$($a$)$ Construct formal proofs of validity for the following argument, using the rule of Attempted: $0$ Conditional Proof:

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

savants." $($Px$,$ Lx$,$ S X $)$

$(2$ 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:

$($i$)$ "Everything comes to every person who waits." $($Px: $x$ is a person. $Zx$:$x$ waits. Ty: $y$ is a thing. Uxy:$x$ comes to $y.)$

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

$(2$ marks$)$

$($d$)$ Prove the validity of the following enthymeme using 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$)$