Chapter12The Categorical Syllogism

12.1 Theory of the Syllogism

We now turn to the theory of the syllogism. A syllogism is an argument composed of three categorical statements, two of which are premises, and the third is the conclusion. The three statements jointly contain three non-logical referring terms (subject terms and predicate terms), each appearing in two of the three statements. The theory of the syllogism has as its job determining which syllogisms are valid. Consider the following example:

There are three terms—“ducks,” “birds,” and “egg layers”—and each appears twice. The word used as the subject term of the conclusion of the syllogism (“ducks”) is called the minor term of the syllogism. The major term of the syllogism is the predicate term of its conclusion (“egg layers”). The third term in the syllogism (“birds”) doesn’t occur in the conclusion at all, but it appears in each of its premises; we call it the middle term.

In order to identify which is the major and which is the minor term, you work backward from the conclusion. The subject term in the conclusion is the minor term and the predicate term in the conclusion is the major term. The middle term is the one mentioned in the premises only but not the conclusion. The premise in the syllogism containing the major term (and a middle term) is called the major premise of the syllogism. The major premise is always written first. The other premise, which links the middle and minor terms, we call the minor premise, and it is written second.

12.2 Moods and Figures

In the example above, all three statements are A statements, which we can represent by “AAA.” Since there are 4 kinds of statements (A, E, I, O), there are 64 possibilities; these were traditionally called the moods of the syllogism.

The figure of a categorical syllogism refers to the four possible arrangements of the middle term. The middle term arrangement is represented as 4 numbered possibilities.

In the example of ducks, egg layers, and birds, the figure is 1, and the categorical syllogism is called AAA1 (fig. 12.1). The combination of mood and figure is known as form. Since there are 4 figures for each mood and there are 64 moods, there are 64 × 4 (= 256) syllogistic forms. Of these forms, only a few are valid; medieval logicians gave each form a mnemonic name to keep track of the valid ones.

12.3 Valid Forms

The syllogism with the form AAA1 is known as “Barbara,” because “Barbara” has three As as vowels. The syllogism with the form EAE1 is known as “Celarent,” the syllogism with the form AII1 is known as “Darii,” and so on.

Fortunately we do not need to remember the fifteen valid forms, nor do we need to apply the complex rules for determining validity that were necessary prior to the development of modern class logic.

12.4 Graphing Syllogisms

Venn diagrams provide us with a concrete and intuitive measure of validity. To determine the validity of a syllogism, we graph its premises on a “trefoil” Venn diagram containing three interlocking circles. For this purpose, we use diagrams with three interlocking circles, as shown in fig. 12.2.

This creates all possibilities for overlap between the three terms but of course does not represent classes in proportion to their size.

Consider the example of the valid form Celarent as shown in fig. 12.3.

We treat the top circle as the middle term, the lower left as the major term, and the lower right as the minor term. First, we graph the major premise, shading out the overlap between M and B. Then we graph the minor premise by shading out the area of W that is not M.

We do not graph the conclusion! Never graph the conclusion. Make a special note of this wherever you plan to do your homework. This method is for checking for validity, which means asking ourselves, If we graph the premises of the syllogism as true, then is it possible for the conclusion to be false? We understand this by inspecting the combination of the two premises. Inspecting the instance of Celarent as shown in fig. 12.3, is it possible for “No W are B” to be false? No. Indeed, it is true. It is represented because the overlap between W and B has already been shaded out. This is a valid argument.

Graphing I and O statements on a three-circle diagram requires thinking a bit differently about representing the existence of something. Let us consider another example:

The major term is “vegetarians,” the minor term is “anarchists,” and the middle term is “bankers” (fig. 12.4).

The middle term is represented by the top circle, with the major to the left and the minor to the right. We graph the first premise by putting an X in the lens between the “vegetarians and bankers” circles. Premise 1 tells us there is “some (at least one) vegetarian” that is “a banker,” but it doesn’t tell us what its relationship is to anarchists. We cannot decide either way, so we put it on top of the line to express our ignorance. However, when we graph the second premise, we shade out the lens between “bankers and anarchists,” which then pushes the X into the remaining space between “bankers and vegetarians.” Remember that we never graph the conclusion. We now look to see whether the conclusion is graphed as a result of the combination of premises. The conclusion states that there are some “anarchists that are not vegetarians.” Is this represented? No, there is no X in the anarchists space at all, nevermind in the anarchist space that is not vegetarian, thus the argument is invalid.

