IV. Examples of Higher-order

1. I find that the things I say to the people I talk to normally follow some rules; for example the words I am writing have grammaticality, semantics, and maybe other properties given by respecting sets of rules. But I am not fully conscious of the order or the rules by which I speak about different topics. In other words, I am now talking about, let's say, self-reference, and I do it in a (I hope) meaningful and grammatical manner. I then change the subject and start to talk about what I have done yesterday in a meaningful and grammatical manner. But between the things I talk about there seems to be no set of rules to which they all should subscribe. This set of rules would be of a higher order. An act of speaking may have internal coherence, but all the acts taken as a set have no evident order; an act of speaking may have coherence internally, but no external coherence. I am not sure if it is possible for an act of speaking to have no internal coherence but have external coherence. Also, the rules governing the external coherence of an act of speaking cannot possibly be grammatical; but I am not sure whether they may or may not be semantical.

2. I've said in my first post that (almost all) concepts do not contain their own description. Language is used to describe the world and only exceptionally itself. This is what interests me the most, because I tend to take a description of something, or the sign for something else to be the thing it describes or signifies. There are some things which I am not sure whether I like or I like to like them.

3. Tim Maudlin, Truth and Paradox:

"[...] suppose that human reasoning capacity can be reduced to some sort of algorithmic procedure, such that sentences about, say, arithmetic that one takes to be provably true with certainty can be characterized as a recursively enumerable set of sentences. Godel's procedure then shows how to identify a sentence which, if the system is consistent, is certain to be both true and not identified by the system as a truth. The idea is now this: We can recognize the Godel sentence of the system as true even though the system itself cannot. Therefore our insight into what is certainly true outruns that of the system. But the system has not been characterized in any way except to say that it is consistent and, in some sense, algorithmic. Therefore our insight is, in principle, more extensive than that of any consistent algorithmic system. Therefore, our insight cannot be the result of any consistent algorithmic system. Therefore, the power of our minds cannot be captured by any algorithm. Therefore we are not like computers. Yet further, according to Penrose, it follows that the physics that governs our brains cannot even be computable, otherwise our insights would, in the relevant sense, be the output of an algorithm [...]."

4. M.C. Escher, Waterfall, 1961

This image plays with the dimensions. From a 3D perspective, the shape is impossible, also tridimensionality is only an ilusion here; but from the two-dimensional perspective you are lead to believe that water falls down from a higher spot. So the depiction of the shapes is based on the fact that it is a 2D depiction imitating a 3D perspective.

III. Tarski's T-Schema and the Liar antinomy

A. The T-Schema
1. This is how Tarski describes the T-Schema in The concept of truth in formalized languages, in the volume Logic, semantics, metamathematics, Oxford at the Clarendon Press, 1956:

1) a true sentence is one which says that the state of affairs is so and so, and the state of affairs is indeed so and so.

(2) x is a true sentence if and only if p.

In order to obtain concrete definitions we substitute in the place of the symbol ”p” in this scheme any sentence, and in the place of ”x” any individual name of this sentence.

The most important common names for which the above condition is satisfied are the so-called quotation-mark names. We denote by this term every name if a sentence (or of any other, even meaningless, expression) which consists of quotation mars, left- and right-hand, and the expression which lies between them, and which (expression) is the object denoted by the name in question.

(3) ”snow is white” is a true sentence if and only if snow is white.
In defining the correspondence between a sentence and a state of affairs, one must specify what particular state of affairs the sentence is corresponding to. In order to do this, we have to use the same sentence whose correspondence relation we are defining to name the state of affairs it corresponds to. Thus, we already presuppose the correspondence relation we are trying to define. 
In saying that "snow is white" is a true sentence iff snow is white we implicitly presuppose that we understand the second occurrence of "snow is white" in the right side of the biconditional and that it can be true or false. 

This way, the T-schema is either a vicious circle or it just says that "snow is white" is true has no different meaning than "snow is white" simpliciter (i.e. it is a deflationary theory of truth).

