Prove that the following arguments are standard by constructing m… - Vidyadip

Question

Prove that the following arguments are standard by constructing metaphorical proof

$(A\ v\ B)\ \rightarrow [D\ \rightarrow\ (P\ \&\ \sim \ Q)]$

$(A\ \&\ J) \rightarrow [(P\ \&\ \sim\ Q)\ \rightarrow\ K]$

$(A\ \&\ J)\ \&\ (\sim\ K\ v\ D)$

$\therefore (D\rightarrow K)\ v\ \sim\ Q$

✓

Answer

$(1)\ (A\ V\ B)\ \rightarrow[D\ \rightarrow\ (P\ \&\ \sim\ Q)]$	$P$
$(2)\ (A\ \&\ J)\ \rightarrow[(P\ \&\ \sim\ Q) \rightarrow K]$	$P$
$(3)\ (A\ \&\ d)\ \&\ ( \sim\ K\ v\ D)$	$P/\therefore (D \rightarrow K) v \sim Q$
$(4)\ A\ \&\ J$	$3,$ Simp.
$(5)\ A$	$4,$ Simp.
$(6)\ A\ v\ B$	$5, $ Add.
$(7)\ D\ \rightarrow (P\ \&\ \sim\ Q)$	$1, 6, MP$
$(8)\ (P\ \& \sim\ Q)\ \rightarrow\ K$	$2, 4, MP$
$(9)\ D\ \rightarrow\ K$	$7, 8, HS$
$(10)\ (D\ \rightarrow\ K)\ v\ \sim\ Q$	$9,$ Add.

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