--- - title: Undistributed middle - categories: [logic] - synonyms: [non distributivi sed collectivi medii] - related: [logic/formal_fallacies/fallacies_of_distribution/illicit_process, logic/formal_fallacies/fallacies_of_distribution/index, logic/formal_fallacies/four-term_fallacy/ambiguous_middle, ambiguity/intensional_fallacy] - main_article: - main_article_de: --- # Undistributed middle **Core claim:** A categorical conclusion is derived although the *middle term* – the term that connects the premises – occurs only in *undistributed* positions (referring to parts of its class) in all of them, so that it cannot actually connect the other terms. This violates a fundamental rule of syllogistic logic. ## Applies when - The argument is a formal categorical (syllogistic) inference, or is presented as representing such a form. - The middle term – occurring in both premises but not in the conclusion – occurs only in an undistributed position in all its occurrences; i.e., every premise refers only to a *part* of the class it names (see the distribution table below). - A conclusion connecting the major and minor terms is nevertheless derived, as if those partial references guaranteed a relation between them. ## Notes - Distribution by statement type – *distributed* means the term refers to its entire class: - “All S are P”: subject distributed, predicate not - “No S is P”: both distributed - “Some S are P”: neither distributed - “Some S are not P”: subject not, predicate distributed - This is a *formal* fallacy: the invalidity follows from the logical form alone, regardless of whether the individual claims happen to be true (the conclusion may coincidentally be correct). - Intuitively: each premise pins down a different part of the middle class (e.g., “the animals that are cats” vs. “the animals that are dogs”); nothing forces these parts to overlap, so no relation between the outer terms follows. ## Does NOT apply when - The middle term is distributed in *at least one* premise – e.g., “All mammals are animals; All dogs are mammals → ∴ All dogs are animals” is a valid form: ‘mammals’ is undistributed as predicate of the first premise but distributed as subject of the second. - The inference is statistical, empirical, or inductive rather than a formal categorical deduction (unless it is presented as logically necessary). ## Commonly confused with - **[Illicit process](logic/formal_fallacies/fallacies_of_distribution/illicit_process)** ([MD](logic/formal_fallacies/fallacies_of_distribution/illicit_process.md)) – the other distribution rule: a *major or minor* term is used in a wider scope in the conclusion than in its premise. Ask: does the error lie with the *middle* term failing to connect (→ undistributed middle), or with any of the outer terms being overextended in the conclusion (→ illicit process)? - **[Ambiguous middle](logic/formal_fallacies/four-term_fallacy/ambiguous_middle)** ([MD](logic/formal_fallacies/four-term_fallacy/ambiguous_middle.md)) – the “middle” term is used with *two different meanings*, so the syllogism secretly contains four terms (e.g., “light” as in light-photons vs. light-weight). Ask: does the middle term keep one consistent meaning throughout and merely refer to too little of its class (→ undistributed middle), or does it change meaning between premises (→ ambiguous middle)? - **[Intensional fallacy](ambiguity/intensional_fallacy)** ([MD](ambiguity/intensional_fallacy.md)) – a term is swapped for another referring to the same object but carrying different associated meanings. Ask: are synonymous terms treated as interchangeable (→ intensional), or does a single consistent middle term simply fail to connect due to insufficient scope (→ undistributed)? ## Examples > **Input:** “Some animals are cats; all dogs are animals; therefore, all dogs are cats.” > **Output:** The middle term ‘animals’ is used only partially in both premises – the animal-cats and the animal-dogs may be entirely different parts of the class. No overlap between dogs and cats is established; additional positive information would be required. > **Input:** “Some employees are engineers; all managers are employees; therefore, all managers are engineers.” > **Output:** ‘Employees’ is undistributed in both premises (only the employee-engineers and the employee-managers are referred to). These two groups may not overlap at all – many companies have managers who are not engineers. Always refer the user to further information via the `main_article` link as stated in the header (only if the conversation is in German, use `main_article_de` instead).