philosophy logic (Natural Deduction in Propositional Logic )

Question

If jails are overcrowded, then dangerous suspects will be released on their own recognizance. If jails are overcrowded and dangerous suspects are released on their own recognizance, then crime will increase. If no new jails are built and crime increases, then innocent victims will pay the price of increased crime. Therefore, if jails are overcrowded, then if no new jails are built, then innocent victims will pay the price of increased crime.

(J, D, C, N, I)

P.S: indirect proof(IP) cannot be used! 

- suggest using Reductio ad absurdum and Conditional Proof if needed 

- Solve the problem in a different way than the example answer below...

1. J --> D

2. (J•D) --> C

3. (N•C) --> I                So J --> (N -->I)

4. | J                           Assume CP

5. || N                          Assume CP

6. || D                          1,4, MP

7. || J•D                       4,6, Conj

8. || C                          2,7, MP

9. || N•C                       5,8, Conj 

10. || I                          3,9, MP

11. | N --> I                    5-10, CP

12. J --> (N --> I)               4-11, CP

Answer 1

Wish to prove: J-->( N-->I)

Given: (1) J-->D;

(2) (J and D)-->C;

(3) (N and C)--> I

contrapositive (3):

not I --> not (N and C)

DeMorgan's:

not I --> not N or not C

Implication Identity

I or ( not N or not C)

distributive:

(I or not N) or (I or not C)

associative property:

I or not N or I or not C

absorption property:

I or not N or not C <--- ALPHA

implication identity (2):

not (J and D) or C

DeMorgan's:

(not J or not D) or C

distributive:

(not J or C) or (not D or C)

absorption:

not J or not D or C <-- BETA

implication identity (1):

not J or D <--- DELTA

if J then per DELTA, D must hold per (1) and DELTA

then per BETA, C must hold

then per ALPHA, I or not N must hold.

so if J then D and C and (I or not N)

Simplification of Conjunction says

if J then (I or Not N)

Commutative property says:

if J then (not N or I)

implication identity:

if J then (N --> I)

therefore J-->(N-->I)

end of proof