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Undistributed Middle

Also Known As: Fallacy of the Undistributed Middle Term
Formal Fallacy ID: undistributed_middle

Definition

A formal syllogistic fallacy where the middle term connecting two premises is never distributed (never refers to all members of its category). This means the two premises might refer to entirely different subsets of the middle term, making the conclusion invalid.

Examples

All dogs are animals. All cats are animals. Therefore, all cats are dogs.

All terrorists want to change society. All activists want to change society. Therefore, all activists are terrorists. (The middle term 'want to change society' is never distributed — it applies to both groups without establishing any exclusive link between them.)

All successful people work hard. All workaholics work hard. Therefore, all workaholics are successful people. (The shared middle term 'work hard' does not distribute across all members of either category, so no valid conclusion about the relationship between the two groups follows.)

Formal Logic Pattern
FOL Pattern
The First-Order Logic formula representing this reasoning pattern's logical structure.
FOL (First-Order Logic) uses quantifiers (∀ = for all, ∃ = there exists), connectives (∧ = and, ∨ = or, ⇒ = implies, ¬ = not), and predicates to capture the essential form of a reasoning pattern. For example, the Ad Hominem fallacy: Person(x) ∧ HasFlaw(x) ⇒ Invalid(Claim(x)). These patterns allow automated verification of logical validity.
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∀x(P(x)→M(x)) ∧ ∀x(S(x)→M(x)) ⇒ ∀x(S(x)→P(x))
Formal Verification:
Formal Verification
Checks whether a reasoning pattern is logically valid or invalid using an automated theorem prover.
Formal verification uses an SMT (Satisfiability Modulo Theories) solver — specifically Z3 — to mathematically check whether an argument's logical structure is valid. Each reasoning pattern is translated into First-Order Logic and tested: Can the premises be true while the conclusion is false? If yes, it's formally invalid. If no, it's formally valid. Many real-world patterns (analogies, heuristics) cannot be fully captured in formal logic — these are marked as not formally decidable, which doesn't mean they're wrong.
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Formally invalid

Verification Steps
Verification Steps
Binary yes/no questions that an AI must answer to detect a reasoning pattern in a text.
Each of the 452 aspects has verification steps — simple yes/no questions designed to systematically detect whether a pattern appears in a text. For ad hominem: "Does the argument attack a person rather than their claim?" For false dichotomy: "Are only two options presented when more exist?" This ensures consistent, reproducible analysis.
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Binary (yes/no) questions an LLM must answer to identify this aspect:

  1. 1

    Does the argument have a syllogistic structure with two premises and a conclusion?

    Type: binary
  2. 2

    Is there a middle term that appears in both premises but not in the conclusion?

    Type: binary
  3. 3

    Does the middle term fail to cover all members of its class in at least one premise (i.e., is it undistributed)?

    Type: binary
  4. 4

    Does the conclusion assert a connection between the two end terms that is not logically guaranteed?

    Type: binary
Deep Dive
The expandable detail section on each aspect page with examples, psychology, and counter-strategies.
The Deep Dive section provides in-depth information about each aspect: a real-world example showing the pattern in action, an explanation of why it works psychologically, practical advice on how to counter it, alternative names, and links to related aspects.
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Related Aspects

← related to
Ambiguous Middle Term

The ambiguous middle term fallacy occurs in syllogistic reasoning when the middle term — the term that connects the two premises but does not appear in the conclusion — is used with two different meanings. Because the middle term does not actually denote the same category in both premises, the syllogism effectively has four terms instead of three, breaking the logical connection that makes the syllogism valid. It is a specific instance of the fallacy of four terms, distinguished by the ambiguity residing specifically in the connecting term.

Hierarchical Context

→ sub type of Formal Fallacies

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