Notes on Venn Diagrams Cynthia Bolton Arizona State University - Summer 2005

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1 Notes on Venn Diagrams Cynthia Bolton Arizona tate University - ummer 2005 Venn Diagrams: An Introduction Venn diagrams are used to pictorially prove the validity or invalidity of syllogisms. A Venn diagram provides a picture of the premises. If the syllogism is valid, the Venn diagram also provides a picture of the conclusion. How To Draw a Venn Diagram General Instructions (1) Draw three interlocking circles. Label the circles,, and. (Each circle represents the subject term, the predicate term, and the middle term of the syllogism.) (2) Diagram the premises. (i) Diagram universal premises first. This essentially means that we will shade before we put an X anywhere. Universal premises begin with the word All or the word No. If both premises are universal, it does not matter which premise we diagram first. And if both premises are particular, both begin with the word some, it does not matter which premises we diagram first. We shade regions we know to be empty. If we have a premise All y are z (where y could be,, or and z could be,, or ), then we will shade the y circle that is outside the z circle. Whatever is left of the y circle will be entirely contained in the z circle. If we have a premise No y are z, then we will shade the region where the y and z circles intersect. There is nothing in that region. (ii) We put an X on a line if the premises do not indicate which side of the line the X falls. We put an X in regions where we know an object to exist. If we have a premise ome y are z, then there is an X where the y and z circles intersect. If we have a premise ome y are not z, then there is an x in the region of the y circle that is outside of the z circle. ince we have three interlocking circles, sometimes the arc of third circle will appear in the region where we are placing an X. If this occurs, we place the X on this

2 line. (3) Read the conclusion. DO NOT DIAGRA THE CONCLUION. If the diagram of the premises give a diagram of the conclusion, then the argument is valid. If the diagram of the premises do not give a diagram of the conclusion, then the argument is not valid.

3 Example 1: rovide a Venn diagram for the following syllogism: All are No are No are tep 1: Draw the three interlocking circles. Label them,, and. tep 2: Diagram one premise. ince both premises are universal, it does not matter which premise we diagram first. We will diagram the premise All are first. We shade everything in the circle that is outside of the circle. Remember, we shade regions we know to be empty. The only part of the circle that remains is also in the circle. tep 3: Diagram the other premise. We will now diagram the statement No are. (For pedagogical reasons, we will pretend that we have yet to diagram All are this way, we can see exactly what the diagram looks like for No are )

4 Again, we are shading regions we know to be empty. The statement No are tells us that nothing can be both an and an. This means that the region between the and the, the and intersection, is empty. Thus, we shade in this region. tep 4: We look at the diagram of both premises. tep 5: We now read the conclusion. The conclusion for this argument is this: No are. This means that nothing can be both an and a. There is nothing in the region where the and intersect. o, the and intersection should be completely shaded in. When we examine our diagram, we see that this is the case. o, we know that this syllogism is valid.

5 Example 2 rovide the Venn diagram for the following argument: ome are ome are not All are tep 1: Draw three interlocking circles. Label them,, and. tep 2: ince both premises are particular, it does not matter which we diagram first. We will start with the first premise: ome are. The statement ome are tells us that there exists at least one object, which we will designate as X, that is both and. The only region where an object can be both and is where the and circles intersect. Note that in this region, there is a line that bisects the intersection in two. On one side of the line, we have a region that intersects the, and circle. On the other side of the line, we have a region that only intersects the and circles. ince we do not know which side of the line the x falls, we place it on the line. tep 3: Now we will diagram the second premise (and for the time being, we will ignore the fact

6 that we have already diagrammed the first premise.) The statement ome are not tells us that there exists an object that is an but not an. tep 4: This is the Venn diagram for our two premises: tep 5: Now we will read the conclusion. The conclusion for this syllogism said All are. If this argument is valid, then the circle, that is outside of the circle, should be completely shaded. But it is not. o, this syllogism is invalid. The premises do not provide a diagram of the conclusion.

7 Example 3: rovide a Venn diagram for the following syllogism: No are ome are not ome are tep 1: Draw the three interlocking circles. Label them,,. tep 2: Diagram the premises. (i) We have a universal premise and a particular premise. This means that we will diagram the universal premise, No are first.

8 (ii) Now, we will diagram the second premise. In this case, it was important to diagram the universal premise first. If we had not, then we would have placed the X on a line. By shading the and intersection, we know that there is nothing in this region. This means that no X will fall in that region; so we put the X on the other side of the line, in an open region. The X falls in the circle alone. tep 3: Now, we read the conclusion. The conclusion says ome are. If the syllogism is valid, then we will have an X in the region where the and circles intersect; but we have no X in this region. Thus, the syllogism is invalid.

