Law of Syllogism in Geometry: Definition & Examples

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  • 0:00 Definition of the Law…
  • 2:06 Examples
  • 5:01 Beware of Being Misled
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Lesson Transcript
Instructor: Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson will explain the law of syllogism and provide several examples showing when it can be used to reach a valid conclusion and when it cannot.

Definition of the Law of Syllogism

In recent years, a satellite television provider made some humorous ads in which a person with cable television eventually had some negative consequences. For example, one went like this:

When you pay too much for cable, you throw things.
When you throw things, people think you have anger issues.
When people think you have anger issues, your schedule clears up.
When your schedule clears up, you grow a scraggly beard.
When you grow a scraggly beard, you start taking in stray animals.
When you start taking in stray animals, you can't stop taking in stray animals.

So, what we end up with is: When you pay too much for cable, you can't stop taking in stray animals.

This sure seems like a strange conclusion, doesn't it? Reaching a conclusion is not always easy. Sometimes there is no correct answer. Even when there is one, it can be hard to know what it is. At the end of the lesson, I'll explain why there is a problem with the conclusion of these ads. But for now, let's consider conclusions in mathematics. Thankfully, when it comes to mathematics, the conclusion is not always so uncertain.

The law of syllogism, also called reasoning by transitivity, is a valid argument form of deductive reasoning that follows a set pattern. It is similar to the transitive property of equality, which reads:

if a = b and b = c then, a = c.

There are also three parts involved in the law of syllogism, and each of these parts is called a conditional statement. A conditional statement has a hypothesis, which follows after the word if, and it has a conclusion, which follows after the word then. A letter is used to represent each phrase of the conditional statement.

Let me introduce the pattern, and then we can look at some examples.

Statement 1: If p, then q.
Statement 2: If q, then r.
Statement 3: If p, then r.

Statements 1 and 2 are called the premises of the argument. If they are true, then statement 3 must be the valid conclusion.

Examples

Now that we know what syllogism is, let's test our knowledge with some examples.

First, an example with a valid conclusion:

Statement 1: If it continues to rain (p), then the soccer field will become wet and muddy (q). This becomes if p, then q.

Statement 2: If the soccer field becomes wet and muddy (q), then the game will be canceled (r). This becomes if q, then r.

Statement 3: If it continues to rain (p), then the game will be canceled (r). This final statement is the conclusion, and becomes if p, then r.

This follows the pattern for the law of syllogism; therefore, it is a valid conclusion.

Now, let's try an example with an invalid conclusion:

Statement 1: If the bank robber steals the money (p), then the sheriff will track him down (q). This is If p, then q.

Statement 2: If the sheriff tracks him down (q), then the bank robber will be arrested (r). This is If q, then r.

Statement 3: If the bank robber steals the money (p), then the bank robber will be rich (s). This is If p, then s.

Instead of building on statement 2, this final statement simply offers another possibility of statement 1. This does not follow the law of syllogism pattern, so statement 3 is an invalid conclusion.

Now, we'll do look at an example that we'll call valid conclusion possible:

Statement 1: If the truck runs over some nails (p), then a tire will go flat (q). If p, then q.

Statement 2: If a tire goes flat (q), then the deliveries will not be made on time (r). If q, then r.

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