# What is the purpose of "x" in the Venn-diagrams depicting categorical propositions?

See the I- and O-diagrams of this article. The "x" does not make sense the way it is formulated. Every member of the intersection is a member of the intersection, obviously.

The "x" cannot be supposed to mean that the intersection is inhabited by members. If the intersection was empty, there would either be no intersection in the diagram, or the intersection would be greyed-out. The existence of the open intersection necessitates the existence of intersectional members.

I guess the "x" is supposed to bring attention to the subset referred to by the diagrammed proposition, and maybe its existence implies that the subsets not enclosing the "x" are potentially uninhabited. This was not explicated in the article as far as I know, and if that's the case, then a far clearer language would simply be to add a third colour.

Consider this: white, grey, black.

If the proposition asserts a (sub)set is empty, that (sub)set is black. If the proposition asserts a (sub)set is inhabited, it is white. If the proposition does not make any assertions about a (sub)set, it is grey. This language would thus produce these Venn diagrams for I and O, the proposition being "Some cars are fast" and "Some people are not rocks":

This language would produce the following diagrams for "All cats are animals" and "No fish are birds":

Not only does this remove the nonsensical "x", but it also removes the homography between white meaning the (sub)set contains all the nouns of the proposition, and white meaning the (sub)set is not addressed by the proposition.

• The x does not stand for an intersection. Aug 28, 2022 at 14:38

The "X" means that the set is not empty. In other words, there exists at least one member of that set. The idea being, I imagine, that putting an "X" in that area indicates that something is in there.

For example, this is the diagram for "some A is B":

Which, of course, means "there exists some X that is both A and B", or "the intersection of A and B is not empty".

The existence of the open intersection necessitates the existence of intersectional members.

It does not. In general, the actual size of the intersection area in a Venn diagram doesn't really mean anything; what matters is what elements are placed in it. And that could very well be "nothing".

I guess the "x" is supposed to bring attention to the subset referred to by the diagrammed proposition, and maybe its existence implies that the subsets not enclosing the "x" are potentially uninhabited.

Correct.

Sure, you could also introduce a third color for this. But I think drawing something inside the set to indicate "this set is not empty"—or in other words, "there's something inside this set"—is clear enough, especially given the examples.

• I have serious problems with this language. Black stays consistent; the proposition claims that the black sets are necessarily uninhabited. It is unambiguous. A white set however, can either mean that the proposition is agnostic towards it, or that the set is the complete set of the noun. This ambiguity and the introduction of the x just complicates things unnecessarily. The notion of being the complete set of the proposition's noun is not something that needs a symbolism either; it can just be inferred from the set names and colors. Aug 28, 2022 at 22:24
• Are there other languages for this and what do people think about this language? Has my tricolor language been used before? Aug 28, 2022 at 22:25
• @user110391 I'm afraid I don't understand what "being the complete set of the proposition's noun" is supposed to mean. But there's no real standard for these diagrams; I imagine the author just made it up to illustrate their point on this one page. The standard notation for this uses symbols, like ∃x.A(x)∧B(x), not colors.
– Draconis
Aug 28, 2022 at 22:42
• The author ought to go back to school The page starts with a description of categorical sentences as "simple sentences composed in a noun-verb-direct object structure" then proceeds with "the verb .... is almost always be"!!! OP needs to find a better source. Aug 28, 2022 at 23:40