## The Logic of Venn Diagrams

This page reviews how to set up Venn diagrams as well as the rules for using Venn diagrams in evaluating argument validity. It also introduces the notion of conditional validity and explains how to use Venn diagrams to evaluate the validity of categorical syllogisms.

In this section, we study how to use Venn diagrams to determine the validity of a categorical syllogism.

### 2.5.1 Basic Setup for Venn Diagrams

In using Venn diagrams to determine the validity of a categorical syllogism, we draw three overlapping circles to represent the minor, middle and major terms. The three circles are divided into seven areas. A categorical syllogism is valid if its two premises together imply the conclusion. That is, if the two premises are true, then the conclusion must be true. Visually in terms of Venn diagrams, this means that if we combine the basic diagrams of the two premises, we would get the basic diagram of the conclusion. To combine the basic diagrams of the premises, we place them on top of the three overlapping circles.

For example, to determine whether the form AOO-2 is valid, we first place the Venn diagram of the major premise, the blue pair of circles, on top of the three circles. Next, we place the Venn diagram of the minor premise, the green pair, on top of the three circles. The illustration shows how the blue pair and the green pair are placed on top of the three circles. Now click the above play button again to see the resulting diagram in black and white.

Next, we try to see if the Venn diagram of the conclusion, the red pair, is already present in the completed diagram. If it is, the argument form is valid; if not, then it is invalid. You will see that a portion of the Venn diagram gradually turns red to illustrate that the red pair is already there in the diagram. This shows that we get the red pair from the blue and the green pairs. This in turn means that we have derived the conclusion from the two premises. As a result, the argument form AOO-2 is valid. Notice we did not superimpose the red pair on the three circles. ### 2.5.2 Rules for Venn Diagrams

We can also view drawing Venn diagrams as a matter of shading some areas and placing Xs within the three circles. In the above example, the Venn diagram for the argument form AOO-2 is completed by shading Area 6 and Area 7, and placing an X in Area 5. Superimposing the blue and the green pairs over the three circles is an easy way to see which areas are shaded and where the X is placed. But to draw Venn Diagrams accurately we need to follow the following two important rules:

1. Shading always goes before placing an X.
2. If one of the two areas in which an X should be placed is shaded, place the X in the other area that is not shaded. If none of the two areas are shaded, put the X on the line between the two areas.

A shaded area means that the area is empty, and no X can be in the area. This is why shading is done first to determine which areas are empty. Placing an X on the line between two unshaded areas means that all we know is that the X is in either of the two areas, but we do not know for sure which one.

You will also learn more about how to apply these two rules by going over examples. Let us first look at the argument form EAE-1. From the illustration, we can see that after shading Area 3 and Area 4 according to the blue pair, and shading Area 5 and Area 6 based on the green pair, the Venn diagram of the conclusion is already present in the three circles, as shown by the part of drawing gradually highlighted in red. Since the diagram in red matches the red pair, the form EAE-1 is valid.

In the next form EAE-3, the part of the diagram in the three circles that is highlighted in red does not match the red pair. This means that the conclusion may not be true given that the premises are true. Consequently, the form is invalid.

The form AAA-1 is one of the most commonly used form in Categorical Logic. The Venn diagram clearly shows that it is valid. The next few examples illustrate how to apply the two rules when drawing the Venn Diagram. In the form OAO-3, we have a pair with a shaded area and another pair with an X.According to Rule #1, we need to draw the shading first. This is why we start with the green pair. We do the shading first to find out which of the seven areas are empty. In this case, we know after the shading that Area 1 and Area 4 are empty. This tells us that we cannot place the blue X (that is, the X in the blue pair) in these two areas. To find out where to put the blue X, we first recognize that it is inside the area α of the blue pair (from now on, we will call the area Blue α for short). In the three circles, Blue α amounts to Area 1 and Area 2. But according to Rule #2, since Area 1 is shaded, X has to be placed in Area 2. This is why in the animation, the blue X shows up in Area 2. As a result, the part highlighted in red matches the red pair (that is, we have an X in Red α). So the form is valid.

In the next example, to decide whether the form AII-1 is valid, we start with the blue pair because it is the pair with a shaded area. After the shading, we know that Area 1 and Area 2 are empty. The green X is inside the β area (that is, Green β). In the three circles, Green β is equivalent to Area 2 and Area 3. Since Area 2 is shaded, we have to place X in Area 3. Consequently, the red pair is present in the three circles (that is, we have an X in Red β), and the form is thus valid.

Now, compare AII-1 with the form AII-2. Since neither Area 2 nor Area 3 is shaded, according to Rule #2, X needs to be placed on the line between the two areas. The resulting drawing highlighted in red does not match the red pair—we do not have an X in Red β. This tells us that AII-2 is invalid.

