~B ~B--> (G-->F) (A-->E) B v (A v G) E v F  How would I go about answering this problem with putting in inference rules to the proof?

Expert Answers
Matthew Fonda eNotes educator| Certified Educator

Another way to prove this is by using rules of derivation:

  1. 1              ~B  [Assumption]
  2. 2              ~B -> (G --> F)  [Assumption]
  3. 3              A --> E  [Assumption]
  4. 4              B v (A v G)  [Assumption]
  5. 1,2           G --> F  [1,2 Modus Ponnens]
  6. 1, 4          A v G  [1, 4 Disjunctive Syllogism]
  7. 7              A  [Assumption / Disjunction elimination]
  8. 3,7           E  [3,7 Modus Ponnens]
  9. 3,7           E v F  [8 Disjunction Introduction]
  10. 10            G  [Assumption / Disjunction elimination]
  11. 1,2, 10     F  [5, 10 Modus Ponnens]
  12. 1, 2, 10    E v F  [11 Disjunction Introduction]
  13. 1, 2, 3, 4  E v F  [6, 7, 9, 10, 12 Disjunction elimination]
neela | Student

We establish or infer the truth of a proposition by the truth tables.

~B is the negation of  proposition B.

When B is true (T), ~B is not true. When B is false( F) , ~B is True (T).

Truth or inference table for  ~B--> (G-->F) . Please do not get confusion for proposition in heading and  the false value, F in the ttruth table:

 ~B   G     F    (G-->F)          ~B--> (G-->F)

   T    T     T        T                        T

   T    T      F       F                         F

   T    F      T        T                       T

   T    F      F        T                        T

   F     T     T        T                        T

   F     T      F       F                        T

   F    F       T      T                         T

Truth table for (A->E)

 A      E           (A->E)

T       T              T

T       F              F

F       T              T

F      F               T

 F     F              T                        

Truth or inference  table for  B v (A v G).

B     A       G         (AVG)           BV((AVG)

T     T        T             T                   T

T     T        F             T                    T

T      F       T             T                    T

T      F       F             F                    T

F      T        T            T                    T

F       T       F            T                    T

F       F      T             T                    T

F       F      F             F                    F  

The truth table  for (EVF) . There proposition F in the heading and in the truth infering table  the false value F. Please do not get confused about the different F's.:

E         F            (E V F)

T         T                  T

T          F                 F

F          T                 T

F          F                 T