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)

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