Induction from Observations

Subsection 8.12.1 Induction from Observations

If we prove by mathematical induction that the sum of the first n positive integers is

\frac{n (n + 1)}{2},

it must be so. No exceptions. It’s over.

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But now let’s look at the real world and consider another reasoning strategy that is typically also called induction. To avoid confusion, we’ll call it empirical induction. When most people, in nonmathematical contexts, use the term “induction”, what they mean is empirical induction.

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In empirical induction, we examine a large and representative set of examples. If we observe a consistent pattern, we generalize and conclude that the pattern will always occur. In other words, we assert something of the form:

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∀x (P(x))

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Unlike mathematical induction, empirical deduction isn’t sound. It can derive conclusions that are false.

For example, we could wander through zoo after zoo and finally conclude that all peacocks have colored feathers. This conclusion is false. But note that a related conclusion, namely that most peacocks have colored feathers, is in fact true.

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If we reason, by empirical deduction, that people seem to get mad if you lie to them, it’s nevertheless possible that we could lie and a particular person won’t get mad.

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But, since we need to act, this sure beats saying “I don’t know.”

Nifty Aside

Dante Alighieri (1265 - 1321) described what would happen in eternity to those who couldn’t make up their minds: They would run around the vestibule of the Inferno (Hell).

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Exercises Exercises

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Exercise Group.

1. Assuming observations that can reasonably be made, indicate, for each of these claims, what role empirical induction could play in attempting to determine the truth of the claim:

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Part 1.

Larger cars weigh more than smaller ones.

Empirical induction would be a good way to tackle the problem and will likely support the claim.
In the process of trying empirical induction we would almost certainly find a counterexample that would refute the claim.
Empirical induction isn’t likely to tell us much about the claim.

Answer.

Correct answer is A.

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Part 2.

$\sqrt{2809}$ = 53.

Empirical induction would be a good way to tackle the problem and will likely support the claim.
In the process of trying empirical induction we would almost certainly find a counterexample that would refute the claim.
Empirical induction isn’t applicable to this problem.

Answer.

Correct answer is C.

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Part 3.

Rain comes to Austin on prime numbered days. (So it rains on the 2nd, 3rd, 5th, etc. of every month.)

Empirical induction would be a good way to tackle the problem and will likely support the claim.
In the process of trying empirical induction we would almost certainly find a counterexample that would refute the claim.
Empirical induction isn’t applicable to this problem.

Answer.

Correct answer is B

Solution.

Explanation: (Part 1) It makes sense to look at a lot of cars. When we do that, we’ll probably find that, at least most of the time, bigger ones weigh more. Maybe, after doing that, we can formulate an explanation for why that is. (Part 2) Empirical induction isn’t appropriate here. We are interested in a single case, not a large number of cases. (Part 3) While it makes sense (to someone who isn’t already laughing) to try to look at the facts to see whether they support the claim, it turns out that the facts do no such thing.