# Colored areas in categorical proposition Venn diagrams

This recent question prompted me to read about the square of opposition and I am confused about the purpose of red and white regions in Venn diagrams from this diagram. I think these Venn diagrams are misleading.

According to the description, the regions colored red signifies it is not empty and the regions colored light red signifies it is not empty in classical logic only. I am uncertain if red regions indicate existential import or not.

If they do indicate existential import, then there is no purpose for white regions? From the linked question, it seems like the conventional interpretation of Aristotelian logic demands that both subject and predicate classes are non-empty for all four categorial proposition types (if we allow all valid immediate inferences to hold). This means all white regions will have at least one member, i.e. not empty under that conventional interpretation. In that case, all white regions should be colored faint red.

• As specified on the diagram red areas are nonempty both in traditional and modern logic. In both views particulars asserting that some S is P means that there is an object that has both properties. Commented Dec 13, 2023 at 7:42

The square shown in the linked diagram is not incorrect as such, but it is describing one particular interpretation of traditional syllogistic logic.

Black areas must be empty, white areas may be empty or non-empty.

In modern logic, the red areas must be non-empty, but the pink areas may be empty or non-empty.

In traditional syllogistic logic, it depends on the way you interpret it. The most common way of understanding syllogistic logic is to suppose that all terms are assumed to be non-empty. Some commentators explain this by claiming that Aristotle is not concerned with talking about things that do not exist. On this view, the red and pink areas must all be non-empty.

A second interpretation is to suppose that the left side of the square has existential import but the right side does not. On this view the pink SaP and red SiP must be non-empty, but the pink SeP and the red SoP may be empty or non-empty.

A third interpretation is to suppose that all of the pink and red areas may be either empty or non-empty. This view separates assumptions about existence from what is being predicated. We might understand this option as expressing how the concepts overlap, but without any commitment to whether the concepts have instances.

I explain these options in a little more detail in my answer to this question.

Concerning your point about whether the predicate must be instantiated: Aristotle himself does not appear to require it. Bear in mind that the rules of immediate inference, i.e. conversion, obversion and contraposition, while taught as part of traditional logic, were later additions. Depending upon which of the above interpretations you adopt, some of the forms of immediate inference do not hold.

• "Bear in mind that the rules of immediate inference, i.e. conversion, obversion and contraposition, while taught as part of traditional logic, were later additions." Are you sure about it? See A's Logic: Methods of Proof: “Perfect” Deductions, Conversion, Reduction: "A direct deduction is a series of steps leading from the premises to the conclusion, each of which is either a conversion of a previous step or an inference from two previous steps relying on a first-figure deduction. 1/2 Commented Dec 13, 2023 at 15:40
• Conversion, in turn, is inferring from a proposition another which has the subject and predicate interchanged. Specifically, Aristotle argues that three such conversions are sound: [...] Aab → Iba. He undertakes to justify these in An. Pr. I.2. From a modern standpoint, the third is sometimes regarded with suspicion. Using it we can get Some monsters are chimeras from the apparently true All chimeras are monsters." 2/2 Commented Dec 13, 2023 at 15:41
• @MauroALLEGRANZA Fair point. In which case, Aristotle is committed to the equivalence of "no S is P" with "no P is S". So the predicate term must also be non-empty, or more accurately, either both subject and predicate are non-empty or both are empty. So the linked diagram is definitely misleading. Commented Dec 13, 2023 at 18:57
• Not very clear... but yes Eab converts to Eba (this is intuitive). See e.g. John Corcoran, A Mathematical Model of Aristotle’s Syllogistic (1973), page 203: "We require the extension of each term to be non-empty because this gives the best fit with Aristotie's inferences and because he seems to require that each meaningful common noun subsume at least one individual (Categories, 2a34-b7)" Commented Dec 14, 2023 at 6:52
• That seems reasonable, at least on the first interpretation. But, e.g., Stephen Read defends the second interpretation that A and I propositions have existential import, but E and O do not. On this view some forms of immediate inference fail. In particular with obversion you cannot go from "some S is not P" to "some S is not-P" since the latter entails a non-empty S, while the former does not. That is why I would favour the third interpretation. Or better still just use modern logic. Commented Dec 14, 2023 at 9:32

The modern point of view is expressed by Bertrand Russell into The Existential Import of Propositions (Mind, Jul.1905, pp. 398-402):

A. All S is P = For all values of x, 'x is an S' implies 'x is a P' .

E. No S is P = For all values of x, 'x is an S' implies 'x is not a P'.

I. Some S is P = For at least one value of x, 'x is an S' and ' is a P' are both true.

O. Some S is not P = For at least one value of x, 'x is an S' and 'x is not a P' are both true.

Thus I and O require that there should be at least one value of x for which x is an S, i.e., that S should exist. I also requires that P should exist, and O requires that not-P should exist. But A and E do not require the existence of either S or P ; for a hypothetical is true whenever its hypothesis is false, so that if 'x is an S' is always false, 'All S is P ' and 'No S is P ' will both be true whatever P may be."

Thus, Existential Import regards universal assertions: A and E.

This is the reason why (see Wiki's diagram) the faded areas are only in the A and E cases ["the faded red areas apply in traditional logic assuming the existence of things stated as S (or things satisfying a statement S in modern logic). In modern logic, this is not assumed so the faded ones do not hold"].

For particular assertions (I and O) both traditional and modern logic agree that "some S is P" ("some S is not P") means that there is an object that has both properties [see Wiki: "red areas are nonempty" and the red areas are only in the I and O cases].