Law of Syllogism – Explanation and Examples

The law of syllogism states that if a first event implies a second and the second implies a third, then the first event implies the third event.

This law is very similar to the transitive property of equality. It can be used with more than three events and is important for making logical arguments make sense in any branch of mathematics.

Before moving on, make sure to review conditional statements.

This section covers:

• What is the Law of Syllogism?
• Deductive Reasoning
• Law of Syllogism Examples

What Is the Law of Syllogism?

The law of syllogism states that it is possible to cut out middle steps in deductive reasoning. That is, if a first thing implies a second thing and that thing implies a third thing, then it is possible to say the first implies the third.

The law of syllogism can be employed with just two statements with three events, or it can be used for strings of hundreds of statements. It just requires that all intermediate conditional statements be true in order for the abridged statement to be true.

Recall that an antecedent is the part of a conditional statement that follows the word “if.” The consequence is the part that follows the word “that.” In a syllogism, the consequence of one statement is the antecedent of another.

Since the truth of an antecedent implies the truth of a consequence in a true conditional statement, it makes sense to string the statements together and cut out the middle parts.

For two statements, in formal logic, this is:

If $P \rightarrow Q$ and $Q \rightarrow R$, then $P \rightarrow R$.

For $n$ statements (where $n$ is a natural number greater than or equal to $2$), this is:

If $P_0 \rightarrow P_1$, $P_1 \rightarrow P_2$ ,…, and $P_{n-1} \rightarrow P_n$, then $P_0 \rightarrow P_n$.

If this sounds familiar, it should. It is very similar to the transitive property of equality, which states that if one thing is equal to a second and the second to a third, then the first is equal to the third.

Validity and Soundness in Proofs

Deductive reasoning – reasoning that uses syllogisms – relies on validity and soundness. A proof that is sound will necessarily be valid, but not vice versa. Ideally, all proofs will be sound.

Validity means that the logical structure of the argument works. That is, it is impossible for all of the premises to be true and the conclusion to be false. But, that can still happen when the premises are false.

This is where soundness comes in. A sound argument is one with a valid structure and true premises.

Law of Syllogism Examples

The law of syllogism is present in all areas of life.

For example, politicians use it to get your vote. They say:

• “If you vote for me for a second term, I will make decisions as I did in my first term.”
• “If I make decisions as I did in my first term, then I will help the homeless.”
• “Therefore, if you vote for me, I will help the homeless.”

They also pop up in children’s literature to show cause and effect. A particularly famous example of this follows a circular argument. That is, it follows an argument where the final event is the same as the first event.

This example is called “If You Give a Mouse a Cookie.” It is about a child who gives a mouse a cookie that then needs milk. Then, through a series of other events, it winds up being thirsty, needing milk and consequently needing another cookie.

Advertisers also use the law of syllogism. For example, they may say:

• “If you use our product, your hair will be shiny.”
• “If your hair is shiny, you will have lots of friends.”
• “Therefore, if you use our product, you will have lots of friends.”

This is how brands associate themselves with positive messages and feelings, even when it may seem like a stretch.

Examples

This section covers common examples of problems involving the law of syllogism and their step-by-step solutions.

Example 1

Can you think of a syllogistic argument that is valid but not sound? That is, an argument where it is impossible for the premises to be true and the conclusion to be false but where the premises are not true?

Solution

There are infinitely many examples of this. This solution will analyze one with animal taxonomy.

Consider the following statements:

1. If it is a fish, then it is a reptile.

2. If it is a reptile, then it is a mammal.

3. Therefore, if it is a fish, then it is a mammal.

This argument structure is valid because the argument structure doesn’t have any faults. That is, if $P$ is “it is a fish,” $Q$ is “it is a reptile,” and $R$ is “it is a mammal,” then this structure follows the law of syllogism. That is, $P \rightarrow Q$ and $Q \rightarrow R$ means that $P \rightarrow R$.

The problem here, however, is that the argument is not sound. This is because the premises are false. If something is a fish, it is not a reptile. For example, consider a clownfish. Similarly, if something is a reptile, it is not a mammal. Consider a python, for example.

Therefore, the structure is valid, but the argument is not sound.

Example 2

Can you think of an argument with a true conclusion but an invalid structure?

Solution

This time, it is required to find a true conclusion but an invalid way of getting to it. Consider another animal kingdom argument.

1. If it can swim, then it is smaller than a cat.
2. If it is smaller than a cat, then it is a fish.
3. If it is a fish, then it can swim.

