Arguments and Validity: Eight (8) Rules of Syllogism in Categorical Logic

An argument consists of two or more propositions offered as evidence for another proposition. In logic and critical thinking, the propositions that are offered as evidence in the argument are called the premises, while the proposition for which the evidence is offered is called the conclusion. Thus, when one gives an argument, one is providing a set of premises as reasons for accepting his or her conclusion. It is important to note that when one gives an argument, one does not necessarily attack or criticize the other. In this way, an argument can also be viewed as a support of someone’s viewpoint.

Types of Arguments

Arguments can either be inductive or deductive. On the one hand, an inductive argument is one in which it is claimed that if the premises are true, then it is probable that the conclusion is true. Hence, even if all of the premises are true, inductive argument or reasoning allows the conclusion to be false. It is also important to note that inductive arguments go from the specific (or particular) to the general. In other words, inductive arguments make broad generalizations from specific observations. Consider the example below.

Example 1:

Ninety percent of the mongo seeds germinate in day 1.
And in day 2, ninety percent of the mongo seeds germinate.
Therefore, ninety percent of the mongo seeds germinate.

Based on the example above, we can also say that inductive arguments are based on observations or experiments.

Deductive arguments, on the other hand, is one in which it is claimed that if the premises are true, then the conclusion is necessarily true. And unlike inductive arguments, deductive arguments proceed from the general to the particular. Thus, a deductive argument or reasoning begins with a general statement or hypothesis and then “examines the possibilities to reach a specific, logical conclusion”. 

Let us consider the example below.

Example 2:

Anybody who kills a person is guilty of a felony.
Jim kills Jack.
Therefore, Jim is guilty of a felony.

Syllogisms

Syllogisms are arguments which consist of three propositions which are so related so that when the first two propositions (that is, premises) are posited as true the third proposition (that is, the conclusion) must also be true. In other words, a syllogism is an argument arranged in a specific manner in such a way that it contains a major premise, minor premise, and a conclusion. Consider the classic example of a categorical syllogism below.

Example 1:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

How do we determine the major premise, minor premise, and the conclusion?

The major premise is the premise that contains the major term, while the minor premise is the premise that contains the minor term. The conclusion is the third proposition whose meaning and truth are implied in the premises.

How do we determine the major term, minor term, and the middle term?

The major term is the predicate of the conclusion, while the minor term is the subject of the conclusion. The middle term is the remaining term which does not (and cannot) appear in the conclusion.

If we look at the example above, then we know that the major term is “mortal” because it is the predicate of the conclusion and the minor term is “Socrates” because it is the subject of the conclusion. The middle term is “man” or “men” because it is the remaining term and which does not appear in the conclusion. As we can see in the example below, the major term is in red color, the minor term in blue, and the middle term in purple.

Rules of Syllogism

Now that we have presented the key concepts in arguments or syllogisms, let us proceed to the determination of their validity. Logicians have formulated eight (8) rules of syllogism, but of course they can be expanded to 10 or reduced to 6. But let us follow what logicians commonly used, that is, the 8 rules of syllogism. It must be noted that all of the 8 rules of syllogism must be met or satisfied for the argument or syllogism to be valid. If at least one of the 8 rules of syllogism is violated, then the argument or syllogism is invalid.

The 8 rules of syllogism are as follow:

1. There should only be three terms in the syllogism, namely: the major term, the minor term, and the middle term. And the meaning of the middle term in the firs premise should not be changed in the second premise; otherwise, the syllogism will have 4 terms.
2. The major and the minor terms should only be universal in the conclusion if they are universal in the premises. In other words, if the major and the minor terms are universal in the conclusion, then they must also be universal in the premises for the argument to be valid. Hence, if the major and minor terms are particular in the conclusion, then rule #2 is not applicable.
3. The middle term must be universal at least once. Or, at least one of the middle terms must be universal.
4. If the premises are affirmative, then the conclusion must be affirmative.
5. If one premise is affirmative and the other negative, then the conclusion must be negative.
6. The argument is invalid whenever the premises are both negative. This is because we cannot draw a valid conclusion from two negative premises.
7. One premise at least must be universal.
8. If one premise is particular, then the conclusion must be particular.

Now, let us apply these 8 rules of syllogism to the arguments below. Let us color the terms to avoid confusion. So, let us assign the color red for the major term, blue for the minor term, and purple for the middle term.

Rule #1 of the 8 rules of syllogism: There should only be three terms in the syllogism, namely: the major term, the minor term, and the middle term.

If we analyze the syllogism above, it would appear that the argument is invalid because it violates rule #1. As we can see, the syllogism above contains 4 terms because the meaning of the middle term “stars” in the first premise is changed in the second premise. The term “stars” in the first premise refers to astronomical bodies or objects, while the term “star” in the second premise refers to celebrities.

