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.

• 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.

• 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.

• 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).

• 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 (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 (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)?

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).