Reference, including self-reference needs a Proxy and so an object cannot refer directly to itself.

With "Mise en abyme" or the Droste effect we can provide a first axiom for SRH:

(1) In Reference there must be at least two entities: the main object or Referent and the Referrer which is a proxy object that in the context stands for the Referent.

In the sentence "he took the ball"

`the ball` is the referrer because obviously this is not an actual ball and you can't throw those words. A ball might look like this picture.

But you can't throw that either. Borrowing from later Wittgenstein we would say that all these objects can be used in many way. In the context of this blog we are most interested in objects taking the place of other objects. We could substitute an apple for the ball and he could still throw it.

So we can construct this odd sentence:

"he took the ⚽"

Yet we all know what it means because the picture ⚽ has taken the place of the object we mean, namely the ball that he took.

We have Quine's "use" and "mention" here. But really those are just specific examples of the more general Wittgenstein "language game" idea. "use" is very subtle and diverse, we can use something in many different ways, often only subtly different.

"I took the symbol ⚽ and then ⚽⚽ meaning that all together there were ⚽⚽⚽ written twice"

In this sentence am I "using" or "mentioning" ⚽? In the last sentence I am actually using the symbol ⚽ itself.

And other examples like "If we kick we can send it to the end of the sentence ⚽"

Here the symbol ⚽ is the actual object of the sentence, and while the implication is that ⚽ obeyed grammatical rules and existed in the right syntactic place (after kick) in the sentence it has under the pressure of semantics been kicked to the end of the sentence. I've no time to unpick this but its a very complicated example of the use of symbols.

However it remains that the process of one object taking the place of another implies that there are 2 objects. A referent and a referrer.

This must then also be true in self-reference, which is just a type of reference.

So the referent and a referrer are not the same object in self-reference. Which means that self-reference is not like an infinity mirror. The referrer by which we set up the self-reference is not the same as the self. We just move from self to a proxy and then back to self again.

Take the example:

"This sentence contains five words"

There are two objects here:

The referrer: "This sentence" and
The referent: "...the characters between the quotes..."

So we can use any referrer, let us call that sentence A. Many things are called 'A' but within the limited context of this page, the "state" of this object to use programming idioms, 'A' can be substituted with the sentence above.

'substitution' is the key operation that explain reference.

Given the rules above and the object

'A'

We can apply substitution to replace A with:

"This sentence contains five words"

Now semantics gives meaning to things in wide rule sets. The sentence is part of the English language and it means that were you to count the words in the sentence you will get a count of 5.

Clearly substitution does not preserve meaning:

While 'A' can be substituted with "This sentence contains five words" to give "'A' contains five words" they do not both have 5 words, they are not the same sentence even though we only substituted referrers with the same referent.

This means that "This sentence" cannot be substituted with A.

"A contains five words" is no longer true,

So while A and "This sentence" are both referrers to the sentence "This sentence contains five words" they cannot substituted.

This is because "This sentence" is used in two ways. It is both a referrer and a part of the referent itself.

We can introduce another dialectic here between Form and Content. "This sentence" and "A" have the same content in that they refer to the "same" sentence, but the Form is different. Indeed we use the form to determine the object. I have written 'A' many times in this page and so there are many objects here, but because they have similar form we call them the same "type of object" or "letter of the alphabet". This is only the first level of meaning however, and we take that "letter of the alphabet|" abstraction and then make a deeper level where it can be substituted by the "This sentence contains five words."

So we can note here that there are Forms with Contents which are themselves Forms with Contents. There is no absolute Form and Content level and Objects and Symbols can occur at any level of abstraction from the physical marks, to membership of an alphabet or set to part of a Linguistic net of meaning.

So returning to:

A = "This sentence contains five words"

This is a swirling mass of similarities, memberships and substitutions. How can "self-reference" be simple!

But we can say that the referrer "This sentence" is bound by another condition that it must have the right form as well as referent to maintain the truth value of the sentence.

Not all referrers are equal!

Perhaps I should introduce some syntax to separate the Quine issue here. 

"This sentence" is just marks on a page.
«This sentence» will  be used to use the sentence.

There is no reference with these characters "This sentence," they may as well be "jdhbstyvsmqwt"

While «This sentence» is semantic and refers to the physical part which looks like "This sentence"

SO «This sentence» is actually one of the shortest self referential statements. Perhaps only smaller is 

«Me»

When we ask what is referring to itself we find the sentence "Me"

Note that "" and «» are operators on the underlying characters and are not part of the sentence themselves.

In these immediate self-reference statements like «Me» the Form does not matter.

 «Moi» 

«मुझे»

It makes no difference. Pop those into the context of the language world's they come from and self-reference is semantically set up.

And if we ask what is «मुझे» referring to we could answer "मुझे". This gets into metaphysics because if someone was to say "मुझे" out loud we would take that to mean a "person" even if a computer said it. I plugged the Windows XP speak recognition into an AIML chatbot once and routed the output to the speech synthesiser. At 2am having a conversation with a computer made me feel like there was someone else in the room. Forget Frankenstein, making the illusion of people is really easy. I don't think we should read too much into a person saying "मुझे" it is just part of the language games, cultures and words we belong too. Ideas of ethereal spirits and souls is completely unnecessary.

Not sure I ever explored into this as deeply before. But hopefully it still remains that the referrer and the referent are different. Not a simple dualism as they can have interconnected forms but they cannot be the same.

"This sentence contains five words"

«This sentence contains five words»

are different.

Consider again that we substitute the sentence for A

A = "This sentence contains five words"

«'A' contains five words» is still correct.

"A contains five words" is just a new sentence like "qhg weqew qweqwe"

I miss one case:

""This sentence contains five words" contains five words"

is a new sentence from the following substitution:

"This sentence" = "This sentence contains five words"

«"This sentence contains five words" contains five words»

Is correct.

Anyway to wind this up the point is that an object cannot really refer to itself. It needs a proxy object that can be substituted for the referent object. These two objects need to be separate.

However it is noted that the Form of the object becomes important if the Referrer is part of the object. This adds conditions that limit the substitution.

I note here Douglas Hofstadter mentions somewhere in Godel, Esher, Bach that self-reference is always limiting. [need find a reference!]

A PDF of this remarkable book is here Godel, Esher, Bach

"But might there not be some vaguely Godelian loop which limits the depth to which any individual can penetrate into his own psyche?" [Godel, Esher, Bach p692]