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Critical Thinking By Example

 Chapter 7: Sufficiency
  Quiz 7.1 Quiz 7.2

 

Material covered in this chapter

  • 7.1 Sufficiency and deductive and inductive arguments

  • 7.2 Fallacies of Sufficiency

 

Recall that we said that with good arguments it is possible to answer “yes” to each of these questions:

1. Assuming that the premise set is acceptable, is the premise set relevant to the conclusion?

2. Assuming that the premise set is acceptable, does the premise set provide sufficient support for the conclusion?

3. Is the premise set acceptable?

With bad arguments it is not possible to answer “yes” to each. In this chapter we will examine arguments where the answer to question 2 is “no”.

It is important to emphasize that in evaluating whether a premise set is sufficient to warrant the acceptance of a conclusion, we assume that the premise set is acceptable. In other words, to say that the premise set is sufficient is to say that, if true, the premise set provides good reason to believe the conclusion.

7.1 Sufficiency and Deductive and Inductive Arguments

Arguments are often divided into two types: deductive and inductive. With deductive arguments, the intention is to make the conclusion certain given that the premises are true. With inductive arguments, the intention is to make the conclusion probable, given the premises are true.

 

Example 7.1: Example of a deductive argument.

All humans are less than 8ft tall. Socrates is a human. So, Socrates is less than 8ft tall.

 

 

Example 7.2: Example of an inductive argument.

99.99% of all humans are less than 8ft tall. Socrates is human. So, Socrates is less than 8ft tall.

 

Example 7.1 is a deductive argument because the intention is to make the conclusion certain, if the premises are true. Example 7.2 is an inductive argument because even if the premises are true, they only make the conclusion highly probable, not certain.

Assessing sufficiency in a deductive argument is a relatively straightforward matter: in a deductive argument a premise set is sufficient only if the argument is valid. If a deductive argument is invalid, then the premise set is insufficient.

Sufficiency is harder to assess with inductive arguments. In part this is because of the fact that, unlike certainty, probability comes in degrees. It will help to examine first why inductive arguments do not claim certainty for the conclusion. Suppose there are 20,000 students at your university, and you randomly survey 9,999 of them. To your surprise, they all answer your questionnaire to the effect that marijuana should be legalized. You use this as a premise in an argument that concludes that the majority of students at your university believe that marijuana should be legalized. Despite the strength of your evidence, your argument is not valid; the conclusion is not certain even if the premises are true. In other words, it is possible that the premise is true and the conclusion is false. For example, suppose the other 10,001 students believe that marijuana should not be legalized, and you were very, very unlucky in whom you randomly sampled. You just happened to get exactly all and only those students who think it should be legalized. On the other hand, even though your argument is not valid, it is still a good argument. Assuming the premise is true, it gives us very strong reason to believe the conclusion is true.

How probable does the premise set have to be to support the conclusion? In practice we often accept different degrees of probability depending on the situation. In some legal jurisdictions (the U.S., for instance) different standards are applied in legal arguments in civil and criminal proceedings to meet the burden of proof. In civil proceedings the standard of proof is often the “preponderance of evidence”. If I am seeking financial compensation from you for my expensive garden gnome collection that I allege you destroyed, I would need to show that it is more probable than not that you were the responsible party. So, in effect, 51% probability that you are guilty is sufficient in civil court. In a criminal trial, the standard is often “beyond a reasonable doubt” and so 51% is not sufficient. In part, the difference in standards is explained by the severity of the penalties attached. In civil trials, the stakes are usually monetary, whereas in criminal proceedings at issue may be prison time, or even capital punishment. Since the costs of a wrongful conviction may be quite different in the two types of cases, we can see why it is good that there are different standards of probability.

Often it is possible to gain more evidence to strengthen an argument. This is not to say that we should always insist on doing everything we can to strengthen the premise set before accepting the conclusion of an argument. Suppose you argue that the best way to leave a building is by taking the east stairs and jumping from the second story. The fact that we could obtain further evidence for your argument by looking up the architectural plans of the building online is irrelevant if the extra time spent doing so would mean that we are engulfed by flames.

