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High School Geometry: Help and Review13 chapters | 162 lessons

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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.

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.

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*.

If this is all the information we have, then we know that statements 1 and 2 follow the pattern for the law of syllogism so far; therefore, we know that a valid conclusion is possible for statement 3 before we even see it. However, for the statement to be truly valid, it must also follow the pattern. For example:

Statement 3: *If the truck runs over some nails (p), then the deliveries will not be made on time (r).* If *p*, then *r*.

Finally, let's take a look at an example where there is no conclusion possible.

Statement 1: *If she smells the flowers (p), then she will begin to sneeze (q).* If *p*, then *q*.

Statement 2: *If she has allergies (r), then she will have sinus troubles (s).* If *r*, then *s*.

Since statements 1 and 2 do not follow the pattern of the law of syllogism, there is no conclusion possible for statement 3 using this law.

Have you figured out what the problem was in the logic of the television ads? The law of syllogism makes an assumption that the first two statements are true. If these are not actually true, then the third statement may not be reasonable. The example of the television ad didn't use exactly the same wording as we have used in our examples, but the idea is the same. The ridiculous conclusions were the results of faulty premises. Let me illustrate this with another outlandish example.

*If I can hold my breath, then I can swim under water.If I can swim under water, then I am a fish.If I can hold my breath, then I am a fish.*

Remember, even though this follows the pattern of the law of syllogism, the conclusion is unreasonable because of some faulty premises. If, for example, you are using the law of syllogism to work a problem or complete a proof, then make sure that your premises are true.

Mathematics requires logical decisions to be made. The **law of syllogism** is a pattern that can be used to help make a logical decision. If you presume that two statements are true and these statements follow the prescribed pattern for the law of syllogism, then there is a logical conclusion that can be reached by using this pattern.

Statement 1: If *p*, then *q*.

Statement 2: If *q*, then *r*.

Statement 3: Conclusion: If *p*, then *r*.

However, don't forget, the pattern will only yield trustworthy results if it is based on trustworthy premises.

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High School Geometry: Help and Review13 chapters | 162 lessons

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