Different Negations of Self-referential Propositions On Page 33, The Liar: an Essay on Truth and Circularity, (Barwise and Etchemendy, 1987) 

Exercise 6 Explain how the claims made by the following sentences
  differ.$$\lnot\downarrow\mathbf{True(this)}$$$$\downarrow\lnot\mathbf{True(this)}$$Is either claim paradoxical? Are both?

Some explanations on notations: $\mathbf{True(this)}$ could be interpreted as the formula inside the parenthesis, which is, $\mathbf{this}$, viz this proposition, is true. However, what this proposition refers to depends on the position $\downarrow$. The first $\mathbf{this}$ refers to the proposition "$\mathbf{True(this)}$", while the second refers to "$\lnot\mathbf{True(this)}$"
Here's how far I understand: The sentence $\downarrow\mathbf{True(this)}$ can be either true or false, which is contrary to the Lair's paradox. Its negation must be neither true nor false. So the first sentence is paradoxical.
The second sentence, in my view, is identical to lair's paradox, thus it's not true or false, either.
I just can't see the difference between these two claims.
EDIT: some explanation on the scope symbol "$\downarrow$" in Barwise et al(1987)

A word of explanation is in order about the scope symbol "$\downarrow$".
  When we provide semantics for L, we will ensure that this automatically
  refers to the proposition expressed by the sentence in
  which it occurs. Thus it will be our formal analogue of the English
  expression "this proposition," when that phrase is used reflexively.
  But even in its reflexive use, this expression is ambiguous. The
  ambiguity emerges in cases like the following.
(2.1) Max has the three of clubs or this proposition is true.
We think the most natural reading here is one in which "this
  proposition" refers to the proposition expressed by the whole of 2.1.
  However, we can also imagine it being used to refer to the proposition
  expressed by the second disjunct alone, in which case it would
  refer to the ordinary Truth-teller proposition. The two readings
  give quite different propositions, with different truth-conditions.
  In the first case, we will say that the scope of "this" is the entire
  sentence; in the second case, its scope is just the second disjunct.
  In our formal language, the two would be disambiguated as follows.
  $$（\text{Max has three clubs}）\lor \mathbf{True(this)}$$
  $$（\text{Max has three clubs}）\lor \downarrow\mathbf{True(this)}$$

 A: Consider:

(1) Snow is white
(2) The proposition (1) is true

(2) seems quite unproblematically true. We understand what it says because we can unpack the reference of "The proposition (1)" thus,

(2*) The proposition (1), i.e. the proposition that snow is white, is true

Contrast

(3) The proposition (3) is true

It is very tempting to say there is something defective about this. It seems we can't unpack the reference. If we try, we get 'The proposition (3), i.e. the proposition (3). i.e. proposition (3), i.e. ...' and the unpacking never bottoms out.
If (3) is defective (and we here assume that it is), it presumably can't be true. Hence

(4) It is not the case that (3)

looks compelling.
Compare

(5) The proposition (5) is not true.

This looks defective for the reason that (3) is: we can't unpack the reference. So (4) being true and (5) being defective must be treated differently.
Now, using Barwise and Etchemendy's notation -- marking the difference between negation 'looking from the outside' and internal negation -- the true (4) corresponds to the true

$\neg\downarrow$True(this proposition)

While the defective (and hence non-true) (5) corresponds to

$\downarrow\neg$ True(this proposition)

So these two, so understood, are indeed distinct.
