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Deductive argumentation is based on propositions that are true or false. Inductive argumentation adds another dimension by introducing the idea that a proposition can be probably true or probably false. The science of the measurement of probability is statistics. Statistical probability is the measure of how probable or improbable a proposition may be calculated to be, on a scale of fractions between zero and one. In making estimates of the probability of a proposition being true (or false), statisticians also usu- ally give a numerical measure of the probability of error, the proportion of times an estimate might turn out to be wrong. Critical argumentation is concerned not so much with the formulas and methods statisticians use to calculate numerical probabilities as with the application of these meth- ods to drawing probable inferences used in everyday reasoning that influence our thinking and deciding about how to proceed in everyday affairs.

As noted in chapter1, statistical generalizations have an exact number representing their probability value, but many inductive generalizations have no exact number attached. Instead, a proposition may be evaluated, in a less precise fashion, as having a ‘high’ or ‘low probability’. Or equiv- alently, it may be said that an event is likely to occur, is unlikely to occur, is very likely to occur, and so forth. To say that a proposition has a high

(low) probability is to say that a high (low) number, representing its sta- tistical value, can be attached to it, even if that number has not yet been calculated.

Probability is a property not only of individual propositions, but also of arguments. An argument is inductively strong where it is improbable that the premises are true and the conclusion is false. ‘Improbable’ means that the probability is low (but it is not specified, in general, exactly how low). It is assumed, as well, that the argument is not deductively valid. Otherwise, deductively valid arguments would simply be a special case of inductively strong arguments where the probability that the premises are true and the conclusion is false would equal zero.

The following two arguments illustrate the difference between deduc- tive and inductive argumentation.

PREMISE: All students who graduated from Godfrey College after 1995 took a course on critical thinking.

PREMISE: Bob was a student who graduated from Godfrey College after 1995.

CONCLUSION: Bob took a course on critical thinking.

This argument is deductively valid, provided, of course, that the first premise is taken as an absolute universal generalization. Now consider another argument.

PREMISE: Most students who graduated from Bohemond College after 1995 took a course on critical thinking.

PREMISE: Elaine was a student who graduated from Bohemond Col- lege after 1995.

CONCLUSION: Elaine took a course on critical thinking.

The second argument is different from the first in that the word ‘most’ is used instead of ‘all’, indicating a probabilistic (inductive) generalization, as opposed to a universal generalization. The second argument is induc- tively strong in the sense that if both premises are true, then it is probable that the conclusion is true. Or in other words, it is improbable that both premises are true and the conclusion false. Thus, inductive arguments are based on probability, not logical necessity. With an inductively strong argument, it is logically possible that all the premises are true and the conclusion false. It’s just that it is not probable.

6. Probability and Inductive Argument 67

Whether one should take a particular inference2_{used in a given case}

of argumentation as inductive or deductive is sometimes difficult to judge, and one has to pay careful attention to the language used. But the criteria for success, and hence the internal structure of each of the two types of argument, is inherently different. This difference of structure is certainly one important indicator in helping us to identify both kinds of arguments. The structure of the argument is the inferential link that joins the premises to the conclusion, and as noted above, there can be two kinds of link. A deductive argument is meant to be strict or tight, implying logical neces- sity in the way the conclusion is supposed to be drawn from the premises. In contrast, an inductive argument, being based on probability rather than necessity, is not meant to be absolute or strict in the same way. An argu- ment could be put forward as inductive, and even if it is very strong by that standard, it remains possible that the premises could turn out to be true and the conclusion false (even if that is improbable). So there is a clear structural difference between the two types of argumentation.

In many cases, the best way to distinguish between instances of the two types of inference is to look at the type of generalization each is based on. If an absolute generalization, such as ‘all’ or ‘every’, is used, then the inference is deductive in nature. If a statistical term giving a number representing a probability value is given, or if a term such as ‘most’, ‘many’, or ‘very few’ is given, then the inference is inductive in nature.

There is a common misconception that deductive argumentation is from the general to the specific, while inductive reasoning always goes from the specific to the general. Many inductive inferences, it is true, go from specific instances to a generalization, as in the following example.

PREMISE 1: Swan one is white.

PREMISE 2: Swan two is white. PREMISE 3: Swan three is white.

PREMISE 4: These three swans are representative of the population of swans in this area.

CONCLUSION: It is probable that the population of swans in this area is white.

2_{An inference, in the technical meaning of the term, is the reasoning process within the}

argument. In a looser sense used here, the term ‘inference’ may be used interchangeably with the term ‘argument’.

This argument represents a very common type of inductive reasoning from specific instances to a wider population of the sort that is called a sampling inference. The sampling inference is in fact so commonly used in and so typical of inductive reasoning that it is easy to presume that all inductive arguments are from specific premises to a generalization as the conclusion. But the example listed above about Bohemond College shows that, in other instances, an inductive argument can go from a general premise to a specific instance as the conclusion.

We have seen many examples above of deductive arguments that go from premises that are generalizations to a conclusion that is specific or from premises that are generalizations to a conclusion that is a general- ization as well. Yet it is also possible to have cases of deductively valid arguments that have specific instances as premise and have a general statement as the conclusion.

PREMISE: This fox ran over that hill.

CONCLUSION: It is possible for foxes to run over hills.

This inference is deductively valid. It would be inconsistent for the premise to be true and the conclusion false. Yet the premise cites a specific instance, while the conclusion is about foxes and hills generally. Contrary to what one might expect then, not all deductive inferences go from the general to the specific. Some go from general premises to general conclu- sions, and some from particular premises to particular conclusions. Still others go from specific instances as premises to a general conclusion.

EXERCISE 2.6

Determine which of the following arguments are deductive in structure and which are inductive in structure.

(a) All super-heroes defeat nasty villains. Batman is a super-hero. Therefore, Batman defeats nasty villains.

(b) Most super-heroes have a faithful sidekick. Batman is a super-hero. Therefore, Batman has a faithful sidekick.

(c) Nearly all super-heroes have a fatal weakness. Superman is a super- hero. Therefore, Superman has a fatal weakness.

(d) Seventy-six percent of Zorro fans are under twelve years old. Ricardo is a Zorro fan. Therefore, Ricardo is under twelve years old.

(e) Every villain meets his match by encountering a good guy. When a villain meets his match by encountering a good guy, in most cases

7. Plausible Argumentation 69

that good guy is a super-hero. Brutus is a villain. Therefore, when Brutus meets his match by encountering a good guy, that good guy will be a super-hero.

(f) This parrot is yellow. This parrot is green. Therefore, some parrots are different in color than others.

(g) This parrot is yellow. That parrot is yellow. Therefore some parrots are yellow.

(h) This parrot is green. That parrot is green. This third parrot is also green. The color of these three parrots is representative of the color of parrots here. Therefore, it is likely that parrots here are green.

(i) Ten balls have been removed from this urn, and all of them are white. The color of the balls removed is representative of the color of the balls in the urn. Therefore, all of the balls in the urn are white.