This appears to be a complex set of questions related to logic and philosophy. Let's break down the information provided and correct any errors present.
1. Translate the following statements into predicate and/or propositional notation, using the symbols above (2 each; 6 total).
(a) Only Socrates among all Greek philosophers was the teacher of Plato.
(b) The only philosopher who was a stonemason was Socrates.
(c) At least one philosopher taught all those philosophers who lived after him.
2. Translate the following predicate formulae into clear, grammatical English (not half-logic!) using the symbols defined above (2 each; 4 total).
(a) ∀x∃y(P x -> T yx)
(b) ∃x(Gx ∧ P x ∧ T xp ∧ ∀y(Gy ∧ P y ∧ T yp -> y = x))
3. In the following, write the given argument as a sequent (5) using the symbol definitions above or your own symbols as required, and prove the resulting sequent (5). (Warning: some premises could contain irrelevant information.)
No Greek philosophers were stonemasons except Socrates.
Heraclitus was a Greek philosopher.
Heraclitus died before Socrates was born, and therefore was not identical to Socrates.
∴ Heraclitus was not a stonemason.
4. Find predicate natural deduction proofs for the following sequents (10 each; 30 total):
(a) ∀xF x ⊢ ∀x(Gx -> F x)
(b) ∀x(F x -> Gx), ∃y∀x(Gx -> Hy) ⊢ ∃y∀x(F x -> Hy)
(c) ∀x(F x -> Gx), −∀x(Hx -> Gx) ⊢ ∃x(Hx ∧ −F x)