How to Spot Logical Fallacies by Drawing Venn Diagrams

How can you prove whether the following arguments are logically (in)valid?

1. All Christians are humans.
   Some humans are atheists.
   Therefore, some Christians are atheists.
2. All humans are persons created in God’s image.
   All persons created in God’s image are beings with moral rights.
   Therefore, all humans are beings with moral rights.
3. Some creationist arguments contain fallacies.
   No arguments that contain fallacies are trustworthy.
   Therefore, no creationist arguments are trustworthy.

All these arguments revolve around categorical statements, which make claims about how one category relates to another. Intuitively, it’s easy to tell that some of these arguments have an illogical “ring” to them. But is there any way to show if a categorical argument is valid?

Enter Venn Diagrams. Venn Diagrams depict overlapping circles, which reveal whether categorical statements work together logically. To see how to use these handy illustrations, let’s go over the different types of statements you can represent with a Venn Diagram.

Drawings That Make a Statement

Every categorical statement mentions two categories.1 The first category is called the subject, represented with the letter S, and the second is called the predicate, represented with the letter P.2 For example, in the statement, All Christians are humans, the subject category is Christians, and the predicate category is humans.

Categorical statements can be drawn as two overlapping circles representing S and P. There are four ways to fill the circles in, representing the basic forms of categorical statements:

Form #1 “All S Are P.”

Example: “All penguins are birds.”

To diagram Form 1 statements, simply shade in all the S circle except where it overlaps with P. The dark shading means nothing exists in that section of the diagram. (As a hint from the critical thinking textbook I used in university, you can think of the shading as meaning, “the lights are out, so nobody’s home.”3) For example, no penguins exist which are not birds.

Form #2: “No S Are P.”

Example: “No reptiles are arctic animals.”

You can diagram Form 2 statements by shading in the overlap between the two circles, showing that no individuals exist in both the S and P categories simultaneously. In this case, no members of the reptiles group also belong to the arctic animals group and vice versa.

Form #3: “Some S Are P.”

Example: “Some Christians are geologists.”

To diagram Form 3 statements, draw an X in the overlap region between the two circles. This shows that at least one member of S is also a member of P. For instance, the X in the diagram below symbolizes that some people are both Christians and geologists.

Form #4: “Some S Are NOT P.”

Example: “Some scientists are not evolutionists.”

You can diagram Form 4 statements by drawing an X which is inside the S circle, but outside the area where S overlaps with P. This shows that at least one member of S is not a member of P. In this case, the X represents scientists who do not subscribe to evolutionary beliefs.

How to Sketch an Argument

With these four types of statements in mind, you can diagram the premises of a categorical argument.4 The resulting diagram will reveal whether the argument’s conclusion is valid. Take for instance, the argument we saw earlier:

Premise 1: Some creationist arguments contain fallacies.

Premise 2: No arguments which contain fallacies are trustworthy.

Conclusion: Therefore, no creationist arguments are trustworthy.

This argument’s premises mention three different categories: creationist arguments, trustworthy arguments, and arguments which contain fallacies. We can represent these categories as three overlapping circles.

Now it’s time to diagram the argument’s premises. For Premise 1, “Some creationist arguments contain fallacies,” we’d draw an X right in the middle of the overlap between the appropriate circles, so that it straddles the border of the trustworthy arguments circle. This shows we’re not certain which side(s) of the border the “some” creationist argument(s) belong to.5

For Premise 2, “no arguments which contain fallacies are trustworthy,” we’d shade in the overlap between the fallacious arguments and trustworthy arguments circles. Then we can examine the diagram.

In a valid argument, we’ll be able to see the argument’s conclusion on the diagram after drawing the premises. So, does the diagram show that no creationist arguments are trustworthy? No. The entire overlap between the creationist arguments and trustworthy arguments circles would have to be shaded to suggest that. Therefore, this is an invalid argument.6 (Keep in mind, though, that not even valid arguments are worth believing unless they also possess all true premises.)

Now let’s try diagramming the premises of the second argument listed above. Premise 1 states, “All humans are persons created in God’s image.” To diagram this, we’d shade in the entire humans circle except where it overlaps with the persons created in God’s image circle. (We’ll ignore the beings with moral rights circle for now, because that category wasn’t named in Premise 1.)

Next, we can diagram Premise 2, which states, “All persons created in God’s image are beings with moral rights.” To illustrate this, we’ll shade in the entire persons created in God’s image circle except where it overlaps with the beings with moral rights circle.

After diagramming these premises, we can already see that the argument’s conclusion, “All humans are beings with moral rights,” is apparent from the illustration. The only unshaded region of the humans circle also overlaps with the beings with moral rights circle. In other words, the diagram shows that the argument’s conclusion logically follows from the premises. Therefore, this argument is valid.

What about the argument which concluded, “All Christians are atheists?” You can draw a quick sketch yourself to prove this argument is as invalid as it sounds.

Categorically Certain

To recap, when a categorical argument has an illogical “ring” to it, drawing some rings of your own lets you prove whether the argument is invalid. Simply sketch three circles, label them with the categories mentioned in the argument, and diagram the argument’s premises. If the argument’s conclusion is apparent from the resulting illustration, the argument is valid. If not, you’ve detected a fallacy. With these handy sketches, you can easily see through invalid categorical arguments which attempt to challenge a biblical worldview—the foundation which makes logic possible.