$($Answer must be in your own hand$-$writing$)$

$($a$)$ Construct formal proofs of validity for the following argument, using the rule of Conditional

Proof:

$-$All professors are learned. All learned professors are savants. Therefore all professors are learned

savants. $)$

$(2$ marks$)$

$($b$)$ Using quantification theory, prove that the following argument is invalid:

"Some pointers are spaniels because some dogs are pointers and some dogs are spaniels."

$(2$ marks$)$

$($c$)$ Symbolize the following sentences, in each case using the indicated symbols:

$(1)$ "Everything comes to every person who waits." ${}_x$:$x$ is a person. $Zx$ : $x$ waits. Ty : $y$ is a thing. $U$

$xy$:$x$ comes to $y.)$

$($ii$)$ Every rose has its thorn. $($Rx: $x$ is a rose. $T_x$ : $x$ is a thorn. Hxy : $x$ has $y.)$

$(2$ marks$)$

$($d$)$ Prove the validity of the following enthymeme wing the indicated symbols:

$-$Any man on the first team can outrun every man on the second team. Therefore no man on the

second

team can outrun any man on the first team." $($Fx: $x$ is a man on the first team. $5x$ : $x$ is a man on

the second team. Oxy : $x$ can outrun $y.)$

$(2$ marks$)$