Question
Determine the validity of the following arguments using the direct method of truth table:
$\sim P\ v \sim Q$
$Q$
$\therefore \sim P$
$\sim P\ v \sim Q$
$Q$
$\therefore \sim P$
| Support Statement | The resulting statement | ||||||
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | |
| $P$ | $Q$ | $\sim P$ | $\sim Q`$ | $\sim P\ v \sim Q$ | $(\sim\ P\ v\ \sim\ Q)\ \&\ Q$ | $\sim P$ | |
| $1$ | $T$ | $T$ | $F$ | $F$ | $F$ | $F$ | $F$ |
| $2$ | $T$ | $F$ | $F$ | $T$ | $T$ | $F$ | $F$ |
| $3$ | $F$ | $T$ | $T$ | $F$ | $T$ | $T^*$ | $T^*$ |
| $4$ | $F$ | $F$ | $T$ | $T$ | $T$ | $F$ | $T$ |
| $1(\sim )$ | $2(\sim )$ | $3, 4(v)$ | $5, 2(\&)$ | As $3$ | |||
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| (P$\rightarrow$Q) & (R v S) |
| (R v S) $\rightarrow$ ~ L |
| L v (M & N) |
| $\therefore$ [(P $\rightarrow$ Q) & M] & ~ L |
| $(A\ \&\ D) \rightarrow C$ |
| $E\rightarrow (B\ v\ D)$ |
| $F \rightarrow (A\ \&\ D)$ |
| $F\ \&\ G$ |
| $C\ v\ (B\ v\ D)$ |
| $(A\ \rightarrow\ B)\ v\ D$ |
| $H\ \rightarrow\ [(A \ \rightarrow\ B)\ \rightarrow\ R]$ |
| $D\ \rightarrow\ E$ |
| $(E\ v\ F)\ \rightarrow\ H$ |
| $E\ v\ F$ |
| $(R\ v\ E)\ \&\ H$ |
| $(A\ \rightarrow\ E)\ \&\ (D\ \rightarrow\ F)$ |
| $B\ \&\ (A\ v\ D)$ |
| $(E\ v\ F)\ \rightarrow\ (B\ v\ D)$ |
| $\sim\ B$ |
| $\therefore D$ |
| $A \rightarrow B$ |
| $\sim B\ v\ D$ |
| $\sim D$ |
| $\therefore \sim A\ \&\ \sim D$ |