Question
Prove that the following arguments are standard by constructing metaphorical proof
| $(A\ v\ B) \rightarrow D$ |
| $A$ |
| $\therefore (D\ \&\ A)\ v\ P$ |
| $(A\ v\ B) \rightarrow D$ |
| $A$ |
| $\therefore (D\ \&\ A)\ v\ P$ |
| $(1)\ (A\ v\ B) \rightarrow D$ | $P$ |
| $(2)\ A$ | $P/\therefore (D\ \&\ A)\ v\ P$ |
| $(3)\ A\ v\ B$ | $2,$ Add |
| $(4)\ D$ | $1, 3, MP$ |
| $(5)\ D\ \&\ A$ | $4, 2.$ Conj. |
| $(6)\ (D\ \&\ A)\ v\ P$ | $5,$ Add |
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| $J\ v\ (J\ \&\ K)$ |
| $J \rightarrow D$ |
| $\sim\ D$ |
| $\therefore K\ v\ M$ |
| $(A\ v\ B)\ \rightarrow [D\ \rightarrow\ (P\ \&\ \sim \ Q)]$ |
| $(A\ \&\ J) \rightarrow [(P\ \&\ \sim\ Q)\ \rightarrow\ K]$ |
| $(A\ \&\ J)\ \&\ (\sim\ K\ v\ D)$ |
| $\therefore (D\rightarrow K)\ v\ \sim\ Q$ |
| $(T\ \&\ P) \rightarrow \sim Q$ |
| $\sim (T\ \&\ P) \rightarrow (R \rightarrow \sim Q)$ |
| $(\sim S\ v\ R)\rightarrow\ \sim\ \sim Q$ |
| $\sim S$ |
| $\therefore \sim R\ \&\ \sim S$ |
| $\sim P\ \&\ (Q\ v\ R)$ |
| $(Q \rightarrow A)\ \&\ (R \rightarrow B)$ |
| $(A\ v\ B) \rightarrow (P\ v\ R)$ |
| $R$ |
| $J \rightarrow K$ |
| $J\ v\ (K\ v\ \sim\ L)$ |
| $\sim K$ |
| $\therefore \sim L\ \&\ \sim K$ |
| $(E\rightarrow F)\rightarrow H$ |
| $\sim J\ v (F\ \&\ G)$ |
| $F \rightarrow(H\rightarrow I)$ |
| $\sim\ \sim J$ |
| $G\ \&\ [(E\rightarrow F) \rightarrow I]$ |
| $P \rightarrow Q$ |
| $\sim Q\ v\ R$ |
| $\sim R$ |
| $\therefore (\sim P \& \sim R)\ v\ S$ |