iLogic

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.

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

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.

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

All S are non-T.

Some P are not S.

Some P are S.

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.

Some P are not S.

and

All S are non-T.

Some P are S.

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

and

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.

Exercise 2.5

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

1.

2.

3.

4.

5.

6.

2. Decide the validity of the following syllogisms. You need to (1) identify the mood and figure; (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.

3. 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 form using the capital letters provided; if necessary, reduce the number of terms and indicate the 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 favoritism are fair-minded. Therefore, people who are not against favoritism are unreasonable. (R: reasonable people, F: fair-minded people, A: people who are against favoritism)

1. Beliefs in magic are at odds with the laws of nature. Beliefs that are at odds with the laws 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 should be legal, O: things that are obscene)
3. Only drugs approved by FDA are safe. Unsafe drugs should be recalled. So some drugs approved by FDA should be recalled. (D: drugs approved by FDA, S: safe drugs, R: drugs that should be recalled)
4. Some genetically altered foods can cause allergic reactions. Not all foods that can cause allergic reactions should be banned. So some genetically altered foods should not be banned. (G: genetically altered foods, R: foods that can cause allergic reactions, B: foods that should be banned)
5. All M-rated video games are unsuitable for young kids. No video games that are gory are 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 be inadmissible in court. Therefore, some eyewitnesses’ testimonies should not be admissible in court. (E: eyewitnesses’ testimonies, R: reliable testimonies, A: evidence that should be admissible in court)
7. All but those who cannot empathize with others are reasonable people. The only reasonable people are open-minded people. Therefore, all people who can empathize with others are open-minded. (E: people who can empathize with others, R: reasonable people, O: open-minded people)
8. Only gamblers are stock investors. Some stock investors are adverse to risk-taking. It follows that some gamblers are not adverse to risk-taking. (G: gamblers, S: stock investors, A: people who are adverse to risk-taking)
9. Few drugs are without side effects. Drugs with side effects should be taken with cautions. So many drugs should be taken with cautions. (D: drugs, S: drugs with side effects, 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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