View source for 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.
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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... :-) --[[User:Army1987|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 ... :)
--[[User:Fubar Obfusco|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?
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Why are ''confusionful'' and ''obfuscating'' autological? [[User:EddEdmondson|EddEdmondson]] 16:24, 15 Jul 2004 (UTC)

:Because it's ''confusionful'' and ''obfuscating'' to put them here. ;) --[[User:Kenny TM~|Kenny TM~]] 15:07, Aug 22, 2004 (UTC)
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Isn't the Grelling-Nelson paradox simply a restatement of [[Russell's paradox]]? --[[User:Wocky|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.


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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.
--[[User:The Dogandpony|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. &ndash; [[User:Smyth|Smyth]]\[[User_talk:Smyth|<sup style="color:gray;">talk</sup>]] 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... &ndash; [[User:GTBacchus|GTBacchus]] 03:24, 12 July 2005 (UTC)

::Is it appropriate to ask whether the word ''appropriate'' is autological? [[User:GTBacchus|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. [[User:Cygnis|Cygnis]] 07:06, 8 June 2007 (UTC)

== 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. [[User:Argyrios Saccopoulos|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?  
:#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.
:#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 [http://publish.uwo.ca/~jbell/chap12.pdf 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. -[[User:GTBacchus|GTBacchus]]<sup>([[User talk:GTBacchus|talk]])</sup> 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. -[[User:GTBacchus|GTBacchus]]<sup>([[User talk:GTBacchus|talk]])</sup> 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. [[User:Argyrios Saccopoulos|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. [[User:Argyrios Saccopoulos|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 [[WP:BOLD|be bold]] in the meanwhile...  -[[User:GTBacchus|GTBacchus]]<sup>([[User talk:GTBacchus|talk]])</sup> 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! [[User:Argyrios Saccopoulos|Argyrios]] 15:42, 4 March 2006 (UTC)

::I like the superscript notation - maybe ''heterological<sup>1</sup>'' is a long<sup>2</sup> word!  And ''short<sup>n</sup>'' is a short<sup>n+1</sup> 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. -[[User:GTBacchus|GTBacchus]]<sup>([[User talk:GTBacchus|talk]])</sup> 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. --[[User:Kingnixon|dan]] 21:15, 27 July 2006 (UTC)

==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. --[[User:Kingnixon|dan]] 18:33, 4 July 2006 (UTC)

== 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.--[[User:Lygophile|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]]. [[User:Argyrios Saccopoulos|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? --[[User:Lygophile|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)


== 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.  '''[[User:DRosenbach|DRosenbach]]''' <sup>([[User_talk:DRosenbach|Talk]] | [[Special:Contributions/DRosenbach|Contribs]])</sup> 12:39, 28 March 2007 (UTC)

:Forget it...I was reading this sentence out of context and thus couldn't properly focus on it.  '''[[User:DRosenbach|DRosenbach]]''' <sup>([[User_talk:DRosenbach|Talk]] | [[Special:Contributions/DRosenbach|Contribs]])</sup> 12:47, 28 March 2007 (UTC)
::Funny...I got stuck on the same point once again, nearly 2.5 years later!  I added a little something in to make it more clear. '''[[User:DRosenbach|<span style="color:#006400">DRosenbach</span>]]''' <sup>([[User_talk:DRosenbach|<span style="color:#006400">Talk</span>]] | [[Special:Contributions/DRosenbach|<span style="color:#006400">Contribs</span>]])</sup> 01:45, 25 August 2009 (UTC)

== 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? [[User:Cygnis|Cygnis]] 07:12, 8 June 2007 (UTC)

: It's not just any word, it's an adjective, you're not reading it right; it is not just saying it's a word, it's saying it is a adjective that can be described with that adjective. --[[User:TiagoTiago|TiagoTiago]] ([[User talk:TiagoTiago|talk]]) 09:20, 1 November 2011 (UTC)

== 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.&mdash;[[User:Goh wz|Goh wz]] 14:30, 15 February 2008 (UTC)

===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).[http://en.wikipedia.org/w/index.php?title=Grelling%E2%80%93Nelson_paradox&oldid=191205607]

== list of autological words ==

i removed this refrence since no page was found and added 'list of paradoxes' in its place. [[User:Constructive editor|Constructive editor]] ([[User talk:Constructive editor|talk]]) 20:00, 12 February 2009 (UTC)

== Paradoxological ==

I propose a neologism to describe a word which in referring to itself simultaneously fails to refer to itself. This is '''paradoxological'''. This word does not further complicate the paradox as it is a heterological word: If paradoxological was paradoxological it would be autological. However in being autological it becomes heterological. In being heterological it remains heterological. [[User:Blacknightshade|Blacknightshade]] ([[User talk:Blacknightshade|talk]]) 22:05, 15 June 2010 (UTC)
:Too bad Wikipedia can't accept original research; yours indeed seems like a great solution. --[[User:TiagoTiago|TiagoTiago]] ([[User talk:TiagoTiago|talk]]) 09:23, 1 November 2011 (UTC)

I would like to suggest that proposed new words be called protologisms instead of neologisms so we can get [[protologism]] off the autologism table.  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/216.110.194.50|216.110.194.50]] ([[User talk:216.110.194.50#top|talk]]) 23:52, 7 August 2017 (UTC)</small> <!--Autosigned by SineBot-->

== Malamanteau? ==

A portmanteau of malapropism and portmanteau. source: xkcd [[User:AdamDavid|AdamDavid]] ([[User talk:AdamDavid|talk]]) 06:17, 12 September 2012 (UTC)

== I don't see a contradiction ==

The representation of a word and the meaning of a word are not the same entity, and do not need to have the same value. The word "heterological" is autological, but the meaning of the word is still heterological. It's like saying that the meaning of the word "purple" changes because it is written in red. The "purple" is red, but it still means "purple."

