$1$ Consider the following logic argument:

$2($a$)$ $ p q $->$r $

$3($b$)$ $ s q $

$4($c$)$ $ t $

$5($d$)$ $ p $->$t $

$6($e$)$ $ p r $->$s $

$7($f$)$ $ q $

$81.$ Create a truth table and find the critical row

$92.$ Identify the values for p$,$q$,$r$,$s and t for the argument to be

valid

$103.$ Use valid argument forms to deduce the conclusion from

the premises and test each one

$[4]$: def implies$($p$,$ q$)$:

return not p or q

print$($f$"\{\text{'}$p$\text{'}\}\backslash$t$\{\text{'}$q$\text{'}\}\backslash$t$\{\text{'}$r$\text{'}\}\backslash$t$\{\text{'}$s$\text{'}\}\backslash$t$\{\text{'}$t$\text{'}\}\backslash$t$\backslash$t$\{\text{'}$a$\text{'}\}\backslash$t$\{\text{'}$b$\text{'}\}\backslash$t$\{\text{'}$c$\text{'}\}\backslash$t$\{\text{'}$d$\text{'}\}$

$\backslash$t$\{\text{'}$e$\text{'}\}\backslash$t$\{\text{'}$f$\text{'}\}")$

for p in $[$True$,$ False$]$:

for q in $[$True$,$ False$]$:

for r in $[$True$,$ False$]$:

for s in $[$True$,$ False$]$:

for t in $[$True$,$ False$]$:

a $=$ implies$($not p$,$ q$)$

b $=$ s or not q

c $=$ not t

$1$

d $=$ implies$($p$,$t$)$

e $=$ implies$($not p and r$,$ not s$)$

f $=$ not q

if all$([$a$,$b$,$c$,$d$,$e$])$:

critical$\_$row $="<-$critical row"

#print$($f$\text{'}\{$p$\}\backslash$t$\{$q$\}\backslash$t$\{$r$\}\backslash$t$\{$s$\}\backslash$t$\{$t$\}\backslash$t$\backslash$t$\{$a$\}\backslash$t$\{$b$\}$

$\backslash$t$\{$c$\}\backslash$t$\{$d$\}\backslash$t$\{$e$\}\backslash$t$\{$f$\}\{$critical$\_$row$\}\text{'})$

else:

critical$\_$row $=""$

flag $=""$

print$($f$\text{'}\{$p$\}\backslash$t$\{$q$\}\backslash$t$\{$r$\}\backslash$t$\{$s$\}\backslash$t$\{$t$\}\backslash$t$\backslash$t$\{$a$\}\backslash$t$\{$b$\}\backslash$t$\{$c$\}$

$\backslash$t$\{$d$\}\backslash$t$\{$e$\}\backslash$t$\{$f$\}\{$critical$\_$row$\}\text{'})$

p q r s t a b c d

e f

True True True True True True True False True

True False

True True True True False True True True False

True False

True True True False True True False False True

True False

True True True False False True False True False

True False

True True False True True True True False True

True False

True True False True False True True True False

True False

True True False False True True False False True

True False

True True False False False True False True False

True False

True False True True True True True False True

True True

True False True True False True True True False

True True

True False True False True True True False True

True True

True False True False False True True True False

True True

True False False True True True True False True

True True

True False False True False True True True False

True True

True False False False True True True False True

True True

True False False False False True True True False

True True

$2$

False True True True True True True False True

False False

False True True True False True True True True

False False

False True True False True True False False True

True False

False True True False False True False True True

True False

False True False True True True True False True

True False

False True False True False True True True True

True False $<-$critical row

False True False False True True False False True

True False

False True False False False True False True True

True False

False False True True True False True False True

False True

False False True True False False True True True

False True

False False True False True False True False True

True True

False False True False False False True True True

True True

False False False True True False True False True

True True

False False False True False False True True True

True True

False False False False True False True False True

True True

False False False False False False True True True

True True

$[6]$: p $=$ False # Meaning that the premise $($not p$)$ is True

q $=$ False # This is the conclusion, $($not q$)$ is True

r $=$ True # Meaning that the premise $($not r$)$ is True

s $=$ False # Meaning that the premise $($not s$)$ is True

t $=$ False # Meaning that the premise $($not t$)$ is True

$[8]$: print$("$By Modus Tollens"$)$

print$($implies$($p$,$t$))$

print$($not t$)$

print$($not p$)$

By Modus Tollens

True

True

True

$3$

$[10]$: print$("$By Generalization"$)$

print$($not p$)$

print$($not p or q$)$

By Generalization

True

True

$[12]$: print$("$By Modus Ponens"$)$

print$($implies$($not p or q$,$r$))$

print$($not p or q$)$

print$($r$)$

By Modus Ponens

True

True

True

$[14]$: print$("$By Conjuction"$)$

print$($r$)$

print$($not p$)$

print$($not p and r$)$

By Conjuction

True

True

True

$[16]$: print$("$By Modus Ponens"$)$

print$($implies$($not p and r$,$ not s$))$

print$($not p and r$)$

print$($not s$)$

By Modus Ponens

True

True

True

$[18]$: print$("$By Elimination"$)$

print$($s or not q$)$

print$($not s$)$

print$($not q$)$

By Elimination

True

True

True

$4$

$[]$: