The fallacy of the undistributed middle () is a

formal fallacy In logic and philosophical logic, philosophy, a formal fallacy is a pattern of reasoning rendered validity (logic), invalid by a flaw in its logical structure. propositional calculus, Propositional logic, for example, is concerned with the meaning ...

that is committed when the middle term in a

categorical syllogism A syllogism (, ''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. In its earliest form (define ...

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 or the major premise. It is thus a

syllogistic fallacy A syllogism (, ''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. In its earliest form (define ...

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 No 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 In propositional logic, affirming the consequent (also known as converse error, fallacy of the converse, or confusion of necessity and sufficiency) is a formal fallacy (or an invalid form of argument) that is committed when, in the context of a ...

and

denying the antecedent Denying the antecedent (also known as inverse error or fallacy of the inverse) is a formal fallacy of inferring the inverse from an original statement. Phrased another way, denying the antecedent occurs in the context of an indicative conditional ...

. However, the fallacy may be resolved if the terms are exchanged in either the conclusion or in the first

co-premise A co-premise is a premise in reasoning and informal logic which is not the main supporting reason for a contention or a lemma, but is logically necessary to ensure the validity of an argument. One premise by itself, or a group of co-premises can ...

. Indeed, from the perspective of

first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

, 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 proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion. An argument is meaningf ...

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 cannot 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 (; January 19, 1809 – October 7, 1849) was an American writer, poet, editor, and literary critic who is best known for his poetry and short stories, particularly his tales involving mystery and the macabre. He is widely re ...

's detective story '' The Purloined Letter'':

References

External links