Let us consider another example.

As we can see, the syllogism above contains only three terms. Hence, this syllogism is valid in the context of rule #1.

Rule #2 of the 8 rules of syllogism: The major and the minor terms should only be universal in the conclusion if they are universal in the premises.

As we can see, the minor term “terrorist” in the conclusion is universal because of the universal signifier “no”. And since the minor term “terrorist” in the second premise is universal because of the universal signifier “no”, then the syllogism above does not violate rule #2 in the context of the minor term. However, the major term “brilliant” in the conclusion is universal because the proposition is negative; as we already know, the predicate terms of all negative propositions are universal. But if we look at the major term in the first premise, it is particular because, as we already know, the predicate terms of all affirmative propositions are particular. In the end, the syllogism above is invalid because it violates rule #2. This is what logicians call the “fallacy of illicit major”.

Let us consider another example.

Because the major term “creative” in the conclusion is particular, as it is a predicate term of an affirmative proposition, then it does not violate rule #2 because the rule is not applicable here. As we can see, rule #2 is applicable only to universal minor and major terms. But if we check the minor term “weird people” in the conclusion, we learned that it is universal because of the universal signifier all. Since the minor term “weird people” is universal in the conclusion, then it must also be universal in the second premise for this syllogism to be valid. If we look at the minor term in the second premise, it is particular because it is a predicate term of an affirmative proposition. Therefore, in the end, the syllogism above is invalid because it violates rule #2. This is what logicians call the “fallacy of illicit minor”.

Let us consider a valid argument below in the context of rule #2 of the 8 rules of syllogism.

The syllogism above is valid in the context of rule #2 of the 8 rules of syllogism because rule #2 is not violated. As we can see, the minor term “Greg” in the conclusion is particular; hence, rule #2 is not applicable. Of course, if a rule is not applicable, then it cannot be violated; and if no rule or law is violated, then the argument is automatically valid. Now, if we look at the major term “liar” in the conclusion, it is universal because it is a predicate term of a negative proposition. But because the minor term “liar” is also universal in the first premise because, again, it is a predicate term of a negative proposition, then this argument satisfies rule #2.

Let us consider another valid argument in the context of rule #2 of the 8 rules of syllogism.

Both the minor and major terms in the conclusion of the syllogism above are particular. For this reason, rule #2 of the 8 rules of syllogism is not applicable. Hence, the syllogism is automatically valid in the context of rule #2 of the 8 rules of syllogism.

Rule #3 of the 8 rules of syllogism: The middle term must be universal at least once.

The syllogism above is valid in the context of rule #3 of the 8 rules of syllogism because the middle term “beans” in the first premise is universal. In fact, rule #3 of the 8 rules of syllogism asks that at least one of the middle terms must be universal.

Let us consider another example.

As we can see, both middle terms in the first and second premise are particular. But because rule #3 of the 8 rules of syllogism asks that at least one of the middle terms must be universal, then the syllogism above is invalid.

Rule #4 of the 8 rules of syllogism: If the premises are affirmative, then the conclusion must be affirmative.

The syllogism above is valid because it satisfies rule #4 of the 8 rules of syllogism. As we can see, both premises are affirmative and the conclusion is affirmative.

Let us consider another example.

The syllogism above is invalid because it does not satisfy rule #4 of the 8 rules of syllogism. As we can see, both premises are affirmative, but the conclusion is negative.

Rule #5 of the 8 rules of syllogism: If one premise is affirmative and the other negative, then the conclusion must be negative.

The syllogism above is valid in the context of rule #5 of the 8 rules of syllogism. As we can see, the first premise is affirmative, the second premise is negative, and the conclusion is negative.

The syllogism above is invalid in the context of rule #5 of the 8 rules of syllogism. As we can see, the first premise is affirmative, the second premise is negative, but the conclusion is affirmative. Hence, it violates rule #5 of the 8 rules of syllogism.

Rule #6 of the 8 rules of syllogism: The argument is invalid whenever the premises are both negative.

Obviously, the above syllogism is invalid because both premises are negative.

Rule #7 of the 8 rules of syllogism: One premise at least must be universal.

The above syllogism is valid in the context of rule #7 of the 8 rules of syllogism because it qualifies the rule. As we can see, the first premise is universal.

Rule #8 of the 8 rules of syllogism: If one premise is particular, then the conclusion must be particular.

The first premise of the above syllogism is particular, and the conclusion is particular too. Therefore, this syllogism is valid in the context of rule #8 of the 8 rules of syllogism.