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Formal Fallacies

Formal fallacies violate the rules of formal logic. They are structurally flawed, regardless of the specific content of the statements.

All formal fallacies are special cases of Non sequitur (Latin for "it does not follow").

1. Affirming the Consequent​

This fallacy has the following form:

  1. If A, then B.
  2. B is true.
  3. Therefore A is true.

Example:​

  1. If it is raining, the street is wet.
  2. The street is wet.
  3. Therefore it is raining.

Why is this flawed?​

There may be other reasons for a wet street (e.g. street cleaning, a burst water pipe). The fallacy lies in inferring the truth of the antecedent (A) from the truth of the consequent (B).

Venn Diagram​

Layer 1mR (rain)N (wet street)

2. Denying the Antecedent​

This fallacy has the following form:

  1. If A, then B.
  2. A is not true.
  3. Therefore B is not true.

Example:​

  1. If someone has a fever, they are ill.
  2. Max does not have a fever.
  3. Therefore Max is not ill.

Why is this flawed?​

There may be other reasons why someone could be ill, even without having a fever. The fallacy lies in inferring the falsity of the consequent (B) from the falsity of the antecedent (A).

Venn Diagram​

The Venn diagram has the same form as in the previous example.

Layer 1mF (fever)K (ill)

3. Quaternio Terminorum (Fallacy of Four Terms)​

This fallacy occurs in categorical syllogisms when a term is used with different meanings, so that the syllogism actually contains four terms instead of three.

Example:

  1. All stars shine in the sky.
  2. Some film actors are stars.
  3. Therefore some film actors shine in the sky.

Why is this flawed? The term "star" is used with two different meanings (celestial body vs. famous person). As a result, the syllogism actually contains four terms instead of three, which makes the logical structure invalid.

4. Fallacy of the Undistributed Middle​

This fallacy occurs in categorical syllogisms when the middle term is not used in full (distributively) in either premise.

Example:

  1. All dogs are mammals.
  2. All cats are mammals.
  3. Therefore all dogs are cats.

Why is this flawed? The middle term "mammals" is not used in full in either premise. The fallacy lies in inferring identity from a shared property (both are mammals).

Venn Diagram​

As you can see, once the non-mammal dogs and non-mammal cats have been excluded (hatching), there is no necessary overlap between dogs and cats, even though both are mammals. Our example m is a dog and a mammal but not a cat.
The information in the premises is not sufficient, however, to rule out animals that are both cats and dogs at the same time.

Layer 1mH (dogs)K (cats)S (mammals)