Logic and
Critical Thinking

Philosophy 4

Gale Justin

u The objective of today’s class is learn how to recognize and evaluate two kinds of inductive arguments.

u Inductive Generalizations

u Arguments based on analogies

u Here is an Inductive Generalization (IG) based on a sample:

This spoonful of soup has enough salt in it. So the soup itself is salty enough.

u In this IG:

u The property we care about is saltiness.

u The sample is the spoonful of soup.

u The whole pot of soup is the population that we are sampling.

u So in general:

u Sample is that part of the whole thing that we want to find something out about.

u Population is the “big thing” that we want to investigate.

u Putting this argument into standard form, it looks like this:

Premiss: The spoonful of soup has enough salt in it.

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Conclusion: So the soup in the pot is salty enough.

u In IG’s:

The premisses and conclusion are based on observation, either our own or someone else’s. The conclusion is a statement about the way the world (or some part of the world) is and the premisses are reports about what we (or someone else) has observed about a sample of that portion of the world we are generalizing about.

u The general form of an IG is:

P: Sample has property P.

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C: Therefore all (or some or some percentage of) the population has property p.

u Special names are given to the propositions or statements that compose a philosophical argument.

u The statements that are regarded as offering reasons are called premisses.

u The one statement that is being recommended on the basis of the reason(s) is called the conclusion.

u With respect to the conclusion of an IG:

u If it begins with “all”, then it is a universal inductive generalization.

u If it begins with “some” or implicitly refers to only some of the population, then it is a non-universal inductive generalization.

u If it begins with a percentage, then it is a statistical inductive generalization.

u Here are examples of the three kinds of conclusions:

u All pine trees keep their needles in winter.

u Some of the wars fought by the U.S. have been won.

u 70% of the U.S. population own more than one car.

u With respect to the premiss of an IG:

u It can be a condensed premiss that refers collectively to all members of the sample.

u It can be expanded into an enumeration of all of the examined cases.

u For example:

Premiss: All of the pine trees that I know of have kept their needles in winter.

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Conclusion: All pine trees keep theirneedles in winter.

u Alternatively:

P1: America won the Revolutionary War.

P2: American won the Spanish American War.

P3: American won the first and second World Wars.

_______________________________

Conclusion: America has usually won its wars.

u Here’s a statistical IG:

Premiss: 80% of Sacramento State students have tried alcohol.

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Conclusion: 80% of U.S. college students have tried alcohol.

u Evaluating an IG:

The most important consideration in evaluating an I.G. is the idea of representivenes. Is the sample representative of the population that you want to generalize about? Sample size is a secondary consideration because if the sample is representative, it need not be very large.

u Note well:

Sample size is not that important, if the sample is representative of the population that you want to generalize about. For example, if you want to know at what heat copper expands, you only need to test two pieces of copper. The first is to determine the temperature at which expansion occurs. The second is needed to check your first test result.

u In general,

u The more diversity there is in the population about which you wish to generalize, the more individuals you will need to sample in order to get a representative sample. But being representative sample is not the same as being a large sample. That is, you cannot make a sample representative just by increasing its size.

u For example,

If I want to know the percentage of democratic leaning voters in the Sac. State student body; and, I have sampled half of the philosophy majors, I will not get a more representative sample by sampling more philosophy majors. I have to sample other majors in order to know what percentage of the student body as a whole is democratic leaning.

u Another example of representativeness:

u One spoonful of soup is representative of the whole pot, if the property that interest you is saltiness.

u A relatively small sample of the U.S. population is sufficient to generalize about the entire U.S. population, if the sample is matched to the population.

u So, as a rule it can be said:

“Size is a less important quality of a sample than its being representative if you want to avoid the charge of hasty generalization. So when you consider a sample, the important question to ask yourself is whether the sample is representative of the population that you want to generalize about.

u Example:

An instructor wishes to know how well a large class has understood a difficult homework assignment. He samples by randomly selecting three students to write out their homework on the board (without looking at each other’s work). They all get it right. The students in the class have understood the homework assignment.

u The I.G.’s structure:

P1: Student A who was randomly selected got his homework assignment right.

P2: Student B who was randomly selected got her homework right.

P3: Student C who was randomly selected got her homework right.

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C: Therefore, all the students in the class got their homework right.

u An alternative structure:

P: All three students who were randomly got their homework right.

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C: Therefore, all the students in the class have understood the homework assignment.

u Ask yourself:

(A) Is this reasoning inductively strong?

(B) Why?

(C) Would the reasoning be equally strong, if the instructor had called on students who raised their hands?

(D) Why or why not?

u A sample can fail to be representative because:

u The sampling procedure is faulty. For example, only cookies from the top of a cookie jar are sampled.

u The sample is too small to have a chance of being representative. For example, only 7 Sacramento State students are sampled to generalize about the student body’s alcohol consumption.

u Analogical Arguments:

u Also called “reasoning by analogy.”

u Are based on analogies, which is a comparison between two or more things. For example, finding an honest person is like finding a needle in a haystack.

u Analogical Arguments:

Starts from the premiss that items are similar in certain respects and goes on to conclude that the items are also similar in some further, as yet unobserved, respect.

u The form of an anological argument:

P1: Just as item of type X has properties F and G so also item of type Y has properties F and G.

P2: Item of type X also has the property H.

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C: So it is likely that item of type Y also has property H.

u For example:

Just as you pay for your groceries, so also you pay for your education. With regard to groceries you decide what to buy. So you ought to be able to decide the courses of your educational training.

u In evaluating an analogical argument, it is useful to distinguish the items being compared:

Let us call the item that we are familiar with, for example grocery shopping, the analogue. Let us call the item that we want to draw our conclusion about, for example, our course of educational training, the primary subject.

u In evaluating an analogical argument:

The most important question to ask is whether there any relevant dissimilarities between the analogue and the primary subject. If these are different in a relevant respect, then the reasoning is not inductively strong.

u A difference is relevant if:

Its presence decreases the probability that the primary subject has the as yet unobserved additional feature that the t the analogue is observed to have.

u For example:

Analogue: grocery shopping

Primary Subject: Education

Argument:

You pay for your groceries and for your education. Since you can choose what groceries you want to buy, you should be able to choose what courses to take.

Are there any relevant dissimilarities between the analogue and the primary subject?

u Homework and Groupwork:

Do Homework Exercise Sheet #10.

Do Groupwork # 7 now.