In general, we may say that the more probable the premises make the conclusion, the better the argument meets the sufficiency requirement. Whether an argument meets the sufficiency requirement will depend on a number of practical matters such as the consequences of accepting an argument, and the cost of obtaining additional evidence.

7.2 Fallacies of Sufficiency

 

Fallacy of Hasty Conclusion

An argument contains this fallacy if the premise set (even if true) is insufficient to warrant the acceptance of the conclusion.

Note: the remaining fallacies of sufficiency are specific forms of this fallacy. On quiz material, you should choose this fallacy only if none of the other six fallacies are appropriate.

Example 7.3

“All the polar bears we saw at the zoo today are white. So, many polar bears are white.”

 

Fallacy of Hasty Generalization

An argument contains this fallacy if the premise set (even if true) is insufficient to warrant the acceptance of the generalization.

Example 7.4

“All the polar bears we saw at the zoo today are white. So, all polar bears are white.”

Fallacy of Affirming the Consequent

This fallacy is committed when the necessary condition in a conditional argument is cited as sufficient for the conclusion.

Note that the fact that it is cloudy is relevant to the conclusion that it is rainy. Imagine someone trying to prove to you that it is rainy. The fact that it is cloudy would provide some evidence that it is rainy, even though the presence of clouds is not in itself sufficient evidence.

Example 7.5

P1: If it is rainy, then it is cloudy.

P2:  It is cloudy.

C: So, it is rainy.

 

Fallacy of Denying the Antecedent

This fallacy is present when the denial of the sufficient condition in a conditional argument is cited as sufficient for the conclusion.

Note that the premise set is relevant to the conclusion in example 7.4 The fact that it is not rainy provides some evidence that it is not cloudy, even though the absence of rain is not sufficient for it not being cloudy.

 

Example 7.6

P1: If it is rainy, then it is cloudy.

P2: It is not rainy.

C: It is not cloudy. 

 

Post Hoc Fallacy

This fallacy occurs when the fact that event A precedes event B in time is said to be sufficient evidence that A causes B.  

Although causes typically proceed effects in time, we can see in example 7.5 that establishing that one event proceeds the other in time is not sufficient to show that there is a causal relationship; in particular, it may simply be a coincident that I washed my car and then it rained.

Example 7.5

P1: I washed my car.

P2: It then rained.

C: Washing my car caused it to rain.

Fallacy of Jumping from Correlation to Causation

This fallacy occurs when the correlation of two event types, A and B, is cited as sufficient evidence that A causes B.

As example 7.6 illustrates, even if two events are perfectly correlated, if unemployment always goes down after the Democrats win, this in itself does not show that a Democratic victory is the cause of reduced employment. For example, it might be the case that there is a business cycle where unemployment regularly goes up and down, and that high unemployment causes the Democrats to win.

Example 7.6

P1: Every time the Democrats win, unemployment goes down.

C: The Democrats being in power causes lower unemployment rates.

 

 

Fallacy of Improper Sampling

This fallacy occurs when the sample cited in the premise set does not properly represent the population cited in the conclusion.

In example 7.7 we can see that the population described in the conclusion is women, that is, the conclusion says something about how women in general view their husbands’ contribution to household chores. The sample, however, is taken from subscribers of Women’s Good Housekeeping Magazine. We have no evidence that those that subscribe to this magazine are representative of women in general. Perhaps subscribers are wealthier than average, or they are in more “progressive” relationships. A better way to support the conclusion would be to take a random sample of women in general.

Example 7.7

P1: In a random telephone survey of subscribers, Women’s Good Housekeeping Magazine found that 93% of all married women believe that their husbands do an equal amount of the household chores.

C: It is false that women think that men do not do an equal amount of the household chores.

 

d
d

d


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