Problem:

$1.$ Consider the following argument:

"Either the tycoons will suppress the masses or the masses will exterminate the tycoons. If the failure of the tycoons to suppress the masses means that they will be exterminated by the masses, then the global political system is rotten. But the global political system is rotten and Marx is correct about the future, if and only if there is going to be general class warfare. Now, either capitalism is doomed, or else the tycoons will suppress the masses and there will be no general class warfare. So$,$ if Marx is correct about the future, then capitalism is doomed." $($Key: S$,$E$,$G$,$M$,$W$,$C$)$

$2.$ This is the standard$/$linear format re$-$representation:

$1.$ Either the tycoons will suppress the masses or the masses will exterminate the tycoons.

$2.$ If the failure of the tycoons to suppress the masses means that they will be exterminated by the masses, then the global political system is rotten.

$3.$ But the global political system is rotten and Marx is correct about the future, if and only if there is going to be general class warfare.

$4.$ Now, either capitalism is doomed, or else the tycoons will suppress the masses and there will be no general class warfare.

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

C$.$ So$,$ if Marx is correct about the future, then capitalism is doomed.

$3.$ This is the formalization and proof of validity:

$1.$ S logical or E

$2.$ space left parenthesis not S superset of E right parenthesis superset of G

$3.$ space left parenthesis G logical and M right parenthesis left right arrow W

$4.$ space C logical or left parenthesis S logical and not W right parenthesis

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

rightwards double arrow L $1.$ space M space left parenthesis $??????$ right parenthesis

rightwards double arrow L $2.$ space not S superset of E space left parenthesis $??????$ right parenthesis

rightwards double arrow L $3.$ space G space left parenthesis $??????$ right parenthesis

rightwards double arrow L $4.$ space G logical and M space left parenthesis $??????$ right parenthesis

rightwards double arrow L $5.$ space W space left parenthesis $??????$ right parenthesis

rightwards double arrow L $6.$ space C logical or not left parenthesis not S logical or not not W right parenthesis space left parenthesis $??????$ right parenthesis

rightwards double arrow L $7.$ space C logical or not left parenthesis not S logical or W right parenthesis space left parenthesis $??????$ right parenthesis

rightwards double arrow L $8.$ space left parenthesis not S logical or W right parenthesis space left parenthesis $??????$ right parenthesis

rightwards double arrow L $9.$ space C space left parenthesis $??????$ right parenthesis

C$.$ space M superset of C space left parenthesis $??????$ right parenthesis

$4.$ Identify which rules were applied.

L$1$

L$2$

L$3$

L$4$

L$5$

L$6$

L$7$

L$8$

L$9$

C

A$.$

Conditional Proof

B$.$

Proof by Contradiction

C$.$

Conjunction Introduction$/$Addition

D$.$

Assumption for Conditional Proof

E$.$

Modus Ponens

F$.$

Disjunctive Syllogism

G$.$

Modus Tollens

H$.$

Reasoning by Cases

I.

Hypothetical Syllogism

J$.$

Implication

K$.$

Double Negation Elimination

L$.$

Reductio

M$.$

Affirming the Consequent

N$.$

DeMorgan's Law

O$.$

Modus Ponens $+$ Biconditional Elimination

P$.$

Disjunction Introduction