Premise 2. Pinocchio claims "my nose grows now" — Michael
There's another angle that can be taken on this, that is of interest to mathematicians, although perhaps to nobody else. That is that Premise 2 is a statement about the velocity of the end of the nose. Velocity is the derivative of position with respect to time.
We can generalise the notion of ordinary derivative to define a left and a right derivative at a point (in time, say). The former is the rate of change (of position of the end of the nose) in the infinitesimally short period immediately before 'now', and the latter is the rate in the infinitesimally short period immediately after 'now' (I know that sounds very woolly, but it can be made precise if one has access to mathematical symbols).
The ordinary derivative only exists if the left and right derivative both exist and equal one another. In the real world, that always happens, because Newton's laws say nothing can instantaneously change its velocity.
But in this magical Pinocchio world we are considering, Newton's laws might not apply. So it is possible that the nose is growing at a constant rate up to time t='now', but stationary after time t. In that case the nose
has no velocity at time t.
So the statement 'my nose is growing now' can be seen as ambiguous, as it can refer to either the left or right derivative of the nose-end's position.
Or it can be seen as meaningless, as the nose-end will have no velocity 'now' if it suddenly stops growing at time t.
If we take it to be ambiguous then when Pinocchio makes the statement, the nose will instantaneously stop growing if it interprets the reference as being to the left derivative, and there will be no contradiction.
But if it interprets it as a reference to the right derivative, it will suffer the same contradiction that occurs in the simpler analysis. However, it's easier to see there why it's a contradiction, because in that case it is a statement about the future. It's equivalent to saying 'my nose will now stop growing if it does not now stop growing', which is as bare a contradiction as 'I am Pinocchio and I am not Pinocchio'.