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pg. 41 #1(a): Are the premises consistent or inconsistent? If inconsistent, derive a contradiction. If consistent, give a true sentential interpretation to prove it.
"If the contract is valid, then Horatio is liable. If Horatio is liable, then he will go bankrupt. Either Horatio will go bankrupt or the bank will lend him money. However, the bank will definitely not lend him money."
Solution: We must check whether or not the premises are consistent, that is, whether or not we can assign truth values to all sentences and make all the premises true. Start by symbolizing the premises:
 $\{1\}$ (1) $V \rightarrow L$ Premise $\{2\}$ (2) $L \rightarrow B$ Premise $\{3\}$ (3) $B \vee M$ Premise $\{4\}$ (4) $\neg M$ Premise
Taking $V$ to be true, $L$ to be true, $M$ to be false, and $B$ to be true yields that the premises are all true. Therefore we have demonstrated that the premises are consistent.

pg. 41 #1(c): Are the premises consistent or inconsistent? If inconsistent, derive a contradiction. If consistent, give a true sentential interpretation to prove it.
"The contract is satisfied if and only if the building is completed by November 30. The building is completed by November 30 if and only if the electrical subcontractor completes his work by November 10. The bank loses money if and only if the contract is not satisfied. Yet the electrical subcontractor completes his work by November 10 if and only if the bank loses money."

Solution: First symbolize the premises:
 $\{1\}$ (1) $C \leftrightarrow B$ Premise $\{2\}$ (2) $B \leftrightarrow E$ Premise $\{3\}$ (3) $L \leftrightarrow \neg C$ Premise $\{4\}$ (4) $E \leftrightarrow L$ Premise
Suppose we would like $C$ to be true. Then Premise 1 forces $B$ to be true and so Premise 2 forces $E$ to be true. We are then forced by Premise 4 to take $L$ to be true. But then Premise 4 suggests that $\neg C$ should be true, contradicting our choice initially that $C$ be true. A similar argument starting with $C$ being false also leads to problems. Therefore we think that these premises may be inconsistent. To show it, we must formally derive a contradiction from the premises 1-4:
 $\{1\}$ (1) $C \leftrightarrow B$ Premise $\{2\}$ (2) $B \leftrightarrow E$ Premise $\{3\}$ (3) $L \leftrightarrow \neg C$ Premise $\{4\}$ (4) $E \leftrightarrow L$ Premise $\{1\}$ (5) $(C \rightarrow B) \wedge (B \rightarrow C)$ 1 T Law of Biconditional Sentences $\{1\}$ (6) $(B \rightarrow C) \wedge (C \rightarrow B)$ 5 T Commutative law of $\wedge$ $\{1\}$ (7) $C \rightarrow B$ 5 T Law of Simplification $\{1\}$ (8) $B \rightarrow C$ 6 T Law of Simplification $\{2\}$ (9) $(B \rightarrow E) \wedge (E \rightarrow B)$ 2 T Law of Biconditional Sentences $\{2\}$ (10) $(E \rightarrow B) \wedge (B \rightarrow E)$ 9 T Commutative law of $\wedge$ $\{2\}$ (11) $B \rightarrow E$ 9 T Law of Simplification $\{2\}$ (12) $E \rightarrow B$ 10 T Law of Simplification $\{3\}$ (13) $(L \rightarrow \neg C) \wedge (\neg C \rightarrow L)$ 3 T Law of Biconditional Sentences $\{3\}$ (14) $(\neg C \rightarrow L) \wedge (\neg C \rightarrow L)$ 13 T Commutative Law of $\wedge$ $\{3\}$ (15) $L \rightarrow \neg C$ 13 T Law of Simplification $\{3\}$ (16) $\neg C \rightarrow L$ 14 T Law of Simplification $\{4\}$ (17) $(E \rightarrow L) \wedge (L \rightarrow E)$ 4 T Law of Biconditional Sentences $\{4\}$ (18) $(L \rightarrow E) \wedge (E \rightarrow L)$ 17 T Commutative Law of $\wedge$ $\{4\}$ (19) $E \rightarrow L$ 17 T Law of Simplification $\{4\}$ (20) $L \rightarrow E$ 18 T Law of Simplification $\{1,2\}$ (21) $C \rightarrow E$ 7 11 T Law of Hypothetical Syllogism $\{1,2,4\}$ (22) $C \rightarrow L$ 19 21 T Law of Hypothetical Syllogism $\{1,2,3,4\}$ (23) $C \rightarrow \neg C$ 15 22 T Law of Hypothetical Syllogism $\{3,4\}$ (24) $\neg C \rightarrow E$ 16 20 T Law of Hypothetical Syllogism $\{2,3,4\}$ (25) $\neg C \rightarrow B$ 24 12 T Law of Hypothetical Syllogism $\{1,2,3,4\}$ (26) $\neg C \rightarrow C$ 25 8 T Law of Hypothetical Syllogism $\{27\}$ (27) $\neg(\neg C)$ Premise $\{27\}$ (28) $C$ 27 T Law of Double Negation $\{1,2,3,4,27\}$ (29) $\neg C$ 28 23 T Law of Detachment $\{1,2,3,4,27\}$ (30) $C \wedge \neg C$ 28 29 T Law of Adjunction $\{1,2,3,4\}$ (31) $\neg C$ 28 30 R.A.A. $\{1,2,3,4\}$ (32) $C$ 26 31 Law of Detachment $\{1,2,3,4\}$ (33) $C \wedge \neg C$ 31 32 Law of Adjunction
We have shown that a contradiction follows from premises 1,2,3, and 4, showing these premises are inconsistent. We did end up at a contradiction at line 29, but this is not the contradiction we are seeking -- notice that it depends on the premise at line 29. To say that we want to "derive a contradiction from premises 1-4" means that the contradiction we are looking for should only list those premises on the left. Lines 30-32 accomplishes this.

pg. 41 #3(a): Prove by reductio ad absurdum:
"If John plays first base, and Smith pitches against us, then Winsocki will win. Either Winsocki will not win, or the team will end up at the bottom of the league. The team will not end up at the bottom of the league. Furthermore, John will play first base. Therefore, Smith will not pitch against us."

