Talk:Grelling–Nelson paradox

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I removed a paragraph stating that, in mathematics, Russell's paradox may be resolved by recourse to an axiom such as Regularity. I removed it because (a) that assertion is more about Russell's paradox than the Grelling-Nelson paradox, and more importantly (b) the assertion is wrong; see the "misconception" part of the Axiom of Regularity article.

Is "mispeled" autological? Since it is easily recognizable as a misspelling of the word misspelled, it could be. However, "mispeled" is certainly not a word in English in its own right -- at least, not in modern written English. Thus, it may be no word at all, and hence neither autological nor heterological.

We could say it's a word in Mispeled Inglish language... :-) --Army1987 19:20, 19 Sep 2004 (UTC)

If I say aloud the sounds "pro-NOON-said wEYEth AH strawn-GEE ack-SOONT," I have indeed said something pronounced with a strange accent ... :) --FOo 01:15, 11 Feb 2004 (UTC)

"But it's not a phrase in English in its own right -- at least, not in modern standard English. Thus, it may be no phrase at all, and hence neither autological nor heterological." Or is it?

Why are confusionful and obfuscating autological? EddEdmondson 16:24, 15 Jul 2004 (UTC)

Because it's confusionful and obfuscating to put them here. ;) --Kenny TM~ 15:07, Aug 22, 2004 (UTC)

Isn't the Grelling-Nelson paradox simply a restatement of Russell's paradox? --Wocky 15:48 26 Dec 2004 (UTC)

My thoughts exactly O_o. Shouldn't this be merged, or at least linked?
Oops. Just noticed it DOES link to Russel's paradox. 'Scuse me.


I really disagree with the notion that ALL words are either autological or heterological. Purple is only heterological because it's written in blue, but if you wrote it in purple, it would be autological. Therefore, this isn't a quality that inheres in the word, but something else entirely.

Say I find a coin on the street and tell myself that the entire universe must be on one side or the other of it. Everything up is on the heads side of the coin, everything down is on the tails side. Then I pick up the coin and press it between my palms. Is this a paradox? How can I be on both sides of the coin at once? Wow, dude, it's cosmic. Whoops, no, it's just nonsense.

You may as well just put the coin in your navel and gaze at it lovingly until the end of time. --The Dogandpony 22:32, 23 Apr 2005 (UTC)

No. The classes autological and heterological refer only to the word itself, not to any particular presentation of it. For most words that are not adjectives, the question of whether they describe themselves is meaningless. Hence we could either class those words as heterological, or limit the classification to adjectives only. Either way, the paradox still works. – Smyth\talk 14:26, 18 Jun 2005 (UTC)
Then is the word printed autological, or heterological? We have to consider the word itself, not a particular printed presentation of it. So is the word printed itself printed? No? Sometimes? Yes, since it has been so many times? . . . Is the word ill-defined autological? How about the word thought, the participle of think? That's a good adjective. Is thought thought when no one's thinking it? As soon as I have the thought that thought is heterological, does my thought make thought autological? I suppose I'm disagreeing that "the word itself" is a well-defined object, with a well-defined set of properties. I'm not really saying it's a problem with the article, which is a good presentation of the paradox, which is a good rendering into terms many non-specialists can understand of an important set theory idea. It's just that language is a virus from outer space, which is no one's fault... – GTBacchus 03:24, 12 July 2005 (UTC)
Is it appropriate to ask whether the word appropriate is autological? GTBacchus 09:02, 12 July 2005 (UTC)
Clearly, they are talking only about adjectives as the article states all adjectives are either autologial or heterological; I'd be inclined to leave it at that instead trying to throw in verbs (transitive or otherwise).
Purple, if written in purple, isn't necessarily autological to begin with. The color it is written in does not define the word, nor is the word meant to define the color used to write it. Cygnis 07:06, 8 June 2007 (UTC)

Contents

[edit] Easily resolvable?

