Consider this argument: 'All mammals are warm-blooded. All dogs are warm-blooded. Therefore, all dogs are mammals.' Is this syllogism valid?
AYes, because the conclusion is actually true and both premises are true
BNo, because the middle term 'warm-blooded' is not distributed in either premise — it appears only as the predicate of A-statements, where predicates are not distributed
CYes, because having two true A-statement premises guarantees a valid A-statement conclusion
DNo, because a valid syllogism requires at least one E-statement (universal negative)
This is the classic fallacy of the *undistributed middle*. 'Warm-blooded' is the middle term (it appears in both premises but not the conclusion). In an A-statement ('All S are P'), only the subject S is distributed — we say something about all S's. The predicate P is *not* distributed: we don't say anything about all warm-blooded things. Since 'warm-blooded' appears only as a predicate in both A-statements, it is undistributed in both premises, and the required logical bridge between dogs and mammals is never established. The argument commits a validity error even though the conclusion happens to be true.
Question 2 Multiple Choice
In the A-statement 'All philosophers are mortal,' which terms are distributed?
ABoth 'philosophers' and 'mortal' are distributed, because the statement makes a universal claim
BNeither term is distributed, because A-statements are affirmative
COnly 'philosophers' is distributed — we speak about every philosopher, but we make no claim about all mortals
DOnly 'mortal' is distributed — the predicate of a universal statement is always distributed
In 'All S are P,' S is distributed: we say something about every member of S. But P is *not* distributed: we are not saying that everything in P is an S — only that the particular S's we're discussing are in P. 'All philosophers are mortal' says something about every philosopher but says nothing about all mortals (many mortals are not philosophers). Confusing this is the source of the undistributed middle fallacy: people assume that if both subjects share the same predicate, they must be related to each other.
Question 3 True / False
A categorical syllogism can be valid even if both of its premises are false — validity is a property of the argument's structure, not the truth of its content.
TTrue
FFalse
Answer: True
Validity means: if the premises were true, the conclusion would necessarily follow. It says nothing about whether the premises actually are true. 'All cats are robots. All robots are sentient. Therefore, all cats are sentient.' is a perfectly valid syllogism (mood AAA, figure 1 — Barbara) even though both premises are false. The conclusion follows necessarily from the premises by the logical structure. This separation of validity from truth is one of the foundational insights of formal logic.
Question 4 True / False
In an E-statement ('No S are P'), neither term is distributed, because the statement makes no positive claim about the members of either class.
TTrue
FFalse
Answer: False
In an E-statement, *both* terms are distributed. 'No S are P' says of every S that it falls outside P (distributing S), and equivalently says of every P that it falls outside S (distributing P). The statement makes a universal claim about all members of both classes — it excludes every S from P and every P from S. This is why E-statements are so logically powerful: they distribute both terms, and both can serve as the middle term in a valid syllogism.
Question 5 Short Answer
What is the middle term in a categorical syllogism, and why must it be distributed in at least one premise for the syllogism to be valid?
Think about your answer, then reveal below.
Model answer: The middle term appears in both premises but not in the conclusion. It is the logical bridge that connects the major term (predicate of the conclusion) to the minor term (subject of the conclusion). For this bridge to work, the middle term must be distributed in at least one premise — meaning at least one premise must make a claim about *all* members of that class. If the middle term is undistributed in both premises, the two premises refer to potentially different subsets of that class, and there is no guarantee they overlap in a way that supports the conclusion.
Consider a Venn diagram: the middle term's circle must fully overlap with at least one other term for the containment relationship to propagate to the conclusion. If 'warm-blooded' appears only as a predicate in two A-statements, we know only that dogs and mammals are each *subsets* of warm-blooded — but subsets of the same set need not overlap with each other. Distribution ensures the middle term spans the full class, creating the necessary logical link.