11. Venn Diagrams

We will cover two areas in this chapter: the technique of the Venn diagram and an application of the Venn diagram.

1A.  The technique of the Venn diagram

1B. An overview of elements

A Venn diagram is a picture of the relationship between categories using circles.  The basic elements in the technique of the Venn diagram are circles, labels, overlapping of circles, shading, the letter x, and the x-bar. 

It should be obvious that the circle represents a category.  A label is placed outside the circle to indicate the identity or name of the category.  If the following circle pictures leprechauns, then the term leprechauns would be placed outside the circle. 

           leprechauns

Thus, "leprechauns" is a word that refers to a group of things (imaginary little green people). The word is not a group of little green people but it refers to this group and fittingly labels the circle that stands for the group.

Distinctive of Venn diagramming is the way circles are overlapped:  

Overlapping is a technique that represents the relationship between categories showing the sub-categories that are formed when categories interrelate.  For example, when we say that "all pirates are dangerous people," we make reference to three categories: pirates, dangerous people, and pirates who are dangerous people.  More specifically we refer to pirates who are not dangerous people, to pirates who are dangerous people, and to dangerous people who are not pirates. So distinct categories are implied when we relate one category to another.  Sub-categories are pictured by the areas on a diagram that are formed by the lines of the intersecting circles.  These areas on the diagram are called regions and they are given specific numbers for identification: 1, 2, 3 from left to right: 

The shading technique is used to indicate that a particular sub-category does not exist; on the diagram it means that there are no members to be found within the shaded region: 

An x is used to indicate that a sub-category does exist and that there are some members in it (at least one):

 The x-bar (two x's connected by a line or bar) is used to indicate that members may exist in one sub-category or another without knowing which.  On the graph, the bar connects x's found in different regions :

The x-bar above indicates that there are some members within the category designated by the left circle but we do not know if these members are in region 1 or region 2.  Thus we cannot be sure about the relationship between the two categories just as we cannot be sure about how these two circles relate to one another given the "fluidity" of the x-bar.  The fluidity is such that we do not have an x in each region; instead, it is actually the case that there is only one x and it in effect slides back and forth on the bar with a question mark beside it asking, "do I belong here or do I belong there?"  However, since shading takes precedence over an x, the location of the x will be resolved when one of the alternative regions is shaded.

 

2B.  An overview of statement Venn diagrams

There are two types: Venn diagrams of categorical statements and Venn diagrams of categorical syllogisms.  This discussion of types will demonstrate how the elements discussed above are put to work as graphs that in effect speak, make statements, and express arguments.  We begin with diagrams of statements.

A E I O statements can each be represented on a Venn diagram. To do so, two intersecting circles are used.  Use is made of two circles because two categories are being related to one another in each categorical statement.  Three regions now emerge and the numbering (1, 2, 3 from left to right) helps us precisely express universal statements by the use of shading and particular statements by the use of the x or x-bar.

  • Universal Affirmations and Denials use shading.

  • Particular Affirmations and Denials use the x (or x-bar).

Intersecting circles, shading, and pinpointing an x are the basics that convert the graph to a mode of speech.  Each type of categorical statement can be expressed by a properly filled out diagram.

1C. The A-form  [All pirates are dangerous people]

On the two circle Venn diagram, the circle on the left represents the subject term and the circle on the right represents the predicate term. The A-form is expressed by shading region 1. 

           Subject Term/Category                        Predicate Term/Category

Shading this region does not represent a reduction in the number of members included in the subject category.  It just means that all the members are compressed into region 2 so that all the members of the subject category are only found in that part of the left circle that overlaps with the right circle.  We might say that all the chickens are chased into the chicken coop of region 2 (No chickens are lost in the process).  When we see that region 2 is within the subject circle, we must conclude that all members of the subject category are included in the predicate category as a slice to a pie.  

In this light, we should be able to visualize the distribution of the respective terms. Region 2 is shared by both categories but the chickens of the subject category are penned in by the radius of the subject term.  Therefore, the subject term (i.e., all the members of the subject category) is only a slice of the pie of membership indicated by the predicate term.  Consequently, the predicate pie is undistributed since we are only concerned with a slice of it (we are only concerned with chickens but other slices of the pie may include turkeys, pheasant, robins, and so forth) while the subject slice is distributed because all the chickens are there.

 

2C.  The E-form [No pirates are dangerous people]

The E-form is expressed on the Venn diagram by chasing the chickens out of the region 2 chicken coop. Shading region 2 does this.

Here are some questions to ask regarding distribution beginning with the observation that the shading indicates denial summarized in the word "no." What service does "no" render in an E statement?  It does double duty serving as both a universal quantifier and as a negative.  It translates to "all ...are not."  So the subject term is distributed (as self-defined by the universal quantifier). Regarding the predicate term, does the visualization help us determine what is being said about all its members?  Yes, by removing all members of the subject term from within the predicate term, we can see that all members in the predicate term are being referred to: we can see that not a single one of them is included in the subject term (any possible overlap of the predicate with the subject is shaded out; denial works both ways: if a is not part of b, then b is not part of a).

