A Lesson in Logic
This week I’m going to look at how to make assumptions in proofs. There are two ways to use assumptions, called a conditional proof and an indirect proof. Today I’ll look at the first, and show you how it works.
As you might have noticed, the proofs are getting longer and longer, partly because of the judicious application of substitution rules. Conditional Proof is a way of shortening these. Since the London 2012 Olympics just started, I’ll use that for my example.
1. If Canada is the best, then it has the most medals and the most golds. (C → (M ∧ G)
2. Therefore, if Canada is the best, then it has the most medals. (C → M)
To show this, we can use a conditional proof. To start, indent the proof further, and make an assumption. This begins what’s called a subproof.
3. |Canada is the best. (C) Assp. (C.P.)
4. |It has the most medals and golds. (M ∧ G) Modus Ponens from 1 and 3.
5. |Canada has the most medals. (M) Simplification from 4.
So by assuming C, you’re able to reach M using only the premises in the original argument and the rules which you already learned. Since that’s the case, it follows that
6. If Canada is the best, then it has the most medals. (C → M) Conditional Proof, 3-5.
Making that assumption gets us from the premises to the conclusion in a much more straightforward way, but you might worry about the ability to just make assumptions in the middle of an argument, and you’re right to do so. There are a few very important rules about just what assumptions can and cannot do. All conditional proof assumptions:
- Must be discharged in the proof. You can’t finish a proof with dangling assumptions, because those assumptions don’t follow from the argument. If it’s for a conditional proof, it has to end in a conditional statement that leads from that assumption to something relevant. It can only show that something implies another, not that something is the case.
- Must be labelled as assumptions. That’s what the Assp. tag is for, and the C.P. after it tells you that it’s for a conditional proof, so you know what to look for.
- Cease functioning as soon as the conditional is reached. That conditional is not considered to be part of the subproof. They cannot function anywhere else in the argument.
Now, these assumptions can be nested. You can have subproofs within subproofs, or multiple subproofs in a single proof, if things get really complicated. The only rules concerning that is that each assumption can only operate within its subproof, and that all assumptions inside a subproof must be discharged before that subproof can be discharged. And that’s conditional proof, though it bears mentioning that there’s another way to prove the example argument without using a conditional proof, using only the rules you’ve already learned. Can you find it? Leave your own proof in the comments.
