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We all know the most famous syllogism introduced by Aristotle:

All men (people) are mortal. Socrates is a man. Therefore Socrates is mortal.

But what if we say:

All mortals are men (people). Alice is mortal. Therefore she's a man.

Assume it to be true that all mortals are men (of course it isn't true in the real world but suppose there is such a world). She clearly belongs to mankind. But she is no man. Two different meanings of the word "man" are used. One encompassing both men and women and one only men. Is there a name for such a syllogism? An inconsistent syllogism maybe?

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    Using the same term to mean different things and thereby getting wrong conclusions is known as an equivocation fallacy in logic, but I don't think there's a specific name for a syllogism with an equivocation fallacy.
    – Hypnosifl
    Commented Jul 9, 2021 at 4:57
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    Your syllogism contains a false premise i.e., that all mortals are men. Even if we correct the wording to say instead, "all mortals are human", it is still false, and so the conclusion is false as well.
    – niels nielsen
    Commented Jul 9, 2021 at 5:23
  • @nielsnielsen Hi! But suppose all mortals are human. Or maybe: all speaking creatures are men. Sophia speaks (god forbid!). Therefore she's a man.
    – Deschele Schilder
    Commented Jul 9, 2021 at 5:36
  • It is formally correct, but one of the premises is false, thus there is no guarantee that the conclusion will be true. You have to review the definition of valid argument.
    – Mauro ALLEGRANZA
    Commented Jul 9, 2021 at 5:56
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    The formal fallacy you present is called officially the fallacy of four terms.There are generally six known rules for determining the validity of a categorical syllogism.Each rule corresponds to a specific fallacy found in categorical syllogisms. Mathematical logic involves symbolization. Aristotelian logic is closer to Rhetoric than mathematics. There are two schools of thought about validity.The older context did not allow false premises from the start. Mathematical logic allows false premises. Soundness is a valid argument with all true premises Go for soundness in categorical syllogisms.
    – Logikal
    Commented Jul 9, 2021 at 10:21

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How is the syllogism in my question called?

Fallacious. It is based on equivocation, that we can can avoid rewriting it as follows:

"All mortals are males. Alice is mortal. Therefore she's a human."

What we get is not a valid syllogism; see Fallacy of four terms.

The definition of "valid syllogism" is about a FORMAL linguistic pattern.

See Barbara:

"Every A is B; every B is C. Therefore: every A is C".

Valid means that every time we plug in terms (predicates, class names) for the schematic letter, IF the premises are TRUE, also the conclusion is TRUE. There is no way to interpret the definition otherwise.

IF we assume that "All mortal are men" is TRUE and Alice is mortal, then necessarily Alice must be a man ("all" means all).

See also Aristotle, On Sophistical Refutations:

[164a] That some deductions are genuine, while others seem to be so but are not, is evident. This happens with arguments... [165b] There are two styles of refutation; for some depend on the language used, while some are independent of language. Those ways of producing the illusion which depend on language are six in number: they are homonymy, ambiguity, combination, division, accent, form of expression. [169a] The error comes about in the case of arguments that depend on homonymy [...] because we are unable to distinguish the various senses (for some terms it is not easy to distinguish, e.g. one, being, and sameness).

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  • But it can be both true that Alice belongs to all men and that she is mortal.But she is still a woman. Which is a kind of man too. A wo-man. So she belongs to man-kind.
    – Deschele Schilder
    Commented Jul 9, 2021 at 9:32
  • @DescheleSchilder the crucial point is that in your syllogism, the word "man" is being used with two different meanings. Therefore, functionally your syllogism has four terms. Compare (from the link Mauro gave): nothing is better than eternal happiness; a ham sandwich is better than nothing; ham sandwiches are better than eternal happiness
    – legionwhale
    Commented Aug 10, 2021 at 13:22
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This syllogism is valid but not sound. "Valid" means the truth of the premises guarantees the truth of the conclusion. "Sound" means all the inferences are valid (i.e. truth-preserving) and all the premises are true (in which case the conclusion is true). The syllogism is unsound because the first premise ("All mortals are men") is false. Aristotle's syllogism is sound because the premises (e.g. "All men are mortals") are true and the inference is valid. The syllogism you presented for consideration would be called the same thing as Aristotle's syllogism (there are many names for it, but I guess it would be "Modus Darii;" same name for same formally valid inference). If the conclusion was derived from two premises which contain a term (e.g. "man") that was used in two different senses in those two premises, that would be the equivocation fallacy. Equivocation is an informal fallacy because it's impossible in symbolic (i.e. formal) logic, because "P" always means only "P," whereas in natural languages any proposition or term of a proposition can be interpreted in multiple senses. But there's no equivocation in the syllogism presented here, because "man"/"men" only appears in one of the premises, so it can't have different senses in different premises. The syllogism just has a false premise and a false conclusion (assuming "Alice" is in fact a woman) that follows from it through a valid inference.

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