A Quartet of Hypothetical Syllogisms

Hypothetical claims are conditionals, or “If . . .then . . .” claims. A syllogism is an argument with two premises. Examining the basic types hypothetical syllogism can help one get a firm grip on how conditionals operate in deductive logic. We want to consider both what can and what can’t be validly inferred from a conditional. So we will look at both valid and invalid syllogisms using conditionals.

Two Valid Forms

An argument with a deductively valid logical form is one where the truth of all of its premises would guarantee the truth of its conclusion. The deductive validity of an argument is determined by its logical form. So once we know that an argument form is deductively valid we know that any argument having that form is also deductively valid. Modus Ponens and Modus Tollens are two deductively valid forms of hypothetical syllogism.

Modus Ponens:

1. If P then Q.
2. P               .                                                 
3. Therefore Q.

Example

1. If Michael studied for the test, then he passed the test.
2. Michael studied for the test.
3. Michael passed the test.

Modus Tollens:

1. If P then Q.
2. Not Q.
3. Not P.

Example

1. If Michael studied for the test, then he passed the test.
2. Michael did not pass the test.
3. So Michael did not study.

These two hypothetical syllogism forms are often used as in logic textbooks as fundamental or basic valid argument forms that can be used to justify further inferences is proving the validity of more complex argument forms.

Two Formal Fallacies

An argument with an invalid argument form commits a formal deductive fallacy. Fallacies are just mistakes in reasoning. Some fallacies, usually ones that are commonplace and particularly easy to commit, are known by name. This is the case with both of our invalid hypothetical syllogisms:

Affirming the Consequent:

1. If P then Q
2. Q             .
3. P

Example

1. If it is raining, Russ is carrying his umbrella.
2. Russ is carrying his umbrella.
3. So it is raining.

Denying the Antecedent:

1. If P then Q
2. Not P    .
3. Not Q.

Example

1. If it is raining, Russ is carrying his umbrella.
2. It is not raining.
3. So Russ is not carrying his umbrella.

To see that these hypothetical syllogisms have invalid argument forms, consider that, knowing that it rains suddenly an often in Seattle, I carry an umbrella most of the time as a matter of habit. So, both of the example arguments for these two formal fallacies often have true premises and a false conclusion. The first premise is generally true. But it is often the case that I am carrying my umbrella and it is not raining.

The names and forms of these two formal fallacies are easy to remember if you know the parts of a conditional. In a standard form “If . . . then . . .” claim, the proposition attached to the “if” is called the antecedent and the proposition that follows “then” is called the consequent. Now notice what the second premise does in each of the fallacious argument forms. The second premise in affirming the consequent does just that, it affirms the consequent of the conditional given in the first premise. Likewise, the second premise in denying the antecedent does just that, it denies the consequent of the conditional given as the other premise.