2. My supervisor said that the T-Schema is only apparently circular because the object-language sentence and the meta-language sentence coincide; but the meta-language may differ, for example:
"snow is white" is a true sentence iff snow is green.
But then one would have to state the truth condition for the meta-language sentence "snow is green" either in a meta-meta-language the coincides with the object-meta-language (a), or not (b):

(a) "snow is green" is a true sentence iff snow is green.
(b) "snow is green" is a true sentence iff snow is black.
And for each expression of meta-language, its truth conditions will have to be defined in a higher-order language ad infinitum, without ever obtaining a final truth condition. One may say that, even though the truth of an expression cannot be completely defined in this way, the T-Schema is still somewhat functional because of this recursion. But in the case that I do not already know the truth condition of any of the expressions used as truth conditions for the ones in the lower-order language, this recursion tells me nothing. 

3. I take the T-schema to intend to say that a sentence like "snow is white" is true if and only if a fact like snow being white is the case. It tries to bypass the fact that "snow being white is the case" is still alinguistic expression, requiring its own definition of meaning or truth.
4. The truth predicate describes the relation of language and the world, but it itself is a part of language, and thus it can refer to itself; it is by nature a part of meta-language. 
To make it even more obvious, let's replace Tarski's quotation-mark names with a different way of referring to a sentence:


The last three words in this phrase make up a true sentence if and only if snow is white.


Just as with the T-Schema, the same sentence is used here as (part of) its own truth condition.
5. In other words, to speak about the truth of a sentence is to mention what it is already being used (the correspondence relation). Mentioning this relation in regards to a certain sentence is trivial, because one always already understands it once it understands the sentence; and this knowledge cannot have a meaningful verbal expression.
6. So the question is, can the correspondence of language to the world be described within language?
7. The same as with Grelling's paradox, to define the truth of a sentence is to say something about a relation between the sentence and something exterior to it, while the sentence already relates to something exterior to itself. Also in common with Grelling's paradox, defining truth seems to lead to recursion.
B. Tarski's treatment of the Liar paradox
6. This is how Tarski analyzes the Liar paradox in The semantic conception of truth and the foundations of semantics, pp. 65-66, in the volume The Philosophy of Language, third edition, edited by A. P. Martinich, Oxford University Press 1996:

To obtain this antinomy in a perspicuous form, consider the following sentence:

The sentence written in this post on line 71 from the top is not true.

For brevity we shall replace the sentence just stated by the letter ‘s.’

According to our convention concerning the adequate usage of the term ‘true’. We assert the following equivalence of the form (T):

(1) ‘s’ is true if, and only if, the sentence written in this post on line 71 from the top is not true.

On the other hand, keeping in mind the meaning of the symbol ‘s,’ we establish empirically the following fact:

(2) 's’ is identical with the sentence written in this post on line 71 from the top.

Now, by a familiar law from the theory of identity (Leibniz’s law), it follows from (2) that we may replace in (1) the expression “the sentence printed in this post on line 71 from the top” by the symbol “ ‘s.’ “ We thus obtain what follows:

(3) ‘s’ is true if, and only if, ‘s’ is not true.

In this way we have arrived at an obvious contradiction.

(I have changed the naming of the paradoxical sentence to something relevant in this context; I counted the lines of text from the top and I don't know if they coincide for every screen, but the point should be obvious.)
8. It seems that initially 's' replaces the sentence written in this post on line 71 from the top, which is: "the sentence written in this post on line 71 from the top is not true".
(2) says that 's' is identical with the sentence written in this post on line 71 from the top -- so 's' is identical with "the sentence written in this post on line 71 from the top is not true". It is not identical with the expression "the sentence written in this post on line 71 from the top". Thus in (3) 's' seems to stand for two different things: the first occurrence in the left side of the biconditional stands for "the sentence written in this post on line 71 from the top is not true", while the second occurrence of 's', in the right side of the biconditional stands for "the sentence written in this post on line 71 from the top" to which is added the predicate "is not true".
Now (3) doesn't even follow the T-Schema, because it adds a truth predicate to the second occurrence of the sentence in the right side of the biconditional; while (1) does -- the sentence "s" is true if and only if s.
9. Nevertheless, the expression "the sentence written in this post on line 71 from the top" is a name for "the sentence written in this post on line 71 from the top is not true", and the name itself is contained in the sentence. So the subject of the paradoxical sentence is actually a name for the whole sentence and replacing it with that which it names yields: 

"the sentence written in this post on line 71 from the top is not true" is not true.
And, again, the replacement can be repeated infinitely.

