$\{1\}$ | (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
$\{1\}$ | (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 |
$\{1\}$ | (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 |