Questions: Middle Term Distribution and Validity Rules

Question 1

Consider: 'All mammals are warm-blooded; all birds are warm-blooded; therefore, all birds are mammals.' Why is this syllogism invalid?

• A. The premises are false — not all mammals are warm-blooded
• B. The conclusion is false — birds are not mammals
• C. The middle term 'warm-blooded' is never distributed — it appears as the undistributed predicate in two A-statements, so the premises only establish overlapping subsets without guaranteeing connection
• D. The syllogism has four terms rather than three, violating the structural requirements of a valid syllogism

Answer: (C)

This is the classic undistributed middle fallacy. 'Warm-blooded' is the middle term, and it appears as the predicate of two A-statements ('All mammals are warm-blooded' and 'All birds are warm-blooded'). In an A-statement, only the subject is distributed — the predicate is not. So the premises establish that all mammals fall within *some* warm-blooded things, and all birds fall within *some* warm-blooded things, but these subsets need not overlap. The middle term never covers all warm-blooded things, so it fails to link mammals and birds. Importantly, the invalidity is about logical form — even if the conclusion happened to be false, the form would still be invalid regardless of content.

Question 2

A syllogism's conclusion is 'No politicians are trustworthy' (an E-statement, which distributes both subject and predicate). What must be true of the premises for this conclusion to be drawn validly?

• A. Both 'politicians' and 'trustworthy' must appear somewhere in the premises
• B. The middle term must appear as the subject in both premises
• C. 'Trustworthy' must be distributed in at least one premise, to avoid illicit process of the predicate term
• D. The conclusion must follow from at least one affirmative premise

Answer: (C)

The second distribution rule states: if a term is distributed in the conclusion, it must be distributed in its corresponding premise. The conclusion 'No politicians are trustworthy' distributes 'trustworthy' (E-statements distribute both terms). So 'trustworthy' must also be distributed in whichever premise it appears in. If the premise only said 'Some trustworthy people are X,' it would only warrant a claim about *some* trustworthy things — but the conclusion makes a claim about *all* trustworthy things. Sneaking in a claim about all when the premise only warranted some is the illicit process fallacy.

Question 3

In an A-statement ('All S are P'), the subject term S is distributed but the predicate term P is not.

• T. True
• F. False

Answer: True

An A-statement claims that every member of S belongs to P — so we are making a claim about all of S (S is distributed). But we are not claiming anything about all of P: other things besides S may also be P. The statement doesn't exhaust or fully cover P's membership. The predicate is undistributed because the claim doesn't range over all P. This asymmetry is why 'All cats are mammals' distributes 'cats' but not 'mammals' — we aren't saying mammals are only cats.

Question 4

For a valid categorical syllogism, the middle term should be distributed in both premises.

• T. True
• F. False

Answer: False

The rule requires the middle term to be distributed in *at least one* premise — not necessarily both. If at least one premise covers all members of the middle term category, that is sufficient to guarantee genuine logical linkage between the major and minor terms. Requiring distribution in both premises would be overly strict and would invalidate many correct syllogisms. The undistributed middle fallacy occurs when the middle term is distributed in *neither* premise.

Question 5

Why does the middle term need to be distributed at least once? What goes wrong logically when it isn't?

Model answer: The middle term's job is to link the major and minor terms by establishing a genuine logical connection. If neither premise distributes the middle term, each premise only makes a claim about *some* members of the middle category — but those 'some' members could be entirely different subsets with no overlap. The first premise links the minor term to part of M; the second links the major term to a potentially different part of M. Without at least one premise covering all of M, there is no guarantee that the two parts overlap, so no valid inference can be drawn.

The classic example illustrates this: 'All cats are mammals; all dogs are mammals.' Both premises talk about some mammals (the mammal-subsets containing cats and dogs respectively), but those subsets could be disjoint subsets of the larger class of mammals. The conclusion 'All cats are dogs' does not follow. Distribution of the middle term in at least one premise is what forces genuine overlap — it guarantees that the middle term is covering its entire extension in at least one premise, ensuring real connection between the other two terms.