Logic at its Limit: The Grelling-Nelson Paradox

[@BirdBrain12]

This may be a separate paradox, but this sounds exactly like the linguistic equivalent to the yes or no question "Is the answer to this question no?"

[@akshay4107]

Who cares

[@leonardoantonio216]

@@akshay4107 Trevor Cronath cares, hence why he pointed it out

[@pierrotA]

@@akshay4107 11:55 We care 😁

[@peaku8129]

@@akshay4107 I care

[@l.w.paradis2108]

Similar in the self-referential aspect, but this is deeper. You can always say that in the quoted passage, "this question" has no clear referent at all. By comparison, 'heterological' is an adjective, and by hypothesis (a hypothesis which accords with intuition), every adjective can be uniquely classified as either heterological or autological, but not both.

[@justin.t.mcclung]

How about a follow up video (or videos) which explain Russell’s Paradox, The Incompleteness Theorem and The Halting Problem, and then show the equivalence of each to the others. This would increase the value of this excellent video exponentially by making it just the first step in a much deeper journey

[@dialectphilosophy]

That's exactly our intention! This is essentially the "introductory" video in a series to come. (Will be some time though, we've got a relativity backlog)

[@snowmanofpoopp]

@@dialectphilosophy also you can do "What the Tortoise Said to Achilles"

[@philipm3173]

@@dialectphilosophy I hope you bring up the catuskoti or tetralemma.

[@ThePixelkd]

Sure. If you put autological in the autological box, it checks out that autological is self-descriptive. The word is 'autological' and it's in the autological box. Check. On the other hand, if you place it in the heterological box, the same thing happens. Seems a bit arbitrary, I admit, but if we put autological in the heterological box, then the statement also appears to be true there - if autological is a heterological word then it doesn't describe itself. My intuition wants to tell me that autological is of course an autological word, but for the life of me, I can't figure out how to prove it in a way that doesn't also work for placing it in the other box. I would really like to see a video analyzing that. I still feel like I am missing something, or that it might be altogether wrong somehow...

[@stevenfallinge7149]

Russel's paradox, the incompleteness theorem, and the halting problem are *not* equivalent to each other. They use analogous methods, but you do not use one get to the other, which is what is usually meant by equivalency. Russel's paradox is simple and can be explained to a 10 year old with little math background. The incompleteness theorem and the halting problem, on the other hand, need a little more background to explain fully, explaining such things as "formula" "proof" "theorem" "tautology" and so on for the completeness theorem, and "Turing machine" for the halting problem, so they require some more work.

[@realbrickbread]

I solved the paradox! Instead of putting heterological and autological *in* the box, you stick them to the side as labels. No strange sorting needed.

[@WWLinkMasterX]

This is sort of how modern mathematics solves Russel's Paradox. ZF set theory has additional rules that say sets can never be members of themselves.

[@NoConsequenc3]

unironically true. The boxes are, for all intents and purposes, arbitrary. Of course resorting those specific box terms yields a silly result - the initial conditions are just as effectively silly.

[@NaN_000]

I can smell contradiction

[@seeker296]

agreed. this paradox is constructed; not natural

[@dbojangles1597]

Well that's one way to understand it. Personally I know a far far deeper solution to this particular problem. And no I'm not speaking on it.

[@Xbob42]

Seems kinda like the issue is just that the sorting issue becomes incoherent when trying to sort the sorters. To me it comes across less a problem of logic, strictly speaking, and more an issue of constructing the thought experiment.

[@gridgaming_]

this channel is amazing, i love the style of narration you use. too unnatural to be described as fully human, but too unique to be generated. really makes the video interesting to listen to. I also love problems like these, it was nice to see this covered as clearly as you did.

[@horrorspirit]

> too unnatural to be described as fully human, but too unique to be generated this is the perfect description for bill wurtz

[@TactfulWaggle]

You have just described the uncanny valley

[@medexamtoolscom]

He just says everything slowly and clearly, that's hardly enough to call inhuman.

[@Virtualmassslave]

dear grido. what has been declared here? becose there are paradoxes, there is no logic. not here no;)

[@MyBeautifulDarkTwistedFantasy6]

LMFAO isn’t that a paradox?

[@nedearbwormback5758]

So this paradox is essentially having two bins, a trash and recycling bin, and all the stuff is sorted into one of the two bins, but then you’re handed the recycling bin itself and asked to throw it away and it's like, "well, you can't throw a bin away into itself, yeah?" Seems like a problem of trying to throw your bin away when you don't got no bin for your bins, I'll tell you hwat.

[@daneo1952]

Yeah, my interpretation of these paradoxes is that we just showed that statements which refer to themselves are not proper statements; not that there is a flaw in logic. "This statement is false" is just not a proper statement, it cannot be true or false, that's all there is to it for me. I feel like your picture describes this quite neatly.

[@maxharasen6548]

you did it, you solved it

[@quorryraphael9980]

but what if you make a bin to put bins in 🤔

[@NXTangl]

@@daneo1952 this is basically the solution that ZFC proposes by forcing sets to be unable to refer to themselves.

[@user-pr6ed3ri2k]

@@quorryraphael9980 what if you want to throw that bin away

[@MinerUser147]

I feel like you can easily sort autological into the autological category. There is only a problem when you try heterological.

[@l554446l]

Yeah, I got confused by that. I was waiting for the explanation why autological can't be categorized as autological.

[@vanlepthien6768]

The "problem" is that sorting "autological" into "heterological" works, too. That breaks the assumption that every word is exclusively one or another, which doesn't make it a paradox, it means that the assumption was bad.

[@MinerUser147]

@@vanlepthien6768 Thanks, I didn't think about that.

[@trinitrojack]

@@vanlepthien6768 How do you sort autological into a heterological category? The word autological is NOT [ a word that describes itself ] ❌ The word autological is NOT [ autological ] ❌

[@l554446l]

​@@vanlepthien6768 Sorry, I still do not understand. Can you help me understand two things? 1. Can you explain how autological fits into heterological? 2. Regardless of #1, the statement in 1:40 is "all words are either autological or not-autological (heterological)." With the operator word "or" that was used, the word does not have to be exclusive to one category but the whole statement would be true as long as at least one of those is satisfied. That is, the statement did NOT say "all words are either... or... but not both." Thanks.

