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

Atter

Conditional Proof:

"All professors are learned. All learned professors are savants. Therefore all professors fare learned savants." $($Px$,Lx,Sx$

$(2$ 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. $Sx$:$x$ is a man on the second team. Oxy : $x$ can outrun $y.)$

$(2$ marks$)$