Question
Determine the validity of the following arguments using the direct method of truth table:
$P \rightarrow\ \sim Q$
$\therefore Q \rightarrow \ \sim P$
$P \rightarrow\ \sim Q$
$\therefore Q \rightarrow \ \sim P$
| Support Statement | The resulting statement | |||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | |
| $P$ | $Q$ | $\sim P$ | $\sim Q$ | $P \rightarrow\ \sim Q$ | $Q \rightarrow\ \sim P$ | |
| $1$ | $T$ | $T$ | $F$ | $F$ | $F$ | $F$ |
| $2$ | $T$ | $F$ | $F$ | $T$ | $T^*$ | $T^*$ |
| $3$ | $F$ | $T$ | $T$ | $F$ | $T^*$ | $T^*$ |
| $4$ | $F$ | $F$ | $T$ | $T$ | $T^*$ | $T^*$ |
| $1(\sim )$ | $2(\sim )$ | $1, 4(\rightarrow)$ | $2, 3(\rightarrow)$ | |||
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| $G \rightarrow H$ |
| $(A\ v\ B) \rightarrow D$ |
| $D \rightarrow [(S\ \&\ T) \rightarrow P]$ |
| $(S\ \&\ T)\ v\ G$ |
| $A\ v\ B$ |
| $(P\ v\ H)\ \&\ D$ |
| $A\ \&\ B.$ |
| $B\rightarrow (D\ v\ E)$ |
| $\sim E$ |
| $D \rightarrow (P \rightarrow \sim Q)$ |
| $\therefore P\rightarrow \sim Q$ |