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If the argument is invalid, show it by choosing truth values for all sentences that make all premises true and the conclusion false. If it is valid, write a formal deduction for it.

pg. 35 #4:
"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 $\neg W$:
$\{1\}$(1) $(W \vee P) \rightarrow I$Premise
$\{2\}$(2) $I \rightarrow C \vee S$Premise
$\{3\}$(3) $S \rightarrow U$Premise
$\{4\}$(4) $\neg C \wedge \neg U$Premise
$\{4\}$(5) $\neg U \wedge \neg C$4 T Commutative Law of $\wedge$
$\{4\}$(6) $\neg C$4 T Law of Simplification
$\{4\}$(7) $\neg U$5 T Law of Simplification
$\{3\}$(8) $\neg U \rightarrow \neg S$3 T Law of Contraposition
$\{3,4\}$(9) $\neg S$7 8 T Law of Detachment
$\{2\}$(10) $\neg(C \vee S) \rightarrow \neg I$2 T Law of Contraposition
$\{1\}$(11) $\neg I \rightarrow \neg (W \vee P)$1 T Law of Contraposition
$\{1,2\}$(12) $\neg (C \vee S) \rightarrow \neg (W \vee P)$10 11 T Law of Hypothetical Syllogism
$\{3,4\}$(13) $\neg C \wedge \neg S$6 9 T Law of Adjunction
$\{3,4\}$(14) $\neg (C \vee S)$13 T DeMorgan's Law
$\{1,2,3,4\}$(15) $\neg(W \vee P)$12 14 T Law of Detachment
$\{1,2,3,4\}$(16) $\neg W \wedge \neg P$15 T DeMorgan's Law
$\{1,2,3,4\}$(17) $\neg W$16 T Law of Simplification


pg. 35 #5:
"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 $L \rightarrow \neg M$:
$\{1\}$(1) $D \vee \neg L$Premise
$\{2\}$(2) $M \rightarrow \neg D$Premise
$\{1\}$(3) $\neg L \vee D$1 T Commutative Law of $\vee$
$\{1\}$(4) $L \rightarrow D$3 T Law of Equivalence of Implication and Disjunction
$\{2\}$(5) $\neg (\neg D) \rightarrow \neg M$2 T Law of Contraposition
$\{6\}$(6) $D$Premise
$\{6\}$(7) $\neg(\neg D)$6 T Law of Double Negation
$\{2,6\}$(8) $\neg M$5 7 T Law of Detachment
$\{2\}$(9) $D \rightarrow \neg M$6 8 C.P.
$\{1,2\}$(10) $L \rightarrow \neg M$4 9 Law of Hypothetical Syllogism


pg. 35 #6:
"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 $\neg R \vee A$. To do it, we will first conclude $R \rightarrow A$ and then apply the equivalence of implication and disjunction:
$\{1\}$(1) $J \rightarrow \neg N$Premise
$\{2\}$(2) $\neg J \rightarrow \neg D$Premise
$\{3\}$(3) $\neg D \rightarrow A$Premise
$\{4\}$(4) $R \rightarrow N$Premise
$\{1\}$(5) $\neg (\neg N) \rightarrow \neg J$1 T Law of Contraposition
$\{6\}$(6) $N$Premise
$\{6\}$(7) $\neg (\neg N)$6 T Law of Double Negation
$\{1,6\}$(8) $\neg J$5 7 T Law of Detachment
$\{1\}$(9) $N \rightarrow \neg J$6 8 C.P.
$\{1,4\}$(10) $R \rightarrow \neg J$4 9 T Law of Hypothetical Syllogism
$\{1,2,4\}$(11) $R \rightarrow \neg D$2 10 T Law of Hypothetical Syllogism
$\{1,2,3,4\}$(12) $R \rightarrow A$3 11 T Law of Hypothetical Syllogism
$\{1,2,3,4\}$(13) $\neg R \vee A$12 T Law of Equivalence of Implication and Disjunction


pg. 35 #7:
"If Algernon is a nuisance, then he is not in jail. If he is in jail, then he is a disgrace. If he is a disgrace, then he is not in the army. Therefore, he is either not in the army or not a nuisance."

Solution: Write the given premises:
$N \rightarrow \neg J$
$J \rightarrow D$
$D \rightarrow \neg A$
This argument wants to conclude "$\neg A \vee \neg N$". However this argument is invalid because we may assign truth values to the sentences that make the premises true but the conclusion false: take $A$ to be true, $N$ to be true, $J$ to be false, and $D$ to be false.

pg. 35 #8:
"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 $\neg E \vee M$. We will do so by deriving $E \rightarrow M$ and applying the Law of Equivalence of Implication and Disjunction:
$\{1\}$(1) $S \vee O$Premise
$\{2\}$(2) $S \rightarrow \neg E$Premise
$\{3\}$(3) $O \rightarrow M$Premise
$\{2\}$(4) $\neg(\neg E) \rightarrow \neg S$2 T Law of Contraposition
$\{5\}$(5) $E$Premise
$\{5\}$(6) $\neg (\neg E)$5 T Law of Double Negation
$\{2,5\}$(7) $\neg S$4 6 T Law of Detachment
$\{2\}$(8) $E \rightarrow \neg S$5 7 C.P.
$\{9\}$(9) $\neg S$Premise
$\{1,9\}$(10) $O$1 9 T Modus tollendo tollens
$\{1\}$(11) $\neg S \rightarrow O$9 10 C.P.
$\{1,2\}$(12) $E \rightarrow O$8 11 T Law of Hypothetical Syllogism
$\{1,2,3\}$(13) $E \rightarrow M$3 12 T Law of Hypothetical Syllogism
$\{1,2,3\}$(14) $\neg E \vee M$13 T Law of Equivalence of Implication and Disjunction