Diagramming Propositions

Let’s agree to use overlapping circles as follows: the circle on the left (or any circle on the left) will be called “S.” “S” stands for the subject term of any statement. The circle on the right (or any circle on the right) will be called “P.” “P” will stand for the predicate term of any statement.

Let’s also agree to do this: call the section on the far left “1,” the section in the middle where they overlap “2,” and the section on the right “3.”

There are four standard form categorical propositions, so there are four ways to use Venn diagrams to turn them into pictures:


This first picture shows a way of saying “All S is P.” With the circle on left representing “S,” a good portion of it has been shaded in. The shading indicates scratching out or eliminating an area. In this picture, the part of S that is not overlapping P has been eliminated, so all that’s left of S is in P, meaning that All S is P.

If we call these three sections from left to right “1,” “2,” and “3,” the section called “1” being filled in means it has been erased, scratched away, eliminated from consideration, so that section 2 is the only section of what had been the circle called “S” (which was made up of sections 1 and 2 together before the shading happened). So all that is left of circle S is section 2, and section 2 is entirely inside circle P (which is made up of 2 and 3).



In this second one, the area that’s been covered up is the overlap of S and P, so it means that everything that’s inside of S is outside of P. As you see, everything to the left of the shaded area is S, everything to the right of the shaded area is P. This represents No S is P.

Section 2 has been shaded, to show that it has been eliminated, so all that remains of circle S is section 1, and all that remains of circle P is section 3, and they do not overlap at all.

The universal statements are pictured by either showing that one category is wholly in the other, or that neither is in the other at all.

These pictures never change; once you memorize how to draw them, you are set. They require no thinking, just repetition of this pattern. However, when we add a third circle, we will have 7 sections instead of 3, and it will be a little more to look at. If you are clear on how to draw the relationship between two circles at a time, you will be fine, because that is all you ever do: focus on two circles at a time.



Particulars, of course, focus at a different level of quantity. We use “x” to represent an individual thing, and since the particular statements say “some S…,” we use an “x” to indicate that there is at least one thing that is a member of the subject class.

Some S is P

It’s pretty clear that this shows that there is at least one thing, and it is a member of both the S class and the P class; it shows up in the area where those two circles overlap.

That is, to show that something occupies both S and P, you place an “x” in section 2.



Our final statement calls for showing an x in the area of S that is not inside of the circle P.

Some S is not P

Placing an “x” in section 1 shows that there is at least one thing inside of S and outside of P.

Commit these representations to memory. All you need to know for diagramming Categorical Syllogisms is how to diagram these four categorical propositions.

Venn diagrams aren’t very useful when we only have one proposition in mind, but when we have a syllogism, we have a series of statements, two premises and one conclusion. Then, using three overlapping circles (one for each category) we diagram the premises using the approach outlined here, as a way to check on the validity of the argument.

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