Lesson 2.2.1

The Law of Detachment

If you know a conditional is true and its hypothesis is satisfied, you can detach the conclusion as a guaranteed fact.

Introduction

The Law of Detachment (also called Modus Ponens) is the most fundamental rule of deductive reasoning. It lets you draw a guaranteed conclusion — no guessing, no patterns, just pure logic.

Past Knowledge

Conditional statements (2.1.3). Hypothesis & conclusion.

Today's Goal

Apply the Law of Detachment to draw valid conclusions.

Future Success

Law of Syllogism (2.2.2) chains multiple detachments. Every proof step uses this law.

Key Concepts

Law of Detachment (Modus Ponens)

If you know:

is true (the conditional), and
is true (the hypothesis is satisfied),

then you can conclude:

Two Requirements

You need both pieces: the true conditional AND the true hypothesis. If you only know the conclusion is true, you cannot conclude the hypothesis (that's the converse error).

Worked Examples

Basic

Simple Detachment

If it snows, then school is cancelled. It is snowing. What can you conclude?

Conditional is true. Hypothesis (“it snows”) is true.

Conclusion: School is cancelled.

Intermediate

Geometric Application

If two angles are vertical angles, then they are congruent. and are vertical angles. What follows?

The theorem (conditional) is true. The hypothesis is satisfied.

Conclusion:

Advanced

Invalid Use

If it rains, the ground is wet. The ground is wet. Can you conclude it rained?

No! You are told the conclusion is true, not the hypothesis. This is the converse error (affirming the consequent). A sprinkler could have wetted the ground.

Answer: No valid conclusion can be drawn.

Common Pitfalls

Affirming the Consequent (Converse Error)

Knowing the conclusion is true does NOT let you conclude the hypothesis. You need the hypothesis to be true, not the conclusion.

Real-Life Applications

Programming & Automation

Every if statement in a computer program is the Law of Detachment in action: if (condition) { action }. When the condition is true, the program “detaches” and executes the action.