Prove that the following arguments are standard by constructing m…

Question

Prove that the following arguments are standard by constructing metaphorical proof

$K\rightarrow\ (W\ \rightarrow\ X)$

$( \sim\ Q\ \&\ \sim\ K)\ \rightarrow\ (\sim\ Y\ v\ \sim\ M)$

$(\sim\ Y\ \rightarrow\ \sim\ Z)\ \&\ (\sim\ M\ \rightarrow\ \sim\ P)$

$(W\ \rightarrow\ X)$

$\therefore \sim\ Z\ v\ \sim\ p$

✓

Answer

$(1)\ Q\ \rightarrow\ (W\ \rightarrow\ X)$	$P$
$(2)\ K\ \rightarrow\ (W\ \rightarrow\ X)$	$P$
$(3)\ (\sim\ Q\ \&\ \sim\ K)\ \rightarrow\ (\sim\ Y\ v\ \sim\ M)$	$P$
$(4)\ (\sim\ Y\ \rightarrow\ \sim\ Z)\ \&\ (\sim\ M\ \rightarrow\ \sim\ P)$	$P$
$(5)\ \sim\ (W\ v\ X)$	$P/\therefore\ \sim\ Z\ v\ \sim\ P$
$(6)\ \sim\ Y\ \rightarrow\ \sim\ Z$	$4,$ Simp.
$(7)\ \sim\ M\ \rightarrow\ \sim\ P$	$4,$ Simp.
$(8)\ \sim\ Q$	$1, 5, MT$
$(9)\ \sim\ K$	$2, 5, MT$
$(10)\ \sim\ Q\ \&\ \sim\ K$	$8, 9,$ Conj.
$(11)\ \sim\ Y\ v\ \sim\ M$	$3, 10, MP$
$(12)\ \sim\ Z\ v\ \sim\ P$	$6, 7, 11, CD$

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