Natural Deduction: Natural Deduction 7.2
Rules of Inference II: Rules of Inference II Constructive Dilemma Simplification Conjunction Addition
Constructive Dilemma: Constructive Dilemma (p É q) . (r É s) p v r _________ q v s
Simplification: Simplification p . q _________ p q
Conjunction: Conjunction p q _________ p . q
Addition: Addition p _________ p v q
Now we have double the rules and double the proving power. : Now we have double the rules and double the proving power.
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K Using one of the rules find a pattern in these premises. The pattern here is MP.
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K 4. D . W 1,3 MP So we draw the conclusion from MP on line 4.
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K 4. D . W Now it looks like there’s nothing else we can do with the first four rules. So, let’s consider using one of the new rules. Conjunctions can always be simplified.
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K 4. D . W 1,3 MP 5. D 4 simp. Now we can use line 5 to find another piece of the code! Looks like we have a use of MP.
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K 4. D . W 1,3 MP D 4 simp. K 2,5 MP So, we now know N is true and K is true. It’s just a matter of conjoining the pieces of the code!
Example: Example 1. N É (D . W) 2. D É K 3. N /N . K 4. D . W 1,3 MP D 4 simp. K 2,5 MP N . K 3,6 conj. We knew the last rule would be a conjunction since the conclusion itself is a conjunction. Use the clues hidden in the proof to your advantage!
Slide 15: Practice some more proofs from 7.2 using the rules. Remember to use the strategies in the text. When in doubt ask for help!