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3.3 Rules

3.3 Rules. Negation: ¬¬p implies p Conjunction: p&q implies p, q Negated conjunction: ¬(p&q) implies ¬pV¬q Ex. p&q, ¬(p&q) invalid. Rules. Disjunction: p Vq has two branches Negated disjuction: one branch: ¬p ¬q Ex. (pVq)Vr ¬((pV(qVr). 3.4 Rule II.

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3.3 Rules

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  1. 3.3 Rules • Negation: ¬¬p implies p • Conjunction: p&q implies p, q • Negated conjunction: ¬(p&q) implies ¬pV¬q Ex. p&q, ¬(p&q) invalid

  2. Rules • Disjunction: pVq has two branches • Negated disjuction: one branch: ¬p ¬q Ex. (pVq)Vr ¬((pV(qVr)

  3. 3.4 Rule II • Conditional: p→q implies ¬pVq, so it has two branches: ¬p; q The negation of conditional: ¬(p→q) implies ¬(¬pVq) implies p&¬q So it has one branch: p ¬q

  4. Rules • Bi-conditional: p↔q implies (p→q)&(q→p); so we have (p→q) (q→p) ¬p q ¬q p

  5. Negated bi-conditional ¬(p↔q) ¬((p→q)&(q→p)) (¬(p→q))V(¬(q→p)) ¬(p→q) ¬(p→q) ¬(¬pVq) ¬(¬qVp) ¬¬p&¬q ¬¬q&¬p p q ¬q ¬p

  6. Example • sV(r&q) s→r ¬(r&m)

  7. 3.5 Other applications • Test for equivalence: Assume ¬(A↔B) A B ¬B ¬A If both branches close, then A and B are equivalent. Ex. p→q implies ¬p→¬q Test for logical truth Assume ¬A, if all branches close, than A is a tautology. Open: contingent or contradiction Ex. pv¬p

  8. Testing for contradiction and safisfiability • Consider A, and assume A itself. • If closed, A is an contradiction • Ex. p&¬p • Open: Satisfiable • Ex. p&r

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