If both of the premises of a categorical syllogism are particular sentences (that is, either I or O statements), then there is no shading in the Venn diagram. The Blue β is equivalent to Area 3 and Area 4 of the three circles. So the blue X needs to be placed on the line between these two areas. The Green β is equivalent to Area 2 and Area 3, and the green X should be placed on the line between them. The resulting diagram shows that we have two Xs on the lines, but not in Red β (Area 3 and Area 6 combined). So the form is invalid.

### 2.5.3 Conditional Validity

Some categorical syllogisms with two universal sentences (i.e., A or E sentences) as premises, but a particular sentence (i.e., an I or O sentence) as the conclusion are conditionally valid. They are valid if a certain set is not empty. For example, the form AAI-1 and EAO-3 are conditionally valid. After the shading is done, notice that in the circle S, three out of four areas (that is Area 2, 5 and 6) are shaded and only Area 3 remains unshaded. Now if the set S is not empty, this would mean that Area 3 cannot be empty. So under the condition that S is not empty we can infer that Area 3 cannot be empty. Consequently, we can place an X in Area 3. (I use a brown X to show that this X does not come from the blue and the green pairs.) As a result, the part of the diagram in red matches the red pair, and the form AAI-1 is valid if the set S is not empty (S ≠ ∅).

In the form EAO-3, after the blue and the green pairs are superimposed on the three circles we can see that in the circle M three areas (Area 1, 3 and 4) are shaded. Now if the set M is not empty, then Area 2 cannot be empty. We indicate this by placing a brown X in Area 2. The resulting diagram highlighted in red matches the red pair, and the form is valid if M ≠ ∅.

The form AEO-3 also has two universal sentences as premises, but a particular sentence as the conclusion. So we need to check to see if it is conditionally valid. If the set M is not empty, then Area 4 cannot be empty. However, even after we place a brown X in Area 4, the resulting diagram highlighted in red does not match the red pair. So the form is simply invalid.

### 2.5.4 Evaluating Categorical Syllogisms

In section 2.4, we learned how to turn a categorical syllogism into the standard form. In this section, we have learned how to use the Venn Diagram to determine if a standard form is valid or not. By combining these two sections, we have a process that enables us to assess the validity of categorical syllogisms written in everyday language. Here is an example that shows how the whole process works.

Some voter-approved propositions are not constitutional. All laws that are unconstitutional should be overturned. So some voter-approved propositions should be over-turned. (V: voter-approved propositions, C: laws that are constitutional, O: laws that should be overturned)

First of all, we paraphrase the argument as

 All non-C are O. Some V are not C. Some V are O.

We then reduce the number of terms to three by applying obversion to the minor premise.

 All non-C are O. All non-C are O. Some V are not C. Obv. Some V are non-C. Some V are O. Some V are O.

The resulting standard form is AII-1. To determine its validity, we draw the Venn Diagram. Notice that the minor term is V, the major term is O and the middle term is non-C. The diagram shows that AII-1 is valid. After completing the whole process, we find out that the written argument is valid.

We saw in section 2.4 that some sentences need to be paraphrased as two categorical sentences. Arguments contain such sentences need to be evaluated using two forms. In the next example

Few politicians are not spin doctors. All spin doctors are untrustworthy. Therefore, not all politicians are trustworthy. (P: politicians, S: spin doctors, T: people who are trustworthy)

The sentence

Few politicians are not spin doctors.

needs to be paraphrased as

Some politicians are not spin doctors, but most politicians are spin doctors.

Recall that we cannot use the quantifier “most”, so we need to replace it with “some”. We thus end up with the sentence

Some politicians are not spin doctors, but some politicians are spin doctors.

After paraphrasing all the sentences we have the argument form:

 All S are non-T. Some P are not S and some P are S. Some P are not T.

If we write “Some P are not S and some P are S” as two sentences, the form would then looks like this:

 All S are non-T. Some P are not S. Some P are S. Some P are not T.

The argument form has three premises. To decide whether it is valid we need to break it apart as two argument forms:

 All S are non-T. All S are non-T. Some P are not S. and Some P are S. Some P are not T. Some P are not T.

We then reduce the number of terms to three and get two standard forms:

 All S are non-T. Obv. No S are T. Some P are not S. Some P are not S. Some P are not T. Some P are not T. EOO-1

and

 All S are non-T. Obv. No S are T. Some P are S. Some P are S. Some P are not T. Some P are not T. EIO-1

The Venn Diagram of EOO-1 tells us that it is invalid. But the next diagram shows that EIO-1 is valid. This means that the original argument is valid because we can get EIO-1 from the original argument by simply tossing out the second premise.

 All S are non-T. Obv. No S are T. Some P are not S. Some P are S. Some P are S. Some P are not T. Some P are not T.