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natural deduction 2

natural deduction 2 - Natural Deduction(2 1 We shall learn...

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1 Natural Deduction (2)

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2 We shall learn the 10 rules of replacement. These rules express the logical equivalence between two statement forms such that one can replace another. We use the symbol :: to signify the relation of logical equivalence.
3 9. DeMorgan’s Rule (DM) ~(p q) :: (~p v ~q) ~(p v q) :: (~p ~q) Example: It is not the case that both Mary and Paul are late :: Mary is not late or Paul is not late. It is not the case that Mary or Paul is late :: Mary is not late and Paul is not late.

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4 You can use truth tables to show that they are equivalent. DM simply requires you to change the connectives “v” and “ ” while shifting the negation symbol outside or inside the parentheses.
5 10. Commutativity (Com) (p v q) :: (q v p) (p q) :: (q p) This is obvious as “v” and “ ” behave similarly to addition and multiplication.

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6 11. Associativity (Assoc) (p v (q v r)) :: ((p v q) v r) (p (q r)) :: ((p q) r) It is based on the fact that the meaning of a conjunction or a disjunction is unaffected by the placement of the parentheses if the same logical operator is used throughout.
7 12. Distribution (Dist) (p (q v r)) :: ((p q) v (p r)) (p v (q r)) :: ((p v q) (p v r)) This applies when a ‘ ’ or a ‘v’ appear in the same statement.

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8 13. Double Negation (DN) P :: ~ ~p No explanation is needed.
9 Difference between implication and replacement Rules of implication apply to whole lines in a proof while rules of replacement apply to whole or any part of a line.

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10 Since we can replace part of a line with rules of replacement, we can first of all replace part of the conclusion line if that line is not already in the premises.
11 Example 1 K (F v B) 2. G K / (F G) v (B G) Working on the conclusion: (F G) v (B G) (G F) v (G B) Com, Com G (F v B) Dist

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12 Now we have obtained (F v B), which is the consequent in the first premise.
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natural deduction 2 - Natural Deduction(2 1 We shall learn...

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