I. "Noun" is a noun

1. I started with an example from one of Douglas Hofstadter's books that caught my eye:

"this sentence contains threee erors."

It actually contains two first-order errors ("threee" and "erors") and the third error is that it says that it contains three errors while having only two. This third error is the kind of complex self-reference I am trying to understand and I'm not yet sure if everybody admits that there can be an error such as this. One can then make the following, more simple, example:

This sentence contains one error.

How many levels of language are there in this example? You must first think that there is no error at all in this sentence; then you take into account the fact that it says that it has one error and it is mistaken. Can one say that it is true because it makes the error of saying it has errors without actually having errors? I still cannot disentangle this example to this day.

2. I then went on to another case of self-reference that seemed a bit easier than the previous one but had equally interesting properties:

"Noun" is a noun.

I first asked, could grammar describe itself? Of course, "verb" is not a verb, but "is a predicate" is a predicate. Every sentence about the grammar of another word or another sentence follows the same grammatical structure that it describes, so the grammar and the meta-grammar would coincide. Not only that, but the meta-grammatical sentences would still be informative or meaningful even though they have the same structure and express the same things that grammar does: it is not at all trivial to say that "noun" is a noun. But is it trivial to say that the meta-grammatical word "noun" is a noun?

"Noun1" is a noun2. "Noun2" is a noun3. "Noun3" is a noun4.

The definition of "noun" is

any member of a class of words that typically can be combined with determiners to serve as the subject of a verb, can be interpreted as singular or plural, can be replaced with a pronoun, and refer to an entity, quality, state, action, or concept

http://www.merriam-webster.com/dictionary/noun

3. This is actually the definition of all the nouns. You can say that "horse" is a member of a class of words that typically can be combined with determiners, etc. So this is the definition of the words that fall into "noun" 's extension. But what is its intension? Nevertheless, the word "noun" falls into its own extension: "noun" is a noun.

When I say that "horse" is a noun, I mean that "horse" follows certain syntactical and semantical rules and that it refers to horses. I am not mentioning merely its typographical or phonetical properties ("horse" has five letters), but its meaning. When I say that horses run wild in the fields, it seems that I am using its meaning.

4. The word "horse" refers to horses, but this information is not contained within its meaning: it does not refer to the fact that it refers to horses; so it is informative and non-trivial to say that "horse" refers to horses. In this way I arrived at the following principle which I have not completeley justified yet:

The higher level mentions what the lower level uses.

So when you use the word "horse", you use it to refer to horses (typically); when you mention the word, you mention what you use it for. Now, given that the higher level must use some words to mention the lower level, this makes possible a new and informative higher level.

5. In the higher level sentence, "horse" is a noun, the word "horse" is mentioned by using the quotation ' "horse" ' and the predicate "is a noun". What is the difference between that which is used and that which is mentioned? The information provided by the higher level sentence ("horse" is a noun) enables me to use the word in sentences such us: horses run wild in the fields. But the higher level information, i.e. that "horse" refers to horses, is not present as information anymore when using the word; it is outside of the meaning of the sentence. It seems to be a rule of language and it is not expressed within the (object-)language; is it semantical or pragmatical?

6. "Noun" is already part of a meta-language (grammar) and so it is used to mention the uses of words. So then when I say that "noun" is a noun, I mention something about the way in which I mention the of use of words; this sentence is meta-meta-linguistical. The normal use of "noun" is that of describing other words, but since it itself falls into its extension, it also describes itself. So "noun" is a noun means that "noun" refers to the fact that it refers to words. It seems that, depending on the predicate used, I can mention uses and I can mention mentions.

7. The point I am trying to make is that, depending on the predicate used to mention a word or a sentence, one can mention the word as a string of letters, or one can also include the meaning:

a. "Man" has three letters.

b. "Man" is synonymous with "human".

(b) is affirming three different things: "man" refers to homo sapines, "human" refers to homo sapiens and that "man" and "human" refer to the same thing. In each of these I take into account the relation of these words to other objects, their references. Thus it seems that I am mentioning each word's particular relation of reference.