Law of Detachment in Geometry: Definition & Examples

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  • 0:01 What Is the Law of Detachment?
  • 2:31 Examples
  • 3:57 Lesson Summary
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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 is about a fundamental logical argument type called the Law of Detachment. The definition is given, and examples of valid and invalid uses of it are provided.

What Is The Law of Detachment?

Sometimes we equate common sense with being logical. In mathematics, logic is more precise. Logic follows a specified pattern of development. One statement leads to a following statement due to a valid application of rules, definitions, theorems, etc. We are going to discuss one of these valid applications today.

The logical argument type we are discussing is commonly referred to as the law of detachment. It also goes by another name, a Latin name, which is modus ponens. Its translation is typically one of the following: the path to affirm, the mode that affirms, or the way to affirm by affirming. There is a pattern that it follows which we will get into in a moment.

First, we need to briefly review what we call the parts of a conditional statement. Suppose you have a conditional statement such as:

If you are driving a long distance, then you will have to make a fuel stop.

In this conditional statement there are two parts. The phrase that follows the word 'if' is called the antecedent or hypothesis. The phrase that follows the word 'then' is called the consequent or conclusion (sometimes the word 'then' is omitted from the conditional). In most cases the letter p is used to represent the hypothesis, while the letter q is used to represent the conclusion. Also, there may be times when the concept of negation may occur. The negation or opposite of a statement is written with the '~' symbol in front of the letter. Negating a statement written in the positive will make it negative, while negating a statement written in the negative will make it positive.

The conditional statement can now be rewritten with the symbols as: If p, then q.

Now, let's get back to the pattern alluded to earlier. The law of detachment has a prescribed pattern. There are two premises (statements that are accepted as true) and a conclusion. They must follow the pattern as shown below.

  • Statement 1: If p, then q.
  • Statement 2: p
    • Conclusion: q

This is what is called a valid logical argument. Again, the first two statements, the premises, are accepted as true. If they are true, then it is logical to come to the valid conclusion.

There are two scenarios for the law of detachment.

Scenario 1

Both premises are given and the conclusion is given. We can judge the validity on whether the pattern is followed.

Scenario 2

Both premises are given and the conclusion is NOT given. We can judge whether a valid conclusion is possible or not based on whether the pattern is being followed so far in the premises.

Examples

Let's look at some examples to illustrate how this works.

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