A Lesson in Logic
Last week I covered using assumptions to discover conditionals in what’s called a Conditional Proof. This week we’ll be using assumptions in a different way, to generate contradictions. This is also the very last rule. In two-value propositional logic, if you’ve been following along and have mastered all of the rules, this is the last one You will have completed the set and be the master of logic! Well, this particular kind, anyway. So, let’s make some contradictions!
This method is called an Indirect Proof, and uses the same system of assumption and subproof that you saw last week. I’ll go over a bit here as well though. The goal of an indirect proof is to use the assumption to generate a contradiction so that you can conclude the negation of the assumption. Today’s example is brought to you by the herculean efforts of champion swimmer Michael Phelps, now the most decorated Olympian ever, with 19 medals to his name. Here’s some thoughts that might have been going through his mind in his last event.
1. If I rest, then I’ll start to lose. (R → ~W)
2. If I rest and start to lose, then I’ll try harder. ((R ∧ ~W) → T)
3. If I try harder, then I’ll start to win. (T → W)
4. Therefore, I won’t rest. (~R)
Now let’s look at the proof.
5. |I will rest. (R), Ass. I. P.
6. |I will start to lose. (~W), Modus Ponens from 1 and 5.
7. |I will rest and start to lose. (R ∧ ~W), Conjunction from 5 and 6.
8. |I will try harder. (T), Modus Ponens from 2 and 7.
9. |I will start to win. (W), Modus Ponens from 3 and 8.
10. |I will start to win and I will start to lose. (W ∧ ~W), Conjunction from 6 and 9.
So now we have a contradiction. He can’t both start to win and start to lose after all. His solution is to not rest, and just try harder immediately.
11. I won’t rest. (~R), Indirect Proof from 5-10.
From this, you can see the goal of the Indirect Proof. If the assumption leads you into a contradiction, then you can conclude that the assumption is false. If the assumption is false, then the negation of the assumption is true. Now, you should be wary about making assumptions, but here are some essential rules. All indirect 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 an indirect proof, it has to end in a contradiction, which is a conjunction of a proposition and its negation.
- Must be labelled as assumptions. That’s what the Assp. tag is for, and the I.P. after it tells you that it’s for an indirect proof, so your readers know what to look for.
- Cease functioning as soon as the contradiction is reached. The negation that follows is not considered to be part of the subproof. This assumption cannot function anywhere else in the argument.
Like conditional proofs, you can nest indirect proofs in each other, and even nest each of the different kinds of subproofs within each other, so a conditional proof might contain an indirect proof, or vice versa. Just remember that every assumption within a subproof must be discharged before you can discharge that subproof.
And that’s it. If you’ve been following along, you know every rule in two value propositional logic. You’re familiar with the laws of logic, the importance of contradictions, all of the rules of induction and substitution, and the basics of using assumptions in proofs. You also know how these things have real world applications that go beyond their usefulness as purely theoretical concepts. If you haven’t been following along, or if you’re a new reader, welcome! You can head over to the lessons in logic introduction to get a better grip on what’s going on here.