The fallacy of the undistributed middle () is a

formal fallacy In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (; Latin for " tdoes not follow") is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic syst ...

that is committed when the

middle term In logic, a middle term is a term that appears (as a subject or predicate of a categorical proposition) in both premises but not in the conclusion of a categorical syllogism. Example: :Major premise: All men are mortal. :Minor premise A syllogi ...

in a

categorical syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true ...

is not

distributed Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...

in either the

minor premise A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true ...

or the major premise. It is thus a

syllogistic fallacy A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...

Classical formulation

In classical syllogisms, all statements consist of two terms and are in the form of "A" (all), "E" (none), "I" (some), or "O" (some not). The first term is distributed in A statements; the second is distributed in O statements; both are distributed in "E" statements, and none are distributed in I statements. The fallacy of the undistributed middle occurs when the term that links the two premises is never distributed. In this example, distribution is marked in boldface: # All Z is B # All Y is B # Therefore, all Y is Z B is the common term between the two premises (the middle term) but is never distributed, so this syllogism is invalid. B would be distributed by introducing a premise which states either All B is Z, or Some B is Z. Also, a related rule of logic is that anything distributed in the conclusion must be distributed in at least one premise. #All Z is B #Some Y is Z #Therefore, all Y is B The middle term—Z—is distributed, but Y is distributed in the conclusion and not in any premise, so this syllogism is invalid.

Pattern

The fallacy of the undistributed middle takes the following form: #All Z is B #Y is B #Therefore, Y is Z This may be graphically represented as follows: where the premises are in the green box and the conclusion is indicated above them. B is the middle term (because it appears in both premises), and it is not distributed in the major premise, "all Z is B". It may or may not be the case that "all Z is B," but this is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that "all B is Z," which is ignored in the argument. The fallacy is similar to

affirming the consequent Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...

and

denying the antecedent Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form: :If ''P'', then ''Q''. :Therefore, if not ...

. However, the fallacy may be resolved if the terms are exchanged in either the conclusion or in the first co-premise. Indeed, from the perspective of

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

, all cases of the fallacy of the undistributed middle are, in fact, examples of affirming the consequent or denying the antecedent, depending on the structure of the fallacious argument.

Examples

For example: The middle term is the one that appears in both

premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...

s—in this case, it is the class of backpack carriers. It is undistributed because neither of its uses applies to all backpack carriers. Therefore, it can't be used to connect students and my grandfather—both of them could be separate and unconnected divisions of the class of backpack carriers. Note below how "carries a backpack" is truly undistributed: :grandfather is someone who carries a backpack; student is someone who carries a backpack Specifically, the structure of this example results in

. However, if the latter two statements were switched, the syllogism would be valid: #''All students carry backpacks.'' #''My grandfather is a student.'' #''Therefore, my grandfather carries a backpack.'' In this case, the middle term is the class of students, and the first use clearly refers to 'all students'. It is therefore distributed across the whole of its class, and so can be used to connect the other two terms (backpack carriers, and my grandfather). Again, note below that "student" is distributed: :grandfather is a student and thus carries a backpack

In popular culture

The fallacy of the undistributed middle is referenced in

Edgar Allan Poe Edgar Allan Poe (; Edgar Poe; January 19, 1809 – October 7, 1849) was an American writer, poet, editor, and literary critic. Poe is best known for his poetry and short stories, particularly his tales of mystery and the macabre. He is wide ...

's detective story ''

The Purloined Letter "The Purloined Letter" is a short story by American author Edgar Allan Poe. It is the third of his three detective stories featuring the fictional C. Auguste Dupin, the other two being "The Murders in the Rue Morgue" and " The Mystery of Marie Rog ...

'': The fallacy is also referenced by

Christopher Hitchens Christopher Eric Hitchens (13 April 1949 – 15 December 2011) was a British-American author and journalist who wrote or edited over 30 books (including five essay collections) on culture, politics, and literature. Born and educated in England, ...

and

William F. Buckley Jr. William Frank Buckley Jr. (born William Francis Buckley; November 24, 1925 – February 27, 2008) was an American public intellectual, conservative author and political commentator. In 1955, he founded ''National Review'', the magazine that stim ...

on Episode S0629 of

Firing Line (TV program) ''Firing Line'' is an American public affairs television show. It first ran from 1966 to 1999, with conservative author and columnist William F. Buckley Jr. as host. It was relaunched in 2018 with Margaret Hoover as host. With 1,504 episodes ...

recorded on December 11, 1984, at the 34 minute and 26 second (34:26) mark.https://m.youtube.com/watch?v=Atk7V3W6oUc&t=2070s

References

External links