I don't agree that the paradox is "resolvable by simply admitting that the words "autological" and "heterological" do not in fact form two well-defined categories into which all adjectives fall." Since when is this a resolution, and who says it is? Can we please get a source? "Heterological" basically means "not autological." Therefore if a word does not fall into the "autological" category, it by definition falls into the "heterological" category. And besides, even if we admit that there are some adjectives that are "fuzzy" (say "lengthy," which is a really long word in Scrabble but not in other contexts) it doesn't damage the paradox, because it isn't a question of whether we can neatly fit the word "heterological" into a category; the point is that putting it in either one results in a logical contradiction. Argyrios 22:40, 2 March 2006 (UTC)

Ok, we have two issues here: does the stated resolution make sense, and can we get a source for it?
  1. To the first question, I think the resolution in the article is correct, albeit unsourced. If a category is "well-defined", that means that one can determine, for every object, whether or not it belongs to that category. The Grelling-Nelson paradox shows that "heterological" represents a category that is not well-defined for all adjectives. That's basically what the above-referenced sentence says. I'm not sure I understand what it is that you don't understand. Simply by defining "heterological" the way we do, we're making a hidden assumption, namely that our definition can be applied to all adjectives without logical contradiction. This assumption turns out to be false.
  2. Now, as to the sourcing issue.... I contributed that analysis when I had a much looser understanding than I do now of WP:NOR. Doing some searching now, the best explanation I'm finding for how we resolve the GNP is here (careful, it's a pdf). In that source, the author says that the GNP occurs because "in the Grelling-Nelson paradox, the definition of the adjective heterological involves considering the concept adjective under which heterological itself falls" (emphasis added). This, the author goes on to say, can be circumvented with Russell's theory of types or something like it, which places restrictions against defining concepts that are susceptible of such semantic "vicious circles". In other words, "heterological" is considered to be not well-defined. -GTBacchus(talk) 23:18, 2 March 2006 (UTC)
So, I'd be delighted if you want to change the article to better reflect that, or some other source. I'd also be interested whether you still think the explanation I'm suggesting doesn't make sense, whether in my words or in the words of the author of some source. I'll take a look later, and if someone else hasn't made the appropriate edits, I'll give it a shot. -GTBacchus(talk) 23:18, 2 March 2006 (UTC)
Thank you for your reply! The above does indeed help. As I had noted from your edit summary that you were merging content from another article, I did not know that you yourself had written it. Apologies for my umbrage. I am by no means a trained logician or mathematician, but I still have a problem with the resolution. Point 1 above seems circular. Please correct me if this is an unfair characterization, but it seems like you're saying "This paradox can be resolved because the way it is defined results in a paradox, and we can't have paradoxes, so we must have screwed up in our definitions." The entire point is that there is a logical contradiction; that's what it means to have a paradox, so the fact that one exists doesn't say anything to me about the potential for a resolution.
In other words, it may be entirely true that, by virtue of resulting in a paradox, "heterological" is not well-defined by the mathematical definition of that word. But by any normal understanding of language, the word is perfectly well defined; it embodies a perfectly clear concept by which any descriptive word can be considered, with no ambiguity as to its meaning. Does the word describe itself? Then it is autological. All words that do not fall into that category are heterological. This is not an ambiguous definition. To use the theory of types to "fix" the system by which we judge words to be self-descriptive or not, you have to arbitrarily make it illegal to coin a word that means "not self-describing." Well, tough shit Russell, the word exists whether you like it or not! ^_^
When you say that anything that results in a paradox is not well-defined because it resulted in a paradox, you could pretend to resolve any paradox just by saying it's not well-defined. But that's not fair. It's circular reasoning. You can't resolve the GNP by denying our ability to define a word that means "not self-describing" -- we clearly can, and just did. And you can't say that the word is ambiguous in meaning because, after defining it, we encounter a paradox. The fact that we encounter a paradox is entirely the point. What is it about the definition itself that makes it an illegal or fuzzy concept? Nothing.
As I said, I may be entirely misunderstanding you, so please correct my thinking if it is in error. Argyrios 01:17, 3 March 2006 (UTC)
Or perhaps I should argue like this: I don't agree that we make the "hidden assumption" that our definition will not result in a logical contradiction. We are simply defining a perfectly valid and clear concept, not making a prediction about where that definition will lead. Of course if you make the assumption that there will be no paradox, it will cause problems for the paradox. By characterizing this as a "hidden assumption," you're making it a premise that there cannot be a paradox, so it's hardly surprising that you reach the conclusion that there isn't one. That's what I mean by circular reasoning. Argyrios 14:15, 3 March 2006 (UTC)

