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Part 1. Symbolize the following sentences using propositional logic (2 pts each)
The statements indicated by the all-caps sentences are the following.
J = Johnson is guilty.
W = Wilson will get the death penalty.
L = Lewis will confess.
T = Johnson pulled the trigger.
S= Johnson is of questionable mental stability.

1. LEWIS will confess or WILSON will get the death penalty, but not both. (L v W) & - (L & W)
2. LEWIS will confess, unless Johnson pulled the TRIGGER and JOHNSON is guilty. L -> T&J
3. Assuming that JOHNSON is not guilty, WILSON will get the death penalty if and only if LEWIS does not confess. -J -> (W -> -L)
4. If Johnson pulled the TRIGGER, JOHNSON is guilty or he is of questionable mental STABILITY. J -> (J v S)
5. WILSON will get the death penalty or he will not get the death penalty. W v -W
6. Johnson is of questionable mental STABILITY just in case JOHNSON is not guilty. S ->-J
7. It is not the case that LEWIS will confess. -L
8. Neither did Johnson pull the TRIGGER nor is JOHNSON guilty. -(T v J)
9. It is not the case that both LEWIS will confess and WILSON will get the death penalty. -(L v W)
10. Johnson pulled the TRIGGER, if and only if, JOHNSON is guilty and Johnson is of questionable mental STABILITY. T <-> (J &S)
Part 2. Construct proofs for the following sequents, using only primitive inference rules (16 pts each)
11. (P & R) → S, S ↔ T, ─T ├ R → ─P
1 (1) P&R -> S A
2 (2) S <-> T A
3 (3) -T A
12. P v Q,   P → (T → S), P → T, S ↔ Q ├ S
13. A→(B→C), B & −C ├ −A
14. R & S, S → (P → Q), Q → P ├   P ↔ Q   
Part 3. Construct a proof for the following sequent, using at least one derived rule (16 pts)
15. C → A, − A, −C → −P ├ − P   (hint: use MT)

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1. LEWIS will confess or WILSON will get the death penalty, but not both.
(L v W) & - (L & W)
2. LEWIS will confess, unless Johnson pulled the TRIGGER and JOHNSON is guilty.
L 🡪 T&J
3. Assuming that JOHNSON is not guilty, WILSON will get the death penalty if and only if LEWIS...
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