II. Grelling's Paradox

1. That "noun" refers to the fact that it refers to words seems to me to be a good expression of the meaning of the sentence "noun" is a noun. So I then moved on to Grelling's paradox, wanting to express the paradoxical sentence in a similar way. What does "heterological" is heterological actually say about the word "heterological"?

Just as "noun" doesn't have any intension, so is "heterological" defined by Grelling:

Let x be a word-property and n the name of x in the language L. Then we define heterological in L as follows:

Def.: n is heterological in L when and only when n has not the property x in the language L

This definition does not say what "heterological" means, it just states the conditions when a word is heterological. In other words, we call a word "heterological" when and only when it does not have the property that it names. So to be able to say that "heterological" is heterological, one must first know the property it expresses. As another thing in common with "noun", the property expressed by "heterological" is a word-property; so "heterological" is part of a meta-language used to describe words.

2. I tried to replace the two occurrences of the word in the paradoxical sentence by expressions such as "the word that denotes the property x..." and by the property it denotes, and I came up with this:

The word that denotes the property of a word not having the property it denotes does not have the property it denotes.

Written in orange is the expression replacing the word "heterological" (I still haven't been able to find out what is the name of such an expression), in green is the semantic content of the predicate "is heterological". The next step would be to replace the second occurrence of "the property it denotes", in green, with the property it actually denotes, which yields:

The word that denotes the property of a word not having the property it denotes does not have the property of not having the property it denotes.

The expression "the property it denotes" can then be replaced again with the property it denotes and so on ad infinitum.

3. I then tried to formalize this, hoping that it would make things clearer. The problem with these reformulations is that I cannot see their paradoxicality. I cannot see how this implies that "heterological" is autological and how this in turn implies that it is heterological. So I tried expressing in the following way, using my own notations and (pseudo)syntax:

a. Hx <=> xDX & ~Xx

i.e. x is heterological if and only if x denotes property X and x has not property X.

b. Hh <=> hDH & ~Hh

i.e. "heterological" is heterological iff "heterological" denotes heterologicality and "heterological" has not the property of heterologicality.

This is the easy way of formalizing it and it is missing the question - what does it mean to say that "heterological" is heterological? The hard way would be to formalize my alternate way of expressing the paradoxical sentence. I can try to replace "H" (heterologicality) with the expression xDX & ~Xx, which yields:

c. Hh <=> hD(xDX & ~Xx) & ~(xDX & ~Xx)h

i.e. "heterological" is heterological iff "heterological" denotes a word that does not have the property it denotes and "heterological" is not a word that does not have the property it denotes. Now if "het" is not a word that does not have the property it denotes, then "het" is a word that has the property it denotes (implying that "het" is autological).

4. This is where I stop trying to formalize since my logic skills end here; but if you go on, you would keep inserting the formula for H into itself forever. I suspected that if you keep doing this, you would see "autological" being used to mention "heterological" and vice-versa, each existing alternately explicit (used) and implicit (mentioned) in the meaning of this paradoxical sentence just as the property of being a noun is implicit in "horses run wild in the fields" and explicit in "horse is a noun". Moreover, the structure of the meaning of Grelling's paradoxical sentence resembles a fractal, just like the Liar sentence.

5. Besides all these, the following fact seems strangely connected with the previous: If I say that "pentasyllabic" is autological, I mean that it has the property it expresses, i.e., "pentasyllabic" is pentasyllabic; and if I say that "autological" is autological I mean that it has the property it expresses, i.e., "autological" is autological, which is the same meaning.

6. Searching for the paradox on the web, I found a similar, but more conclusive, treatment of the problem by Alan Rhoda:

http://www.alanrhoda.net/blog/2006/01/neologism-paleologisms-and-grellings.html

He says that heterologicality and autologicality are meta-properties of exemplifying certain first-order properties. One or the other of these two meta-properties supervene depending on whether a certain first-order property is exemplified or not. Thus, "short" is autological or heterological depending on what "short" expresses. Using his example, if "short" means "less than an inch", then "short" is short and it is autological; if "short" means "less then a millimeter", then "short" is not short and it is heterological.

