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#1:
"If either wages or prices are raised, there will be inflation. If there is inflation, then either Congress must regulate it or the people will suffer. If the people suffer, Congressmen will be unpopular. Congress will not regulate inflation, and Congressmen will not be unpopular. Therefore, wages will not rise."

Solution: Our goal is to conclude ¬W:
{1}(1) (WP)IPremise
{2}(2) ICSPremise
{3}(3) SUPremise
{4}(4) ¬C¬UPremise
{4}(5) ¬U¬C-commutativity on 4
{4}(6) ¬C-elimination on 4
{4}(7) ¬U-elimination on 5
{3}(8) ¬U¬Scontrapositive of 3
{3,4}(9) ¬S-elimination on 7 and 8
{2}(10) ¬(CS)¬Icontrapositive of 2
{1}(11) ¬I¬(WP)contrapositive of 1
{1,2}(12) ¬(CS)¬(WP)syllogism on 10 and 11
{3,4}(13) ¬C¬S-introduction on 6 and 9
{3,4}(14) ¬(CS)DeMorgan's law on 13
{1,2,3,4}(15) ¬(WP)-elimination on 12 and 14
{1,2,3,4}(16) ¬W¬PDeMorgan's law on 15
{1,2,3,4}(17) ¬W-elimination on 16


#2:
"Either logic is difficult, or not many students like it. If mathematics is easy, then logic is not difficult. Therefore, if many students like logic, mathematics is not easy."

Solution: Our goal is to conclude L¬M:
{1}(1) D¬LPremise
{2}(2) M¬DPremise
{1}(3) ¬LD-commutativity on 1
{1}(4) LDequivalence of implication and disjunction on 3
{2}(5) ¬(¬D)¬Mcontrapositive of 2
{6}(6) DPremise
{6}(7) ¬(¬D)double negation of 6
{2,6}(8) ¬M-elmination on 5 and 7
{2}(9) D¬Mrule of conditional proof on 6 and 8
{1,2}(10) L¬Msyllogism on 4 and 9


#3:
"If Algernon is in jail, then he is not a nuisance to his family. If he is not in jail, then he is not a disgrace. If he is not a disgrace, then he is in the army. If he is drunk, he is a nuisance to his family. Therefore, he is either not drunk or in the army."

Solution: Our goal is to conclude ¬RA. To do it, we will first conclude RA and then apply the equivalence of implication and disjunction:
{1}(1) J¬NPremise
{2}(2) ¬J¬DPremise
{3}(3) ¬DAPremise
{4}(4) RNPremise
{1}(5) ¬(¬N)¬Jcontrapositive of 1
{6}(6) NPremise
{6}(7) ¬(¬N)double negative of 6
{1,6}(8) ¬J-elimination on 5 and 7
{1}(9) N¬Jrule of conditional proof on 6 and 8
{1,4}(10) R¬Jsyllogism on 4 and 9
{1,2,4}(11) R¬Dsyllogism on 2 and 10
{1,2,3,4}(12) RAsyllogism on 3 and 11
{1,2,3,4}(13) ¬RAlaw of equivalence of implication and disjunction on 12


#4:
"Either John and Henry are the same age, or John is older than Henry. If John and Henry are the same age, then Elizabeth and John are not the same age. If John is older than Henry, then John is older than Mary. Therefore, either Elizabeth and John are not the same age or John is older than Mary."
Solution: The goal is to derive ¬EM. We will do so by deriving EM and applying the Law of Equivalence of Implication and Disjunction:
{1}(1) SOPremise
{2}(2) S¬EPremise
{3}(3) OMPremise
{2}(4) ¬(¬E)¬Scontrapositive of 2
{5}(5) EPremise
{5}(6) ¬(¬E)double negative of 5
{2,5}(7) ¬S-elimination of 4 and 6
{2}(8) E¬Sconditional proof on 5 and 7
{9}(9) ¬SPremise
{1,9}(10) Omodus tollendo tollens on 1 and 9
{1}(11) ¬SOconditional proof on 9 and 10
{1,2}(12) EOsyllogism on 8 and 11
{1,2,3}(13) EMsyllogism on 3 and 12
{1,2,3}(14) ¬EMequivalence of implication and disjunction on 13