[@fireballferret8146]

Tenletters is my favourite autological "word" ...it's been living rent free in my brain over a decade and I finally have a word for it now, thanks!

[@SoulDelSol]

That's not a word.

[@j3ffn4v4rr0]

@@SoulDelSol Word.

[@JustAnotherCommenter]

A word is a word but a sentence is not a sentence

[@gavinschutte]

Thank you for explaining the Russell's Paradox using language as a substitute. I've always struggled with maths, and when we did it in Philosophy, I had no idea what was going on XD

[@sumdumbmick]

you mean pseudomenos logoi, right? Russell is credited for it because Russell was an aristocrat, not because he actually did anything worth a damn.

[@unclejuju12]

I love how you brought together all 3 of these paradoxes. They are like the NP complete set in that if we can solve one of the logical paradoxes we unlock all of them lol. Amazing video!

[@HackersRUs]

Rather than like the NP complete set, they are exactly the undecidable set of problems.

[@dimiturtabakov1108]

I'm glad you mentioned Russel's paradox (more easily digestible in the form of "the barber paradox") and Godel's incompleteness, cause they popped into my head and it occurred to me that a lot of paradoxes are a mere product of our ability to say "A equals not A."

[@dimiturtabakov1108]

@@shadowfax333 So, we hear (or read) a combination of words, which we habitually associate with some assigned meanings and that usually works for us - it helps us navigate reality; but, on occasion, words can be arranged in such a way as to suggest reality is wrong. And since, by definition, reality can't be wrong, it must be our perception or description of it that is faulty. I.e. paradoxes are like optical illusions for the mind. Neat :)

[@NoConsequenc3]

@@dimiturtabakov1108 yup! Paradoxes are the result of a irrational mind attempting to rationalize the world. To be fair it's really, really good at it - but sadly that's never going to be enough lol

[@visancosmin8991]

You ignore consciousness, that's why you get paradoxes.

[@glenliesegang233]

"This is not a pipe" Rene Magritte's painting, hints at how conceptual pictures come with exceptions which only seem paradoxical based on rigid frames.

[@dimiturtabakov1108]

@@visancosmin8991 I don't know, that one sounds a touch too Chopra-esque for my liking. My thinking was that paradoxes are akin to optical illusions and if a creature with eyes (e.g. a fly) can experience an optical illusion, a creature with the capacity to understand language can experience a paradox. Consciousness is a bit too Ill-defined and thus have small to negligible explanatory powers.

[@rjstegbauer]

I loved how you pulled together Russell's, Godel's and Turning's paradoxes!

[@bootstrapperwilson7687]

Sod Turning, what about Turing?

[@silkwesir1444]

@@SigFigNewton even more than that. they are essentially the same paradox taking on different "disguises".

[@Megaritz]

@@SigFigNewton There is a video using category theory to show how these are all related to each other: "What A General Diagonal Argument Looks Like (Category Theory)" by Thricery.

[@Alorand]

There are three categories: 1) Autological 2) Heterological 3) One of those self-referential paradoxes Category 3 can not be grouped with other categories. If you want to sort between 1 and 2 you don't start from all words, but from all words in the combined bin 1+2 which is first separated from box 3.

[@mikicerise6250]

There are only two kinds of words, paradoxoids, which create a self-referential paradox when describing themselves, and all other words.

[@faran_iqbal]

@@mikicerise6250 Then how about a word like "non-paradoxoid" ?

[@JustAnotherCommenter]

@@faran_iqbal That would be heterological.

[@JustAnotherCommenter]

Alorand, category 3 is just a restatement of category 1. In fact, it is the definition of category 1. So this kind of method of classification does not work.

[@QuicksilverSG]

@11:05 "There's no way to escape bivalent classification, which leaves us reckoning with a terrible truth: logic is broken." Not quite. As you demonstrate, bivalent logic leads to logical paradoxes, and is indeed broken. However, we can resort to a different system of logic which does not lead to such paradoxes: Intuitionistic Logic. In this system of logic, your assumption that all propositions can be sorted into two mutually exclusive categories (AKA bivalent logic) is not taken for granted. Instead, one must first demonstrate that such a sorting process can be completed for all propositions under consideration. As you demonstrated with the words "autological" and "heterological", all attempts to sort these words into those two categories fail to produce consistent results. Hence, such categorization fails to meet the requirements of Intuitionistic Logic, and the conclusion is that those words cannot be categorized in that manner. In formal logic terms, Intuitionistic Logic does not accept the Law of Excluded Middle, which is a hidden assumption of bivalent logic. This is the principle on which many Reductio Ad Absurdum logical arguments are made, i.e. that by showing the opposite of an assertion to be contradictory, you have proven the assertion to be true. Such attempts to apply bivalent logic to real-world situations often fail to produce satisfactory conclusions, due to the intricate ambiguity of human motivations. Intuitionistic Logic provides an alternative to bivalent logic that often corresponds more closely to human intuition. One way to visualize the "excluded middle" of bivalent logic is to consider the dividing line between mutually exclusive categories @9:57 to be an irreducible thing in itself. In bivalent logic, the dividing line itself does not exist, since there can be nothing other than its two mutually exclusive categories. In real life, however, the existence of such a dividing line can often be more significant than the two sides it separates.

[@reingp]

Why isn't autological cathegorized as autological? testing both hypothesys: 1. The word [autological] is a word that [describes itself]. > True 2. The word [autological] is not a word that [describes itself]. > False What am I missing?

[@GuyAtTheSix]

Keep up the good work. Your videos are truly amazing and inspirational.

[@stevemenegaz9824]

Logic is in the eye of the beholder. You choose your axioms carefully, minimizing the number to the minimum necessary and hopefully admitting the right ones required to resolve paradoxes yet not produce contradictions. When I studied Axiomatic Set Theory, there was a whole list of paradoxes that were problems, but not necessarily problematic towards getting a working theory of Set Theory. For example, Russell's paradox was just a example of admitting too much and was easily fixed with the Axiom Schema of Separation. I am not sure if Grelling Nelson is a hurdle to set theory, but as more of a mathematician than a philosopher, I only pick the battles I have to fight.We have evolved from ZF to ZFC to Von Neumann -Bernays Goedel to Grothendieck Tarski to ? By Goedel Incompleteness, we know we can not close the box. We just hopefully have enough to get er done for our particular application. .I wouldnt call logic broken, That is the way Mother Nature is and we should learn to work with it.

