Deductive and Inductive Reasoning 1

Deductive and Inductive Reasoning 1

Deductive and Inductive Reasoning[1]

Deductive Arguments: Valid and Invalid, Sound and Unsound

A valid deductive argument is an argument such that if the premises are assumed true, it is impossible for the conclusion to be false. In such arguments the conclusion follows with strict necessity from the premises. Conversely, an invalid deductive argumentis a deductive argument such that if the premises are assumed true, it is possible for the conclusion to be false. In these arguments the conclusion does not follow with strict necessity from the premises.

A sound argument is a deductive argument that is valid and has all true premises. Both conditions must be met for an argument to be sound, and if either is missing, the argument is unsound. A sound argument is what is meant by a “good” deductive argument in the fullest sense of the term.

Compare the following two deductive arguments with the chart below (next page):

All automakers are computer manufacturers.

Union Carbide is an auto maker.

Therefore, Union Carbide is a computer manufacturer.

All banks are financial organizations.

Wells Fargo is a financial organization.

Therefore, Wells Fargo is a bank.

Note that the above arguments are in the form of a deductive syllogism, which consists of a major premise, a minor premise, and a conclusion. Each premise, respectively, consists of two terms: the major term and the middle term, and the minor term and the middle term. The conclusion connects the minor and the major term.

Below are the four kinds of premises, with examples of each:

universal affirmative - All humans are mortal.

universal negative - No humans are mortal.

particular affirmative - Some humans are mortal.

particular negative - Some humans are not mortal.

In addition, the middle term can be configured in four ways: upper left/lower right, upper right/lower right, upper left/lower left, and upper right/lower left. All these variations allow for 256 possible variations in the form of the syllogism, but only 24 of them are valid deductive syllogisms, which can be reduced to two standard forms.[2]

Table 1.1 Deductive Arguments

Valid / Invalid
True premises, true conclusion / All wines are beverages.
Chardonnay is a wine.
Therefore, Chardonnay is a beverage.
[sound] / All wines are beverages.
Chardonnay is a beverage.
Therefore, Chardonnay is a wine.
[unsound]
True premises, false conclusion / None exist. / All wines are beverages.
Ginger ale is a beverage.
Therefore, ginger ale is a wine.
[unsound]
False premises, true conclusion / All wines are soft drinks.
Ginger ale is a wine.
Therefore, ginger ale is a soft drink.
[unsound] / All wines are whiskeys.
Chardonnay is a whiskey.
Therefore, Chardonnay is a wine.
[unsound]
False premises, false conclusion / All wines are whiskeys.
Ginger ale is a wine.
Therefore, ginger ale is a whiskey.
[unsound] / All wines are whiskeys.
Ginger ale is a whiskey.
Therefore, ginger ale is a wine.
[unsound]

Inductive Arguments: Strong and Weak, Cogent and Uncogent

An inductive argument is one in which the premises are claimed to support the conclusion in such a way that if they are assumed true, then based on that assumption, it is only probable that the conclusion is true. If the premises do in fact support the conclusion in this way, the argument is said to be strong. Thus, a strong inductive argument is such that if the premises are assumed true, then based on that assumption, it is probable that the conclusion is true. Conversely, a weak inductive argument is such that if the premises are assumed true, then based on that assumption it is not probable that the conclusion is true.

Here are two examples of inductive arguments. The first is weak, the second strong:

This barrel contains one hundred apples.

Three apples selected at random were found to be ripe.

Therefore, probably all one hundred apples are ripe.

This barrel contains one hundred apples.

Ninety apples selected at random were found to be ripe.

Therefore, probably all one hundred apples are ripe.

As is evident from these examples, strength and weakness, unlike validity and invalidity, generally admit of degrees, but as with validity and invalidity, strength and weakness are only indirectly related to truth and falsity. The central question in determining strength or weakness is whether the conclusion would probably be true if the premises are assumed true.

A cogent argument is an inductive argument that is strong and has all true premises; if either condition is missing, the argument is uncogent. A cogent argument is the deductive analogue of a sound deductive argument and is what is meant by a “good” inductive argument without qualification.

Compare the following two inductive arguments with the chart below:

All meteorites found to this day have contained gold.

Therefore, probably the next meteorite to be found will contain gold.

During the past fifty years, inflation has consistently reduced the value of the American dollar. Therefore, industrial productivity will probably increase in the years ahead.

Table 1.2 Inductive Arguments

Strong / Weak
True premise
Probably true conclusion / All previous American presidents were men.
Therefore, probably the next American president will be a man.
[cogent] / A few American presidents were Federalists.
Therefore, probably the next American president will be a man.
[uncogent]
True premise
Probably false conclusion / None exist. / A few American presidents were Federalists.
Therefore, probably the next American president will be a
Federalist.
[uncogent]
False premise
Probably true conclusion / All American presidents were television debaters.
Therefore, probably the next American president will be a television debater.
[uncogent] / A few American presidents were Libertarians.
Therefore, probably the next American president will be a television debater.
[uncogent]
False premise
Probably false conclusion / All previous American presidents were women.
Therefore, probably the next American president will be a woman.
[uncogent] / A few American presidents were Libertarians.
Therefore, probably the next American president will be a Libertarian.
[uncogent]

[1] From Patrick Hurley, A Concise Introduction to Logic, 5th ed. , p. 42-48.

[2] See Howard DeLong, A Profile of Mathematical Logic, p.14-24.