Ambiguous Middle

Alias:

Ambiguous Middle Term
Four-Term Fallacy
Taxonomy: Logical Fallacy > Informal Fallacy > Ambiguity > Equivocation > Ambiguous Middle < Four-Term Fallacy < Syllogistic Fallacy < Formal Fallacy < Logical Fallacy

Form:

Any validating form of categorical syllogism with an ambiguous middle term. For a short introduction to categorical syllogisms, see the entry for syllogistic fallacy.

Example Counter-Example
All human fetuses are human.
Any human is a being with a right to life.
Therefore, all human fetuses are beings with a right to life. All dog fetuses are canine.
Any canine is an animal that must be on a leash.
Therefore, all dog fetuses are animals that must be on a leash.
Exposition:

A categorical syllogism is, by definition, an argument with three categorical terms occurring within it. Each such term occurs in two statements in the argument, and the middle term is the one that occurs in both premisses but not in the conclusion.

Since each term occurs twice in a syllogism, if any term is ambiguous it is possible that it occurs with two different meanings. If the syllogism would be otherwise valid, it is said to commit the syllogistic four-term fallacy―that is, a single word may ambiguously stand for two terms. In effect, such an argument has four terms, which violates the definition of "categorical syllogism". Moreover, if a word or phrase in such an argument ambiguously represents two terms, the argument commits the informal fallacy of Equivocation.

For instance, in the Example above, the three terms are "human fetus", "human", and "being that has a right to life". "Human" is the middle term, because it occurs in both premisses. In the first premiss, it is an adjective meaning "of or belonging to human beings", whereas in the second it is a noun meaning "human being". Thus, the Example really has four terms rather than three.

The Counter-Example is an argument of the same form that is obviously fallacious, thus showing that the Example also commits the fallacy. In the Counter-Example, "canine" is the middle term and it is clearly used in two different senses in the premisses: in the first premiss, it is an adjective meaning "of or belonging to a dog", while in the second it is a noun meaning "dog".

Exposure:

Ambiguous Middle is an unusual fallacy that has both a formal and informal aspect that can be seen in the Taxonomy, above, which shows it as a type of both formal and informal fallacy. It is formal in that it is a subfallacy of the syllogistic fallacy of Four Terms, since Ambiguous Middle really has two middle terms instead of one. It is informal in that it is a subfallacy of Equivocation, since the fact that there are two middle terms is disguised by using a single word or phrase ambiguously.

Source: William L. Reese, Dictionary of Philosophy and Religion (Humanities Press, 1980), p. 169.

Reader Response:

Jim wrote in with a criticism of the Example used above―note that I subsequently changed the Counter-Example by replacing "organs" with "fetuses":

When someone says in an argument that "All human fetuses are human," I believe it is clear to most people that what they mean is "All human fetuses are humans," that is "are human beings," but have simply made a grammatical error that has become acceptable through common usage. To say that the term "human" in this statement is ambiguous is to be purposely obtuse. That you know that the word "human" in the first premise is meant to be taken as a noun and not as an adjective becomes clear in your counter example where you chose to substitute "dog organs" for "canine organs" to make the counter example clearer/more intelligible. The premise "All canine organs are canine" sounds silly whereas the premise "All human fetuses are human" does not. Your "counter example" becomes:
All dog organs are dogs.
Any dog must be on a leash.
Therefore, all dog organs must be on a leash. and is wrong because the first premise is wrong, not because of equivocation.
A couple of points in reply:

You seem to be under the misimpression that the argument in the Example is someone else's argument, but it is a cooked-up example, that is, it's my argument. However, it is based on some real-life arguments that I have encountered, but cleaned-up and made more explicit. For instance, the argument used as an Example of Equivocation is essentially the same as the above Example. What I have done is to take the argument and turn it into a categorical syllogism in order to illustrate the syllogistic fallacy of Ambiguous Middle, which is an equivocation occurring in a syllogism. Unfortunately, arguments as they occur in the raw often don't make good examples, especially of categorical syllogisms. I use real examples whenever possible, but the point of the examples and counter-examples is to help people understand the nature of the fallacy, and real world arguments are often poor examples. For this reason, the examples that I use are often too obviously fallacious; they would never fool anyone.
Arguments which commit a fallacy of ambiguity have two meanings. On one meaning, the argument is valid, but one of the premisses is false or controversial. On the other meaning, the premisses are uncontroversially true, but the argument is invalid. Here are the two meanings of the Example:
All human fetuses are the fetuses of human beings.
Any human being has the right to life.
Therefore, all human fetuses have the right to life.
The premisses of this argument are both uncontroversially true, but it is invalid.
All human fetuses are human beings.
Any human being has the right to life.
Therefore, all human fetuses have the right to life.
This argument—which is the way in which you are interpreting the Example—is a perfectly valid syllogism of the form Barbara, but it has a controversial first premiss. No one who disagreed with the conclusion would be likely to agree with the first premiss. For this reason, despite the fact that the argument is valid, it commits a different fallacy, namely, Begging the Question, that is, it has a question-begging first premiss.
The way in which ambiguous arguments are persuasive is by combining both these meanings into one, so that they seem to be both valid and to have uncontroversial premisses. But it is only by switching back and forth mentally between the two meanings of the ambiguous term that such an argument will seem to be sound.