# Existential import in "No A is B"

I have doubts about the existential import in categorial propositions of the form "No A is B" under Aristotelian interpretation. In "No A is B", it is assumed that both A and B are not empty, right?

However when I take a look at the "No A is B" Venn diagram in the square of opposition diagram from Wikipedia, it tells me that in "No A is B", only A is guaranteed to be nonempty and we don't know whether B is empty or not. I think this interpretation is wrong because "No A is B" is logically equivalent to its converse "No B is A" so both A and B should be nonempty. In other words, the correct Venn diagram for "No A is B" should be something like

Am I wrong?

• Hi. Could you please clarify how this diagram tells you that A is guaranteed to be nonempty? When I click your link I am redirected to a diagram which is not the one in your post, and in this diagram the legend says that red areas are nonempty, but no part of A is red in the subdiagram representing "no A is B"
– Stef
Dec 11, 2023 at 19:52
• @Stef In term logic, the faded red area indicates A is not empty. (In modern logic, recolor the faded red area with white to remove existential import.) You can find more details here: en.wikipedia.org/wiki/Square_of_opposition Dec 12, 2023 at 0:37
• (I might be misunderstanding the diagram...) Dec 12, 2023 at 2:04
• I think you're definitely misunderstanding the diagram. That's the diagram for "Some (but not all) A are B -- and vice versa." Dec 12, 2023 at 16:04
• Aristotle goes through categorical propositions which discuss this and more I highly recommend reading it. I believe that "no a is b" does have existential import because there must be some a to which b could be assigned. Dec 16, 2023 at 13:14

The issue is a little bit tricky...

Existential import concerns

"the question whether a universal or A proposition such as "all buttercups are blue" implies the existence of its subject, i.e. whether it implies the existential proposition "blue buttercups exist".

[the modern historian] Terence Parsons argues that ancient philosophers did not experience the problem of existential import as only the A (universal affirmative) and I (particular affirmative) forms had existential import : "Affirmatives have existential import, and negatives do not. The ancients thus did not see the incoherence of the square as formulated by Aristotle because there was no incoherence to see."

He goes on to cite a medieval philosopher William of Moerbeke (1215–35 – c. 1286): "In affirmative propositions a term is always asserted to supposit for something. Thus, if it supposits for nothing the proposition is false. However, in negative propositions the assertion is either that the term does not supposit for something or that it supposits for something of which the predicate is truly denied. Thus a negative proposition has two causes of truth."

Trying to read it in modern terms, we have that an A is ∀x(Ax → Bx) while an E is ∀x(Ax → ¬Bx) and they have the same logical form (regarding quantifiers).

Thus, if we assume that A must be non-empty, we have to conclude that also B (in the first case) and ¬B in the second must be not-empty.

But for Aristotelian logic, "No A is B" is a basic statement of Syllogism and it is not "reducible" to an A form using a "not-B" predicate.

For Aristotelian logic, what matters for subalternation (to derive from A the corresponding I) is the subject A; in principle, the "complement of B can be not-empty while B itself being empty [re-write the predicate "Mortal" as "not-Immortal"; if we assume as domain the universe of creatures, we have that "No Philosopher is not-Immortal" is equivalent to the plain "Every Philosopher is Mortal", and we have that "not-Immortal" is instantiated, while "Immortal" is not].

It's arbitrary, but you probably don't want to give B any existential import because it would prevent you from saying things like, "no horse is a unicorn."

In modern mathematical usage, it would not have any existential import for either A or B.

In Aristotelian logic, if it is interpreted as having any existential import, it's only for A.

• That's exactly the example that came to my mind. Dec 12, 2023 at 16:06

I think you are correct. In traditional logic there are several forms of immediate inference called obversion, conversion and contraposition. Under the rules of conversion, "no A is B" is logically equivalent to "no B is A". So it would be reasonable to say that either existential import applies to both A and B or to neither.

That said, there are different interpretations of how Aristotelian logic should be understood. It is possible to treat all of the A, E, I and O propositions as having existential import, or just the A and I, or none of them. Under some of these interpretations, the rules of immediate inference don't all apply.

If you want to see a longer answer on existential import in Aristotelian logic, you might be interested in this answer.

• Perhaps I don't understand what it means for a proposition to "have existential import." Suppose A and B are respectively subject and predicate terms for some proposition with existential import. Existential import refers to the fact that A is nonempty? Or both A and B is nonempty? Dec 11, 2023 at 15:32
• It might depend on context. If you allow that all the rules of conversion and obversion hold, then it would have to be both. Otherwise, just A. If you adopt the fairly common interpretation of Aristotelian logic that it is not concerned with empty terms at all, then both A and B are assumed to exist. Dec 11, 2023 at 18:13

It is better to refer to the formulas with all-quantor and existence-quantor from SEP: 1.1 The Modern Revision of the Square. The formulas cover also the case S and/or P the empty predicate.

You find these formulas also in the wikipedia article, in the last column of the table below your diagram.

Quantification is always over the underling domain of discourse.

I have doubts about the existential import in categorial propositions of the form "No A is B" under Aristotelian interpretation.

Then you can stop here. Existential import is a notion entirely foreign to Aristotle's theory of syllogisms.

The notion was invented by 19th century philosophers and recycled by Bertrand Russell as a critique of Aristotle's syllogistic. Aristotle would have been very surprised.