 

3C. The I-form [Some pirates are dangerous people]

The I-form is expressed by placing an x in region 2.

An important point needs to be made at this juncture about x-claims (or particular claims) given the fact that two circle Venn diagrams form the basis for the three circle diagrams that picture arguments. Modern method in the use of Venn requires the placement of an x on a diagram by at least one premise in order to be able to read a particular claim ("some ...are" or "some... are not") from the Venn visualization of the concluding statement.  Under the title of "the universals rule" for a valid categorical argument, we discussed modern interpretation on this point (if the premises are both universal then for validity the conclusion must also be universal; you must stay on track).  Applied in  the technique of Venn diagramming, this comes to visualization in the fact that two universal premises will both use shading and no x.  Consequently, it is impossible for an x to show up in the visualization of the concluding statement (which is the relationship between the minor term, or left top circle, and the major term, the top right circle as explained below). 

With respect to distribution, the subject term is undistributed as is evident in the fact that the x represents the particular quantifier "some."  The predicate term is also evidently undistributed as seen in the restriction of the x within a slice of the predicate pie.  The members that make up the "some" portion of the subject term are in turn merely a portion of the predicate term as well.

 

4C. The O-form [Some pirates are not dangerous people]

The O-form is expressed by placing an x in region 1.

One way to read this is to observe that there are some members in the subject category and those members stand outside of membership in the predicate category. Again, we can use the visualization to help us see the distribution of terms.  The subject term is self-defined as undistributed (x = some and some = non-distribution).  But what is the distribution of the predicate? In saying that some pirates are not dangerous people are we saying something about all members in the category of dangerous people?  By looking at the diagram we can see that the answer is yes because we can see that every member of the predicate term (dangerous people) is not related to the portion of pirates spoken of in the subject term.  The x is a slice of the subject showing that the subject is undistributed but this x-slice is outside the radius of the predicate circle showing that the predicate is distributed (something is said about all members in the predicate category, namely, they are all not included in the undistributed subject term represented by the x). 

 

3B.  An overview of argument Venn diagrams 

1C. Three intersecting circles

Three intersecting circles are used to represent categorical syllogisms because all arguments of this type have exactly three terms.  There are three statements (two premises and a conclusion) so each term must be used twice.

2C. Labeling and locating terms

Labeling correctly is the first step in the process of expressing arguments in the form of a Venn graph. Labels obviously identify the categories being interfaced in each claim.  But the location of the respective terms adds something to the identification.  The location of the terms on the diagram indicates the function that the term has in the argument.  That is, the top left circle pictures the minor term of the argument, the top right circle pictures the major term of the argument, and the bottom circle pictures the middle term of the argument (if the middle term is pirates then the bottom circle must be labeled as such because it represents not only the category pirates but it also represents the middle term): 

3C. Numbering the regions

As soon as we add a third term to the diagram we automatically create a number of sub-sets or sub-categories pictured as seven regions on the diagram with an eighth region outside of the diagram. The eighth region represents the universe of discourse in which all the relationships expressed in the argument exist.  Or to put it another way, region 8 represents all other things in the universe besides the three things interfaced in a particular argument. These regions are numbered from left to right as you move down the graph:

Numbering the regions gives each one a name to facilitate identification so that arguments can be expressed by means of the three circle diagram in a standardized way.

 

4C. Perceiving relationships on a three circle Venn diagram

Many more relationships than three are brought into being by the interrelationship of three categories. Visualized this means that each region pictures or shows a relationship between a sub-set of one category and the two other categories each pictured by a whole circle (region 1 is a sub-set of the left top circle and it stands in a relation to the whole top right circle and in a relation to the whole bottom circle).  In this way, what is said of each sub-set in relation to each of the other whole categories is expressed diagrammatically. 

For example, let's make the top left circle represent textbooks, the top right represent fun books, and the bottom circle represent best sellers. To read the relationships one by one, we need to focus on a region, then relate that region to two whole circles and thus to two categories distinct from the category of which it is a part. Accordingly, region 1 represents the sub-set of textbooks, namely, it represents textbooks that are neither best sellers nor fun books. Region 6 pictures the sub-category consisting of fun books that are best sellers but not textbooks, and so forth.

 

2A.  The application of the Venn diagram

Venn diagrams are useful in letting us see the relationship between categories.  They can thus help us grasp matters like immediate inference (we can see the exact and full contradiction, for example, between an E claim and an I claim).  But they also aid the mind in grasping and evaluating categorical arguments because the statements of arguments appear before our eyes on the three circle diagram. We will now discuss the procedure that must be followed to properly use a three circle Venn diagram.  Afterwards, we will briefly explain the principle of deduction that underlies this procedure.

