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Chapter 10: INDUCTIVE REASONING

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Chapter 10: INDUCTIVE REASONING

•   93% of Chinese have lactose intolerance.
Lee is Chinese.
Lee has lactose intolerance.

•   It has never snowed in Palpa in the last 50 years.
It is not going to snow in Palpa this year.

•   Hari is a sous-chef.
Sous-chefs generally have good kitchen skills
So, Hari can probably cook well.

•   These arguments are of course not valid. Lee might be among the 7% of Chinese who can digest lactose. Snow might fall in Palpa this winter due to unusual changes in global weather. But despite the fact that the arguments are invalid, their conclusions are more likely to be true than false given the information in the premises. If the premises are indeed true, it would be rational for us to be highly confident of the conclusion, even if we are not completely certain of their truth. 

•   In other words, it is possible for the premises of an invalid argument to provide strong support for its conclusion. Such arguments are known as inductively strong arguments. We might define an inductively strong argument as one that satisfies two conditions: 

a)      It is an invalid argument.

b)      The conclusion is highly likely to be true given that the premises are true. 

•   Recall that a valid argument can have false premises. The same applies to an inductively strong argument. The two arguments given earlier remain inductively strong, even if Lee is not Chinese, or it turns out that it snowed in Palpa last year. 

•   When we say the conclusion is highly likely to be true given that the premises are true, it does not mean "it is highly likely for the conclusion and the premises to be true." Consider this argument: 

         Someone somewhere is eating bread right now.
         Someone somewhere is eating rice right now.

•   The argument is not inductively strong because the fact that someone is eating bread gives us no reason to believe that someone is eating rice. There is no evidential connection between them, which is what is required when the conclusion is highly likely to be true given that the premise is true. What we should do is imagine a situation in which the premises are true, and then ask ourselves how likely it is that the conclusion is true in the same situation.

1.      INDUCTIVE STRENGTH

•   Although inductively strong arguments are invalid, they are crucial for science and everyday life. We often have to make predictions about the future based on past experiences. Our past experiences can never logically guarantee that our predictions are correct, but they can tell us what is more likely to happen. Our lives would be completely paralysed if we did not plan our actions on the basis of probability. As Bishop Butler (1692-1752) said in a famous quote, "Probability is the very guide of life."

•   The inductive strength of the argument is a measure of the degree of support that is provided. Unlike validity, inductive strength is not an all-or-nothing matter. An argument is either valid or not valid, and there is no such thing as a partially valid argument. In contrast, the inductive strength of an argument is a matter of degree, as can be seen in this example: 

•   x% of Chinese have lactose intolerance.
Lee is Chinese.
Lee has lactose intolerance.

•    Intuitively, whether the premise supports the conclusion depends crucially on the value of the variable x. If x is 100%, the argument is obviously deductively valid. If x is 99.999%, then the argument is invalid but inductively very strong. If x is 70%, the argument is still strong but less so. If x is 10%, then the premises are too weak to support the conclusion.

•   We can give a mathematical definition of inductive strength in terms of the conditional probability of the conclusion given the premises. Inductive strength will then vary from 0 to an upper limit of 1, which corresponds to deductive validity. Suppose we have an argument with premises P1, P2, …, Pn and conclusion C. The inductive strength of the argument is then the conditional probability of the conclusion given the conjunction of all the premises, or in mathematical notation:
Pr (C | P1and P2 … and Pn)

•   As an illustration, consider this argument: 
Sita bought just one lottery ticket. 
The lottery has 1000 tickets, only one of which will be the winning ticket. 
The winning ticket will be chosen randomly. 
Sita will not win the lottery. 

•   Since Sita has bought only one ticket, she will lose the lottery as long as any of the remaining 999 tickets are chosen. The conditional probability of the conclusion given all the facts about the lottery is therefore 0.999, which is very high, and so this is an inductively strong argument. On the other hand, if we change the conclusion of the argument to Sita will win the lottery, the inductive strength of the argument will be a rather low 0.001.

2.      DEFEASIBILITY OF INDUCTIVE REASONING 

•   Adding new premises to a valid argument will not make it invalid. If all Chinese have lactose intolerance and Lee is Chinese, then it follows that Lee has lactose intolerance. Our conclusion will not change by additional information such as Lee is a chain-smoking philosopher with peculiar sleeping habits. However, new premises can increase or decrease the inductive strength of an argument. Consider this argument: 

            Regina fell off the roof of a 50-story building. 
         Regina is dead. 

•   This argument as it stands is inductively strong since it is rather unlikely for someone to survive such a fall. But suppose we discover some new information:

         Regina fell off the roof of a 50-story building.
         Regina landed on a big tent on the ground floor of the building. 
            Regina is dead. 

•   Now the argument becomes weaker than before because it is less clear that Regina must die from the fall. After all, there are cases where people managed to survive after falling from tall buildings. But wait, there is more to come: 

Regina fell off the roof of a 50-story building.
Regina landed on a big tent on the ground floor of the building. 
The roof of the tent is fixed with sharp sticks pointing upward. 
Regina is dead.

•   Now the situation is again different and the argument is stronger, perhaps even stronger than in the beginning when we are told only that Regina has fallen. The reasoning is defeasible when the corresponding argument is rationally compelling but not deductively valid. The truth of the premises of a good defeasible argument provides support for the conclusion, even though it is possible for the premises to be true and the conclusion false. For example, noting that all penguins observed so far cannot sustain flight, we conclude that no penguin can fly. But this conclusion might turn out to be wrong if we discover a new species of flying penguins tomorrow. Old evidence providing strong support for a theory might fail to do so when new evidence comes in. 

3.      CASES OF INDUCTIVE REASONING 

•   There are different types of inductive reasoning. Here are some main ones: 

Induction based on statistics: We rely on statistics to make generalisations about groups of things, and to make predictions about particular cases. For example, we might have seen lots of spiders, and they all produce silk, and so we conclude this is true of all spiders, including those which have not been observed. 

Induction based on analogy: These are arguments where two objects A and B are very similar, and so we conclude that something that is true of A ought to be true of B as well. Suppose a chemical is discovered to be toxic to mice. By analogy we suspect it will be harmful to human beings as well, given the biological similarities between the two. This is again a form of induction since the conclusion does not logically follow.

Induction based on inference to the best explanation:Very often we do not have enough evidence to prove that something must be true. Sometimes the evidence can also be conflicting and point to different conclusions. What we can do is to consider the alternative theories available and pick the one that on balance has the most evidence supporting it, all things considered. For example, when we leave our home we might notice that the street is wet. This might be because it has just rained, but it is also possible that somebody has just washed the street. But you notice that some cars passing by are also wet, so you conclude it is most likely that it rained. 

4.      DEDUCTIVE AND INDUCTIVE ARGUMENTS?

•   In philosophy a deductive argument is contrasted with an inductive argument. Inductive arguments also have premises and a conclusion. The difference is that with a deductive argument, the conclusion must be true, and an inductive argument generally means that the conclusion is only probable (likely). 

•   A deductive argument is simply a valid argument, and an inductive argument is an inductively strong argument. But then it is no longer the case that every argument is either deductive or inductive. 

END OF THE PART

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