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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. $$(\text{Max has three clubs})\lor \mathbf{True(this)}$$ $$(\text{Max has three clubs})\lor \downarrow\mathbf{True(this)}$$

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What does the $\downarrow$ notation signify? –  Arthur Fischer Dec 12 '12 at 11:28
    
@ArthurFischer: I've added the relevant content from the book. I hope it is helpful. –  Metta World Peace Dec 12 '12 at 12:23
    
This notation is confusing. For my own analysis of the (Simplified) Liar Paradox, which also uses a True predicate, see: dcproof.com/SimplifiedLiar.htm –  Dan Christensen Dec 12 '12 at 19:36
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1 Answer 1

up vote 2 down vote accepted

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.

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Thank you for your answer. I'm confused about your usage of "defective", "is not the case", and "is not true". It seems to me, in this context, law of excluded middle, is assumed, but clearly you avoid the word "false" for some reason. –  Metta World Peace Dec 12 '12 at 14:31
    
I think the point is that a meaningless sentence is neither true nor false, c.f. "Colourless green ideas sleep furiously." That is not to say self-referential propositions are necessarily meaningless, but it does require some care to make them meaningful... –  Zhen Lin Dec 12 '12 at 14:36
    
@ZhenLin: So your point is, in Peter Smith's argument, "defective" et al are interchangeable with apparently meaningless, or potentially meaningful? It looks fine to me. –  Metta World Peace Dec 12 '12 at 14:49
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"Defective" was (intentionally) unspecific, to allow for different stories about what, exactly, is amiss with the likes of (3), stories which imply that (3) lacks a determinate truth-value. –  Peter Smith Dec 12 '12 at 15:39
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