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I am not sure which formalization is right [1] or [2]:

'The teacher of Plato does not exist.'

[1] ∃x(Tx,p ∧ ∀y[Ty,p → y=x] ∧ ¬∃y[y = x])

[2] ∃x(Tx,p ∧ ∀y[Ty,p → y=x] ∧ ¬∃z[z = x])

Is it possible to use the variable 'y' again with the existential quantifier, like in [1] or is [2] the correct formalization? Can anybody help? Thanks in advance.

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Whether you use y or z makes no difference because the scopes are different. I'm not sure either is a good symbolization though. They are both always false; they say, "There exists a unique person who teaches Plato and doesn't exist." It's a contradiction.

It might be better to interpret it as "No one taught Plato": ¬∃x(Tx,p)

Or as "Plato did not have a unique teacher": ¬∃x(Tx,p ∧ ∀y[Ty,p → y=x])

The problem is really the original statement, which is oddly worded. To say "the teacher of Plato" presupposes that such a teacher exists. But then the rest contradicts that. Because of the contradiction, it doesn't seem charitable to read it literally (as you did).

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