philosophy logic (Natural Deduction in Propositional Logic)

Question

If government deficits continue at their present rate and a recession sets in, then interest on the national debt will become unbearable and the government will default on its loans. If a recession sets in, then the government will not default on its loans. Therefore, either government deficits will not continue at their present rate or a recession will not set in. (C, R, I, D)

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

- suggest using Reductio ad absurdum and Conditional Proof if needed 

Answer 1

Given: 1) (C and R) --> (I and D);

2) R --> not D

Prove: not C or not R

If (C and R) then (I and D) must hold.

By simplification, D must hold.

By (2), not R must hold by contrapositive <-- ALPHA

on the other hand, if R holds, then not D must hold per (2).

Then I and D does not hold, so per contrapositive of (1),

not (C and R) must hold

Hence not C or not R <--- BETA

so in either case, not C or not R or not R

the proof follows by absorption