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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| $G \rightarrow H$ |
| $(A\ v\ B) \rightarrow D$ |
| $D \rightarrow [(S\ \&\ T) \rightarrow P]$ |
| $(S\ \&\ T)\ v\ G$ |
| $A\ v\ B$ |
| $(P\ v\ H)\ \&\ D$ |
| $R\ \rightarrow\ (A\ \&\ B)$ |
| $P\ v\ \sim\ (S\ \&\ T)$ |
| $\sim\ T\ \&\ \sim\ P$ |
| $(A\ \&\ B)\ \rightarrow\ (S\ \&\ T)$ |
| $(\sim\ R\ \&\ \sim\ T)\ v\ D$ |