"Supervene" is the keyword for me here. "Pentasyllabic" is both pentasyllabic and autological. These are two different propositions: there is a difference between saying that "pentasyllabic" is autological and saying that "pentasyllabic" is pentasyllabic. But if the word we predicate autologicality to is "autologicality", we do not get two different propositions. Autologicality is a property of the word "pentasyllabic" only insofar as it expresses pentasyllabicity, and pentasyllabicity cannot be autological, just as shortness cannot be autological. This suggests that heterologicality and autologicality are not properties of words in themselves and that they are not properties of the properties expressed by these words. Rather, they are properties of certain relations between words and the things expressed by them. Nevertheless, it seems difficult to accept this given we predicate autologicality or heterologicality of words and not of the relations between words and a properties.

Alan Rhoda then goes on to say that:

As meta-properties, heterologicality and autologicality are properties of exemplifying a certain property, namely, the property that 'heterological' and 'autological', respectively, express.

which I do not understand. As meta-properties of what word or thing are heterologicality and autologicality properties of exemplifying a certain property, namely, the property that "heterological" and "autological", respectively, express?

Simplifying, he seems to say that -- as metaproperties, heterologicality and autologicality are properties of exemplifying the property that "heterological" and "autological", respectively, express. But "heterological" and "autological" express heterologicality and autologicality, respectively. So what is he actually saying?

I assume that he is referring to our two words, and that he's saying that, for each of the two words:

As metaproperties of "het", heterologicality and autologicality are properties of "het" exemplifying autologicality and heterologicality, respectively.

As meta-properties of "aut", heterologicality and autologicality are properties of "aut" exemplifying heterologicality and autologicality, respectively.

He then arrives at a similar point as the one I made above, but without using any alternate formulation of the paradoxical sentence:

For just as 'short' is neither determinately short or not short until we fix the meaning of the term, so neither are 'heterological' and 'autological' determinately heterological or autological until we fix the meanings of those terms. But when we try to do so, we find that the meaning is continually deferred. Thus, heterologicality is the property of exemplifying heterologicality, which is the property of exemplifying the property of exemplifying heterologicality, which is the property of exemplifying the property of exemplifying the property of exemplifying heterologicality, and so on.

I'm still not sure what is the thing that is said to have heterologicality in his exposition; but I assume that it is the word "het".

7. The "heterological" paradox is not at all the only way of arriving at the same contradictions. I've found a much more transparent way of recreating the paradox in a paper by Marc Cohen: http://faculty.washington.edu/smcohen/120/SecondOrder.pdf

a. We start with some undetermined thing that has the property of being a cube:

Cube(b)

b. We say that the property of being a cube has the property of being a shape.

Shape(Cube)

c. We add the property of commonness which is had by a property if it is exemplified by at least different two things:

Common(P) ↔ ∃x ∃y (x ≠ y & P(x) & P(y))

So, for example, the property of being a shape, if it is exemplified by the property of being a cube and by the property of being a sphere, is common. Of course, the property of being a shape isn't the only one common; the property of being human is also exemplified by at least two things. So if a property exemplified by at least two different things is common and if the property of being common is exemplified by at least two different properties, then "common" is common.

Now, besides just being common, the property of being common has this special higher-order property -- it exemplifies itself. We will call this property "extraordinariness":

Extraordinary(Common) ↔ Common(Common)

And all the properties that do not exemplify themselves are called "ordinary":

∀P (Ordinary(P) ↔ ¬P(P))

The paradox arises from the question: is the property of being ordinary ordinary? If we say that "ordinary" is ordinary we mean that it does not exemplify itself; but it just did. If we say that "ordinary" is extraordinary, we mean that it does exemplify itself, and therefore "ordinary" is ordinary.

