Law of Syllogism

In geometry, the law of syllogism is used in logical reasoning to draw conclusions from given statements. The word syllogism means deduction or inference in Greek. It is like a chain rule and similar to the transitive property; that is, if \(a = b\) and \(b = c,\) then \(a = c.\)

According to the law of syllogism in geometry, if two conditional statements are true, then:
 

• If p ≥ q (if p, then q)
   
• If q ≥ r (if q, then r)
   
• Therefore, p ≥ r (if p, then r)
   

Example 1: The strength of logical reasoning appears repeatedly in geometric proofs. When we substitute one statement for another, we are effectively applying the law of syllogism.

• If \(∠A\) is supplementary to \(∠B,\)
   
• If \(∠B = 115°,\)
   
• Then \(∠A = 65°.\)

Example 2: Let us think of a triangle where all of its sides are of the same length. Since the triangle's sides are equal in length, it is an equilateral triangle. We can now apply the law of syllogism to the triangle.
 

• All equilateral triangles are equiangular.
   
• Triangle ABC is an equilateral triangle.
   
• Triangle ABC is equiangular.

The law of syllogism is the fundamental principle in logical reasoning that allows us to conclude from two conditional statements. It has three parts: the first two are premises, and the last one is the conclusion. The conditional statement that follows the word “IF” is the hypothesis, and the inference follows after the word “THEN”.

The syllogism follows the pattern, 

• Statement 1: If P, then Q
   
• Statement 2: If Q, then R
   
• Statement 3: If P, then R

Here, statements 1 and 2 are the premises. If both the premises are true, then the conclusion (statement 3) is true. 

For example,
 

• If a triangle is equilateral, then its sides are equal.
   
• If all the sides of a triangle are equal, its angles are 60°
   
• Then, in an equiangular triangle, all the angles are 60°

Since both conditions describe equivalent properties in Euclidean geometry, the conclusion depends on how the definitions are applied. If interpreted differently, the syllogism may not hold.

We can draw a conclusion using the law of syllogism by linking the ‘if’ statements with the ‘then’ statement, where the end of the first statement matches with the start of the second statement, and skipping the middle.
Symbolically, if \(p\rightarrow q\) and \(q  \rightarrow r\) are both true, then we can conclude that \(p \rightarrow r\).

A conclusion can be drawn using the law of syllogism by understanding that, if the first statement leads to a second, and the second statement leads to a third, then the first statement leads to the third. 

For example, 
 

• If it is raining, the ground will get wet.
   
• If the ground gets wet, then the plants will grow. 
   
• If it rains, the plants will grow.
   

The conclusion here is that “if it rains, the plants would grow,” which directly links the first and third statements.

A syllogism has two statements, a major and a minor premise. While the major premise represents a general statement, the minor premise applies it to a specific case. There are three types of syllogism: 

Conditional Syllogism

A conditional syllogism uses statements linked together with the “if-then” condition to reach a new conclusion. 

For example, 
 

• If it is a Tuesday, I will have a math class. 
   
• If I have a math class, I would need my pencils. 
   
• If it is a Tuesday, I would need my pencils.
   

Categorical Syllogism
 

A categorical syllogism consists of two premises and a conclusion, where the three propositions are connected by categories such as "all," "some," "no," or "some not."

For example,
 

• All men are mortal.
   
• Arnold is a man.
   
• Arnold is mortal.

Disjunctive Syllogism
 

A disjunctive syllogism is a logical chain containing two premises and an implication in the form of either-or sentences.

For example,
 

• Either the light is on, or the room is dark. 
   
• The light is not on. 
   
• Therefore, the room is dark.

The Law of Syllogism is one of the easiest and fun topics in mathematics. Here are some tips and tricks to help students master the laws of Syllogism.
 

• Teachers can start teaching the law of syllogism using everyday examples. Give them an easy example to help them understand, such as,
  
  If I finish homework, I can watch TV.
  If I can watch TV, I will watch cartoons.
  
  Ask them to guess the conclusion. The conclusion is that “If I finish homework, I will watch cartoons.”
   
• Use arrow diagrams for a simple explanation. Draw arrow maps like \(p \rightarrow q \rightarrow r \rightarrow s.\) The visual method helps in learning the law of syllogism naturally.
   
• Teachers can teach the trick “middle steps cancel.” The second statement is usually cancelled out when the first two are true. This trick will save students a lot of time.
   
• Parents and teachers can use different colors for the three statements to help learners identify and differentiate the conclusion and statements.
   
• Students should practice fun chain challenges that combine many syllogisms related to geometry.
   
• Students should focus on daily exercises rather than a single long lesson. Try speaking out the reasoning verbally.

The law of syllogism is used in real-world situations to make decisions. These are some of its applications:
 

• Research: By linking observations to established theories, researchers use syllogisms. 
   
• Geometry Proofs: The law of syllogism is used to prove mathematical relationships, mainly in geometry.
   
• AI Decisions: In AI, we use this law to make decisions.