You ask good questions, Argyrios.

I agree, there's something sketchy about calling it a "resolution", when we actually just declare the word illegal, after the fact. Maybe the paradox isn't resolved, but simply avoided, if we just invalidate any words that are susceptible to this sort of thing. That's how mathematicians deal with the problem; they use something called the axiom of regularity which says that things like "the set of all sets not containing themselves as elements" (which is the set theoretic version of the word heterological) is excluded from discussion. Nevertheless, this is English, and not mathematics, and the word heterological is defined, even if it turns out that its definition leads to a paradox. I guess the semantic paradox in English isn't resolved; it just highlights something about natural language and logic.

I still maintain that when we define heterological to mean "not autological", we're making an assumption. I think we're assuming at that point that every word - including autological and heterological themselves - falls unambiguously in or out of the category "autological". You say that the word heterological "embodies a perfectly clear concept by which any descriptive word can be considered, with no ambiguity as to its meaning." That's not really true for "any descriptive word," because autological and heterological are descriptive words.

Here's a different angle: One could resolve the GNP in a Russellian spirit by making the following distinction. When an adjective describes another adjective, it is actually something different from an adjective - it's a meta-adjective. Some words are like long, which can be either an adjective ("Marco Polo took a long journey") or a meta-adjective ("Sesquipedalian is a long adjective"). The words autological and heterological, on the other hand, can only ever function as meta-adjectives. Now, here's the catch: autological and heterological are applied to adjecives - not to meta-adjectives, so they can't be pointed at themselves, and we're safe.

Maybe that's no more satisfying than what's already in the article. Either way, it's still unsourced, because I'm just saying stuff here. A citation would be a Good Thing; I'll work on that, and whoever else wants to be bold in the meanwhile... -GTBacchus(talk) 03:51, 4 March 2006 (UTC)

I'll also look for stuff. I seem to remember first encountering this paradox in Godel, Escher, Bach. Hofstader's take on the paradox might be worth including if I can get another copy of the book. It does still seem impossible to formulate a general rule that will resolve the paradox; you still have to make it an individual, unique case. Suppose we make the rule that words functioning as meta-adjectives must only be used to describe adjectives, not other meta-adjectives. But that invalidates perfectly legal strings like "Heterological is a long word," long functioning here as a meta-adjective. Yes, you say, but long has the capacity to function both as an adjective and a meta-adjective, so we make the new rule that words capable of functioning only as meta-adjectives cannot be used to describe other meta-adjectives. Very well -- coin a new word "long¹," which only functions in the meta capacity, and the problem remains. So say that we make the rule that words functioning only as meta-adjectives cannot be applied to themselves. Then that invalidates legal strings such as "short¹ is a short¹ word." Anyway, I won't change it yet, but I'll try to take a look at that book and see what insights it might contain. Cheers! Argyrios 15:42, 4 March 2006 (UTC)
I like the superscript notation - maybe heterological1 is a long2 word! And shortn is a shortn+1 word for all n! Anyway, whether we try to patch the language that way, or some other way, there'll still be some word like heterological lurking somewhere in the new system; I think it's inevitable. -GTBacchus(talk) 22:12, 4 March 2006 (UTC)
the problem with this, though, is that while it solves problems for philosophy texts, it is rather meaningless for natural language. having identical words with closely related definitions strikes me as a cheating way around giving the same word more than one definition. i also have doubts about splitting adjectives and meta-adjectives, but this probably isn't the place for that debate. --dan 21:15, 27 July 2006 (UTC)

