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.