The problem of induction
6.2 Extending the problem Extending the problem
6.2 Extending the problem
Our examples were about what we should believe about the future. But the problem of induction applies to any nondeductive inference. That is, the problem of induction applies to any argument where the truth of the premises do not guarantee the truth of the conclusion. A few more examples will bring out the importance of the problem. First, we can extend the argument from beliefs about the future to beliefs about the past: why do you believe that the law of gravity held in the distant past?
It is tempting to think that the past had to be a particular way in order to get to where we are now. But it didn’t. The world could have popped into existence 5 minutes ago, with all the memories and dinosaur fossils included.
Why do you think this didn’t happen?
It is tempting to answer that you remember 10 minutes ago, so the world must be more than 5 minutes old. Well you think you remember 10 minutes ago. But those “memories” consist of structures in your brain which could have been made 5 minutes ago. Why do you think this didn’t happen?
Furthermore, we can extend the argument from beliefs about other times to beliefs about other places. For example, the law of gravity holds around here. Why should we believe that it holds in remote parts of the universe?
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We would need to assume that far away regions of space resemble nearby regions of space. Why should we believe this?
There is an underlying pattern in these worries. We want to make an inference from what we are observing to what we are not observing . What we are observing is described in our evidence. The problem is whether the evidence justifies claims not described in our evidence. Put in this way, we can see that the problem extends to sampling inferences, and these will play an important role in the next chapter. Here’s a quick preview.
Suppose you have a sample of objects of some type F, of which a certain proportion have some property, say, being green. It is natural to take this as evidence that the whole population of Fs has the same proportion of green objects.
1. x% of observed Fs are green justifies
Conclusion: x% of unobserved Fs are green
But why should we believe that the proportion of green objects in the sample is probably the same as the proportion of green objects in the rest of population? We would need a Principle of Uniformity taking us from the sample to the rest of the population (where G = green):
1. x% of observed Fs are green
2. Principle of Uniformity-Sampling : Samples resemble the rest of the population
justifies
Conclusion: x% of unobserved Fs are green
But once again, this principle of uniformity is in need of justification.
And things get even worse (here we preview Chapter 12). Consider that we directly experience only our sensations, and use them to infer the existence of objects. Then our belief in such objects is based on an inference from what is experienced to what is unexperienced:
1. I am having a red experience justifies
Conclusion: There is a red object in front of me To defend this inference, we need something like:
1. I am having a red experience
2. Principle of Accuracy : Experiences resemble reality3 justifies
Conclusion: There is a red object in front of me
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And it is unclear how the Principle of Accuracy can be justified. Without it, it seems we can have no justification to believe any hypothesis that goes beyond our immediate experience. So you may have justification to believe you are having a white experience, or are feeling cold, but that is as far as it goes. The King of Hy-Brasil was justified in believing that his ankles were wet and cold, but not that his island was sinking. You have no justification to believe that you are not the only agent in the world, or hallucinating, or that you are a butterfly dreaming that you are a human.
As Russell put it:
It is therefore important to discover whether there is any answer to Hume [i.e. the sceptic] . . . If not, there is no intellectual difference between sanity and insanity. The lunatic who believes that he is a poached egg is to be condemned solely on the ground that he is in a minority. (Russell 1946, p. 646)
What would an answer to Hume look like? What’s needed is a justification for the Principles of Uniformity. Answers can be divided into those which offer a justification based on our evidence (6.3 and 6.4), and those which offer a justification that is independent of our evidence (6.5). It might be useful to connect this with B ayes Theorem:
PH |E PH P E|H
( )
=( ) ( )
P E( )
Justifications based on our evidence will focus on P(E|H), and justifications independent of our evidence will focus on P(H).4
6.3 Circularity 6.3 Circularity
Could evidence justify our belief that nature is uniform? We saw in 5.3, and in a different way in 6.1, that there is a problem with any argument that starts with our evidence and concludes that nature is uniform. The evidence used would be about the observed; the conclusion that nature is uniform is partly about the unobser ved. So such an argument would go from (observed) evidence to a conclusion about the unobserved. So it would need to assume that nature is uniform. And for this reason the argument would fail. We are trying to show that what is observed can give us justification about the unobserved. So it is circular to assume that what is observed can give us evidence about the unobserved.
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Nevertheless, some philosophers have argued that the circularity needed is acceptable. I will sketch an argument associated with Max Black (1954).5 Recall an argument from earlier:
Premise-Circular Argument
1. In the past, the future resembled the past
2. Principle of Uniformity : The future will resemble the past justifies
Conclusion: In the future, the future will resemble the past (Principle of Uniformity).
Call this type of argument premise-circular because the conclusion appears as a premise. Black offers an alternative argument that is not premise-circular.
The key is to understand the Principle of Uniformity, not as a premise, but as a rule of inference.
First, we need to introduce the concept of a rule of inference . A rule of inference is a rule that tells us when we should make an inference from premises to conclusion. For example, the deductive rule of affirming the antecedent (modus ponens ) licenses the following inference:
1. A 2. A→ B Conclusion: B
We might express the rule itself as:
Affirming the antecedent Infer from “A” and “A→ B” to “B”
Notice that the rule itself is not a premise of the argument. Instead, it licenses the inference from premises to conclusion.
Now consider the following inductive rule of inference, which we’ll call R:
R Infer from “in the past, X” to “in the future, X” (where X is a type of event)
This is the Principle of Uniformity reformulated as a rule of inference. It is sufficient to justify the kind of inferences we want, for example, where X = Sunrise:
Premise: In the past, the sun has risen justifies
Conclusion: In the future, the sun will rise
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The obvious worry is whether we are justified in using R. At this point, Black argues that R can be justified by using R:
Rule-Circular Argument
Premise: In the past, R has been successful justifies
Conclusion: In the future, R will be successful
In case you’ve forgotten what R is, let’s make it explicit:
Rule-Circular Argument
Premise: In the past, inferring from “in the past, X” to “in the future, X”
has been successful justifies
Conclusion: In the future, inferring from “in the past, X” to “in the future, X” will be successful
This argument is rule-circular, that is, the argument uses a rule, R, whose correctness is asserted by the conclusion.6 (Well not quite. The conclusion only talks about the future. But that’s the bit we’re interested in. We already know R was successful in the past.) R supports itself. And Black claims that rule- circularity is acceptable.