© 2015 McGraw-Hill Higher Education. All rights reserved. 1

Chapter 11

Inductive ReasoningPart 1

Arguments from Analogy Generalizing from Samples

Statistical Syllogisms

© 2015 McGraw-Hill Higher Education. All rights reserved. 2

Arguments from Analogy• Something has an attribute; :. A similar thing has that attribute.Examples:

Bill likes hunting; therefore his brother Sam does too. Darby is an excellent dog-sitter. Therefore she would be

an excellent baby-sitter.

© 2015 McGraw-Hill Higher Education. All rights reserved. 3

Bill likes hunting; therefore his brother Sam does too.

Darby is an excellent dog-sitter. Therefore she would be an excellent baby-sitter.

• Premise-analogue: Bill• Conclusion-analogue: Sam• Attribute of interest: liking hunting

• Premise-analogue: Darby’s performance as a dog sitter

• Conclusion-analogue: Darby’s performance as a baby-sitter

• Attribute of interest: being excellent at the task

© 2015 McGraw-Hill Higher Education. All rights reserved. 4

The more numerous and diversified the similarities are between the premise-analogue and the conclusion-analogue the stronger the argument.

2 is stronger!

I. Bill likes hunting; therefore his brother Sam does too.2. Bill and his brother Sam both like fishing, playing poker, and going to football games.Bill also likes hunting.:. Therefore Sam does too.

© 2015 McGraw-Hill Higher Education. All rights reserved. 5

The more numerous and diversified the differences are between the premise-analogue and the conclusion-analogue the weaker the argument.

If Bill and Sam are different ages, live in different parts of the country, have different professions, have different marital status, that weakens the argument.

Bill likes hunting; therefore his brother Sam does too.

© 2015 McGraw-Hill Higher Education. All rights reserved. 6

Given more than one premise-analogue, the more numerous and diversified the premise analogues, the stronger the argument.

2 is stronger!

1. Bill likes hunting; therefore his brother Sam does too.

2. Bill, his father, his uncle, and his sister like hunting; therefore his brother Sam does too.

© 2015 McGraw-Hill Higher Education. All rights reserved. 7

Given more than one premise-analogue, the fewer the contrary premise-analogues the stronger the argument, and the more the contrary premise-analogues, the weaker the argument.

2 is weaker!

1. Bill, his father, his uncle, and his sister like hunting; therefore his brother Sam does too.

2. Bill, his father, his uncle, and his sister like hunting, but despite the fact that there are others in the family who don’t like hunting, it is likely Sam likes hunting too.

© 2015 McGraw-Hill Higher Education. All rights reserved. 8

In medical research• Research involving animals is applied

analogically to humans

• Jurists search for legal precedents (analogous cases) to guide current decisions. Stare decisis.

ARGUMENTS FROM ANALOGY are especially important in three areas:

In historical reasoning

• Our understanding of current events is enhanced by comparing them to past events

In the law

© 2015 McGraw-Hill Higher Education. All rights reserved. 9

Generalizing from a Sample• An argument that all, or most, or some percentage of

a sample have an attribute; therefore all, or most, or that percentage of a population have that attribute.

Examples: So far I’ve like all of Professor Stooler’s lectures;

therefore I will like all of his lectures. Most Pit Bulls I know are sweet dogs; therefore most Pit

Bulls are sweet dogs.

© 2015 McGraw-Hill Higher Education. All rights reserved. 10

So far I’ve like all of Professor Stooler’s lectures; therefore I will like all of his lectures.

Most Pit Bulls I know are sweet dogs; therefore most Pit Bulls are sweet dogs.

• Sample: Stooler lectures I’ve heard so far• Population: All Stooler lectures I will hear• Attribute of interest: Being liked by me

• Sample: Pit Bulls I know• Population: Pit Bulls• Attribute of interest: being sweet dogs

© 2015 McGraw-Hill Higher Education. All rights reserved. 11

The more atypical the sample, the weaker the generalization

If the Pit Bulls I know were all trained to be show dogs, the argument is weaker.

Most Pit Bulls I know are sweet dogs; therefore most Pit Bulls are sweet dogs.

© 2015 McGraw-Hill Higher Education. All rights reserved. 12

The less diversified the sample, the weaker the generalization

First argument is weaker!

I know three Pit Bulls from the same litter. They are sweet dogs. Therefore most Pit Bulls are sweet dogs.

I know three Pit Bulls from different litters. They are sweet dogs. Therefore most Pit Bulls are sweet dogs.

© 2015 McGraw-Hill Higher Education. All rights reserved. 13

Generalizations based on samples too small to accurately mirror the population are inherently weak.

“Hasty generalization!”

I know three Pit Bulls from the same litter. They are sweet dogs. Therefore most Pit Bulls are sweet dogs.

