Question I

$($a$)$ Constract formal proofs of validity for the following argument. using the rule of Conditional Proof.

"All proffessors are learned. All learned proffessors are Savants. Thorefore all professors are learned Savants." $($Px$,$ Lx$,$ Sx$)$

$($b$)$ Using quantification theory. prove that the following arguments is invalid:

"Some pointers are spaniels bocause some dogs are pointors and some dogs are spaniels."

$($c$)$ Symbdize the following sentonces$,$ in each case using the Indicated symbols:

$($i$)$ Everything comes to exrery person who coaits $"($Px: $x$ is a person.

$zx$:$x$ waits Ty:y is a thing. Uxy: $x$ comes to $y.)$

$($iii$)$ Every rose has its thorn. $($Rx: $x$ is a rose. Tx: $x$ is a thom.

$Hxy$: $x$ has $y.$

$($d$)$ Prove the validity of the following enthymeme using the indicated symbols.

"Any man on the first time can outrun every man on the second team. Therefore no men. on the second team can outrun any man on the first team. $"$C $x$:$x$ is a man on the first team Sx: $x$ is aman on the second toam. Oxy:x can Outrun y$.)$