey said they would vote for the Rhinoceros Party. Therefore, 41% of the total electorate will vote for the Rhinoceros Party.

or, like the first example,

Apple #1 is Grade A.

Apple #2 is Grade A.

Apple #3 is Grade A

Apple #N is Grade A,


Therefore, all the apples are Grade A.

II. Causal reasoning or inductive elimination

Some of these inductive methods are sometimes called "Mill's Methods" because they are discussed in detail by John Stuart Mill, a 19th century British philosopher, in his System of Logic.

a) Method of Agreement: When two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which all instances agree is the cause of the given phenomenon.

E.g., If A and B and C produce P, and

If A and D and E produce P, and

If A and F and G produce P,

We may conclude (inductively) that A is the cause of P.

b) Method of Difference: If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common, save one, that one occurring only in the former, the circumstance in which alone the two instances differ, is the cause of the phenomenon.

E.g., If A and B and C produce P, and

If A and B alone do not produce P,

We may conclude (inductively) that C is the cause of P.

c) Joint Method of Agreement and Difference: When 2 or more instances in which the effect occurs have only one circumstance in common AND 2 or more instances in which it does not occur have nothing in common (except the absence of that circumstance), the circumstance in which alone the two sets of instances differ is the cause of the effect.

E.g., A + B + C + D > E

A + B + H + G > E

A + G + J + L > E

not A + G + M + N > not E

not A + O + C + R > not E

Therefore, A > E

In all cases, we must ensure that we are considering all of the relevant factors that are involved in the production of the effect. Even if we are careful, we can never be certain that we have considered them all, which is why the argument is inductive (and not deductive).

d) Mill also mentions the method of concomitant variation by direct and inverse variation and the method of residues. We need not discuss these here.

III. Analogy

In an inductive argument that uses analogy, we draw a conclusion about some thing, based on relevant similarities that that thing has to another.

In general, these arguments have the following form:

1. Objects of one kind are known to be similar in certain respects to objects of another kind.

2. Objects of the first kind are known to have some additional characteristic.

3. Therefore, on the basis of their similarity (under #1), we conclude that objects of the second kind will have this additional characteristic.

The crucial question is whether, in step #1, the similarities noted between the two things are relevant to the issue. Relevant similarities strengthen an analogy; relevant dissimilarities weaken an analogy.

For example:

1. Rats and humans are similar on the basis of their physiology.

2. Rats, when injected with chemical C, have a large percentage of heart disease.

3. Therefore, when humans consume chemical C, they will have a large percentage of heart disease.

There is no foolproof way of ensuring relevance in the similar characteristics. But we can have some rules of thumb: knowledge of the general area under consideration will help us distinguish between significant and insignificant elements and, generally, getting used to using analogy, and seeing how far it may be useful.

Exercises on Causal Reasoning:

Which inductive 'method' is being used in each of the following examples? How strong is the inference being drawn and justify your answer.

1. When asked if certain comic books depicting violence had any effect on their actions, many juvenile delinquents have said "yes." In reply, the Comics Code Authority has pointed out that most children who read comic books which depict violence do not become juvenile delinquents, and it therefore concludes that such comic books exert very little influence upon the behavior of its youthful readers.


2. In studying British genius, Havelock Ellis found that most of the famous men whose lives he looked into were first-born children. He therefore concluded that certain environmental factors contributed significantly to the development of outstanding ability.

3. Several years ago 16 children died and 250 other persons were hospitalized in Tijuana as a result of poisoning. First, certain drug products were suspected of having become contaminated, but an investigation showed that many people who were unaffected had consumed these products. Then suspicion fell on the bread that the afflicted parties had been known to eat. An investigation revealed that parathion, a deadly pesticide used in Northern Mexico against the boll weevil, had been stored in a certain warehouse along with flour and sugar, which were distributed to about nine bakers and used to make bread and sweet rolls. Mexican officials concluded that the insecticide had become mixed with the flour and sugar and was responsible for the poisonings.

4. Just before the last war, a poll showed the Prime Minister's popularity to be at 39 percent. Two days after he sent in troops, another poll showed his popularity to be at 48 percent. His action evidently made a favorable impression on the Canadian people.

5. In his autobiography Lord Asquith relates an incident concerning a member of Parliament named Kinglake, whose speeches contained excellent content but whose voice was so poor that his speeches made little impact. One day he delivered a particularly brilliant speech, which as always was received apathetically. The next day the second Sir Robert Peel, after getting permission, concluded his own speech with the identical words of Kinglake's conclusion and received a standing ovation. Asquith concludes that Peel's delivery made all the difference.

6. Because of certain deviations in the predicted orbit of Uranus, Leverrier concluded that a theretofore undiscovered planet was exerting a pull on Uranus and causing the deviation. His conclusion subsequently led to the discovery of the planet Neptune.

