2.2: Counterexamples

Counterexamples

• A valid argument is one where: if the premises are true, the conclusion must be true.
   
• When judging validity, we assume the premises are true, and see if the conclusion really follows (ignoring all other real-world knowledge).
   
• An argument is invalid if it's possible for the premises to be true and the conclusion false.

Counterexamples

• A counterexample to an argument is a hypothetical situation in which all of the premises are true, but the conclusion is false.
   
• If you can think of a counterexample for an argument, you know that it is invalid.
   
• (Although, if you can't think of one, you can't necessarily conclude that the argument is valid.)

Counterexamples

Can you think of a counterexample to the following argument?

1. All dogs are fluffy.

2. Fido is fluffy.

3. Therefore, Fido is a dog.

Counterexamples

Can you think of a counterexample to the following argument?

1. Alice is smarter than Bob.

2. Bob is smarter than Carol and Dave.

3. Therefore, Alice is smarter than Dave.

Counterexamples

Can you think of a counterexample to the following argument?

1. If Alice goes to work, she will either take the bus or ride her bicycle.
2. Alice is going to work.
3. So, she will ride her bicycle.

Counterexamples

Can you think of a counterexample to the following argument?

1. If Alice takes the bus to work, she will be late.
2. Alice does not take the bus to work.
3. So, Alice is early.

Counterexamples

Can you think of a counterexample to the following argument?

1. Every human being has been to the moon.
2. The moon is a star.
3. Therefore, every human being has been to a star.

Counterexamples

• When coming up with a counterexample, you can imagine anything - but don't "cheat"!

1. If Dave forgets his umbrella, he'll get wet.
2. Dave did forget his umbrella.
3. So, Dave will get wet.

"What if Dave in sentences 1 and 2 refers to David Letterman, but Dave in sentence 3 refers to David Duchovny?" - CHEATING!

• You have to assume that words have the meaning that is obviously intended.

Key Concepts

• Counterexample