[@NoConsequenc3]

good reply

[@NaN_000]

I don't agree

[@b43xoit]

to category theory?

[@Chance57]

Eye of the beholder indeed. Is the word "beautiful" autological? To some, perhaps. If there's room for debate then framing it as a simple yes or no is "not even wrong."

[@Jupa]

But it makes me think about the development of AI. Surely with an unclosed box, things that entail complete self autonomy without human intervention. An extreme example is to recreate a human being robot, a realistic example would be fully autonomous vehicles becoming a universal standard. Relying on algorithms, but the inherent incompleteness of the box means that we can never rely on algorithms to live harmoniously with mother nature without human subjective input. I'm not a mathematical major, or anything close. But I am fascinated by it. I also think that these paradoxes should remind us of your end conclusion 'learning to work with nature' in a world that seemingly feels more off-balanced despite algorithms and artificial intelligence becoming embedded in our modern lives. I think people are pushing a little too far and maybe we should relax and work with the world, not figure out ways to make the world pander to us. I know this is very basic but I hope you know what I am trying to say

[@TheRenaSystem]

I knew where this was going from the beginning but still loved every second! Amazing vid!

[@georgetsironis9577]

I agree that heterological creates a paradox but the word autological is autological by the definition of 1:40. You only check if it fits in the autological definition and it does. Heterological words are the words that do not fit in the autological definition.

[@johncaemmerer7094]

The assertion that all words, mathematical expressions, etc., can be sorted into two mutually exclusive categories such as "true" and "false" is being made here without proof, or rather, it rests on unexamined basic principles of logic that are postulated at the start based on some form of intuition or common sense. In this traditional logic you treat words that define a category as if they were like bins, and in this case you imagine having the autological bin and the heterological bin. As others have pointed out below, in this scenario asking whether the word autological is autological is like asking whether the autological bin (or the heterological bin) belongs in the autological bin or the heterological bin. The answers are neither correct nor incorrect because it's meaningless to ask the question in the first place, just as it would be to ask whether one should put a physical bin in itself or in the other bin. The correct third category therefore is not "neither", as suggested in the video, because that category still assumes the question is meaningful. A correct name of the third category is "words for which it is not meaningful to ask whether they are autological or heterological". This category does not fit anywhere in the Venn diagram because it is the equivalent of saying "statements for which it is not meaningful to locate them in a Venn diagram depicting binary truth and falsehood". I think this is what Ludwig Wittgenstein was trying to show, namely that Aristotelean logic -- the logic of categories with unambiguous definitions and sorting procedures -- does not give us a very useful model at all of human language, or large parts of it in any case. I think he actually demonstrated this pretty convincingly, and yet many people (including philosophers and especially popularizers of philosophy) persist in treating these sorts of things as unsolved dilemmas.

[@ArtemisiaSayakaRandazzo]

So far this is one of the most interesting and accurate channels I've ever seen in youtube.

[@raymoncada]

Is it?

[@TheOnlyGeggles]

@@raymoncada Well, it's not perfectly accurate, for example Russel's paradox is NOT equivalent to Gödels incompleteness theorem, but is instead slightly weaker, but it would require a deeper understanding of the mathematics involved, than I would expect from a popsci (or popmath apparently now) KZbin channel. But I would argue that it is more accurate than most other channels in this niche. At least for now, we'll see where they go from here, I suppose.

[@l.w.paradis2108]

@@TheOnlyGeggles He did not actually show that assuming the word 'autological' to be sortable (as being either autological or heterological) leads to a contradiction by the same reasoning as assuming 'heterological' to be sortable must lead to a contradiction. I don't see how it would be by the same reasoning. 'Heterological' is easy.

[@TheOnlyGeggles]

@@l.w.paradis2108 I think he considers it paradoxical not only for a word to not be sortable into either box, but also to have a word fit into both boxes (since one box is supposed to be the negation of the other, hence their contents should be disjoint).

[@l.w.paradis2108]

@@TheOnlyGeggles Yes, that would be -- but where is the demonstration?

[@seismicdna]

Wasn’t the set theoretic answer to actually make a new type of object called classes which were basically collections of objects which could not be included in other sets? So in a way, even though it’s not resolved, the problem is side-stepped by creating a slightly different system which avoids the problematic recursion/self-reference. I wonder what the linguistic analog to classes are.

[@dialectphilosophy]

Indeed the solution to Russell's paradox was simply to disallow certain objects as sets. But as you state, it's really more of a side-step, and many feel that the essential mystery inherent to the paradox is still open to interpretation.

[@noirox4891]

@@dialectphilosophy And, even more importantly, Gödel's Incompleteness theorem tells us that this new, extended system will still contain such an unresolvable self-reference! It may be a different one, but as long as we leave enough freedom for basic logic, there will be unprovable or inconsistent statements.

[@DanielTanios]

@@noirox4891 That's not quite correct. Propositional logic, for example, is consistent, complete and decidable. Same for Primitive Recrusive Arithmetic. Godel's incompleteness result explicitly only applies to formal systems strong enough to give rise to Peano Arithmetic. Even then, although Godel's results have been accepted and well-understood by mathematicians for decades, their implications remain hotly contested and there are many subtleties involved (eg: en.m.wikipedia.org/wiki/Hilbert%27s_second_problem#Modern_viewpoints_on_the_status_of_the_problem).

[@noirox4891]

@@DanielTanios Ah, I am realising maybe I misread the comment (Or replied to the wrong one? My reply seems completely unrelated in retrospect), I was referring to the case where we already have a sufficiently strong formal system. In that case, it is, if i remember correctly, impossible to introduce a new axiom such that the system is no longer strong enough for the Incompleteness Theorem to apply. And it frankly blew my mind when I understood that when starting with a complex system, no new axioms (And thus also not the naive process of finding unprovable statements and adding them as axioms) will ever produce a consistent, complete system. That is what I wanted to share here, apologies for any confusion!

[@visancosmin8991]

@@dialectphilosophy The essential mystery is that you don't take consciousness into account.