7. To determine the effect of fluorinated water on teeth, the neighbouring communities of Newburgh and Kingston, New York, conducted an experiment. For ten years the residents of Newburgh drank water containing 1 part sodium fluoride to 1 million parts of water while the residents of Kingston drank water containing little or no fluorine. Before the test began, 1,000 children in each community had their teeth carefully checked so that the control groups used had approximately the same number of cavities. After ten years it was found that the children of Newburgh had approximately 40 percent fewer cavities than those of Kingston. The result was attributed to the fluorine.

8. To test his vaccine against anthrax, Pasteur inoculated twenty-four sheep and then injected them and twenty-four other sheep with anthrax microbes. Two days later the vaccinated sheep were still healthy while the others were dead or dying. Pasteur was satisfied that his vaccine worked.

Exercises on generalization

Suppose that on a recent visit to a small village in southern France, a village named "Fromage," you noticed that the first six Fromagians whom you met ate cheese with white wine. You therefore concluded that most Fromagians eat cheese with white wine. Would this inference be made stronger or weaker by the following alterations? Why?

1. The six Fromagians were members of the same family.

2. The six Fromagians were members of the same economic and social class.

3. We observed 100 Fromagians eating cheese with white wine.

4. The six Fromagians were all Catholic and we saw them on Friday.

5. We remembered seeing many residents of the Cote d'Azur eating cheese with white wine.

6. The six Fromagians were all Catholic and they were observed on different days of the week.

7. Instead, we concluded that most Frenchmen ate cheese with white wine.

8. We remembered seeing many Parisians eating cheese with white wine.

9. We saw one Fromagian who ate cheese with red wine.

10. We saw one Fromagian who ate snails with white wine.

Exercises on analogy

Determine which of the following selections contain inductive analogies and which contain analogies for explanation or vividness. Appraise the inductive analogies by applying the tests discussed here.

1. There should be ethical limits on the operation of our economy. Our competitive economy is like a vehicle with a motor, but no driver--the more powerful the motor, the more dangerous the vehicle.

Albert Jacquard, J'accuse l'economie triomphante

1. Speaking of Federal appropriations to the States, James M. Beck, Solicitor General under President Harding, compared them "to that tragedy on the ocean seas when the Titanic was struck by a submerged ice floe. After the collision, which was hardly felt by the steamer at the time, the great liner at first seemed to be intact and unhurt and continued to move. But a death wound had been inflicted under the surface of the water ... The power of appropriation is such an ice floe ... and has inflicted a similar fatal wound to the good ship Constitution."

As quoted by Lindsay Rogers, "Speaking of Books: Metaphors"

2. When a squid injects its ink into the water to confuse an enemy or its potential prey, it is obfuscating - that, clouding the water in order to prevent clear sight. Many editorial writers act like the frightened squid. They confuse and cloud, obfuscate, by introducing issues which are not germane to the question being discussed.

Curtis Bradford and Hazel Moritz, The Communication of Ideas

3. Much of the revulsion against the use of atomic weapons arises because the very newness makes it seem more horrible. A careful cataloguing of the injuries resulting from the use of the automobile would also be impressive but any proposal to outlaw the automobile would be considered ridiculous.

R. E. Lapp. Must We Hide?

4. Concerning the illegal faking of football injuries to get a time-out, the sports writer Whitney Martin once wrote: "Efforts have been made to defend the faking of injuries by pointing out it has been done hundreds of times, which is the same as saying a speeder isn't guilty of exceeding the speed limit because others do it and get away with it."

5. The difference between a composition that is not planned and one that is well planned is the difference between a pile of stones and a house made of stone. A pile of stones has no organization; it is a mere heap. A stone house has organization: the stones have been put into place according to a design; they are parts of a whole.

Donald Davidson, American Composition and Rhetoric

6. Gentlemen, I want you to suppose a case for a moment. Suppose that all the property you were worth was in gold, and you had put it in the hands of Blondin, the famous rope walker, to carry across the Niagara Falls on a tight rope. Would you shake the rope while he was passing over it, or keep shouting to him, "Blondin, stoop a little more! Go a little faster!" No, I am sure you would not. You would hold your breath as well as your tongue, and keep your hands off until he was safely over. Now the government is in the same position. It is carrying an immense weight across a stormy ocean. Untold treasures are in its hands. It is doing the best it can. Don't badger it! Just keep still, and it will get you safely over.

Abraham Lincoln

7. Running a government is like running a ship; we need a strong hand at the helm. -- Thomas Carlyle

8. In arguing that the cure of mental illness should be stressed rather than the exact diagnosis of it, Dr. William Menninger, the well-known psychiatrist, once stated: "One does not have to know the cause of a fire to put it out."

9. A Monarchy is a merchantman which sails well, but will sometimes strike on rock, and go to bottom; a republic is a raft which will never sink, but then your feet are always in the water. -- Fisher Ames