6.1. Standard -Form Categorical Syllogism

Consider the following argument:

No logic students are irrational people.

Some politicians are irrational people.

Therefore some politicians are not logic students.

This argument is an example of a standard-form categorical syllogism. A syllogism is a deductive argument in which a conclusion is inferred from two premisses. A categorical syllogism consists of three categorical propositions that together contain exactly three terms, each of which occurs in exactly two of the constituent propositions. A categorical syllogism is in standard form when its premisses and conclusion are all standard-form categorical propositions (A, E, I, or O), and are arranged in a specified standard order.

6.1. A Major, Minor, and Middle Terms

The conclusion of a standard-form categorical syllogism is the key to defining its elements:

• The predicate term of the conclusion is called the major term.
• The subject term of the conclusion is called the minor term.
• The third term, the one that occurs in the two premisses but not in the conclusion, is called the middle term.
• The premiss that contains the major term is called the major premiss.
• The premiss that contains the minor term is called the minor premiss.

In a standard-form categorical syllogism, the major premiss is stated first, the minor premiss second, and the conclusion last.

In the example that opened this chapter, the conclusion is "Some politicians are not logic students," so "logic students" is the major term and "politicians" is the minor term. "Irrational people," which appears in the premisses but not the conclusion, is the middle term. The major premiss is "No logic students are irrational people." The minor premiss is "Some politicians are irrational people."

 Summary Table:

The parts of a standard-form categorical syllogism

6.1. B Mood

Categorical syllogisms can be defined in terms of the type of the categorical propositions that compose them. In the example that opens this chapter [[EXAMPLE 6.1.B.1]], the major premiss is an E proposition (universal negative), the minor premiss is an I proposition (particular affirmative), and the conclusion is an O proposition (particular negative) (see Chapter 5). These three letters—EIO—represent the mood of the syllogism. All categorical syllogisms can be classified in terms of such a three-letter mood.

6.1. C Figure

Another variable defining the structure of a categorical syllogism is its figure, or the position of the middle term in the two premisses. Because the middle term can occupy one of two positions—subject or predicate—in each premiss, there are four possible different figures. These are conventionally labeled one through four.

• In the first figure, the middle term is the subject of the major premiss and the predicate of the minor premiss.
• In the second figure, the middle term is the predicate of both major and minor premisses.
• In the third figure, the middle term is the subject of both the major and minor premisses.
• In the fourth figure, the middle term is the predicate of the major premiss and the subject of the minor premiss.

 Summary Table

The Four Figures

Taken together, mood and figure completely describe the form of any standard-form categorical syllogism. The form of the example that opens this chapter [[EXAMPLE 6.1.C.5]], for instance, is EIO-2. The expression EIO indicates the mood of the syllogism and the number 2 indicates that the syllogism is in the second figure. Because there are 64 possible moods and 4 different figures, there are 256 distinct forms that standard-form syllogisms may assume. As we will see, however, only a few of these forms represent valid deductive arguments.

6.2. The Formal Nature of Syllogistic Arguments

The mood and figure of a syllogism uniquely determine its form and the form of a syllogism determines whether the syllogism is valid or invalid. Thus any syllogism of the form AAA-1

All M is P.

All S is M.

? All S is P.

is a valid argument, no matter what terms we substitute for the letters S, P, and M. In other words, in syllogisms of this and other valid forms, if the premisses are true, then the conclusion must also be true. The conclusion could be false only if one or both premisses were false.

Conversely, any argument in an invalid syllogistic form is invalid, even if both its premisses and its conclusion happen to be true.

A syllogistic form is invalid if it is possible to construct an argument in that form with true premisses and a false conclusion. Thus a powerful way to refute an argument in an invalid form is to counter it with an analogous argument—an argument in the same form—with obviously true premisses and an obviously false conclusion.

Although this method of logical analogy can demonstrate that a syllogistic form is invalid, it is a cumbersome tool for identifying which of the 256 possible forms is invalid. What’s more, the inability to find a refuting analogy does not conclusively demonstrate that a valid form is valid. The rest of the chapter is devoted to an explanation of more effective methods for testing syllogisms.