[@Alex-5d-space]

Thank you for such a great option to gradually reveal this issue. Your approach to visualization and explanation gave even more confidence that it is important to study physics and even more important to learn how to feel and imagine in your head. When the answer is logically built, then the picture in the head develops, and new questions and assumptions arise. Very nice and interesting explanation. I really look forward to new releases and look through the previous ones on your chanel 🙏

[@RealWorldMusicTheory]

Well, I might be too naive. But isn‘t this just an issue of imprecise use of language? I don‘t want to oppose Turing or Gödel, but this particular language example isn‘t a paradox in my understanding. To me it seems the whole issue stems from neglecting the meta description levels. Words describe the world. But there is nothing „doggy“ about a “dog“ - the word doesn’t look, sound or smell like a dog. We adjust assign that symbol “dog“ to the concept of a dog. Then we can go to the first meta level, in which we can use words to describe words. But again there is nothing particularly “nounsy“ about a “noun“. We just assign that symbol the concept of noun. Same for heterological. If I now go to the second meta level, I can describe the words that describe words. On that level we assign the meta(1st level) word the concept of “autological”. So now the logic is “hereological on meta level 1 is autolocigal using the terms of meta level 2” or short “heterological(1) is autological(2)”. Likewise “heterological(1) is NOT heterological(2)”. There is no logical contradiction, if we’re explicit about the meta level we use. Of course we can play that meta level game ad infinitum. But we can do that with any word: “noun(1) is a noun(2)”, “noun(2) is a noun(3)”. The only peculiarity about heterological and autological is that they alternate from meta level to meta level. But that doesn’t break logic imo. Please help me, if I’m wrong here. I’ve just learned to be precise with the usually pretty imprecise human language. As I said at the beginning, I don’t think my explanation holds true for Gödel or Turing (I wouldn‘t dare to!). But I think it can resolve this particular example with words. Likely this language example is not comparable to the logical problems these mathematicians unearthed.

[@d3consultancyservice12]

Hi there! For me, it is simple: when you define some categories of objects, the definitions themselves do not belong to any categories, they are outside the “universe” of your categories; and it seems common sense; the apparent paradox occurs if you are using as objects “words”; this is a particular case in which the “definitions” are made of same “substance” as are the elements inside your categories; using boolean logic without taken into account the context, give rise often to paradoxes, because you are ending by comparing things that are not comparable, (in the sens that it is no comparison yet defined), like apples with pears; one of the great common nonsense in theoretical physics, is the self-interaction of particles (it works but surely for wrong reason); this kind of nonsense is happening when we extrapolate concepts beyond the limits of validity; or we are mixing the scales of applicability; another apparently logical conclusion is to say that because we are made of elementary particles that are governed by quantum laws, therefore behaving purely non-deterministic, we as a collection of particles, we behave consequently; therefore, the is no possible free will; but there is emergence that create levels above level, and from level to level the concepts change, we can not compare one concept from one level with another concept from a level above/below; logic is just a tool to be used in a well-defined context, you go outside the context, your logic is becoming nonsense

[@rossevans11]

Not quite. In this example you have bivalent sets, one representing true, and the other representing false. In such a system, any logical statement should be sortable into one or the other. This is basically what Godel proved, that any axiomatic system can have true statements whose truth value cannot be determined within the system. The only way out of the paradox is to define axioms for each case, the problem is, in mathematics, there are an infinite number of true statements which are unprovable in any given system.

[@d3consultancyservice12]

@@rossevans11 "This is basically what Godel proved, that any axiomatic system can have true statements whose truth value cannot be determined within the system" what i'm saying is equivalent with that, and even more, doesn't make any sense to construct systems in which the 'truth' can't be determined; that is not a paradox, it's a guide for the good scientist; logic it's not absolute, it's contextual; as well as time is not absolute etc etc etc

[@MugenTJ]

It’s similar to a categorical error. The contradiction exists when you attempt to treat the word as the category itself.

[@plasmarob741]

I'm glad you pointed out Grelling-Nelson is just Russell’s Paradox. I feel from a higher level of analysis, it's all ultimately the same problem. A computer scientist myself, I know what you'll cover next and I'm excited to see each subject made approachable for the average audience! Keep at it! I'll save my musings on the answers for another time.

[@PunmasterSTP]

That was indeed a good twist at the end, and it was a phenomenal video overall. I'm glad content like this exists somewhere on KZbin.

[@XetXetable]

The examples given at the end aren't really the same; rather, they are all theorems proven via a diagonal argument. This makes them special cases of Lawvere's fixed point theorem, but that's not the same thing as them being "translations" of each other into different domains. Other examples include Cantor's theorem about the uncountability of the reals (and varients there of), the non-definability of satisfiability, Tarski's theorem on the undefinability of a truth predicate, the non-enumerability of computable total functions, Borodin’s Gap Theorem in complexity theory, the Knaster-Tarski theorem in preorder theory, (the existence of) Kripke’s theory of truth, Brouwer’s fixed point theorem and the Ascoli theorem in topology, Helly’s theorem in distribution theory, Montel’s theorem from complex function theory, and Nash’s equilibria theorem from game theory are all, similarly, fixed point theorems proved via a similar scheme. This pattern is pretty common. The first to use it was Cantor in the proof of the theorem bearing his name, in which he remarked (originally in German); "This proof appears remarkable not only because of its great simplicity, but also for the reason that its underlying principle can readily be extended." Perhapse diagonal arguments are the true topic of this video, and the claim at the end that these theorems are essentially translations of eachother is a rationalization for not naming the thing itself. If you actually go through the task of proving the theorems formally, you'll realize that the bulk of the work is in finding/constructing either suitable epimorphisms for the argument to go through (thus concluding that a fixed point must exist) or finding a suitable endomorphism without a fixed point (thus concluding that an epimorphism doesn't exist). The actual diagonal argument itself is, usually, the easiest part of the proof, however unintuitive a newbie might find it.

[@deadalus1991]

This comment is as interesting as the video lol

[@Julian-tf8nj]

I found a mind-blowing paper discussing the *Lawvere's fixed point theorem* , and how many famous diagonal arguments can be derived from it: www.uibk.ac.at/mathematik/algebra/staff/fritz-tobias/ct2021_course_projects/lawvere.pdf My hat off to Category Theory!! 😄

[@argfasdfgadfgasdfgsdfgsdfg6351]

There is a simple logical error in your argument: You can only use a definition once it is fully defined. Same applies to Goeddel's incompleteness or the liar's paradox. What you are doing is basically a recursion to something that was never defined in the first place. In the autological case, just replace the word 'itself' by the definition of 'autological'. You will get: " The word 'autological' is a word that describes 'a word that describes 'a word that describes 'a word that describes' a word that describes'....." and so on and so on. Logic is not broken, you are just not making logical assumptions.

