Question
Prove that the following arguments are standard by constructing metaphorical proof
| $P \rightarrow Q$ |
| $P\ \&\ \sim S$ |
| $\therefore Q\ \&\ \sim P$ |
| $P \rightarrow Q$ |
| $P\ \&\ \sim S$ |
| $\therefore Q\ \&\ \sim P$ |
| $(1)\ P \rightarrow Q$ | $P$ |
| $(2)\ P\ \&\ \sim Q$ | $P/ \therefore\ Q\ \&\ \sim P$ |
| $(3)\ P$ | $2,$ Simp. |
| $(4)\ Q$ | $1, 3, MP$ |
| $(5)\ \sim\ Q$ | $2,$ Simp. |
| $(6)\ \sim\ P$ | $1, 5, MT$ |
| $(7)\ Q\ \&\ \sim\ P$ | $4, 6,$ Conj. |
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| $R\ \rightarrow\ (A\ \&\ B)$ |
| $P\ v\ \sim\ (S\ \&\ T)$ |
| $\sim\ T\ \&\ \sim\ P$ |
| $(A\ \&\ B)\ \rightarrow\ (S\ \&\ T)$ |
| $(\sim\ R\ \&\ \sim\ T)\ v\ D$ |
| $(R\ S) \rightarrow (F\ E)$ |
| $\sim (R\ S) \rightarrow J$ |
| $(F\ E) \rightarrow \sim H$ |
| $\sim\ \sim H$ |
| $\therefore \sim (J\ v\ G) \& \sim\ \sim H$ |
| $(A\ v\ D) \rightarrow (R\ \&\ S)$ |
| $E\ v\ (A\ v\ D)$ |
| $F\ \&\ \sim E$ |
| $(R\ \&\ F)\ v\ K$ |
| $R \rightarrow S$ |
| $P \rightarrow Q$ |
| $P\ v\ Q$ |
| $Q \rightarrow R$ |
| $\therefore (P \rightarrow R) \& (R\ v\ S)$ |