Solution: To prove this by contradiction, first symbolize the given premises:
 $\{1\}$ (1) $(J \wedge S) \rightarrow W$ Premise $\{2\}$ (2) $\neg W \vee T$ Premise $\{3\}$ (3) $\neg T$ Premise $\{4\}$ (4) $J$ Premise
The conclusion of this argument is $\neg S$. If we want to prove this using reductio ad absurdum, we will add $\neg(\neg S)$ as a premise and derive a contradiction, after which we will apply rule RAA to conclude $\neg S$:
 $\{1\}$ (1) $(J \wedge S) \rightarrow W$ Premise $\{2\}$ (2) $\neg W \vee T$ Premise $\{3\}$ (3) $\neg T$ Premise $\{4\}$ (4) $J$ Premise $\{5\}$ (5) $\neg(\neg S)$ Premise $\{5\}$ (6) $S$ 5 T Law of Double Negation $\{4,5\}$ (7) $J \wedge S$ 4 6 T Law of Adjunction $\{1,4,5\}$ (8) $W$ 1 7 T Law of Detachment $\{1,4,5\}$ (9) $\neg (\neg W)$ 8 T Law of Double Negation $\{1,2,4,5\}$ (10) $T$ 2 9 T Modus tollendo ponens $\{1,2,3,4,5\}$ (11) $T \wedge \neg T$ 3 10 T Law of Adjunction $\{1,2,3,4\}$ (12) $\neg S$ 5 11 RAA

pg. 41 #4(a): Prove by reductio ad absurdum:
"Either logic is not difficult, or not many students like it. If mathematics is easy, then logic is not difficult. Therefore, if many students like logic, then mathematics is not easy."
Solution: First note the premises in the argument:
 $\{1\}$ (1) $D \vee \neg L$ Premise $\{2\}$ (2) $M \rightarrow \neg D$ Premise
The argument would like to deduce $L \rightarrow \neg M$ from these premises. We are asked to do it using reductio ad absurdum, meaning we will introduce the negation of the conclusion, $\neg (L \rightarrow \neg M)$, as a premise and find a contradiction from it, allowing us to use rule RAA to deduce the conclusion we seek:
 $\{1\}$ (1) $D \vee \neg L$ Premise $\{2\}$ (2) $M \rightarrow \neg D$ Premise $\{3\}$ (3) $\neg (L \rightarrow \neg M)$ Premise $\{3\}$ (4) $L \wedge \neg( \neg M)$ 3 T Law of Negation of Implication $\{3\}$ (5) $L$ 4 T Law of Simplification $\{3\}$ (6) $\neg (\neg M) \wedge L$ 3 T Commutative law for $\wedge$ $\{3\}$ (7) $\neg(\neg M)$ 6 T Law of Simplification $\{3\}$ (8) $M$ 7 T Law of Double Negation $\{2,3\}$ (9) $\neg D$ 2 8 T Law of Detachment $\{1,2,3\}$ (10) $\neg L$ 1 9 T Modus tollendo ponens $\{1,2,3\}$ (11) $L \wedge \neg L$ 5 10 T Law of Adjunction $\{1,2\}$ (12) $L \rightarrow \neg M$ 3 11 RAA

pg. 41 #4(b): Prove by reductio ad absurdum:
"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: First we symbolize the premises:
 $\{1\}$ (1) $S \vee O$ Premise $\{2\}$ (2) $S \rightarrow \neg E$ Premise $\{3\}$ (3) $O \rightarrow M$ Premise
We are being asked to prove $\neg E \vee M$ from these premises using reductio ad absurdum. To do this, we will introduce $\neg (\neg E \vee M)$ as a premise and then arrive at a contradiction:
 $\{1\}$ (1) $S \vee O$ Premise $\{2\}$ (2) $S \rightarrow \neg E$ Premise $\{3\}$ (3) $O \rightarrow M$ Premise $\{4\}$ (4) $\neg (\neg E \vee M)$ Premise $\{4\}$ (5) $\neg(\neg E) \wedge \neg M$ 4 T DeMorgan's law $\{4\}$ (6) $\neg(\neg E)$ 5 T Law of Simplification $\{4\}$ (7) $\neg M \wedge \neg(\neg E)$ 5 T Commutative law of $\wedge$ $\{4\}$ (8) $\neg M$ 7 T Law of Simplification $\{3\}$ (9) $\neg M \rightarrow \neg O$ 3 T Law of Contraposition $\{2\}$ (10) $\neg(\neg E) \rightarrow \neg S$ 2 T Law of Contraposition $\{2,4\}$ (11) $\neg S$ 6 10 T Law of Detachment $\{3,4\}$ (12) $\neg O$ 8 9 T Law of Detachment $\{1,2,4\}$ (13) $O$ 1 11 T Modus tollendo ponens $\{1,2,3,4\}$ (14) $O \wedge \neg O$ 12 13 T Law of Adjunction $\{1,2,3\}$ (15) $\neg E \vee M$ 4 14 RAA