1. Venn Diagram Technique for testing syllogisms
We have used two-circle Venn diagrams to represent standard-form categorical propositions. In order to test a categorical syllogism by the method of Venn diagrams, one must first represent both of its premises in one diagram. That will require drawing three overlapping circles, for the two premises of a standard-form syllogism contain three different terms-minor term, major term, and middle term.

2. Let’s start with this example:
No doctors are professional wrestlers.
All cardiologists are doctors.
 No cardiologists are professional wrestlers.
Since this argument, like all standard-form categorical syllogisms, has three category-terms (in this case, “cardiologists,” “professional wrestlers,” and “doctors”), we need three interlocking circles rather than two to represent the three categories. The minor term will go on the top left, the major term will go on the the top right, and the middle term will be the bottom circle.

3. The diagram for our example is as follows:
The first premise states that no doctors are professional wrestlers. To represent this claim, we shade that part of the Doctors circle that overlaps with the Professional wrestlers circle, as follows:

5. The second premise states that all cardiologists are doctors
The second premise states that all cardiologists are doctors. To represent this claim, we shade that part of the Cardiologists circle that does not overlap with the Doctors circle:

6. We now have all the information we need to see whether the argument is valid. The conclusion tells us that no cardiologists are professional wrestlers. This means that the area where the Cardiologists and Professional wrestlers circles overlap is shaded, that is, empty. We look at the diagram to see if the area is shaded, and we see that it is indeed shaded. That means that the conclusion is implicitly “contained in” (i.e. follows logically from) the premises. Thus, the argument is shown to be valid.

7. Let’s look at a second example:
All snakes are reptiles.
All reptiles are cold-blooded animals.
 All snakes are cold-blooded animals.

8. Next, we diagram the first premise
Next, we diagram the first premise. The premise states that all snakes are reptiles. We represent this information by shading the area of the Snakes circle that does not overlap with the Reptiles circle.

9. Next, we diagram the second premise
Next, we diagram the second premise. The second premise states that all reptiles are cold-blooded animals. We represent this claim by shading that part of the Reptiles circle that does not overlap with the Cold-blooded animals circle.

10. Finally, we look to see if the information contained in the conclusion is depicted in the diagram. The conclusion tells us that all snakes are cold-blooded animals. This means that the part of the Snakes circle that does not overlap with the Cold-blooded animals circle should be completely shaded. Inspection of the diagram shows that this is indeed the case. So, the argument is valid.

11. Let’s look at a third example:
Some Baptists are coffee-lovers.
All Baptists are Protestants.
 Some Protestants are coffee-lovers.
Notice that this example includes two “some” statements. Diagramming “some” statements is a little trickier than diagramming “all” or “no” statements. As we have seen, “some” statements are diagrammed by placing Xs rather than by shading. Most mistakes in Venn diagramming involve incorrect placement of an X.

12. To avoid such mistakes, remember the following rules:
1. If the argument contains one “all” or “no” statement, this statement should be diagrammed first. In other words, always do any necessary shading before placing an X. If the argument contains two “all” or “no” statements, either statement can be done first.
2. When placing an X in the area, if one part of the area has been shaded, place the X in the unshaded part. Examples: x

13. When placing an X in an area, if one part of the area has not been shaded, place the X precisely on the line separating the two parts. Example: x

14. Back to the third example:
Some Baptists are coffee-lovers
All Baptists are Protestants.
 Some Protestants are coffee-lovers.
First, we draw and label our three circles:

15. Next, we need to decide which premise to diagram first
Next, we need to decide which premise to diagram first. Should it be the “some” premise or the “all” premise? Rule one states that we should start with the “all” premise:

16. Now we can diagram the first premise, which states that some Baptists are coffee-lovers. To represent this claim, we place an X in the area of the Baptists circle that overlaps with the Coffee-lovers circle. Part of this area, however, is shaded. This means that there is nothing in that area. For that reason, we place the X in the unshaded portion of the Baptists circle that overlaps with the Coffee-lovers circle, as follows:

17. Finally, we inspect the completed diagram to see if the information contained in the conclusion is represented in the diagram. The conclusion states that some Protestants are coffee-lovers. This means that there should be an X in the area of the Protestants circle that overlaps with the Coffee-lovers circle. A glance at the diagram show that there is an X in this area. Thus, the argument is valid.

18. So far, all the categorical syllogisms we have looked at have been valid. But Venn diagrams can also show when a categorical syllogism is invalid. Here is one example:
All painters are artists.
Some magicians are artists.
 Some magicians are painters.
First, we draw and label our three circles:

19. Since the premise begins with “all” and the second premise begins with “some,” we diagram the first premise first. The first premise states that all painters are artists. To depict this claim, we shade that part of the Painters circle that does not overlap with the Artists circle:

20. Next, we enter the information of the second premise, the claim that some magicians are artists. To represent this claim, we place an X in that portion of the Magicians circle that overlaps with the Artists circle. That area, however, is divided into two parts (the areas here marked “1” and “2”), and we have no information that warrants placing the X in one of these areas rather than the other. In such cases, we place the X precisely on the line between the two sections, as follows:

21. The X on the line means that we have no way of knowing from the information given whether the magician-who-is-an-artist is also a magician-who-is-a-painter.
The conclusion states that some magicians are painters. This means that there should be an X that is definitely in the area where the Magicians and Painters overlap. There is an X in the Magicians circle, but it dangles on the line between the Artists circle and the Painters circle. We don’t know whether it is inside or outside the Painters circle. For that reason, the argument is invalid.