[@IDMYM8]

0:45 There's one important question I have. If anybody is reading please look because I want to make things a bit clear. At 0:45 We see the definition of Heterological says, "Denoting any word which does not describe itself." This will mean the word "Autological" is not 🚫 Heterological, thus not falling into paradox. If the definition of the word Heterologival was, "A word that descries other than itself", then only the paradox will occur for the word "Autological". Am I correct? By which I clearly mean that the word "Heterological" is the only world that is paradoxical and not "Autological". ???

[@Oler-yx7xj]

What always feels weird with Russel's paradox and the liars paradox is that they seem to be about some very edge cases (set that contains sets according to a rule which itself talks about containment, the truthiness of statement that itself talks about its truthiness, whether program halts while the program is itself about halting), that for most things formal logics and other stuff should work fine. It would be interesting to see videos showing examples that are not so recurrent. On PBS Infinite series there was a video about an unmeasurable set, that seem a good example.

[@dialectphilosophy]

We miss PBS Infinite Series 😢

[@Reddles37]

I kind of think of it like a buffer overflow or something in a program. At first it might seem fine since the program works fine with most input, and you probably wouldn't even notice the problem without careful debugging. But a hacker only needs this one flaw in your program and next thing you know they have it executing arbitrary code. Something sort of similar should be possible in pure logic, where you use these paradoxes to generate a logical inconsistency and then add a chain of valid logical statements to propogate that inconsistency to something more important.

[@lawrencedoliveiro9104]

Self-reference is fundamental to understanding reality itself. Which is why attempts to ban it (e.g. Russell’s theory of types) are doomed to failure. One way to come to grips with Russell’s Paradox is to look at a proof attempt as a computer program, a.k.a. an algorithm. If you remember the definition of an algorithm, it must terminate after a finite series of steps. But in the case of the Paradox, the assumption that the proposition is true leads to the conclusion that it is not true, which leads to the conclusion that it is true, which leads to ... so you have, in computer science terms, an “endless loop”. You only get a final answer when the procedure terminates, which it never does. As I recall in my brief exposure to denotational semantics, this outcome is denoted by the “bottom” symbol, “⊥”.

[@SporeMystify]

@@lawrencedoliveiro9104 That sounds a lot like creating that "neither" category the video mentioned

[@lawrencedoliveiro9104]

@@SporeMystify The “neither” category represents a different answer returned after a finite number of steps.

[@DynestiGTI]

Probably one of the most underrated educational channels on KZbin. I hope to one day see you gain many more subscribers. I love that you acknowledge past KZbin videos that have done the same topic and try to do something unique or better rather than regurgitate the same thing again to jump on the bandwagon like other popular channels do. I'm still highly anticipating your followup videos on SR and GR.

[@dialectphilosophy]

Appreciate the support and thank you for watching! We will be returning to Relativity soon...

[@domenicobarillari2046]

As a practicing physicist ,and someone who also loves other areas of inquiry, all I can do whenever I watch another DIALECT product is yell BRAVO, and BRAVO once again!! Keep it up folks! I share these all of the time. best regards, D. Barillari

[@visancosmin8991]

Physical world doesn't exist. "Physical world" is just an idea in consciousness. See my paper "Meaning and Context: A Brief Introduction".

[@GamingBlake2002]

@@visancosmin8991 Whether materialism or idealism is true has no bearing whatsoever on working physics, so this point is irrelevant. I read your paper, and it's a joke. Three pages in, you say the following: "Thus, the first requirement for anyone that wishes to understand reality, is to be aware of how consciousness creates everything that we see and generally experience around us, creation which, of course, is not to be understood as if consciousness creates “material” objects outside ourselves, but creates the appearance of such objects inside itself. If this first requirement is not met, no amount of rational arguments can make one see. Thus, before continue reading, the reader must make sure he meets this first requirement." So in order to even have the right to read your paper, the reader must accept your conclusion that you have no intention of giving any arguments for? This paragraph is just a long-winded acceptance of your failure as a philosopher. You can't argue your position, so you won't even try.

[@visancosmin8991]

@@GamingBlake2002 Give 1 single example of something outside consciousness.

[@visancosmin8991]

@@GamingBlake2002 Also, why are you angry ? Lack of girlfriend ? So many vargins on za internet.

[@GamingBlake2002]

@@visancosmin8991 There's no use in answering your first question, as you'd simply reject any answer I give that conflicts with your worldview and assert that all that can be experienced is a construct of consciousness. As for your latter comment, I'm not angry. When I called your paper a joke, I wasn't simply insulting it; I truly found it humorous! So thanks for the laugh. Anyway, you're the one resulting to personal, non-academic insults, so you're clearly the one who's angry. I think you'd find more utility in actually developing philosophy skills than in insulting people who point out your lack of them.

[@d.-_-.b]

When the phrase "Everything in moderation" applies to itself that means you're allowed to splurge on some things, but then you're not being moderate on those particular things, which falsifies the phrase.

[@alazif7973]

This is a good paradox. Going to go out on a limb, but since this is philosophy of language we can reach out and ask what makes a word a word in the first place. Perhaps the issue with heterological is not that it inherently breaks logic, but rather it is a word for something illogical in concept. Lets say if you made a name for a shape that is a square-circle, like a squarcle. Does making a word for it make its definition exist? No. Likewise, perhaps heterological maybe doesn't describe a real thing, but rather a non-thing. Just because we understand what the word translates to doesn't mean its meaning is a logical existence, like the squarcle. So this problem might not be a problem at all, and just a confusion. Autological might have the same issue, but in a sort of reverse "it is tautology" meaning, so it is in fact saying nothing and thus doesn't have a meaning itself beyond itself to categorize.

[@spiralgaming8940]

Waiting for more SR and GR videos because you left us with tons of questions (which causes lots of sleepless nights!) I felt that SR is often taken as so obvious (It is, at some point because it's mathematics is not that hard, highschool maths) however lacking in intuitive explanation of the far reaching ideas which I think is one of the holes we have in understanding in GR. Do you have any other view to understand SR. Just like yours and @ScienceClicEN 's approach to GR.

