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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| $(T\ \&\ P) \rightarrow \sim Q$ |
| $\sim (T\ \&\ P) \rightarrow (R \rightarrow \sim Q)$ |
| $(\sim S\ v\ R)\rightarrow\ \sim\ \sim Q$ |
| $\sim S$ |
| $\therefore \sim R\ \&\ \sim S$ |
| $\sim P\ \&\ (Q\ v\ R)$ |
| $(Q \rightarrow A)\ \&\ (R \rightarrow B)$ |
| $(A\ v\ B) \rightarrow (P\ v\ R)$ |
| $R$ |
| $P \rightarrow Q$ |
| $\sim Q\ v\ R$ |
| $\sim R$ |
| $\therefore (\sim P \& \sim R)\ v\ S$ |
| $(A \leftrightarrow B) \rightarrow (D \leftrightarrow E)$ |
| $(D \leftrightarrow E) \rightarrow\ \sim H$ |
| $\sim\ \sim H$ |
| $\sim (A \leftrightarrow B) \rightarrow F$ |
| $(F\ v\ G)\ \&\ \sim\ \sim H$ |
| $X \rightarrow Y$ |
| $Y \rightarrow Z$ |
| $(X \rightarrow Z) \rightarrow (Y \rightarrow P)$ |
| $(Y\ V\ P) \rightarrow Z$ |
| $\therefore Z\ v\ Q$ |