G.RL.1.3: Valid Arguments & Counterexamples

Does That Make Sense?

In geometry, we build arguments (proofs!) to show things are true. But how do we know if an argument is logically solid? And how do we show a general statement is *false*?

This lesson looks at checking if an argument's structure is valid and how to use counterexamples to disprove statements.

1. Valid Logical Arguments

A logical argument is valid if the conclusion logically follows from the premises (the starting statements/facts). It's about the structure of the argument.

Important: Validity doesn't mean the premises or conclusion are actually *true* in the real world! It just means *if* the premises were true, the conclusion *would have to be* true.

Example of a Valid Argument (Law of Detachment / Modus Ponens):

Premise 1: If it is raining ($P$), then the ground is wet ($Q$). ($P \rightarrow Q$)
Premise 2: It is raining ($P$).
Conclusion: Therefore, the ground is wet ($Q$).

This structure is valid. If the premises are true, the conclusion must be true.

Example of an Invalid Argument (Converse Error):

Premise 1: If it is raining ($P$), then the ground is wet ($Q$). ($P \rightarrow Q$)
Premise 2: The ground is wet ($Q$).
Conclusion: Therefore, it is raining ($P$).

This structure is invalid. The ground could be wet for other reasons (sprinkler!). The conclusion doesn't *necessarily* follow.

Practice: Valid or Invalid?

Read the argument. Does the conclusion logically follow from the premises? (Focus on the structure, not whether the statements are true!)

1. Premise 1: If a shape is a square, then it is a rectangle. Premise 2: Shape A is a square. Conclusion: Shape A is a rectangle.

2. Premise 1: If you live in Oklahoma, you live in the USA. Premise 2: You live in the USA. Conclusion: You live in Oklahoma.

3. Premise 1: All dogs bark. Premise 2: Fido is a dog. Conclusion: Fido barks.

4. Premise 1: If it's a holiday, the school is closed. Premise 2: The school is closed. Conclusion: It's a holiday.

2. Counterexamples: Proving Something False

To prove a general statement or a conditional statement is false, you only need to find one single example where the statement doesn't hold true. This example is called a counterexample.

For a conditional statement "If P, then Q", a counterexample is an instance where P is true, but Q is false.

Example 1:

Statement: "All birds can fly."

Counterexample: A penguin (it's a bird, but it cannot fly).

Example 2:

Statement: "If a number is odd, then it is divisible by 3."

Counterexample: The number 5 (it's odd, but it's not divisible by 3).

Example 3:

Statement: "If a shape has four sides, then it is a square."

Counterexample: A rectangle (it has four sides, but it's not necessarily a square).

Practice: Find the Counterexample!

For each statement, choose the option that serves as a counterexample.

1. Statement: "If an animal lives in water, then it is a fish."

2. Statement: "If $x^2 = 25$, then $x = 5$."

3. Statement: "All prime numbers are odd."

Good Arguments & Finding Flaws