Law of Syllogism (Definition & Examples)

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Logic in geometry

Logic is a learned skill; it is as much a branch of mathematics as it is a kind of philosophy, or reasoning. Logic in geometry allows you to see connections and patterns, to make leaps of understanding from the single event to universal truths.

Logic is an attempt to use strict rules of thinking to reach reliable results, or conclusions, about claims, or premises. Here is a string of logical thinking:

You can neatly summarize that by saying that 15 minutes of study each night will pay off in higher grades on your geometry quizzes and tests.

Syllogism definition

Within logic, various types of arguments, premises, and conclusions can be formed. A syllogism is a method of reasoning by drawing a conclusion from two premises.

The particular pattern of a syllogism is that the first, major premise shares something with a second, minor premise, which in turn leads to a conclusion, like this:

1. I am creeped out, but also fascinated, by all spiders.

2. That enormous tarantula is a spider.

3. I am creeped out, but also fascinated, by that enormous tarantula.

Syllogism examples

What is truth?

A syllogism can present faulty premises. The conclusion to any faulty premise is automatically invalid, like this example:

1. All animals have four legs.

2. A snake is an animal.

3. All snakes have four legs.

That makes no sense, since the major premise is wrong. Spiders have eight legs; snakes have none; birds have two. Anything built from that incorrect, major premise (that all animals have four legs) is, then, invalid.

A syllogism can also have a faulty conclusion from valid premises. Look at this, and spot the problem:

1. Most people get nervous when they tell lies.

2. You appear nervous.

3. You must be lying about something.

The major and minor premises are fine; most people really do get nervous when they tell lies, and you really could appear nervous. But the conclusion is faulty, because the minor premise could be explained by dozens of other things: you are running late; you dressed hastily and your shoes don't match; your coach is thinking of benching you for the big game.

The structure of a syllogism

In a syllogism, the major premise is broad and wide, like saying, "All triangles have three sides and three interior angles." The major premise is often a conditional statement, beginning with "If."

The minor premise scales down that premise to something local, exact, or familiar: "This is a three-sided polygon." It can also be a conditional statement beginning with "If."

The conclusion connects the universal truth of the major premise to the immediate example of the minor premise: "Then this three-sided polygon is a triangle." Conclusions often begin with "Then."

The law of syllogism is also known as reasoning by transitivity. It is similar to the transitive property of equality, which says if this whatsit is like that doohickey, and that doohickey is like this thingamabob, then this whatsit is like this thingamabob:

1. If a = b

2. and if b = c

3. then a = c

Taking the same example from earlier and recasting the premises as conditional statements, we could write:

1. If all triangles have three sides and three interior angles,

2. And if this is a three-sided polygon,

3. Then this three-sided polygon is a triangle.

The law of syllogism provides for two conditional statements ("If …") followed by a conclusion ("Then …"). Logicians usually assign letters to these parts of the syllogism:

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 a valid conclusion.

Syllogism in geometry examples

The power of logic is seen over and over in geometric proofs. When you substitute terms, for example, you are following the law of syllogism:

1. If ∠A is supplementary to ∠B

2. and if ∠B = 115°

3. then ∠A = 65°

Perhaps without even noticing, you solve many steps in geometric proofs using the law of syllogism. The law of syllogism directs you to use deductive reasoning, which allows you to work down to specific examples from generalized postulates and theorems.

Suppose you have two horizontal, parallel lines and a point on the top line.

Euclid's Parallel Postulate tells us that for every line and a point not on that line, only one line can contain that point and be parallel to the line. The law of syllogism can help you to apply that postulate:

1. If a point not on a line can be in only one line parallel to that line,

2. And if Point B is on a line parallel to line DE,

3. Then only one line parallel to line DE contains point B

It is reasonable to simplify that same set of statements while preserving the law of syllogism, to better see the pattern of a = b, b = c, a = c:

1. A point not on a line can be in only one line parallel to that line.

2. Line AC, containing Point B, is parallel to line DE.

3. Line AC is the only line parallel to DE that contains Point B.

Extended syllogisms examples

You can extend syllogisms to build a series of premises and conclusions:

1. If I study each subject 15 minutes a night, then I will get good grades (if p then q)

2. If I get good grades, then I will get into good colleges (if q then r)

3. If I study each subject 15 minutes a night, then I will get into good colleges (if p then r)

Your premises must connect to ensure a valid conclusion. If your minor premise (if q then r) had been, "If I am smart, then my parents will be proud," no valid conclusion can emerge. The minor premise is unrelated to the major premise.

Lesson Summary

The gift of comedy writers is to wring a surprise out of the everyday, and one way to do that is to take logic and stand it on its head. Consider this odd leap: "If it rains today, then I better buy Band-Aids." The comically sad tale behind that? "If it rains today, then my dog will get wet, and, once inside, he'll shake water off, which will get the cat wet. If the cat gets wet, then she'll get angry and scratch me. I better buy Band-Aids." That is a syllogism.

Now that you have worked through this lesson, you are able to recognize and explain the law of syllogism as used in geometry (If p, then q; if q, then r; if p, then r), apply the law of syllogism to generate valid conclusions from valid premises, and identify and discern invalid conclusions or faulty premises in logic.