Deductive Reasoning

Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture. 

Let’s learn about those laws of logic.  

Laws of Logic 

Law of Detachment

If the hypothesis of a true conditional statement is true, then the conclusion is also true.  

Law of Syllogism

Solve Examples 

Law of Detachment 

Example 1: 

Use the Law of Detachment to make a valid conclusion in the true situation. 

1. If two segments have the same length, then they are congruent. You know that BC = XY. 
2. Mary goes to the movies every Friday and Saturday night. Today is Friday. 

Solution: 

1. Because BC− = XY− satisfies the hypothesis of a true conditional statement, the conclusion is also true. So, BC− ≅ XY-. 

2. First, identify the hypothesis and the conclusion of the statement. The hypothesis is “If it is Friday or Saturday night,” and the conclusion is “then Mary goes to the movies.” 

“Today is Friday” satisfies the hypothesis of the conditional statement, so you can conclude that Mary will go to the movies tonight. 

Law of Syllogism 

Example 2: 

If possible, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. 

1. If Rick takes chemistry this year, then Jesse ill be Rick’s lab partner. If Jesse is Rik’s lab partner, then Rick will get an A in chemistry. 

2. If x² > 25, then x² > 20. If x > 5, then x² > 25.          

3. If a polygon is regular, then all angles in the interior of the polygon are congruent. 

If a polygon is regular, then all of its sides are congruent. 

Solution: 

1. The conclusion of the first statement is the hypothesis of the second statement, so you can write the following new statement. 

If Rick takes chemistry this year, then Rick will get an A in chemistry. 

2. Notice that the conclusion of the second statement is the hypothesis of the first statement, so you can write the following new statement. 

If x > 5, then x² > 20. 

3. Neither statement’s conclusion is the same as the other statement’s hypothesis. You cannot use the Law of Syllogism to write a new conditional statement. 

Use Inductive and Deductive Reasoning 

Example 3: 

What conclusion can you make about the product of an even integer and any other integer? 

Solution: 

Step 1: 

Look for a pattern in several examples. Use inductive reasoning to make a conjecture. 

(–2)(2) = –4, (–1)(2) = –2, 2(2) = 4, 3(2) = 6, (–2)(–4) = 8, (–1)(–4) = 4, 2(–4) = –8,  
3(–4) = –12. 

Conjecture: Even integer × Any integer = Even integer. 

Step 2: 

Let n and m each be any integer. Use deductive reasoning to show the conjecture is true. 

2n is an even integer because any integer multiplied by 2 is even. 

2nm represents the product of an even integer and any integer m. 

2nm is the product of 2 and an integer nm. So, 2nm is an even integer. 

The product of an even integer and any integer is an even integer. 

Reasoning from a Graph 

Example 4: 

Whether the statement is the result of inductive reasoning or deductive reasoning. 

Explain your choice. 

1. The northern elephant seal requires more strokes to surface the deeper it dives. 
2. The northern elephant seal uses more strokes to surface from 60 feet than from 250 feet. 

Solution: 

1. Inductive reasoning is based on a pattern in the data. 
2. Deductive reasoning, because you are comparing values that are given on the graph. 

Questions to Solve 

Question 1:

Make a valid conclusion in the situation. (Hint: Use Law of Detachment) 

If the measure of an angle is 90 degrees, then it is a right angle. The measure of ∠ A is 90 degrees.  

Solution: 

Since the hypothesis “measure of an angle is 90 degrees” is true, the conclusion is true.  

The valid conclusion is “∠ A is a right angle.” 

Question 2: 

Write the statement that follows from the pair of statements that are given. (Hint: Use Law of Syllogism) 

If a rectangle has four equal side lengths, then it is a square. If a polygon is a square, then it is a regular polygon. 

Solution: 

Since the conclusion in the first statement is the same as the hypothesis in the second statement. So, we can write: 

“If a rectangle has four equal side lengths, then it is a regular polygon.” 

Question 3: 

Decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning. 

