The Categorical Syllogism: Determining the Validity of Arguments

A SYLLOGISMis typically a three-proposition deductive argument—that is, a mediate inference that consists of two premises and a conclusion. Among the various forms of syllogism, the so-called categorical syllogism is perhaps the most popular as it serves as the heart of the Traditional Aristotelian Logic. Until today, the categorical syllogism appears in most, if not all, Logic syllabi as many essential lessons in correct reasoning can be learned from the topic.

In a categorical syllogism,all the propositions used are categorical propositions, hence the label ‘categorical.’ So let’s start by reviewing the four types of categorical statement.

The Four Types of Categorical Proposition

The statement that relates two classes or ‘categories’ is called a categorical proposition. The classes in question are denoted respectively by the subject term (S) and the predicate term (P). In effect, this type of proposition gives a direct assertion of agreement or disagreement between the two terms.The proposition asserts that either allor part of the class denoted by the subject term is included in or excluded from the class denoted by the predicate term. Here are some examples of categorical statement.

1. All dogs are mammals.

2. No acids are bases.                                                                                 

3. Some philosophers are mathematicians.

4. Some Americans are not cheaters.

The first example asserts that the whole class of dogs are included in the class of mammals; the second declares that the entire class of acids are excluded from the class of bases; the third states that a part of philosophers are included in the class of mathematicians; and the last one claims that a part of the class of Americans are excluded from the class of cheaters.

Categorical statements have the so-called quantity and quality. The “quantity”of a categorical statement is either universal or particular, depending on whether the statement makes a claim about all members or just some members of the class denoted by the subject term. The quantity of the subject term determines the quantity of the whole proposition. Since the propositions which take the form “All S are P” and “No S are P” obviously assert something about every member of class S, they are considered universal.

   e.g. Allballs are round objects; No plants are rational being.

Statements that take the form “Some S are P” and “Some S are not P” logically assert something about “at least one” members of class S and hence, are particular.

   e.g. Some politicians are females; Some government officials are not corrupt.

Note that the quantity of a categorical statement can be determined through the quantifier of the subject term: “All” and “No” imply universal quantity, while “some” indicates particular quantity.

The “quality” of a categorical statement is either affirmativeor negative, depending on whether the statement affirms or denies class membership. Propositions which have the form “All S are P” and “Some S are P” are affirmative because in these forms, the subject asserts the predicate.

   e.g. All rocks aresolid objects; Some animals are carnivores.

On the other hand, the predicate is denied or negated by the subject in the statements which have the form “No S are P” and “Some S are not P.” Hence, they are negative.

   e.g. No women are priests (=All women are not priests); Some animals are notaquatic creatures.

Generally, the copula indicates the quality of a proposition. If the copula is affirmative (am, is, are), the entire proposition is affirmative, regardless of whether or not the terms are negative. Similarly, negative copula (am not, is not, are not) necessarily makes the whole proposition negative.

Combining the quality and quantity of propositions results in four structures of statements known as the four standard forms of categorical propositions:

1. Universal Affirmative in which the whole of the subject class is included in the predicate class.

      e.g. All reptiles are animals.

            All men are mortals.

2. Universal Negative in which the whole of the subject class is excluded in the predicate class.

      e.g No vegetarians are carnivores.

           No moons are planets.

3. Particular Affirmative in which a part of the subject class is included in the predicate class.

      e.g Some congressmen are smokers.

             Some medicines are drugs.

4. Particular Negative in which a part of the subject class is excluded in the predicate class.

      e.g Some dogs are not Dalmatians.

           Some students are not athletes.

Since the early Middle Ages, the four types of categorical statements have been designated by letters:

A for Universal Affirmative (All S are P)

E for Universal Negative (No S are P)

I for Particular Affirmative (Some S are P)

O for Particular Negative (Some S are not P)

Components of categorical syllogism

A categorical syllogism is a form of argument that is composed of three categorical propositions. That is, its two premises and conclusion are all categorical statements.

The three categorical propositions in a categorical syllogism contain a total of three different terms, each of which appears twice in distinct propositions. The following is an example of a categorical syllogism:

All amphibians are cold-blooded vertebrates.

All frogs are amphibians.

Therefore, all frogs are cold-blooded vertebrates.

As you would notice from our example, a categorical syllogism consists of three and only three propositions, and three and only three terms.

The terms used in our example are amphibians, cold-blooded vertebrates, and frogs. In a categorical syllogism, the predicate term of the conclusion is called the major term (usually represented by P) while the subject term is the minor term(S). In our example therefore, the major term is cold-blooded vertebrates whereas the minor term is frogs.

The third term in the syllogism does not appear in the conclusion, but is employed in both premises. It is the term that connects, relates, or mediates the two other terms, hence called middle term (M). The middle term in our example is amphibians.

Notice that aside from the conclusion, we have two premises in a categorical syllogism. One of them contains the major term and is thus called the major premise. The other premise, which contains the minor term, is referred to as the minor premise. “All amphibians are cold-blooded vertebrates” is the major premise in our example. The remaining premise is the minor premise as it contains the minor term.

Standard logical form

One thing about categorical syllogism is that its validity depends solely upon its structure or logical form, not upon its contents. Thus, in determining whether or not a particular categorical syllogism is valid, restating the argument in standard form is vital. 

A categorical syllogism is in standard form if it meets the following four conditions:

1. All three statements are standard-form categorical propositions.

This means that each statement in the argument has a proper quantifier, subject term, copula, and predicate term (Q-S-C-P).

2. Each term appears twice in the argument.

The major term appears once in the conclusion and once in the major premise; the minor term, in the conclusion and the minor premise; and the middle term, in both premises.

3. Each term is used in the same sense throughout the argument. The possibility of equivocation must be ruled out.

4. The major premise is listed first, the minor premise second, and the conclusion last.

Note that our example (about ‘amphibians’) is already in standard form since it meets all these conditions.

Figure and mood

The variety of structures in which a standard categorical syllogism may occur can be labeled by stating its form. The form of a categorical syllogism is the combination of its figure and mood. The moodsimply refers to the types of categorical propositions (A,E, I, or O) used in the syllogism, listed in the order in which they appear in standard form. For instance, a syllogism with an E proposition as its major premise, an I proposition as its minor premise, and O proposition as its conclusion has the mood EIO. In our example (about ‘amphibians’), the mood is thus AAA.

Now, you would also notice that each syllogistic mood, say EIO, can have four distinct versions depending on the arrangement of the major, minor, and middle terms in the premises. It is because a particular term can either be the subject or the predicate of the proposition. Thus, we need to supplement the labeling system with mentioning its figure, that which identifies the four distinct ways the middle terms are arranged in the syllogism:

Figure 1: the middle term is the subject term of the major premise and the predicate term of the minor premise

Figure 2: the middle term is the predicate term of both premises

Figure 3: the middle term is the subject term of both premises

Figure 4:  the middle term appears as the predicate term of the major premise and the subject term of the minor premise

Going back to our example, its logical form (mood and figure) is therefore AAA-1. Considering all possible moods and figures, there are exactly 256 distinct forms of categorical syllogism (four types of major premise multiplied by four kinds of minor premise multiplied by four kinds of conclusion multiplied by four possible figures).

Also Check Out:
Reasoning and Debate: A Handbook and a Textbook by Jensen DG. Mañebog

See next:
The Validity of Categorical Syllogism (searchable through the search engine of this site)