Rules of Replacement 2

# Rules of Replacement 2 - because the S and R are together...

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Natural Deduction Natural Deduction 7.4 © 2006 Kevin J. Browne

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Rules of Replacement II Transposition Material Implication Material Equivalence Exportation Tautology
Transposition Transposition p p q q ~ ~ ~ ~

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Material Implication p p q q v v ~ ~
Material Equivalence Material Equivalence p = q (p q) . (q p) p p = (p . q) v (~p . q )

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Exportation Exportation p p (q (q r) (p . q) (p . q) r
Tautology Tautology p p v v p p . .

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An Example An Example 1. (S . K) R 2. K /S R Here we seem unable to use any of the rules of inference immediately. Let’s try a rule of replacement… exportation? But this leads to a problem! Exportation gives us: S (K R) This can’t be right

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Unformatted text preview: because the S and R are together in the conclusion but they’re separated now. An Example An Example 1. (S . K) ⊃ R 2. K /S ⊃ R But, S . K is a conjunction. (commutativity) An Example An Example 1. (S . K) ⊃ R 2. K /S ⊃ R 3. (K . S) ⊃ R 1 comm. Now we can perform exportation with no problem. An Example An Example 1. (S . K) ⊃ R 2. K /S ⊃ R 3. (K . S) ⊃ R 1 comm. 4. K ⊃ (S ⊃ R) 3 exp. This allows us to use a rule of inference. MP An Example An Example 1. (S . K) ⊃ R 2. K /S ⊃ R 3. (K . S) ⊃ R 1 comm. 4. K ⊃ (S ⊃ R) 3 exp. 5. S ⊃ R 2,4 MP...
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Rules of Replacement 2 - because the S and R are together...

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