6.3. Venn Diagram Technique for Testing Syllogisms

As we saw in Chapter 5, two-circle Venn Diagrams represent the relationship between the classes designated by the subject and predicate terms in standard-form categorical propositions. If we add a third circle, we can represent the relationship among the classes designated by the three terms of a categorical syllogism. We use the label S to designate the circle for the minor term (the subject of the conclusion), the label P to designate the circle for the major term (the predicate of the conclusion), and the label M to designate the circle for the middle term. The result is a diagram of eight classes that represent the possible combinations of S, P, and M.

With this diagram we can represent the propositions in a categorical syllogism of any form to determine whether or not that form yields a valid deductive argument. To do this, we diagram the premisses and then examine the result to see if it includes a diagram of the conclusion. If it does, we know that the premisses entail the conclusion—that together they say what is said by the conclusion—and that the form is valid. If not, we know that the conclusion is not implied by the premisses, and the form is invalid.

Let’s see how this works for a syllogistic argument in the form AAA-1:

All M is P.

All S is M.

? All S is P.

To diagram the major premiss, "All M is P," we focus on the two circles labeled M and P. In Boolean terms, this means that the class of things that are M and not P is empty (MP¯=0). We diagram this by shading out all of the M circle that is not contained in (or overlapped by) the P circle.

To diagram the minor premiss, "All S is M", we shade out all of S that is not contained in (or overlapped by) M.

Combining these two diagrams gives us a diagram of both premisses— "All M is P" and "All S is M"—at the same time.

Examining this diagram reveals that the shaded areas include the region of S that is outside of P, and that the only unshaded region of S falls within the circle for P. In other words, this diagram of the premisses includes, without any modifications, a diagram of the conclusion: "All S is P," or SP¯=0. The premisses "say" that the conclusion, so the form is valid.

Now consider syllogisms in the form AAA-2:

All P is M.

All S is M.

? All S is P.

A Venn Diagram of the premisses looks like this:

If the conclusion were valid, all of the S circle that does not overlap with the P circle should be shaded. But this is not the case. Part of S that is outside of P—the SP¯M region—is not shaded. The conclusion says that something stronger than is expressed by the premisses. Syllogisms in this form are invalid, as an example [[EXAMPLE 6.3.1]] with obviously true premisses and an obviously false conclusion confirms.

When diagramming syllogistic forms with one universal and one particular premiss, it is important to diagram the universal premiss first before inserting an x for the particular premiss. Consider the form AII-3:

All M is P.

Some M is S.

? Some S is P.

Diagramming the universal premiss first and the particular premiss second, we get:

If we had diagrammed "Some M is S" before "All M is P," we could have placed an x into either the SMP¯ or the SMP regions. Diagramming "All M is P" first, however, shows us that SMP¯ is empty, leaving SMP as the only choice for the x that represents "Some M is S." But an x in SMP is also a diagram of the conclusion, "Some S is P." Thus the premisses entail the conclusion, and this syllogistic form is valid.

One more case illustrates a final important point about the construction of Venn Diagrams. Consider the form AII-2:

All P is M.

Some S is M.

? Some S is P.

Diagramming the universal premise gives us:

Turning to the particular premiss, "Some S is M," we run into a difficulty. The overlapping areas of circle S and P contain two regions. One of them, SPM, is included in circle P, but the other one, SP¯M isn’t. Putting the x within either of these regions would express more information than the premiss provides. In cases like this, we put the x right on the line that separates the two regions. We then get:

Does this diagram of the premisses include a diagram of the conclusion? If it did, there would have to be an x in a region of overlap between S and P, either SPM¯ or SPM. The shading in SPM¯ indicates that it is empty. The x on the line between SPM and SP¯M indicates that either of these classes must have a member, but it does not tell us which. The premisses do not tell entail the conclusion, and the form is invalid.

Summary Table

Creating Venn Diagrams of Categorical Syllogisms

1. Label the circles of a three-circle Venn Diagram with the syllogism’s three terms.
2. Diagram both premisses. If one premiss is universal and the other particular, start with the universal premiss.
3. If a particular premiss does not indicate on which side of a line between two regions to place an x, place the x on the line.
4. Inspect the resulting diagram of the premisses to see whether or not it also captures the conclusion. If it does, the syllogism is valid; if it doesn’t, the syllogism is invalid.