The conclusion here is true. Fish can swim.

Following the line of reasoning, however, the law of syllogism states that the valid conclusion is “if it can swim, then it is a fish.”

This statement is, of course, false. For example, people can swim, and people are not fish.

Thus, the valid argument is not sound, but the invalid argument yields a true conclusion.

Why is this important? In a mathematical argument, the process is just as important as the conclusion. Using correct reasoning is the only way to trust any conclusion drawn.

Example 3

Use the law of syllogism to come to a conclusion based on the given statements.

A. All widgets are gizmos.

B. If it is a thing, then it is a widget.

Solution

First, it is necessary to convert the first statement to a conditional statement. It becomes “If it is a widget, then it is a gizmo.”

While A and B now have a common event (widget), they are not in the correct order used for the law of syllogism. Thus, swap them. The new statements are:

A. If it is a thing, then it is a widget.

B. If it is a widget, then it is a gizmo.

Now, using the law of syllogism, the conclusion is “if it is a thing, then it is a gizmo.”

These conditional statements are nonsense, so it is impossible to determine whether they are true or false.

Example 4

Use the law of syllogism to draw conclusions based on the following statements.

A. If it is a flying elephant, then it has stripes.

B. If it has stripes, then it is a zebra.

C. If it is a zebra, it is not a horse.

D. If it is not a horse, then it only eats cookies.

Solution

There is one main conclusion to be drawn from these statements, but there are several others as well.

The main conclusion combines the first event with the last. Namely, “if it is a flying elephant, then it only eats cookies.”

But, it is also possible to pick other events and conclude later ones. These are:

• “If it is a flying elephant, then it is a zebra.”
• “If it is a flying elephant, then it is not a horse.”
• “If it has stripes, then it is not a horse.”
• “If it has stripes, then it only eats cookies.”
• “If it is a zebra, then it only eats cookies.”

Note that this is mostly absurd. There are a few true conclusions (flying elephants are not horses and things with stripes are not horses). But, the prior untrue premises prevent the argument from being sound despite a valid structure and true conclusions.

Example 5

Is this argument sound? Why or why not?

1. If it is Kentucky, then it is an American State.
2. If it is an American State, then it is in North America.
3. If it is in North America, then it is in the Northern Hemisphere.
4. If it is in the Northern Hemisphere, then it borders at least two other states.
5. Therefore, if it is Kentucky, then it borders at least two other states.

Solution

Here, once again, the argument has a valid structure and a true conclusion. Yet; it is not sound because some of its intermediate arguments are false.

This structure is valid because $P_0 \rightarrow P_1$, $P_1 \rightarrow P_2$, $P_2 \rightarrow P_3$, $P_3 \rightarrow P_4$. Therefore, $P_0 \rightarrow P_4$. This is a basic application of the law of syllogism.

Yet, the fourth statement is not true. There are states in the Northern Hemisphere that border fewer than two other states. For example, Hawaii has no land borders.

Thus, the argument is valid but not sound.

Practice Problems

1. Use the law of syllogism to make the following into one statement:
   A. If it is sunny, then it is raining.
   B. If it is raining, I will bring my umbrella.
2. Use the law of syllogism to make the following into one statement:
   A. If it is a prime number greater than two, then it is odd.
   B. If it is odd, then it is not divisible by $2$.
   C. If it is not divisible by $2$, then it is one less than an even number.
3. Use the contrapositives of these statements and the law of syllogism to make a new conditional statement.
   A. If it is a square, then it is not a triangle.
   B. If it is not a triangle, then it does not have three sides.
4. Jayla reads the following and says the argument is invalid. What is her mistake?
   A. If it is a planet in the solar system, then it has at least six moons.
   B. If it has at least six moons, it is not Earth.
5. Explain why this is true but invalid.
   A. If it is a mammal, then it is not a fish.
   B. If it is not a fish, then it is a cat.
   C. Therefore, if it is a cat, then it is a mammal.

Answer Key

1. “If it is sunny, then I will bring my umbrella.”
2. “If it is a prime number greater than two, then it is one less than an even number.”
3. “If it is a triangle, then it is not a square” and “If it has three sides, then it is a triangle.” Therefore, “If it has three sides, then it is not a square.”
4. The argument is valid in structure, but the first statement is not true. Therefore, it is not sound.
5. The conclusion is true, but it does not follow if one assumes the previous statements to be true.