[@vasyakalistrov8184]

i caught some sleepless nights too after dialekt's GR videos

[@dialectphilosophy]

We think you'll be resting easier after our next couple videos...

[@hugoballroom5510]

I have a feeling this is just a digression.

[@natywubet2175]

@@dialectphilosophy this gives me a lof of hope. No doubt you are the best

[@user-pr6ed3ri2k]

69thliker

[@EneldoSancocho]

A usual way to escape this problem: the question makes no sense, but in practice it doesn't matter. For example, the statement A = "this statement is false" can't be true nor false, so if we apply the logistic axiom B (just to give it a name) that says that any proper statement is either true or false, then B implies A is not a proper statement, this way you avoid going insane

[@davidioanhedges]

You have described Gödel's incompleteness theorem, there will always be some statements that cannot be proved within the system, as they make no sense within the system, the two solutions are ignore the statement, or pick an answer and add it as an axiom (realising there will be always another statement you cannot prove) Note the statements usually cannot be ignored, as they are fundamental and many other solutions rely on them ...

[@EneldoSancocho]

@@davidioanhedges Not I haven't, the statement A can't be true nor false, because A => not A, i can't pick a truth value for A unless I pick both True and False, but then i can't use logic anymore. Gödel says there are statements that can't be proofed true nor false, but they don't generate a contradiction.

[@EneldoSancocho]

@@davidioanhedges Gödel statements like that are easy to create, but the interesting ones are that which seem to talk about the same axiomatic system we consider

[@EneldoSancocho]

@@davidioanhedges Something I don't understand: lets say the Goldbach (G) is proven to be unprovable ( don't know if that is the correct word). If we consider an axiomatic system that incorporates some kind of arithmetic, logic axioms. Then the statment A1="G can't be proved to be true or false" implies A2="there's no counterexample of G" (otherwise we could prove G is false), then A2 implies G is true which implies A1 is false. In short: A1=>A2=>G=>not A1 I guess i'm missing something, but i heard many times that G could be unprovable, and it seems that is a contradiction. ???

[@EneldoSancocho]

@@davidioanhedges By the way, excuse my bad english. And I'm not a doctor it's the name of a songs, so honesty i don't really know what i'm talking about

[@danielvarga_p]

Gödel incompleteness in action?! Thank you it is a really great work!

[@petersansgaming8783]

I think this fits more into Russell's paradox (at least intuitively). Of course all of them suffer from being self referential.

[@visancosmin8991]

@@petersansgaming8783 And this happens because people talk about objects by ignoring the subject that thinks the objects. Once you take into account the subject, there is no paradox left. It becomes trivial that no-thing = every-thing, or in other words that I am God.

[@dogcreator7439]

Nice.

[@argfasdfgadfgasdfgsdfgsdfg6351]

I really don't get what all the fuss is about. This is just the liar's paradox and it can simply be resolved by only using definitions once they are fully defined.

[@visancosmin8991]

@@argfasdfgadfgasdfgsdfgsdfg6351 lol. You are not conscious of consciousness.

[@julianemery718]

I like how clear this is, how you break this problem into smaller parts after first showing the overall idea, and how you showed different ways of approaching the problem.

[@doomtho42]

To me, this “paradox” actually just exemplifies the fallacy of attempting to define a concept in terms of itself. In a way it’s a bit like trying to plug a power strip into itself. The thing is, logic can only function within a framework of fundamental rules or axioms (in our power strip analogy, this framework would be akin to a power source of some sort, e.g. a battery). So this “paradox” is essentially just what happens when you try to perform a logical operation on the framework of the system of logic itself.

[@artdadamo3501]

Self-referential words and phrases (i.e., words and phrases which refer to themselves) commonly create paradox. Example: the barber of the village is a man and cuts the hair of every man in the village who does not cut his own hair. Who cuts the barber's hair?

[@jayanthony6375]

the barber because he not only cuts the hair of everyone in the village who cant cut his hair; nowhere is the restraint that he cannot also cut the hair of those who can cut their own hair... or did i miss something?

[@feynstein1004]

A second barber? 😅

[@pyropulseIXXI]

@@jayanthony6375 Yes you obviously did miss something. The barber only cuts the hair of those that do not cut their own hair. By cutting his own hair, he would have to not cut his own hair. Pretty simple

[@pyropulseIXXI]

@@feynstein1004 there is only one barber

[@ozymandiasultor9480]

@@jayanthony6375 The barber must cut the hair of every man in the village who does not cut their hair by themself, but must not cut the hair of the person who is cutting his hair. When it comes to him, he can cut only the hair of the person who does not cut his hair himself, so he can't cut his hair, then again, he cuts the hair of everyone who does not cut his hair, and there is the paradox...

[@tim40gabby25]

I have 2 boxes at home, one containing anything that can be imagined, the other - and still empty - box containing that which cannot be imagined. The act of placing anything into my empty box reclassifies it as that which can be imagined, so the empty box remains empty. This has puzzled me.

[@tomholroyd7519]

Is it Sylvan's box?

[@tim40gabby25]

@@tomholroyd7519 no, but I see what you did there :)

[@TheOnlyGeggles]

The same thing happens with the boxes called "natural" and "supernatural". People saying that science is incomplete, because it cannot study the supernatural, makes no sense, because if it _could_ study something "supernatural", it would immediately cease to be supernatural, going into the natural box instead.

[@khajiithadwares2263]

Is there any merit to considering there are more than 2 boxes {of containment} at any given time? For example numbers: We consider what we can count in our daily lives and place numbers in pile A (odd numbers) and pile B (even) Our experience with numbers doesnt need consider negative numbers since those are only numbers we imagine, they are non-numbers, unless we imagine them. We talk about apples, but we dont talk about the absence of apples. Its similar to how we consider the lack of life (ghosts, cemeteries, cold heartbeat) as supernatural, yet all of those are based on people (ghosts of people, cemeteries for people, warm hb of a person) I think putting things into 2 boxes only partains to measuring presentable physical quantities. 4 boxes are needed to also measure a passing through time or their temporal fleeting qualities. So the two-boxes paradigm is false. (or at least incomplete).

[@tim40gabby25]

@@khajiithadwares2263 interesting, though doesn't address my question.