9 Example 4 rovide a Venn diagram for the following syllogism: All are ome are not ome are not At this point, we presumably know that we need to first draw three interlocking circles. o, let us proceed directly to diagramming the premises. tep 1: Diagram the universal premise first. This means that we will diagram the first premise: All are. tep 2: Diagram the particular premise: ome are not. In this case, it did not matter that we diagrammed the universal premise first. We still had to place the X on the line. We know that the X is in the circle outside of the circle; but we do not know whether this X also falls in the circle as well. tep 3: Read the conclusion. The conclusion is this: ome are not. If the syllogism is valid, then we will have an X in the circle outside of the circle. The problem is that the premises only allow us to infer one X; and this X is not even in the circle.

10 Example 5: rovide a Venn diagram for the following syllogism: ome are ome are not ome are Diagramming the remises: tep 1: ince both premises are particular, it does not matter which we start with. We will tart, for the sake of convenience, with the first premise: ome are The premise, ome are, tells that there exists at least one thing that is both and. o, we put the X in the and intersection. But there is a line going through this region (it is part of the arc of the circle). We put the X on this line because we do not know whether the X is also an as well as an and or whether it is just an and. tep 2: We will now diagram the premise: ome are not. The premise ome are not tells us that there is an X in the circle that is not in the circle. o, the X goes in the crescent outside of the circle. There is a line inside this region (it is part of the arc of the circle). We put the X on the line because we do not know whether the X is an alone or whether it can also be a.

11 tep 3: We now read the conclusion: ome are. If the syllogism is valid, then we are guaranteed that there is an X in the region where the and circles intersect. We are not guaranteed that there is such an X. It is possible that one of our X s (or both) could fall in the region where the and circles intersect. But it is also possible that one X would fall into the circle outside of the circle and the other X could fall in the region where just the and circles intersect. Thus, this syllogism is invalid.

12 Example 6 rovide a Venn diagram for the following syllogism: ome are All are. ome are tep 1: We diagram the universal premise first. This means we start out with the second premise: All are. To diagram All are, we shade all of the circle that is outside of the circle. Whatever is left of the circle will also be in the circle. tep 2: We diagram the second premise: ome are. The premise, ome are tells us that there is an X where the and circles intersect. There is a line in this region (it is part of the arc of the circle). The region on one side of this line is shaded (there is nothing in this region it is empty) while the region on the other side is open. We put the X in the open region. tep 3: The conclusion says this: ome are. If there is an X where the and circles intersect, then this syllogism is valid. There is an X in this region. o, the syllogism is valid.

13 Example 7 lease provide a Venn diagram for the following syllogism: ome are not No are. No are tep 1: We will diagram the universal premise first. This means we will diagram No are. No are tells us that the intersection between the and circles is empty. o, we shade this region. tep 2: We now diagram the premise ome are not The premise ome are not tells us that there is an object in the circle that is not in the circle. There is a line (part of the arc of the circle) in this region of the circle. ince we do not know whether our object falls in the region alone or in the region as well, we place the X on the line. tep 3: We read our conclusion. Our conclusion is No are. If our syllogism is valid, then then the region where the and circles intersect will be shaded in. Only half of this region is shaded in; so, we know our syllogism is invalid.

14 Example 8 lease provide a Venn diagram for the following syllogism: No are ome are not ome are tep 1: We diagram the universal premise, No are first. The premise No are tells us that nothing can be both an and a. This means that the region where the and circles intersect is empty. tep 2: We now diagram the particular premise, ome are not. The premise ome are not tells us that there is an object that is an but not an ; so we place the X in the region of the circle that excludes the circle. ince this region has a line going through it (this line is part of the arc of the circle), we place the X on the line. The X may be an alone or it may be both an and. tep 3: We read the conclusion: ome are. ince our X may fall on the side of the line where it is an alone, and not a, our conclusion does not follow from our premises. Thus, the syllogism is invalid.

15 Example 9 lease provide a Venn diagram for the following syllogism: All are All are All are tep 1: ince both premises are universal, it does not matter which we diagram first. For the sake of convenience, we shall diagram All are first. Whatever is left of the circle should be included in the circle. We shade the rest. tep 2: We will now diagram the premise All are. Whatever is left of the circle should be included in the circle. We shade the rest. tep 3: We read the conclusion: All are. If this syllogism is valid, then whatever is left of the circle should be fully contained in the circle. But part of the circle that is outside of the circle is unshaded. Thus, this argument is invalid.

16 Example 10 lease provide a Venn diagram for the following syllogism: ome are not All are. No are tep 1: We will diagram the universal premise, All are first. Whatever is left of the circle should be contained in the circle. We shade the rest. tep 2: We diagram the particular premise ome are not. This premise tells us that there is an X that is in the circle that is not in the circle. In this region of the circle, there is a line (it is part of the arc of the circle). We put the X on the line because we are not sure whether the X is also an or whether it is an alone. tep 3: We read the conclusion: No are. ince the intersection between the and circles is not completely shaded in, this syllogism is invalid.