Euler's Circles & Venn Diagrams

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A Rough Diagrammatic Expression of Categorical Propositions…

These circles represent the set of objects that are members of a given term.
One circle represents the “set” S (subject) and the other the “set” P (predicate).
*S represents at least one member of the “set” S.

Euler’s Circles

Diagrams can visually represent categorical propositions in Aristotelian-Scholastic logic.

One method comes from the mathematician Leonard Euler (1707-1783). He used two circles, one circle representing everything that belongs to the subject (S) and a second circle representing everything that belongs to the predicate (P).

SAP (all S is P)

Each member of S belongs to P.
So, the circle S is in the circle P.

SEP (no S is P)

Each member of S does not belong to P.
So, the circles are entirely apart.

SIP (some S is P)

At least one member of S belongs to P.
Notice the circles overlap.
S appears in the common area.

SOP (some S is not P)

At least one member of S does not belong to P.
Notice the circles overlap.
S appears outside the common area.

These are obviously similar to the “Rough Diagrammatic Expression.” However, it has limitations. One downside is that I and O propositions are drawn similarly. Another downside is that there’s no easy way to use these diagrams when evaluating syllogisms.

Venn Diagrams

A more popular method comes from John Venn (1843-1923). Two circles overlapping — and as long as we can shade in areas and place X’s where needed — is sufficient to represent all four categorical propositions. This tends to be more powerful than “Euler’s Circles.”

The left circle represents all members of the subject (S) and the right circle represents all members of the predicate (P).

There are three areas: area 1, area 2, and area 3.
(A fourth area is possible: outside of both circles.)

Members that exist in the overlap (i.e., area 2) belong to both the class S and the class P.

A “filled in” or “shaded in” area here indicates that there is nothing there. An “X” indicates that at least one thing exists there. (It gives “existential import.”) An area not “shaded in” signifies possibility: members may or may not be there.

SAP (all S is P)

Notice that area 2 has an X in it, area 1 is shaded in, and area 3 is not shaded in.

Area 2 has an X in it because members of S are present in P. Area 1 is shaded in because there is no S outside of P.

SEP (no S is P)

Notice that area 1 has an X in it, area 2 is shaded in, and area 3 is not shaded in.

Area 1 has an X in it because members of S are outside of P. Area 2 is shaded in because there is nothing in S that contains a P.

SIP (some S is P)

Notice that area 2 has an X in it, area 1 is not shaded in, and area 3 is not shaded in.

Area 2 has an X in it because there is at least one member of S that is a P. Also, note that area 1 is not shaded in because it’s unknown if all S is a P. That’s just a possibility.

SOP (some S is not P)

Notice that area 1 has an X in it, area 2 is not shaded in, and area 3 is not shaded in.

Area 1 has an X in it because there is at least one member of S that is not a P. Also, note that area 2 is not shaded in because it’s unknown if no S is a P. That’s just a possibility.

A Technical Note.

In these Venn diagrams, I followed a contemporary convention. You’ll notice that for universal propositions the X has a circle around it. This indicates that these universal propositions have “existential import.” Modern logic, by default, takes it that they don’t.

“All men are mortal,” from a Boolean viewpoint, is interpreted as hypothetical: “if there exists such a thing as a man, then that being is mortal.” First-order predicate logic would symbolize this: “(∀x)(Hx → Mx).” Only particular propositions are taken to imply the existence of anything, while universal propositions don’t.

Is the modern standpoint a superior interpretation? I’m not entirely sure. Is there a necessary need to introduce the notion of “existential import” at all? We can entertain propositions in our minds without considering that import.

No unicorns exist. Granted. But why does the proposition “some unicorns are male” imply existence whereas “all unicorns are one-horned horses” does not imply existence? The former proposition can perfectly well be entertained without having it imply existence!

It’s also the case, in the Scholastic tradition, that “existence” is an analogous term. I “exist.” My height “exists,” even though my height is an accident that presupposes the substance me existing. Finally, ideas “exist” in the mind. Does a unicorn “exist”? No, not in the usual sense of the term. It does, nevertheless, “exist” in my mind.

Regardless, in Aristotelian-Scholastic logic, we don’t treat universal propositions as hypothetical. For example, when “all men are mortal” is true, it follows that “some men are mortal” is true. Modern logic would claim that inference engages in an “existential fallacy.” It’s not fallacious on the Aristotelian-Scholastic account, since it doesn’t treat an A or E proposition as hypothetical.

My purpose with the X with a circle around it is to make sure we can apply these “contemporary” Venn diagrams to derive all possible conclusions that are traditionally associated with Aristotelian-Scholastic logic. The present concern isn’t to affirm or deny any theory of existential import. That requires a deep dive into the philosophy of logic.

A More Complicated Look at Euler’s Circles

Another way to represent Euler’s Circles is with five figures. These visual diagrams, although not as practical as Venn Diagrams, help us consider how different content can be contained in a categorical proposition. Think of it as a thought experiment.

These circles represent the members of the subject (S) and the predicate (P) respectively.
Observe that S is within P for Figure 1. Observe that P is within S for Figure 5.

What can represent an E proposition? There’s only one possibility, namely, Figure 3. There is good reason for that: both subject and predicate terms are distributed with the absolute exclusion of S from P implying the absolute exclusion of P from S.

Try to consider each figure before you think about the A, I, E, and O propositions.

The specific content of a proposition can be called its “materiality.”
The type of proposition can be called its “form.”

For example…

The form of the proposition “some men are mortal” is I (because it is an affirmative, particular proposition). The matter of this proposition is the concrete judgement that “some men are mortal.”

What can Figure 1 represent materially? In the diagram all of S is contained within P. The most natural conclusion is to declare that it can represent an A proposition. That’s obviously correct, but is that all? No! It can represent the material content of some possible I propositions. (“Possible” doesn’t mean “necessary” or “every.”)

This is because the truth of SIP doesn’t prohibit the truth of SAP. With that said, on the other hand, if we are given the truth of SIP, we cannot logically infer from that fact alone the truth of SAP.

To return to our example, “some men are mortal.”
Think about its matter.
Its matter matches the diagrammatic representation of Figure 1.

When we understand the matter of a proposition, we understand its specific content. It’s about the relation between men and mortality. Since we know — for independent reasons outside of formal logic — that all men are mortal and that there are more mortal beings than there are men, we understand that Figure 1 represents the material content of this I (affirmative, particular) proposition.

Not all SAP propositions materially match Figure 1, though.
Some fall under Figure 2.

For example, “all squares are equilateral rectangles.”
Formally this is an A proposition. Its matter is represented with Figure 2. The two circles exactly coincide — or really are identical. (And this makes a good biconditional proposition: “X is a square if and only if X is an equilateral rectangle.”)

Let’s consider: “some rectangles are squares.”

It’s a good example because it highlights how these five figures show differing extensions of the subject and predicate terms. The material content of this example goes with Figure 5. It’s formally an I proposition.

The predicate’s materiality is of being a square, and that happens to be within the subject’s materiality of being a rectangle. Hence the P circle is within the S circle. This is because all squares are in fact rectangles but only some rectangles are in fact squares.

Additional Examples. . .

Euler’s Circles & Venn Diagrams