We only started learning identity as well as "only one," "at least one," etc. today. So if my answers are wrong, please provide me in the right direction.
Dionysius and Pseudo-Dionysius cannot be identical. For Dionysius was a member of the Court of Aeropagus, and Pseudo-Dionysius wasn't.
d = Dionysius, p = Pseudo-Deionysius, a = the Court of Aeropagus, M_ _ = _ is a member of _
Mda v ¬ Mdp ⊢ d $\ne$ p
Cisco loves Tessa and so does Rudi. Rudi is an acrobat and Cisco is not. Therefore at least 2 people love Tessa. Assume a domain of people.
c = Cisco, t = Tessa, r = Rudi, L_ _ = _ loves _ , A_ = _is an acrobat
Lct ∧ Lrt, Ar ∨ ¬Ac ⊢ ∃x∃y(Lxt ∧ Lyt)
Of all idealists, only Berkeley has won enduring philosophical fame. Only one bishop of Cloyne has won enduring philosophical fame, and he was an idealist. Therefore Berkeley was bishop of Cloyne.
I_ = _ was an idealist, B_ _ = _ is a bishop of _, W _ = _ has won enduring philosophical fame, b = Berkeley, c = Cloyne
Ib ∧ Wb ∧ ∀((Ix ∧ Wx) → x = b), ∃x(Ix ∧ Wx ∧ Bxc ∧ ∀y((Iy ∧ Wy ∧ Byc) → x = y)) ⊢ Bbc
The $\lor $ should be an $\land$, and it should be $Mpa$, not $Mdp$. So: $Mda \land \neg Mpa \vdash d \not = p$
Again, the $\lor$ should be a $\land$. Also, add $x\not = y$ to the conclusion
Drop the $Iy$ from the second premise