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I was messing around with liar and similar sentences, and noticed a recurring issue with indexical quarantine with respect to the examples at stake. Meaning that something seemed to "go wrong" when you reformulated these weird sentences (not just negative examples like, "This sentence is unknown," but positive ones like, "This sentence ought to be true") as questions: "Is this sentence true?" is not true or false, because it is not truth-apt in the first place; but it is not false because untrue, so that we see that untruth and falsity are not identical. Or, "Assume that this sentence is true," cannot be complied with, inasmuch as truth is not a property of imperative sentences. So otherwise, you have to say something like, "Is, 'This sentence is true,' true?" and, "Assume that, 'This sentence is true,' is true."

So these weird sentences are "indexically quarantined" in logical space. There are different ways to cash out "resolving the liar paradox": the "simplest" to describe, if not to accomplish, is to show that the liar sentence is not true and false; the next would be to show that, even if it is true and false, in the case of the liar sentence this is not an actual contradiction; or, at "worst," we might just show that even if the liar sentence is a counterexample to the LNC, yet for some reason this status doesn't actually contribute much, if anything, to the question of the LNC as covering arbitrarily many other sets of truths. There might be something about the liar sentence that confines inferences from it (or to it) to their own "logical hell" (hereafter Lell) if you will. That is the idea on offer here: the indexical quarantine of these sentences consigns them to Lell.

Now, "who else" has been damned for all eternity to burn in the fires of Lell? This is the question of my post: is, "Ask this question?" an indexically quarantined sentence?

Firstly, again, answering yes or no directly yields an unwell-formed sentence, "Yes, ask this question," or, "No, don't ask this question." Yet neither of those are questions, so... However, even if we suppose that, "Yes (no), (don't) ask that question," with "that" understood to refer back to "this question," is a sufficient answer, it remains that by asking "this question" in the first place, the local inquirer "defeats the purpose" of asking imperative questions. In other words, since asking the question is tantamount to complying with a yes-answer to the question, then asking that question involves not asking for the sake of a reply that would then be used as part of a reason why the question would or would not be asked.

"Answer this question correctly?" illustrates the problem some more: it seems that these indexically quarantined questions might have no possible correct answers, after all. They are their own empty circles of Lell. Does the same go for, "What is a correct/incorrect answer to this question?" (or, haha, "What if this question only has incorrect answers?")?

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    "Is this sentence true?" is not true or false, because it is not truth-apt in the first place" No, it is neither true nor false because it is question. I guess you mean something else but if you make a self-referential question, the question definitely ought not to be whether the question is true. Instead, for example, you could ask "Do you know the answer to this question?" This can be construed as self-referential and we don't know what to reply because it doesn't really make sense and we can't even say no because this would be an answer we know. Commented Dec 30, 2021 at 11:08
  • (1/3) Is indexical quarantine excluding any self-perspective? J. Ismael claims such is unavoidable in physics, "Most of the examples of self-reference in logic are static: people are looking at logical relations among eternal propositions. But I’ve been interested here in physics and interested in representation as an embodied activity. We are assuming the universe is a field of events, and self-reference is arising because some of what the system is representing is also what the system is doing. That will create interference between the two levels – the representational and the physical"
    – J Kusin
    Commented Dec 31, 2021 at 3:56
  • (2/3) "The funny business is all at the semantic level. It doesn’t place constraints on what a system can do in physical terms. It places constraints on what an embedded system can truthfully represent: on what it can know, on what is there to be known from its perspective." iai.tv/articles/…
    – J Kusin
    Commented Dec 31, 2021 at 3:56
  • (3/3) See her example of the computer with the goal of answering any physical fact, "It is not, however, hard to find a factual question that it can’t answer truthfully. Ask it ‘is the answer to this question that’s about to be displayed in the output channel ‘no’?"..."the computer can’t stabilize the fact that the answer is meant to describe (the word that appears on the screen) independently of giving the answer, and the thing is set up so that no matter which answer it gives, what it does in giving it conflict with what it says"
    – J Kusin
    Commented Dec 31, 2021 at 4:02

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Short Answer

is, "Ask this question?" an indexically quarantined sentence?

Depends on the context "Ask this question?" is produced in. In semiotics and linguistics, deixis can occur in several forms, and the syntax of the question is not the only determinant of indexicality. Brainy people tend to forget that sometimes symbols are rooted in broader contexts of language or physical experience itself.

Long Answer

In the question, "Ask this question?", this is simply a word that on its own ambiguously expresses indexicality. The context of the sentence if one observes the use-mention distinction does make it a self-reference as Speakpigeon has noted. The construction of such self-references is trivial.

In computer science, they can be done by creating a class with a data member that holds a pointer to the address of the object itself. In fact, in most contemporary language, an object can obtain its own address by using the 'this operator'. To wit:

this.ExecuteMethod();

which is used within the definition of the class itself.

In mathematics, in graph theory, you can have a graph of one node with one edge whose source and destination are the same.

Now, there is a second case, and that's "Ask this question?" doesn't occur in your question, but in some other sentence, such as in a screenplay where two characters are rehearsing for a trial where a lawyer has to ask a question. We can create an artifact:

Lawyer, pointing to notepad: Ask this question?

In this alternative context, "Ask this question" is not self-referential because the screenplay has the character pointing to a question, and therefore this indexicality is rooted in spatial deixis which is the sort of reference involved in ostensive definition. Clearly, in the context of the pointing, the 'this' in this case is not the sentence itself, as it was when you used the sentence, but some other actual, read instantiated or reified question.

So, the question of to what does 'this' refer when you present ANY sentence with 'this' used in it will always depend on the ultimate context of the language.

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