1.1 Deductive Argument

An argument consists of two or more premises and a conclusion. Since Aristotle, arguments have been divided into two categories: deductive and inductive. This chapter focuses on deductive arguments.

A deductive argument is one in which the truth of the premises guarantees the truth of the conclusion. In other words, if the premises are true, the conclusion must logically be true. Formally, this can be expressed as: If P, then Q. P. Therefore Q. The first statement, known as the major premise, announces a general rule. The second sentence, known as the minor premise, relates to a specific fact. The conclusion follows from the premises, which makes it a valid argument.

The classical example of a deductive argument is a syllogism, a common example of which is a definition. Formally, a syllogism is structured:

Major premise: B = A

Minor premise: C = A

Conclusion: C = B.

To be specific:

Major premise: A (B) mammal is (A) a warm-blooded vertebrate animal that possesses hair or fur and secretes milk.

Minor premise: A (C) platypus is (A) a warm-blooded vertebrate animal that possesses hair or fur and secretes milk.

Conclusion: A (C) platypus is (B) a mammal.

Note that the definition is compound since it requires that the living being in question meet several requirements: be warm-blooded, have a skeleton, possess hair or fur and secrete milk. Three of these are mandatory requirements; the other requirement consists of two alternatives. What matters for the truth of the deductive argument is that these requirements completely define what constitutes a mammal, such that they are present in all mammals. This is called the requirement of distribution, which means that all of the characteristics of the class are distributed to each member of the class, so that the definition generates a clear yes/no answer when it is applied.

What happens if the two terms in the major premise are not completely distributed; i.e., the definition is incomplete? You can end up with arguments like this:

Major premise: All cats are mammals.

Minor premise: All dogs are mammals.

Conclusion: All cats are dogs.

This is obviously an invalid deduction and yet it appears to follow the form of a syllogism. What’s the problem? The major premise is not completely distributed, since there are mammals that aren’t cats. If the term cats comprised all mammals, dogs could not be mammals, and thus the minor premise would be false. When the premises are false, the conclusion must also be false.

An argument that is valid and contains true premises is classified as sound. Soundness is the true test for arguments, which means that all deductive arguments must be separately evaluated for the truth of the premises as well as their compliance with logical form.

Now let’s examine another syllogism that meets the distribution requirement:

Major premise: New York City consists of five boroughs: Manhattan, Brooklyn, Queens, Staten Island, and the Bronx.

Minor premise: Kirk lives in New York City but does not live in Brooklyn, Queens, Staten Island, or the Bronx.

Conclusion: Kirk lives in Manhattan.

This is commonly called process-of-elimination reasoning, and formally referred to as argument by residue. Since the list of possibilities is complete, and there are only five alternatives, if four are excluded, then it follows that the fifth alternative must apply. And since both premises are true, then the argument is sound.

Let’s go back to Kirk, who lives in Manhattan borough. Most New Yorkers know that Manhattan borough consists of three islands: Manhattan Island (population 1,626,159), Randall’s Island (population 38,595) and Roosevelt Island (population 11,722). So what about the following deductive argument:

Major premise: Manhattan borough consists of three islands.

Minor premise: Kirk lives in Manhattan borough.

Conclusion: Kirk lives on an island.

Most New Yorkers would agree, but they would be wrong–because the major premise is incompletely distributed! Turns out that over a century ago, the Harlem River, which forms the northern boundary of Manhattan, was rerouted to improve access to the Hudson River. The part of Manhattan known as Marble Hill was separated from the island of Manhattan and became part of the mainland. But Marble Hill (population 70,296) remained part of Manhattan despite its geographic relocation to the mainland.  Therefore, the argument above is unsound because the definition of Manhattan borough does not fully describe the contents of that borough.

Using major premises that are fully distributed makes it possible to create a sound argument by residue:

Major premise: Manhattan borough consists of Manhattan, Randalls, and Roosevelt Islands, plus Marble Hill on the mainland.

Minor premise: Kirk lives in Manhattan but does not live on an island.

Conclusion: Kirk lives in Marble Hill.

Since the premises are true–and the major premise is distributed throughout all of Manhattan borough– this is a sound argument.

When people make arguments, they often leave out one or both premises because they assume the listener knows what they are. An argument with one or both premises unstated is known as an enthymeme. For example, “I passed my road test, so I can get my driver’s license.” Left unstated is the major premise that the road test is the last (and usually the most difficult) test that must be passed to obtain a driver’s license. Or “he’s a libertarian, therefore he’ll be against these new restrictions on public conduct.” The major premise is that libertarians oppose restrictive laws, but it is not true since there are restrictions that libertarians support.

Sometimes both the major and minor premises are known, so all that must be stated is the conclusion. If someone says, “It’s time to change the baby’s diaper,” or “it’s time to let the dog out,” is there any question that what the premises are? You might argue about the facts (“no, I’m cooking cabbage,” or “he just likes to chew on the leash”) but that would not affect the validity of the argument, just the soundness. You may not have thought that a casual comment like “It must be time for dinner” is an argument, but it is, even if the premises are uncertain! For one person, “dinner time” could be defined as “when I am hungry at the end of the day,” and for another, “whenever we are eating our main meal of the day.”