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

Attemp

Conditional Proot:

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

savants. $(Px,Lx,5x)$

$(2$ marks$)$

$($b$)$ Using quantufication 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. is a person. $Zx$:$x$ waits Iy $y$ is a thing Uxy : $x$ comes to $y)$

$($Ii$)$ Every rose has its thorn. $($Rx: $x$ is a rose. $Tx$ : $x$ a a thom. Hry $x$ 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. $Sx.$ is a man on the second team. Oxy : $x$ can outrun $y)$

$(2$ marks$)$