Here is another example using all I statements:

The middle term is “cheats,” so that is represented by the top circle; the major term is used car sales people, represented by the left-hand lower circle, and the minor term is bankers, represented by the right-hand lower circle. We graph the first premise by putting an X on the centre line in the middle of the lens formed by the “cheats” and “used car sales people” circles. We graph the second premise by putting a Y on the line in the centre of the lens formed by the “cheats” and “bankers” circles. We put the “X” and the “Y” on the line to express our ignorance about its relationship to other classes. Remember that we never graph the conclusion. We look to see if the conclusion is represented. Is there at least one banker who is a used car sales person? No.

We know for sure the argument is invalid because although we know that there is someone (X) who is a used car sales person and a cheat, we don’t know whether that person is a banker. And although we know that there is someone (Y) who is a cheat and a banker, we don’t know whether that person is a used car sales person. If we could know that X and Y were the same person, then the argument would be valid, but the premises do not authorize us to make that claim. In order to be valid, we need to see that there is in fact some banker that is also a used car salesperson, and we just don’t know that.

12.5 Enthymemes

We have seen enthymemes before, but we can also identify them in categorical syllogisms. In Chapter 7, we introduced enthymemes as follows: an enthymeme is an argument in which a required premise is not stated explicitly but is assumed implicitly as part of the argument. This section discusses how to identify an enthymeme in a categorical syllogism.

Consider these examples:

Example 1 can be reconstructed as a syllogism, working from the conclusion backward. The conclusion is “So they need food.” The “they” is referring back to “humans,” so the conclusion is “Humans need food.” Is this a “some” or an “all” statement? It is making a universal rule: “All humans need food.” Remember that we turn the predicate into a class, so we would transform this into “All humans are food needers.” Since this is the conclusion, “humans” is the minor term, and “food needers” is the major term. This makes “animals” the middle term. We get one premise above, “Humans are animals,” which translates to “All humans are animals.” So we have identified one premise and the conclusion:

• Premise: All humans are animals.
• Conclusion: All humans are food needers.

How do we find the suppressed premise? Recall the work we did on syllogisms identifying transitivity. What would it take to connect the two statements? Both statements say something about humans. But what is the connection between being an animal and being a food needer? It is not explicit. We have to make it explicit by adding a premise. It would be too weak to say that “Some animals are food needers,” so an “all” statement makes more sense. Is it “All animals are food needers” or “All food needers are animals.” Well, we should identify that plants are food needers too, even if they eat differently. So we have to say that “All animals are food needers” is the suppressed premise, which turns out to be our major premise. The syllogism turns out to be a Barbara figure.

• P1: All animals are food needers.
• P2: All humans are animals.
• _______________
• C: All humans are food needers.

Example 2 requires even more of our translation skills from the previous units: “Humans are fools so they regret their wasted lives.” Immediately you should notice that the syllogism has four terms: “Humans,” “fools,” “people who waste their lives,” and “people who are regretful.” The following chart offers interpretations to consider for translating this argument:

Since it has three premises and four terms, it must be reconstructed as a pair of syllogisms where the conclusion of the first syllogism is a premise in the second (fig. 12.6). The pair could either make a claim about some humans, as in the conclusion of the first interpretation or about all humans, as demonstrated in the second interpretation.

Here the conclusion of the first argument, that all (or some) men are lifewasters, forms a premise in the second argument, producing two valid syllogisms chained together to form a larger argument.

12.6 Rules for Using Venn Diagrams to Determine Validity

1. 1. Identify the premises and conclusion. Determine that there are two premises and a conclusion. If there appears to be only one premise, then the argument may be an enthymeme with an implicit premise, and if there appears to be three premises, the argument may be a chained enthymeme, in which two arguments are joined together by an implicit statement that is the conclusion of one argument and a premise in the other.
2. 2. Identify the three referring terms. The predicate term of the conclusion is the major term; the subject term of the conclusion is the minor term. The middle term appears only in the two premises. If there are four terms, the argument is a fallacy of ambiguity (a fallacy of four terms) or a chained enthymeme.
3. 3. Place each statement in standard categorical form and if you want, you can abbreviate the terms with a capital letter, but you must explain which term corresponds to which letter.
4. 4. Formalize the argument by placing the major premise first. Place the abbreviated version of the minor premise second. Place the conclusion last under the line.
5. 5. Diagram the argument. First, draw three intersecting circles, with one on top, and make sure to label them so that the lower left circle is labelled with the letter that stands for the major term, the top centre circle is labelled with the letter for the middle term, and the lower right circle is labelled for the minor term. Then graph the two premises on the diagram. Use different colours or crosshatching so that you can see each premise independently. Do not graph the conclusion. Make sure to graph particular premises by putting the X on the line if there is a line dividing the space where the X goes. If one side of the line is shaded by the graph of a universal premise, you must move the X into the remaining open space.
6. 6. Test the argument for validity. Examine the diagram you have made. Look to see whether the graph for the conclusion is present. If it is, the argument is formally valid, if it is not present, the argument is invalid.