eighteen proof rules

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18 Terms

1
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modus ponens

  1. p → q

  2. p ∴ q

2
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modus tollens

  1. p → q

  2. ~q ∴ ~p

3
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hypothetical syllogism

  1. p → q

  2. q → r ∴ p → r

4
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disjunctive syllogism

  1. p v q

  2. ~p ∴ ~q


OR

  1. p v q

  2. ~q ∴ ~p

5
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constructive dilemma

  1. p v q

  2. p → r

  3. q → s ∴ r v s

6
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simplification

  1. p . q ∴ p

OR

  1. p . q ∴ q

7
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conjunction

  1. p

  2. q ∴ p . q

8
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addition

  1. p ∴ p v q

OR

  1. p ∴ q v p

9
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double negation

  • p :: ~~p

  • ~~p :: p

10
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commutation

  • (p v q) :: (q v p)

  • (p . q) :: (q . p)

11
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association

  • (p v (q v r)) :: ((p v q) v r)

  • (p . (q . r)) :: ((p . q) . r)

12
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de morgan’s law

  • ~(p . q) :: ~p v ~q

  • ~(p v q) :: ~p . ~q

13
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contraposition

  • (p → q) :: (~q → ~p)

14
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distribution

  • (p . (q v r)) :: ((p . q) v (p . r))

  • (p v (q . r)) :: ((p v q) . (p v r))

15
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exportation

  • ((p . q) → r) :: (p → (q → r))

16
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redundancy

  • p :: (p . p)

  • p :: (p v p)

17
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material equivalence

  • (p ←> q) :: ((p → q) . (q → p))

  • (p ←> q) :: (p . q) v (~p . ~q))

18
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material implication

  • (p → q) :: (~p v q)