Ordinary(Ordinary) ↔ ¬Ordinary(Ordinary)

8. We can discard the previous steps up to the property of commonness. The fact that there are some properties which do not exemplify themselves makes possible that there are some properties that do exemplify themselves. It is interesting that the paradox always arises from the property of not exemplifying itself, be it heterologicality, ordinariness, or "not including itself". They all are properties that, when exemplified by some object, say that the object does not exemplify itself. But when we ask is "ordinary" ordinary? we mean to ask whether the property of not exemplifying itself does not exemplify itself. The problem seems to be with the indexical word "itself". The second occurrence of "itself" refers to "the property of not exemplifying itself"; but what does the first occurrence of "itself" refer to? It seems that these properties are so defined as to always be predicates and never subjects. We say that "shape" is ordinary, i.e. that it does not exemplify itself, i.e. that "shape" is not a shape. But if we take the property of not exemplifying itself out of context, how do we define it so that it can become a logical subject?

The same problem appeared in Grelling's paradox: a word is said to be heterological if it does not have the property it refers to. I transformed "heterological" from a predicate, which is its normal use, into a subject by defining it as the word that denotes a word not having the property it denotes.

Now, if "the property of not exemplifying itself" has no meaning out of a well defined context, how come we are able to say that if "ordinary" is ordinary, then it is extraordinary; and that if "heterological" is heterological, then it is autological? Just as Alan Rhoda explained above, if heterologicality and ordinariness always defer their meanings, how are we able to make hypotheses about them?

9. Jay Newhard explains this in a paper ("Grelling's Paradox", you can find it on Jstor) and uses the indexical behavior of "heterological" as a solution to the paradox; according to him, "het" is an indexical predicate:

Since x is the word-property designated by the word of which heterologicality is predicated, [an intensional definition of heterologicality] defines "heterological" as a predicate which takes its semantic content in part from the property designated by the word-property designator of which it is predicated [...] Put another way, the semantic content of "heterological" is sensitive to the semantic context.

He says that affirming both that "heterological" is heterological and that "heterological" is autological leads to a semantic failure -- i.e. the paradoxical sentences do not actually express a proposition, even though they have semantic content. He formalizes the paradox as follows:

The intensional definition of heterologicality: n is heterological = n has not the property D(n)

The semantic content of "n is heterological" is: n has not the property D(n)

The semantic content of occurrences of "heterologicality": not the property D(n)

n is a word type and D(n) is a word-property denoted by it.

So, if I understood correctly, this would yield:

"heterologicality" is heterological = "heterologicality" has not the property D("heterologicality")

which in turn, yields:

"heterologicality" has not the property D(not the property D(n))

Leaving n uninstantiated.

Newhard holds that the semantic failure occurs only when applying this predicate to predicates that are context sensitive, but not for the other predicates.

10. These two forms of the paradox together with the Liar have the following thing in common: they refer to a relation which ought to be between them and something exterior to them; in other words, they refer to themselves-referring-to-something-else or they refer to their use.

11. As Jay Newhard says, "heterological" takes its semantic content in part from the word-property denoted by the word of which it is predicated. So the predicate "heterological" takes its semantic content in part from the property "heterologicality"; also it is both used and mentioned:

"heterologicality" has not the property heterologicality.

"has not the property D(n)" is the semantic content of the predicate "heterological".

So the complex predicate "has not the property heterologicality" is the semantic content of the predicate "is heterological" when predicated to the word "heterological". In other words, the predicate "heterological" is used here. It is also mentioned in the subject, " "heterologicality" "; and, since "heterologicality" is both subject and predicate in this sentence, the predicate "is heterological" ascribes the property "heterologicality" to the word "heterological". So what about the word designating the property ascribed by the predicate "is heterological" to the subject "heterological"? Is it used or mentioned? I incline to believe that, in the expression "the property heterologicality", the word "heterologicality" is mentioned, and the expression should actually be the property "heterologicality".

12. I incline to believe that whenever a word designating a property is the subject of a predicate, the word is only mentioned. Instead of saying "the property yellow", one should say the property "yellow". But this conflicts with the standard view on quotation, which says that whenever we quote a word we are either looking at its surface aspects (typographical, phonetical) or its meaning. So when we mention a word like "yellow" we are allowed to say that, for example, "yellow" has six letters, or that "yellow" denotes a color. This shows that simultaneously quotations mention words and that they are used; they are used to mention words, and we can use them in different ways to have different mentions.

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.