Complete the following natural deduction proof The given num

Complete the following natural deduction proof. The given numbered lines are the argument\'s premises and the line beginning with a single slash is the argument\'s conclusion. Derive the argument\'s conclusion in a series of new lines. (Modus Ponens, Modus Tollens, Pure Hypothetical Syllogism, and Disjunctive Syllogism are the only ones that can be used).

Solution

Consider the given premises

~ I R

R ~ C

(~I ~ C ) ~ Z

Z V [ ~ I R ) ( ~ C Z )

And the conclusion is ~ R

We show that the conclusion ~ R logically follows from the given premises

Proof :

1. ~ I R                                        Premise

2. R ~ C                                       Premise

3. ~ I ~ C                                    ( 1 ) , ( 2 ) , Chain 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

Thus , ~ R is logically follows from the given premises.