1. The rule at your school is that you must attend all of your classes in order to participate in sports after school. You played in a soccer game after school on Monday. Therefore, you went to all of your classes on Monday. 
2. For the past 5 years, your neighbor has gone on vacation every July 4th and asked you to feed her hamster. You conclude that you will be asked to feed her hamster next July 4th. 

Solution: 

1. “You played in a soccer game after school on Monday” is a fact. On the basis of this fact, we reached the conclusion that you went to all of your classes on Monday.  So, it is deductive reasoning.   
2. “For the past 5 years, your neighbor goes on vacation every July 4th and asks you to feed her hamster” is a pattern. On the basis of this pattern, we reached the conclusion. So, it is inductive reasoning.  

Question 4: 

In parts a to d, use the true statements below to determine whether you know the conclusion is true or false. Explain your reasoning. 

1. If Arlo goes to the baseball game, then he will buy a hot dog. 
2. If the baseball game is not sold out, then Arlo and Mia will go to the game. 
3. If Mia goes to the baseball game, then she will buy popcorn. 
4. The baseball game is not sold out. 

a. Arlo bought a hot dog.  

b. Arlo and Mia went to the game. 

c. Mia bought a hot dog.  

d. Arlo had some of Mia’s popcorn. 

Solution: 

Let’s first solve part b.  

a. Statement 4 is a true fact. That means the hypothesis in statement 2 is true. So using the law of detachment, the conclusion “Arlo and Mia will go to the game” is true.  

b. Using the law of syllogism, the parts of the conclusion in statement 2 is the same as hypothesis in statement 1 and 3.  

So, we can say that If Arlo and Mia go to the baseball game, Arlo will buy a hot dog, and Mia will buy popcorn.  

Since “Arlo and Mia will go to the game” is true (as we solved in part b), the conclusion written in the just above statement is true.  

So, “Arlo bought a hot dog” is true.  

a. In part a, we proved that “Mia will buy popcorn” is true. So “Mia bought a hot dog” is false.  

b. There is nowhere mentioned as a hypothesis or a conclusion that Arlo had some of Mia’s popcorn, so we will take it false.  

Answers: 

1. True 
2. True 
3. False  
4. False  

Key Concepts Covered  

• Inductive reasoning is reaching a conclusion based on a series of observations or some patterns. It is about making conjecture. The conjecture may be true or false.  
• Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to reach a conclusion. It is about showing that conjectures are true or false.  
• Law of Detachment: If the hypothesis of a true conditional statement is true, then the conclusion is also true.  
• Law of Syllogism: 

Exercise

In questions 1 to 4, decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning.

• Angela knows that Walt is taller than Peter. She also knows that Peter is taller than Natalie. Angela reasons that Walt is taller than Natalie.
• Josh knows that Brand X computers cost less than Brand Y computers. All other brands that Josh knows of cost less than Brand X. Josh reasons that Brand Y costs more than all the other brands.
• For the past three Wednesdays, the cafeteria has served macaroni and cheese for lunch. Dana concludes that the cafeteria will serve macaroni and cheese for lunch this Wednesday.
• If you live in Nevada and are between the ages of 16 and 18, then you must take driver’s education to get your license. Anthony lives in Nevada, is 16 years old, and has his driver’s license. Therefore, Anthony took driver’s education.

Use law of detachment to make a valid conclusion in the situation.

• If you get a hit, then your baseball team will win. You hit a home run.
• If Rylee gets promoted, then Callie will also be promoted. Rylee is promoted.
• Any time Kendra runs in a cross-country race, if she runs a strong race, then she wins. In the cross-country race last Saturday, Kendra ran her best race.

Use the law of Syllogism to write the statement that follows from the pair of statements that are given.

• If Moose is hungry when he goes to the pizza shop, then he’ll finish a whole pizza.
  If Moose eats a whole pizza, then he goes through a pitcher of soda.
• If you mail the payment by noon, then it will arrive by tomorrow. If your payment arrives by tomorrow, then you won’t be charged a late fee.
• If Estelle takes her broker’s advice, she’ll invest in stock X. If Estelle invests in stock X, she’ll earn 50% on her investment by next year.