CA 208 Logic Ex4

1
CA 208 Logic Ex4

• Commutativity
• (P?Q) ? .................
• Associativity
• P?(Q?R) ? (P?Q)?R
• Distributivity
• P?(Q?R) ? ..................
• Idempotency
• (P ? P) ? P
• Absorption
• (P ? T) ? ................

• Commutativity
• (P?Q) ? (Q?P)
• Associativity
• P?(Q?R) ? ................
• Distributivity
• P?(Q?R) ? (P ? Q) ? (P ? R)
• Idempotency
• (P ? P) ? .................
• Absorption
• (P ? ?) ? P

2
CA 208 Logic Ex4

• De Morgan
• ?(P?Q) ? .............
• Double Negation
• ? ? P? ...........
• The Falsum/Absurd ?
• (P ? ? P) ? ?

• De Morgan
• ?(P?Q) ? (? P ? ? Q)
• The Verum/True T
• (P ? ? P) ? ......

3
CA 208 Logic Ex4

• Given our definition of the syntax of
  propositional logic, which of the following are
  formulas of propositional logic and which are not
  (and why)?
• A
• ?A
• ??A
• ?(?A)
• ??A ?A
• ? AA
• A?B
• (A?B)
• ((A?B))
• A?B?C
• (A?B)?C
• ((A?B)?C)
• ?C
• (?C)
• ??

4
CA 208 Logic Ex4

• Prove the following in the natural deduction
  calculus (proof system)
• A,B - (B?A)
• A,B - (A?A)
• A,B - ((A?B) ?C)
• (A?B),(B ?C) - C
• (C ?B), (B ?A) - (C ? A)
• A, (A ?(B?C)), (C?(D?E)), (B?(F?E)) - E
• A, ((B?A) ?C), (C?E) - E
• - (C?C)