1B. The procedure

After correct labeling, there are two steps to follow in expressing argument on a Venn diagram.  First, the premises are placed on the diagram.  Each premise translates to a standard type (AEIO) and can therefore be visualized.  Attention needs to be given to only two circles at a time (this is a matter of perception: as you look at the three circles you may temporarily block out one category circle to stress the relationship between the two that are in view in a given AEIO statement). The minor premise is the relation between the minor term and the middle term so it is  put on a three circle diagram by relating the top left circle to the bottom circle.  The major premise is the relation between the major term and the middle term so it is put on a three circle diagram by relating the top right circle to the bottom circle.  Second, the attempt is made to read the conclusion from the visualization. Notice that the concluding statement of the argument in front of us is never directly placed on the diagram.  We only place the premises on the diagram and then look to see if the conclusion is thereby expressed. If the conclusion can be read from the diagram then the argument is deductively valid.

 

2B. The principle

Venn application is based on the principle in a deductively valid argument that what is implicit in the premises becomes explicit in the conclusion.  Thus, when you put the premises onto the diagram, one ingredient after another necessary to express the conclusion are put in place.  If the argument is deductively valid, then the conclusion will be clearly evident like a perfectly browned pie coming freshly baked from the oven.

For example, consider an argument made up of the following three terms: P (the minor term), Q (the major term), and T (the middle term).  

  • All Q are not T

  • All P are T

  • So, All P are not Q

Correct labeling will identify the top left circle as P, the top right circle as Q, and the bottom circle as T. The major premise says, "all Q are not T," and it is expressed on the diagram by shading regions 5 and 6.  The minor premise, "all P are T," causes shading of regions 1 and 2.  Now the question becomes, "does the diagram say the conclusion, does it say that all P are not Q?  It does, so the argument is DV.  How it does so is seen in the fact that all the members of P are in region 4 and none are in the overlapping regions (2, 5) because of the shading.  Finally and obviously region 4 is outside of Q (not within the category of Q).  

Note that when we try to read the concluding statement from the diagram we only look at the relation between the top two circles (between minor term and major term).  So the fact that region 4 is also within the T circle is not of concern.  T is the middle term which never occurs in the conclusion so it is left out of consideration in reading the conclusion.  Reference to T was necessary to bring about the shading of region 2 (determined by the relation of P to T) and the shading of region 5 (determined by the relation of Q to T).  But then, having done its work between the premises (as the between term or middle term), it falls out of view.

 

Venn Exercises1

A. Use the techniques of Venn diagrams (circles, labels, x's, shading) for the following exercises.

1. In a visualization, express the claim that some pirates exist.

2. In a visualization, express the claim that there are no leprechauns.

B. Translate the following statements onto a Venn diagram.  Standard categorical form must be presupposed in this process. Use the standard techniques.

1. Some pirates are happy.

2. No dogs eat grass.

3. All cats eat grass.

4. Not all hockey players are dangerous.

5. Only logic students love Venn diagrams.

6. No one can come unless drawn.

 

Venn Exercises2

For more exercises see Churchill, pp153-154 (A, 1-10) pp194- 195 (1-15) pp202-203 (B, 1-6) p.211 (C, 8; use two Venn diagrams)

 

A. Place the following on an appropriate Venn diagram.

1. All M are not S 

 

2. All K are T

 

3. All P are S 

 

4. All P are not M

 

5. Some J are not Q 

 

6. Only Y are B

 

7. There are no bricklayers who are also preachers.

 

B. Determine if the following are DV or ~DV and prove your answer by means of a Venn Diagram

1. All M are not S

Only S are P

.. All P are not M

 

 

2. No O are T

Only P are O

.. All P are not T

 

 

 

3. Most P are not M

All E are M

.. Some P are not E

 

 

Venn Exercises3

A. Define

1. Venn diagram

2. region

3. shading

4. x-bar

B. Truth Values

1. t f After you place the conclusion onto the Venn diagram, you can then proceed with premise placement and interpretation for validity.

2. t f The minor term, which is in the predicate position in the conclusion, is represented by the left top circle in the Venn diagram.

3. t f If you know that the major term is in the subject position of a premise then you know that this premise is the minor premise of the argument.

C. VALIDITY

1. Explain the procedure for determining validity via Venn.

 

2. On what principle is reading validity from a Venn diagram based? Explain.

 

D. VENN

1. Is the following DV or ~DV? Prove by a Venn diagram

 

 

2. EXPLAIN HOW THE FOLLOWING COMMITS THE FALLACY OF FOUR TERMS. (Note: the argument refers to Japan forcibly taking possession of China in the late 1930's):

 

 

 

3. Fill in the missing enthymeme and iron out matters of standard form so the argument is DV and prove your answer by using two Venn Diagrams:

 

All philosophy students are responsive students. All responsive students are teachable persons. No teachable persons are unintelligent persons. So, All philosophy students are intelligent persons.