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2.5 Venn Diagrams

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

2.5.1 Basic Setup for Venn Diagrams

In using Venn diagrams to determine the validity of a categorical syllogism, we draw threeoverlapping circles to represent the minor, middle and major terms. The three circles aredivided into seven areas.

A categorical syllogism is valid if its two premises together imply the conclusion. That is, if thetwo 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 basicdiagram of the conclusion. To combine the basic diagrams of the premises, we place them ontop of the three overlapping circles.

For example, to determine whether the form AOO-2 is valid, we first place the Venn diagramof the major premise, the blue pair of circles, on top of the three circles. Next, we place theVenn diagram of the minor premise, the green pair, on top of the three circles. Click on theplay button to view the illustration.

I L O G I C

The animation shows how the blue pair and the green pair are placed on top of the threecircles. 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 thecompleted diagram. If it is, the argument form is valid; if not, then it is invalid. Click on theplay button of the next illustration. You will see that a portion of the Venn diagram graduallyturns red to illustrate that the red pair is already there in the diagram. This shows that we getthe red pair from the blue and the green pairs. This in turn means that we have derived theconclusion from the two premises. As a result, the argument form AOO-2 is valid. Notice wedid 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 Xswithin 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 theblue 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 thefollowing 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 theother area that is not shaded. If none of the two areas are shaded, put the X on theline between the two areas.

A shaded area means that the area is empty, and no X can be in the area. This is why shadingis done first to determine which areas are empty. Placing an X on the line between twounshaded areas means that all we know is that the X is in either of the two areas, but we donot know for sure which one.

Use the following interactive illustration to become familiar with these two rules.

You will also learn more about how to apply these two rules by going over examples. Let usfirst look at the argument form EAE-1.

From the animation, we can see that after shading Area 3 and Area 4 according to the bluepair, and shading Area 5 and Area 6 based on the green pair, the Venn diagram of theconclusion is already present in the three circles, as shown by the part of drawing graduallyhighlighted 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 redpair. 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 Venndiagram 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. Accordingto Rule #1, we need to draw the shading first. This is why we start with the green pair. We dothe shading first to find out which of the seven areas are empty. In this case, we know afterthe 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 firstrecognize that it is inside the area α of the blue pair (from now on, we will call the area Blue αfor 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 Xshows up in Area 2. As a result, the part highlighted in red matches the red pair (that is, wehave 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 pairbecause 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 2is shaded, we have to place X in Area 3. Consequently, the red pair is present in the threecircles (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 theline between the two areas. The resulting drawing highlighted in red does not match the redpair—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 Ostatements), 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 beplaced 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 thatwe have two Xs on the lines, but not in Red β (Area 3 and Area 6 combined). So the form isinvalid.

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 areconditionally valid.

After the shading is done, notice that in the circle S, three out of four areas (that is Area 2, 5and 6) are shaded and only Area 3 remains unshaded. Now if the set S is not empty, this wouldmean that Area 3 cannot be empty. So under the condition that S is not empty we can inferthat Area 3 cannot be empty. Consequently, we can place an X in Area 3. (I use a brown X toshow that this X does not come from the blue and the green pairs.) As a result, the part of thediagram 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 thecircle M three areas (Area 1, 3 and 4) are shaded. Now if the set M is not empty, then Area 2cannot be empty. We indicate this by placing a brown X in Area 2. The resulting diagramhighlighted 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 theconclusion. 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 brownX in Area 4, the resulting diagram highlighted in red does not match the red pair. So the formis 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 thissection, 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 thevalidity of categorical syllogisms written in everyday language. Here is an example that showshow the whole process works.

Some voter-approved propositions are not constitutional. All laws that areunconstitutional should be overturned. So some voter-approved propositions shouldbe 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 thewritten argument is valid.

We saw in section 2.4 that some sentences need to be paraphrased as two categoricalsentences. Arguments contain such sentences need to be evaluated using two forms. In thenext 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 aretrustworthy)

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 thusend 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 lookslike 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 apartas 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 originalargument 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.

Exercise 2.5

I. Use the Venn Diagram to determine whether the categorical syllogism is valid,conditionally valid or invalid.

1.