[@maxtseluyko9588]

Paradoxically, Hegel solved it even before it was formulated,but because he used no formal notation, his writings (Wissenschaft der Logik specifically) were perceived as non-sensical by the later logicians and noone studies Hegel these days.

[@rafaelborobia2559]

Hegel's dialectics were definitely better at understanding the problems with logic and its limitations. Marx's dialectical materialism helps us even more to note this problems and surpass them. I recommend Lefevre's "Logique formelle, logique dialectique" in his struggle to 'aufheben' the classical formal logic.

[@delvish9622]

Language is relational and as such no word describes itself. Words are symbols that represent objects or concepts but the symbols are arbitrary, they just have to be consistently applied in their usage. Point is, a word being autological depends on the symbol selection in cases like "polysyllabic" because another language could just as easily express it in one syllable. So right there labeling something as autological is entirely dependent on the arbitrary selection of symbols. Autological is the name given to words that appear to describe themselves, but actually don't because symbol selection is arbitrary. Or the illusion of autological appears again when you apply an object oriented description to concepts, for example the word "inanimate" is meant to be applied to objects, and it makes no sense to apply it to a concept. We'd have to have an example of an animate concept in order to use it in a way that offers a meaningful distinction. So we have a concept that in order to be correct requires symbol selection that allows it to be correct, and requires applying other concepts incorrectly. All we can really conclude is that autological is an incorrect concept, an illusion that arises from our symbols. Heterological too is an incorrect concept because its definition implies an opposite state(autological) which undermines what language actually is. It's a bit like taking the phrase "the map is not the territory" which is a statement of fact, turning that into a concept represented by a symbol(heterological) and then assuming there must be a valid inverse of that concept such that "the map IS the territory" and denoting it with another symbol(autological). All you've done is trick yourself into elevating a falsehood onto equal footing as an objective fact and then wonder why your system of logic breaks.

[@dialectphilosophy]

We go back and forth on this issue a lot. Your opening insight that language is relational is certainly very apt and definitely at the heart of the issue, but can no word describe itself? That's a harder case to make. Certainly no word can define itself, but if we treat words as objects themselves, then attributes of those objects can indeed be described by the word which corresponds to the object. "Green" will describe the word green if we treat it as a physical object and color it green. "Polysyllabic" will describe the word polysyllabic if we treat it phonetically. Changing up the words may change the relations but it doesn't change the fact that some words will always describe themselves. Indeed, such a method is essentially employed by Gödel's incompleteness theorem, in which logical statements with one-free variable are encoded via natural numbers (the "word" is made into an "object") and then those natural numbers are fed back into the logical statements which they correspond to as inputs (we ask "does the word describe itself?") And then a paradox is constructed therefrom. How valid is this process however? Does it ultimately mean anything? It's still unclear to us.

[@delvish9622]

@@dialectphilosophy Here's the thing though, words aren't actually objects. And promoting them to objects is irrational, if you don't think so I propose an experiment, try to sit in the word "chair". This desire to promote concepts into objects is at the root of a lot of human errors in my opinion, for example "spacetime". And it's no surprise to me that at every turn it generates paradoxes. We're making the mistake of presuming our abstractions are somehow the things themselves.

[@billy-cg1qq]

Omg, I haven't felt like this after watching a Math video in so long! My eyes teared up, and I almost cried! Very nice video! I wish I could learn Math and all the wonders of the universe.

[@JackPullen-Paradox]

I think that you did a very good job with this paradox. The graphics were very good, too. Didn't pay attention to how long it took, which means that it was very engrossing. So all-in-all, probably better than very good. Closer to excellent!

[@RyanLynch1]

amazing video! just one suggestion: i wish you had shown the case for why "autological" could apply to either category

[@JosephVozzo]

The issue with Autological is actually that it can fit into both categories, not that it can't fit into either.

[@allstar4065]

@@JosephVozzo Prove it

[@vilmernyberg193]

@@allstar4065 Autological is a word that describes it self (autological) Autogical is a word that does not describe itself (heterological)

[@TiagoCavalcanti-ji6hu]

@@vilmernyberg193 So, in reality, does Autological describe itself or not?

[@sophiahan8182]

@@TiagoCavalcanti-ji6hu Yes

[@GrimIkatsui]

I loved this video. I'm familiar with the other paradoxes, but this is the first time I've seen them in this form (and I'm a linguist!). Seeing this perspective on the problem gave me a new idea to consider, to possibly crack this paradox. Known: An autological word is one that describes itself. A polysyllabic word is one that has multiple syllables. Consequence of idea: Autological describes words that describe themselves. Polysyllabic describes words that have multiple syllables. Takeaway: The key difference here is that "autological" describes words that describe a property, while "polysyllabic" simply describes words with a property. In fact, autological and heterological are the only two words you've mentioned that describe what a word describes, as opposed to what it is. This in and of itself is of little consequence, but paired with the fact that "autological" describes words that describe themselves the mechanism for confusion becomes apparent. Attempting to navigate the classification of the word autological means confronting this self referential nature, whereby its description of itself can change in relation to its previously discerned self. This can be done by noting that "autological" describes a dynamic relationship between itself and its definition while autological words describe static relationships with their respective meanings. For this to be reflected in the definition of autological one can simply write "An autological word is a word that always describes itself." Conclusion: Because autological does not always describe itself it is heterological. Heterological on the other hand does not always describe itself and this is its definition, so it is autological. Though these words may now be sorted into the two bins, they are unique and may also be sorted into a new bin based on this property. I call them extralogical words. This is because I could just as easily have navigated their dynamic properties by using a different selector, and they do still appear to break logic. Instead of "An autological word is a word that always describes itself." I can change it to "An autological word is a word that is capable of describing itself." With this alternative definition/logical selector, Autological is an autological word because it sometimes describes itself. Similarly, heterological is also an autological word given this definition. Which do you think makes the most sense? Should they both be autological, or should only one be because they have opposite meanings? This is why I chose the definition I did, but I see validity in both of them.

[@HackersRUs]

I don't think you did anything. If heterological does not always describe itself, then it is autological. If it is autological, it always describes itself. We have arrived at the same problem. Similarly, you can still conclude autological is autological, and that autological is heterological

[@parkerstroh6586]

@@HackersRUs wait but if you can describe heterological as autological, then it is not heterological by this new exclusive definition. I think he might’ve made an interesting point

[@GrimIkatsui]

​@@HackersRUs Yes, the word heterological is autological because it always sometimes describes itself. In order for it to then switch back to being heterological as you propose it must sometimes always sometimes describe itself, but this is not what we observe.

