ln2-page8

# ln2-page8 - E.g on this last one at line 11 we could have...

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p p 6 6 I The basic structure of an indirect proof is to assume the opposite of what we’re trying to prove, and derive a contradiction. I.e., to show p , we assume p , and show 6 . Let us take as our example: 1. 2. 3. A ! B A ! ⇠ B SHOW: A Pr Pr We set this up by assuming P and adding a new SHOW: line. 1. 2. 3. 4. 5. A ! B A ! ⇠ B SHOW: A A SHOW: 6 Pr Pr ID Ass DD We can show 6 by showing any contradiction. (Any contradiction will do. It need not be the opposite of what we assumed.) 1. 2. 3. 4. 5. 6. 7. 8. A ! B A ! ⇠ B SHOW: A A SHOW: 6 B B 6 Pr Pr ID Ass DD 1,4 ! O 2,4 ! O 6,7 6 I Having shown that a contradiction results when we assume A is true, we can safely conclude that A is false, by ID. 1. 2. 3. 4. 5. 6. 7. 8. A ! B A ! ⇠ B SHOW: A A SHOW: 6 B B 6 Pr Pr ID Ass DD 1,4 ! O 2,4 ! O 6,7 6 I Indirect Derivation is very powerful. It can be used for almost any type of problem, and even when what we’re trying to prove is complex. Here are some additional examples. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. ( P \$ Q ) _ ⇠ A A ! P Q SHOW: A A SHOW: 6 P ⇠⇠ A P \$ Q P ! Q P 6 Pr Pr Pr ID Ass DD 2,5 ! O 5 DN 1,8 _ O 9 \$ O 3,10 ! O 7,11 6 I (There are quite often multiple ways of doing these problems.
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Unformatted text preview: E.g., on this last one, at line 11, we could have used 7 and 10 to get Q instead of 3 and 10 to get ⇠ P . This would also have given us a different contradiction (lines 3 and 11). Either way you do it is Fne.) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. P ! ( N & M ) Q _⇠ N ⇠ ⇠ ⇠ SHOW: ⇠ ( P & ⇠ Q ) P & ⇠ Q ⇠ ⇠ ⇠ SHOW: 6 P ⇠ Q N & M ⇠ N N 6 Pr Pr ID Ass DD 4 &O 4 &O 1,6 ! O 2,7 _ O 8 &O 9,10 6 I There are, strictly speaking, two forms of indirect proof. The above are all examples of assuming that something is true in order to prove that it is false. We can also assume that something is false in order to prove that it is true. This form works almost exactly the same. Here are two examples. 1. 2. 3. 4. 5. 6. 7. 8. P _ Q Q ! P ⇠ ⇠ ⇠ SHOW: P ⇠ P ⇠ ⇠ ⇠ SHOW: 6 Q P 6 Pr Pr ID Ass DD 1,4 _ O 2,6 ! O 4,7 6 I 8...
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