Counterexamples

A break from proofs and rules for a note about counterexamples. I’ve talked about them before briefly, but before we go any further, it’s important to talk a bit more about soundness, validity, and counterexamples. Counterexample is one of the most elegant ways to demonstrate an argument’s invalidity, but it rests on a solid understanding of what constitutes validity, and what constitutes soundness. 

Validity

An argument is deductively valid if it follows the right form. Proofs are designed to show an argument’s validity, so any argument which you can do a proof for is valid. These are arguments which use the rules of inference as well as the substitution rules which you’ll learn over the next few months, in order to reach conclusions which follow necessarily from true premises. Her’es an example of a valid argument.

1. (B ∧ G) I read books and I play games.
2. (G → E) If I play games, then I enjoy games.
3. (E) Therefore, I enjoy games.
4. (G) I play games                            Simplification from 1
5. (E) I enjoy games                          Modus Ponens from 2 and 4

Soundness

Here’s the kicker. It isn’t enough for an argument to be valid for it to be acceptable. Soundness is best expressed as validity + true premises. The above argument is not only valid, it’s sound. I do read books and play games, I play games because I enjoy them, and so it’s true to conclude that I enjoy games. The goal of logic is to construct sound arguments, not merely valid ones.

Counterexamples

Counterexamples rely on validity. If there’s an instance where all of the premises can be true, but the conclusion is false, then you’ve found a counterexample. We’ve found a few of these over the course of learning rules of inference, thanks to the news. If there’s a counterexample to an argument, then the conclusion doesn’t follow from the premises. The best way to find them is to use truth tables, whhich will consider every permutation of true and false in the premises.

Thanks for following along so far, we’ve got a long way to go before we start getting into more complex proofs and fallacies, but here’s an argument to work on. Can you find a counterexample for it?

1. (W → M) If snow is white, then the cat is on the mat.
2. (R ∧ M) Rainbows are pretty and the cat is on the mat.
3. (W) Therefore, snow is white.