Deduction

Deductive reasoning, or deduction, involves analyzing valid forms of argument and drawing out the conclusions implicit in their premises. There are several different forms of deductive reasoning, as used in different forms of reasoning problems.

In conditional reasoning the reasoner must draw a conclusion based on a conditional, or “if…then,” proposition. For example, from the conditional proposition “if today is Monday, then I will attend cooking class today” and the categorical (declarative) proposition “today is Monday,” one can infer the conclusion, “I will attend cooking class today.” In fact, two kinds of valid inference can be drawn from a conditional proposition. In the form of argument known as modus ponens, the categorical proposition affirms the antecedent of the conditional, and the conclusion affirms the consequent, as in the example just given. In the form known as modus tollens, the categorical proposition denies the consequent of the conditional, and the conclusion denies the antecedent. Thus:

If today is Monday, then I will attend cooking class today. I will not attend cooking class today. Therefore, today is not Monday.

Two other kinds of inference that are sometimes drawn from conditional propositions are not logically justified. In one such fallacy, “affirming the consequent,” the categorical proposition affirms the consequent of the conditional, and the conclusion affirms the antecedent, as in the example:

If John is a bachelor, then he is male. John is male. Therefore, John is a bachelor.

In another invalid inference form, “denying the antecedent,” the categorical proposition denies the antecedent of the conditional, and the conclusion denies the conclusion of the conditional:

If Othello is a bachelor, then he is male. Othello is not a bachelor. Therefore, Othello is not male.

The invalidity of these inference forms is indicated by the fact that in each case it is possible for the premises of the inference to be true while the conclusion is false.

It is important to realize that in conditional reasoning, and in all forms of deductive reasoning, the validity of an inference does not depend on whether the premises and the conclusion are actually (in the “real world”) true or false. All that matters is whether it is possible to conceive of a situation in which the conclusion would be false and all of the premises would be true. Indeed, there are valid inferences in which one or more of the premises and the conclusion are actually false:

Either the current pope is married or he is a divorcé. The current pope is not a divorcé. Therefore, the current pope is married.

This inference is valid because, although the premises and the conclusion are not all true, it is impossible to conceive of a situation in which all of the premises would be true but the conclusion would be false. Examples such as these demonstrate that the validity of an inference depends upon its form or structure, not on its content.

Reasoning skills are often assessed through problems involving syllogisms, which are deductive arguments consisting of two premises and a conclusion. Two kinds of syllogisms are particularly common.

In a categorical syllogism the premises and the conclusion state that some or all members of one category are or are not members of another category, as in the following examples:

All robins are birds. All birds are animals. Therefore, all robins are animals.

Some bachelors are not astronauts. All bachelors are human beings. Therefore, some human beings are not astronauts.

A linear syllogism involves a quantitative comparison in which each term displays either more of less of a particular attribute or quality, and the reasoner must draw conclusions based on the quantification. An example of a reasoning problem based on a linear syllogism is: “John is taller than Bill, and Bill is taller than Pete. Who is tallest?” Linear syllogisms can also involve negations, as in “Bill is not as tall as John.”