Natural Deduction 2

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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 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!