This sample is too small to accurately mirror the entire Pit Bull Population

Note: If a population is known to be homogeneous, then it is safe to generalize from a small sample.

© 2015 McGraw-Hill Higher Education. All rights reserved. 14

Generalizations based on atypical samples are inherently weak even if they aren’t small.

“Generalizing from Exceptional Cases”! (Biased Generalization)

I know twenty Pit Bulls, all from the same parents. They are sweet dogs. Therefore most Pit Bulls are sweet dogs.

This sample is atypical!

© 2015 McGraw-Hill Higher Education. All rights reserved. 15

Scientific Generalizations

• Samples, populations, and attributes are clearly specified with a SAMPLING FRAME• Steps are taken to minimize bias (skew) in samples• Calculations are done using statistical mathematics

© 2015 McGraw-Hill Higher Education. All rights reserved. 16

Basic Concepts of Scientific Generalizing (1)

• True proportion—the proportion of a population that actually has an attribute of interest• Sample proportion—the proportion of a sample that

has an attribute of interest• Random Selection Process—a method for insuring

that every member of a population has an equal chance of being in the sample

© 2015 McGraw-Hill Higher Education. All rights reserved. 17

Basic Concepts of Scientific Generalizing (2)

• Error margin—the range of random variation of a sample proportion across multiple random samples• Confidence level—the probability that the random

variation of a sample proportion from random sample to random sample will fall within the error margin• Statistically significant—from a statistical point of

view, probably not due to chance

© 2015 McGraw-Hill Higher Education. All rights reserved. 18

Scientific Generalizations In a large population of widgets there is a TRUE

PROPORTION that are broken A random sample of that population will yield a SAMPLE

PROPORTION that approximates the true proportion Sample proportions vary randomly from sample to

sample (The “error margin”)

The larger the random sample (n), the more likely it is that the sample proportion will be close to the true proportion

© 2015 McGraw-Hill Higher Education. All rights reserved. 19

In a large population of widgets there is a TRUE PROPORTION that are broken

A random sample of that population will yield a SAMPLE PROPORTION that approximates the true proportion

Sample proportions vary randomly from sample to sample (The “error margin”)

The larger the random sample (n), the more likely it is that the sample proportion will be close to the true proportion

Scientific Generalizations

© 2015 McGraw-Hill Higher Education. All rights reserved. 20

In a large population of widgets there is a TRUE PROPORTION that are broken

A random sample of that population will yield a SAMPLE PROPORTION that approximates the true proportion

Sample proportions vary randomly from sample to sample (The “error margin”)

The larger the random sample (n), the more likely it is that the sample proportion will be close to the true proportion

Higher confidence levels require wider error margins!!!

Scientific Generalizations

© 2015 McGraw-Hill Higher Education. All rights reserved. 21

PUBLIC OPINION POLLS involve scientific sampling

How much we can take from them depends on what two things?

1. How well does the sample represent the population?

2. Are questions neutral?

Mathematical calculations done by trained pollsters are uncontroversial

© 2015 McGraw-Hill Higher Education. All rights reserved. 22

Reject error margin and confidence level indicators that exceed the strength of the supporting evidence.

• “Exactly,” “around,” “about” “approximately,” etc. are “error margin indicators”

• “Possibly,” “maybe,” “probably,” “almost certainly, etc. are “confidence level indicators”

Everyday generalizations are usually based on small and biased samples.

© 2015 McGraw-Hill Higher Education. All rights reserved. 23

Informal confidence-level indicators:

Informal error-margin indicators

• Possibly• Quite possibly• Probably• Very probably• I’d bet the farm

• Exactly• Around• Approximately• About• Give or take• In the ballpark of

© 2015 McGraw-Hill Higher Education. All rights reserved. 24

In real-life generalizations, your “confidence level” depends on your “error margin”

A safe bet matches a high confidence level with a wide error margin!

Don’t “bet the farm” a generalization holds, unless you give yourself a wide error margin.

© 2015 McGraw-Hill Higher Education. All rights reserved. 25

The Statistical Syllogism• An argument that applies a general statement to a

specific case.Examples:

Most teachers are Democrats.York is a teacher.:. York is a Democrat.

© 2015 McGraw-Hill Higher Education. All rights reserved. 26

Most teachers are Democrats; York is a teacher.:. York is a Democrat

• Premise-population: teachers• Conclusion-individual: York• Attribute of interest: being a Democrat

© 2015 McGraw-Hill Higher Education. All rights reserved. 27

The higher the proportion of the premise population that has the attribute of interest, the stronger the argument.

Stronger argument!

80% of teachers are Democrats; York is a teacher.:. York is a Democrat

90% of teachers are Democrats; York is a teacher.:. York is a Democrat