6.4 Syllogistic Rules and Syllogistic Fallacies

This section presents six rules for valid syllogisms and the fallacies that result from violating them.

Rule 1. Avoid four terms.

A categorical syllogism by definition has only three terms. If it has four, it is not a categorical syllogism. Sometimes, however, two terms masquerade as a single term. This happens when a word or phrase with more than one meaning is used in a different sense in two propositions. The syllogism appears to have only three terms, but because one term plays two roles, it actually has four.

Because it involves playing on ambiguity, the fallacy that results from violating this rule is called the fallacy of equivocation.

Rule 2. Distribute the middle term in at least one premise.

A proposition distributes a term if it asserts something about every member of the class the term designates. (See Summary: Quantity, Quality, and Distribution, Chapter 5 [[LINK TO THIS SUMMARY TABLE IN CHAPTER 5]].) The term "philosopher" is distributed in the proposition "All philosophers are thinkers," but the term "thinkers" is not. Because it is the middle term that links the terms of the conclusion, a syllogism cannot be valid unless either the subject or the predicate of the conclusion is related to the whole of the class the middle term designates. If that is not so, each of the terms of the conclusion might be connected to a different part of the middle term, and not necessarily connected with each other.

To violate this rule is to commit the fallacy of the undistributed middle.

Rule 3. Any term distributed in the conclusion must be distributed in the premisses.

A premiss that asserts something about only some members of a class (that does not distribute the term that designates that class) cannot entail a conclusion that asserts something about every member of that class (that does distribute the term). Therefore, when the conclusion of a syllogism distributes a term that is undistributed in the premisses, the syllogism is invalid.

To violate this rule is to commit the fallacy of illicit process. This fallacy takes two forms:

• Illicit process of the major term (illicit major) occurs when the major term is distributed in the conclusion but not in the premisses.
• Illicit process of the minor term (illicit minor) occurs when the minor term is distributed in the conclusion but not in the premisses.

Rule 4. Avoid two negative premises.

A negative (E or O) categorical proposition denies that a certain term applies to a class, in whole or in part. Suppose now that we are dealing with two negative premisses in a categorical proposition. One would say that the middle term M does not apply to the subject term S (or vice versa). The other premisses would say that M does not apply to P (or vice versa). Together, they tell us nothing about the relationship between P and S. As a result, all syllogisms with two negative premisses must be invalid.

To violate this rule is to commit the fallacy of exclusive prmisses.

Rule 5. If either premise is negative the conclusion must be negative.

An affirmative conclusion asserts that one of two classes—S or P—is contained in the other, in whole or in part. It can only be derived from premisses that assert the existence of a third class—M—that contains one of the classes in the conclusion and is itself contained in the other. An affirmative conclusion, then, can only follow from affirmative premisses.

To violate this rule is to commit the fallacy of drawing an affirmative conclusion from a negative premiss.

 RULE 6 From two universal premises no particular conclusion may be drawn.

This rule is based on the modern Boolean interpretation of categorical propositions according to which particular propositions have existential import but universal propositions do not. Following this interpretation, a particular conclusion cannot follow from universal premisses. In traditional, Aristotelian logic, this rule did not apply.

To violate this rule is to commit the existential fallacy.

6.5. Exposition of the 15 Valid Forms of the Categorical Syllogism

Applying these rules to the 256 possible forms for standard-form syllogisms and eliminating those that violate one or more of the rules leaves exactly 15 valid forms. Each of these has a traditional name. The following table lists the valid forms with their traditional names and links to an example for each.

Summary Table

The 15 Valid Forms of the Standard-Form Categorical Syllogism

6.6 Deduction of the 15 Valid Forms of the Categorical Syllogism.

This section of advanced material in the full text describes how to apply the syllogistic rules to the set of all possible standard-form categorical syllogisms to eliminate invalid forms and identify the 15 valid forms.