This seems rather obvious. Surely there's a source that would use this to criticize the paradox. —&nbsp;[[User_talk:trlkly|trlkly]] 06:11, 13 September 2012 (UTC)

I'd expand this criticism to point out that not only does the word "purple" continue to mean "purple" regardless of which ink is used to print it, the question of whether the word "purple" is autological or heterological can be definitively answered with "that question doesn't make sense".

The error is in assuming that just because "autological" and "heterological" express opposite meanings about what a word refers to, all words that refer must be one or the other. In fact, the two terms only make sense in a restricted set of words - though it is obviously entertaining to misapply them and see what happens.  [[User:TooManyFingers|TooManyFingers]] ([[User talk:TooManyFingers|talk]]) 18:54, 20 March 2021 (UTC)

In other words, I'd say that calling a word autological can sometimes be neither true nor false, but a category error.  [[User:TooManyFingers|TooManyFingers]] ([[User talk:TooManyFingers|talk]]) 23:06, 20 March 2021 (UTC)
*Russell, father of type theory, would be proud of this resolution. :-) I think there is also an argument to be made that the problem only exists in classical logic, where the [[principle of excluded middle]] is an axiom. In other (minimal or constructive) logics, where it is left out, we can define "homological" and "heterological" so they have subtly different meanings from "not heterological" and "not homological", respectively. [[User:Hairy Dude|Hairy Dude]] ([[User talk:Hairy Dude|talk]]) 12:16, 21 November 2021 (UTC)

== The Word 'Ineffable'? ==

The definition I'm seeing for 'ineffable' is "too great or extreme to be expressed or described in words." But if we have a word which means "too great…to be expressed in words," wouldn't the word contradict its own existence?  <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Jrayk|Jrayk]] ([[User talk:Jrayk|talk]] • [[Special:Contributions/Jrayk|contribs]]) 02:58, 15 April 2013 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

== It's Russell's Paradox--only the names have been changed! ==

For better or worse, natural language allows us to define things that cannot '''logically''' exist. Logically, just like there can exist no '''set''' of all those sets that are not '''elements of''' themselves, there can exist no '''word''' that '''describes''' all those words that do not describe themselves. 

'''Russell's Paradox:'''  ~EXIST(r):[Set(r) & ALL(a):[Set(a) => [ElementOf(a,r) <=> ~ElementOf(a,a)]]]

We can substitute:

* Set → Word

* r → heterological

* ElementOf → Describes

'''Grelling's Paradox:'''  ~EXIST(heterological):[Word(heterological) & ALL(a):[Word(a) => [Describes(heterological,a) <=> ~Describes(a,a)]]]

--[[User:Danchristensen|Danchristensen]] ([[User talk:Danchristensen|talk]]) 04:31, 11 February 2021 (UTC)

== Another pair of words? ==

I have a feeling that this subject, along with the subject of Russell's paradox, might paradoxically become simpler by postulating a pair of adjectives '''wnkrz''' and '''zrknw''' in English such that one of them means "May appear to be pointless mental wankery but is actually a significant philosophical endeavour" and the other one means "May appear to be a significant philosophical endeavour but is actually pointless mental wankery", using them to attempt a solution of the Newly Generalized Grelling–Nelson Paradox, and then during the attempt, attempting to solve each successive metaparadox that continues to arise from what is now clearly an inherently self-referential form of self-reference? Underneath our feet, it may be turtles all the way down; above our intellects though, it's metaparadoxes all the way up! :) [[User:TooManyFingers|TooManyFingers]] ([[User talk:TooManyFingers|talk]]) 13:44, 15 July 2023 (UTC)

== "example"/"counterexample" ==

Consider [[Autological word|autological words]] (which are words that describe themself like "word, English, written, etc." Also consider that its opposite is a heterological word.

Now, imagine I am writing a list of autological words. Would it be wrong to place the word "example" on there? After all, if it is on a list of autological words, the word "example" must be an example, right? Now, keep your interpretation in mind. Based on ''this'' anwser, what about the word "counterexample"? If you had it on a list of examples, for "counterexample" to be true, you're contradicting the terms of the list. However, if you say it's not true, then isn't it a counterexample and therefore self-describing?

My understanding is that this is an example of the Grelling–Nelson paradox, but I'm not a linguistics expert. I'm not even sure if this subreddit is the best place, but I don't know where else to ask. [[User:Electricmaster|Electricmaster]] ([[User talk:Electricmaster|talk]]) 18:25, 11 October 2023 (UTC)