Natural Deduction Proof 1 NI R ID 2 Rp NC 3 NI DNC NZ Soluti

Natural Deduction Proof

1 NI R ID 2 Rp NC 3 (NI DNC) NZ

Solution

Answer :

Consider the given premises

˜ T R

R ˜ C

( ˜ I ˜ C ) ˜ Z

Z V [ (˜ I R) ( ˜ C Z ) ]

And the conclusion is ˜ R

Now we prove that ˜ R is logicallows from the given premises.

Proof :

Answer :

Consider the given premises

˜ I R

R ˜ C

( ˜ I ˜ C ) ˜ Z

Z V [ (˜ I R) ( ˜ C Z ) ]

And the conclusion is ˜ R

Now we prove that ˜ R is logicallows from the given premises.

Proof :

1.    ˜ I R                    Premise

2.   R ˜ C             Premise

3.   ˜ I ˜ C                  ( 1 ) , ( 2 ) , Chaun rule

4.   ( ˜ I ˜ C ) ˜ Z    Premise

5.   ˜ Z                           ( 3 ) , ( 4 ) , Modus Ponen\'s rule

6.   Z V [( ˜ I R) ( ˜ C Z ) ]     Premise

7. ( ˜ I R) ( ˜ C Z )                ( 5 ) , ( 6 ) , Disjunctive Syllogism

8. ( ˜ C Z )                                  ( 1 ) , ( 7 ) ,Modus Ponen\'s rule

9. ˜ ( ˜ C )                                       ( 5 ) , ( 8 ) , Modus Tollen\'s rule

10. C                                             ( 9 ) Law of double negation

11.   ˜ R                                        ( 2 ) , ( 10 ) Modus Tollen\'s rule

Hence , ˜ R is logically follows from the given premises