[edit] autological

i've always thought autological was the more interesting of the two. heterological can't seem to go either way, but autological can go both! if you say it is, then it is, and if you don't, then it isn't. but neither way seems to be wrong or cause any problems. --dan 18:33, 4 July 2006 (UTC)

[edit] ah, well..

this is just the exact same paradox as saying "i allways lie". or answering "yes" when someone asks "are you just saying 'no' to everything i ask?". i like the concept of autological, i didnt know that word, but the paradox is just bloated up as more interesting then it is. id rather jsut see an artical about autological words, like the list of these words, but then as an actual artical, but hey, thats just me probably.--Lygophile 16:11, 9 November 2006 (UTC)

This paradox is structurally most similar to Russell's Paradox. Aside from being a paradox, it has no essential similarity to the Liar's Paradox. Argyrios 17:59, 9 November 2006 (UTC)
russel's is a summary based on self-exclusion. grelling-nelson's is a summary of which in alternate contexts multiple words can have their inclusion based on their exclusion, including one in the context of merely the summary itself which word itself refers to that context. the liars paradox comes in different forms, and the two versions i mentioned above (though i dont know if the second would count as 'liar', also i formulated it reversed:)) are confermations of the exclusion or non-existence of positives (such as said confermations), much like 'nothing is certain' which is also false if it is treu in the exact same construct (xcept that non-certain does not exclude a potentiallity as non-existend does, so it isnt necessarily treu if it is false, so its an impossible statement rather than a treu paradox), and just like 'I am analphabetic' (other versions of the liar paradox would just be 'i didnt write this'). i find this very similar in construct to both russels and grelling nelsons, all actually saying the exact same thing: 'I belong to the non-I group'. but since i have never before given these paradoxes much thought, do these logical paradoxes all just say: 'I am not I', or are there other possibilities? --Lygophile 11:00, 13 November 2006 (UTC)
hmm, i just realised i used a lot of phrases that arent actual full paradoxes but just imply a paradox making it impossible. ok bad examples, still though, in life you come across many such paradoxes (usually of the liar type), whats so special about this one? hmm...maybe just that its a logically undisputable paradox? (if it is that)


[edit] Paradox?

If the answer is 'no', "heterological" is heterological (again leading to a contradiction).

I don't necessarily understand why the sentence above results in a paradox. DRosenbach (Talk | Contribs) 12:39, 28 March 2007 (UTC)

Forget it...I was reading this sentence out of context and thus couldn't properly focus on it. DRosenbach (Talk | Contribs) 12:47, 28 March 2007 (UTC)

[edit] Foo?

To test if the (imaginary) word "'foo" is autological one can ask: Is "foo" a foo word? If the answer is 'yes', "foo" is autological. If the answer is 'no', "foo" is heterological.

I think this should be cut. Let's say the (imaginary) word is an actual word; why does that acceptance suddenly make it an adjective to fall under the "autological" or "heterological" analysis? Cygnis 07:12, 8 June 2007 (UTC)

[edit] Another attempted solution

I've removed the following attempted solution from the main namespace since it appears to violate WP:NOR, but the editors of this article may find this useful. It's from an anonymous IP, though.—Goh wz 14:30, 15 February 2008 (UTC)

[edit] I believe I have solved this "paradox"

If the definition of "heterological" is interpreted as an adjective whose abstraction of which describes the adjective, then we can thus reason that "heterological" is in such conxtext an extraneous word, and has an abstraction of which is null, thus excluding it from comparisons (much like dimensional fallacies such as kilometers/per kilometers).[1]