Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 8.1.2 What Do Real Proofs Look Like?

When we’re reasoning about the world around us, our effective arguments never look like the proofs we’ve just been constructing.

Imagine arguing that Mary must drive me to the store by saying, “Step 1. John or Mary must drive me to the store. Premise. Step 2. If John drives me to the store, he will be late for work. Premise. … .” By the time we got to, “Step 6. Mary must drive me to the store. Disjunctive Syllogism from Steps 4 and 5,” no one would be awake, much less following the reasoning.

When we’re reasoning about less trivial problems, the need for a more concise language is even more clear.

Consider the following argument:

We need to improve the quality of our schools. Doing that costs money. The only way to get the money will be to raise taxes. But voters are generally unwilling to raise taxes unless they clearly understand what benefit they’ll get if they do. So we need to launch a public awareness campaign.

Most people would agree that this argument is valid (i.e., the logic is correct). In most communities, though, there would likely be disagreement on whether it’s sound (in other words, does it start with premises that are true). People may dispute the truth of one or more of the premises.

Notice that we can agree that the argument is valid, even though I’ve left out several steps, including some noncontroversial premises. For example, I haven’t mentioned that the only way to raise taxes is for the voters to approve of doing so. Nor have I made it explicit that a public awareness campaign could change the minds of any voters.

The key is that, in a complex world, if I want to convince you of something, I have to make a short argument that is focused on those things that are both relevant and nonobvious. Rarely, in discussions of how to wash dishes, do we mention the effect that gravity would have on a glass if we let go of it.

The world of mathematicians and computer scientists is no different. Even the more formal proofs that are their bread and butter don’t look like the ones we’ve just been writing:

• They don’t have four columns; they’re written in clear English.

• They don’t generally use all the premises that are available (many of which may be irrelevant).

• They’re often a lot shorter.

In this chapter, we’ll make the transition to writing these kinds of proofs. As we do so, we must keep in mind that, if challenged, we must be able to fill in the details of any omitted steps. Thus, the request, “I don’t see how you got from statement x to statement y. Please explain,” is fair game. In fact, it is in response to this very question that we often realize that one person has assumed premises that the other person does not accept.

Me: So now we’ve shown that xy = 2z. Dividing both sides by y, we have that x = 2z/y.

You: Wait. How do you know that you can divide both sides by y?

Me: Oops. I guess I’d have to know that y  0. But I do know that. Let me add it as a premise.

Me: It’s going to snow tomorrow. So everyone will be wearing boots.

You: How do you know about the boots?

Me: Well, I’ve assumed that everyone will check the weather and that they’ll want to stay warm and that they have boots.

Big Idea

An English proof is a concise explanation of a valid argument. But, when you write an English proof, you must be prepared to defend every step if you’re challenged to do so.