2.

3.

4.

5.

6.

II. Decide the validity of the following syllogisms. You need to (1) identify the mood andfigure; (2) draw Venn Diagrams; (3) determine whether the syllogism is valid,conditionally valid or invalid.

1. All P are M.No M are S.  

2. Some P are not M.All M are S.  

No S are P. Some S are not P.

3. No P are M.All M are S.  

4. All P are M.All S are M.  

No S are P. All S are P.

5. No M are P.No M are S.  

6. Some P are M.All M are S.  

No S are P. Some S are P.

7. Some P are M.All S are M.  

8. All P are M.Some S are not M.  

Some S are P. Some S are not P.

9. All P are M.Some M are S.  

10. Some P are not M.Some M are S.  

Some S are P. Some S are not P.

11. No P are M.Some M are S.  

12. All P are M.All M are S.  

Some S are not P. All S are P.

13. Some M are P.No M are S.  

14. Some M are not P.All S are M.  

Some S are not P. Some S are not P.

15. Some P are M.Some S are not M.  

16. Some P are not M.No M are S.  

Some S are P. Some S are not P.

17. All P are M.All M are S.  

18. All P are M.All S are M.  

Some S are P. Some S are P.

19. No P are M.All M are S.  

20. All P are M.No M are S.  

Some S are not P. Some S are not P.

III. Use Venn Diagrams to determine whether the following syllogisms are valid or invalid.You need to (1) paraphrase the argument into the standard categorical argument formusing the capital letters provided; if necessary, reduce the number of terms and indicatethe operation(s) used; (2) identify the mood and figure; (3) draw the Venn Diagram; (4)decide whether the syllogism is valid, conditionally valid or invalid.

Example:

All reasonable people are fair-minded. Only people who are against favoritismare fair-minded. Therefore, people who are not against favoritism areunreasonable. (R: reasonable people, F: fair-minded people, A: people who areagainst favoritism)

1. Beliefs in magic are at odds with the laws of nature. Beliefs that are at odds with thelaws of nature are superstitions. Therefore, beliefs in magic are superstitions. (M:beliefs in magic, O: beliefs that are at odds with the laws of nature, S: superstitions)

2. Whatever is immoral should be illegal. Whatever is obscene is not moral. Therefore,whatever is obscene should not be legal. (M: things that are moral, L: things that shouldbe legal, O: things that are obscene)

3. Only drugs approved by FDA are safe. Unsafe drugs should be recalled. So some drugsapproved by FDA should be recalled. (D: drugs approved by FDA, S: safe drugs, R: drugsthat should be recalled)

4. Some genetically altered foods can cause allergic reactions. Not all foods that can causeallergic reactions should be banned. So some genetically altered foods should not bebanned. (G: genetically altered foods, R: foods that can cause allergic reactions, B: foodsthat should be banned)

5. All M-rated video games are unsuitable for young kids. No video games that are goryare suitable for young kids. It follows that many M-rated video games are gory. (M: M-rated video games, S: games suitable for young kids, G: gory games)

6. Not all eyewitnesses’ testimonies are reliable. All unreliable testimonies should beinadmissible in court. Therefore, some eyewitnesses’ testimonies should not beadmissible in court. (E: eyewitnesses’ testimonies, R: reliable testimonies, A: evidencethat should be admissible in court)

7. All but those who cannot empathize with others are reasonable people. The onlyreasonable people are open-minded people. Therefore, all people who can empathizewith others are open-minded. (E: people who can empathize with others, R: reasonablepeople, O: open-minded people)

8. Only gamblers are stock investors. Some stock investors are adverse to risk-taking. Itfollows that some gamblers are not adverse to risk-taking. (G: gamblers, S: stockinvestors, A: people who are adverse to risk-taking)

9. Few drugs are without side effects. Drugs with side effects should be taken withcautions. So many drugs should be taken with cautions. (D: drugs, S: drugs with sideeffects, C: things that should be taken with cautions)

10. Not all Democrats are fiscal liberals. Only fiscal liberals support universal health care.It follows that some Democrats do not support universal health care. (D: Democrats, F:fiscal liberals, S: supporters of universal health care)

 

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