Law of Detachment

While much of symbolic logic revolves around finding the truth value of statements, we can also use it to write new truthful statements in certain situations. One of these situations involves the Law of Detachment, which states that if the following two statements are true:

1. If p, then q

2. p

That we can derive a third true statement:

3. q.

The Law of Detachment feels very abstract in general terms, so this article will explore some practical applications of the rule to help us understand it. Let's get started!

The Law of Detachment in action

The best way to understand how the Law of Detachment works is to look at an example. If both of the following statements are true, use the Law of Detachment to write a third true statement:

1. If you are a penguin, then you live in the Southern Hemisphere.

2. You are a penguin.

The first step is choosing which statement is p. Your first instinct might be to make statement 1 p because it appears first, but it's not a better fit for the "if p, then q" formatting above. Instead, let's make statement 2, "You are a penguin," statement p. That means that statement q will be, "You live in the Southern Hemisphere."

We can now rewrite the statements as:

1. If p, then q

2. p.

We know that statement p is true and that statement q is conditional on statement p being true. By the Law of Detachment, we can now write a third true statement, namely that which we labeled q above:

3. You live in the Southern Hemisphere.

Logic problems like this seem challenging, but they're rather intuitive once we wrap our heads around what they are asking us. The most important thing is to read each statement carefully so you know exactly what information you're working with. Likewise, don't let funny or nonsensical statements throw you off. The example above hinges on the premise that you are a penguin, so you have to take that as an undisputed fact even though you are clearly not a penguin in real life.

Practice Questions

a. Use the Law of Detachment to write a new, true statement provided the following statements are both true:

1. If it is Billy's birthday, then he will wear a blue shirt.

2. Today is Billy's birthday.

Our first move is to assign statements p and q. The first statement is going to be our "if p, then q" statement, so let's assign statement 2 as statement p. That means statement q will be the other half of our conditional, namely that Billy will wear a blue shirt. Now, we can apply the Law of Detachment:

1. If p, then q

2. p.

Since statement p is true and statement q is true if statement p is true, we can use the Law of Detachment to certify statement q as true:

3. Billy will wear a blue shirt.

b. Use the Law of Detachment to write a new, true statement provided the following statements are both true:

1. The election for class president will be held tomorrow.

2. If the election for class president is held tomorrow, Emily will win.

The first step here is determining what statements p and q are. This time, the first statement makes sense as p since the conditional "if p, then q" is statement 2. Thus, statement p is "the election for class president will be held tomorrow" and statement q is the conditional "Emily will win." Next, we apply the Law of Detachment:

1. If p, then q

2. p.

Since statement p is true and statement q is true if statement p is true, we can use the Law of Detachment to write a third true statement. It's the one we named statement q above:

3. Emily will win.

c. Can you use the Law of Detachment to write a third true statement provided the statements below are both true?

1. James is the tallest student in his class.

2. Mary earned the highest score on the most recent math test.

We need a conditional statement to apply the Law of Detachment in symbolic logic because we need an "if p, then q." However, neither statement above is conditional. It's true that James is the tallest student in his class and that Mary earned the highest score on the most recent math test, but we cannot deduce a third true statement using this information and the Law of Detachment. Therefore, the answer to the question is no.

Topics related to the Law of Detachment

Symbolic Logic

Law of Syllogism

Biconditional Statement

Flashcards covering the Law of Detachment

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Law of Detachment

Introduction to Proofs Practice Tests

You can trust Varsity Tutors to deepen your student's understanding of the Law of Detachment

Many students draw the right conclusion in Law of Detachment problems without realizing that they've applied the Law of Detachment, and that approach will work as long as the Law of Detachment is the only concept tested. However, symbolic logic usually involves exploring multiple ideas. A 1-on-1 math tutor could help the student in your life differentiate the Law of Detachment from similar concepts such as the Law of Syllogism, increasing their understanding of symbolic logic as a whole. Contact the Educational Directors at Varsity Tutors today to learn more about what makes tutoring such a powerful educational tool.