[@HackersRUs]

@@GrimIkatsui Ah, then the problem is that if you change the definition, they are no longer the words that we care about. The definitions themselves are important not that they are tied to any one word. Changing the definitions is meaningless; then it's just a different word with the same name. Keep in mind if you break this, you've broken the halting problem because of the equivalence.

[@GrimIkatsui]

@@HackersRUs This change in definition retains the original meaning for every other word, and it was prompted by logical analysis of the original definition's failure to consider how some words operate differently. To be more precise, words are not static, but the original definition for autological assumed they were. I don't see how forcing ourselves to assume something false is more helpful.

[@withjoe1880]

Just use three categories: 1) Category A (Ex. Autological) 2) Category B (Ex. Heterological) 3) Labels (The words "heterological" and "autological") Here we define labels as the two words that we are using to divide all other words. Then, by definition these two words are always labels. Alternatively, we might use 4 categories: 1) Category A 2) Category B 3) Category A and Category B 4) Neither Category A nor Category B These cannot be combined into fewer categories without losing meaning. The problem here is with language. There is no problem with the logic. Language is limited by human understanding, perception, and capabilities and cannot fully describe anything uniquely. Any language is merely an abstraction of the ideas and concepts that make up reality and the limitations of these languages appear as paradoxes. There is an even more simple paradox in the English language: "This statement is false." If it is false, it is true... Etc. If it is true, it is false... Etc.

[@bigboibebop]

10:46 why does this feel like a metaphor for most societal disputes

[@gaurisingh8394]

Never commented on a video before, but this one truly blew my mind. I'm actually just trying to think about this all over again. Speechless. Excellent work!

[@Pencil0fDoom]

Popped his YT cherry! Masseltov bro.

[@TiagoCavalcanti-ji6hu]

Welcome. It's an interesting road, to say the least.

[@WWLinkMasterX]

I saw really cool video about Category Theory that explained how these are all examples of "Diagonal Arguments." The Liar's Paradox and Cantor's Diagonal Argument are also examples.

[@exofbounds]

Great stuff! There is one thing worth pointing out. Logic alone (i.e. first-order logic) cannot create such paradoxes. Loosely speaking. The problem arises when something like naive set theory comes into the scene. Which assumes all mathematical entities (in set theory) are in one (mathematical) universe. And they can interact with each other freely without restriction. The core lesson we have learned is that this is not true. We can define structures beyond one universe. Collapse those universes lead to collapse of logic.

[@TheOnlyGeggles]

Indeed, this is called "impredicativity". Having an infinite (cumulative) hierarchy of (type) universes avoids the issue entirely

[@lawrencedoliveiro9104]

So there must be walls between different compartments of reality? But when we look around at reality, we see no such walls -- there are no such separate compartments. In other words, self-reference is an inescapable part of reality itself.

[@TomKaitchuck]

New foundations provides another solution to this in set theory. It is also the one taken by most Programming Languages: things cannot be used in a definition until they have themselves been defined.

[@lawrencedoliveiro9104]

@@TomKaitchuck Except most languages allow recursive definitions--definitions which refer (directly or indirectly) to themselves.

[@TomKaitchuck]

@@lawrencedoliveiro9104 That's fine. It doesn't lead to contribution or ambiguity. The result of running such a function is not ambiguous, Because the inner and outer invocations are distinct and can return different results. Recursion can always be replaced with a loop. The only way you run into contradictions is when the definition of a function depends on the result of the invocation currently running. In languages where you compile first and run later, this isn't even expressible. If on the other hand the language has access to its own interpreter or can compile and run code on the fly, it is possible to construct the halting problem, even if the language doesn't support recursion or loops.

[@GotPinkBlackandBlue]

Love the video! Excellent job on the illustrations and animations! You really got me thinking, and I came up with a counter augment. 10:00 "There isn't any other option, yet these words stubbornly refuse to adhere to any such bivalent classification. Leaving us with a headache inducing paradox" is a very bold assertion. Using occam's razor, would it not be simpler to say 1:30 "all words are either autological or not-autological (heterological)" is a false statement, rather than concluding logic was broken. I present you with my own bivalent classification: things that are "categorizable into a binary" or are "non-binary".

[@cainwilson8564]

They simply exist in a superposition. Paradoxes are a perfect example of the phrase “more than the sum of its parts” as they exist outside of their given options.

[@Minalkra]

I love that you explained in great detail why heterological is a paradoxical case and then tell us that autological is a similar case without explaining the autological case for us. Which, let's be frank, is the more interesting one. EDIT: Grammar.

[@cooldawg2009]

Yea, I dont see a contradiction For “Autological” in the logical setup he has created 1. Autological is Autological - True 2. Autological is NOT Autlogical - False I dont see the contradiction.

[@vishnurajagopalUHD]

yes, I was wondering about the same, if auto is auto, then it is right, and if auto is hetero, then it does not describe itself which means it is hetero, which is also right?

[@johnv4994]

@@cooldawg2009 The problem is that we are asserting that all words are EITHER autological or heterological. For the word "autological", the contradiction is that it is BOTH autological and heterological, while for "heterological", it's contradictory

[@cooldawg2009]

@@johnv4994 the WORD Autological is Heterological, bc it does not decribe itself, it means words that describe themselves which Autological does not. Therefore, Autological is NOT Autological. I dont see contradiction Can you spell out how the word Autological is Autological?

[@johnv4994]

@@cooldawg2009 Assume "autological" is autological. This means "Autological is a word that describes itself". The logic holds up, as that sentence just means "Autological is autological" which we assumed. The problem is that the logic is STILL consistent regardless of whether we assume the word "autological" is autological or heterological. In other words, "autological" can fit in both bins, which doesn't make sense. (I'm assuming the first part of your reply is supposed to be an "Assume 'autological' is heterological" example)

[@garryschniderham8291]

This paradox is like the machine that simulates it's self to see if it works and then gives out the opposite Boolean value

[@andrewwafae]

That's precisely what the solution to the halting problem he mentioned is..

[@9erik1]

Nice video. I had forgotten about this paradox. While it's tempting to go along with others and relate this paradox to Russell's paradox/Godel's 2nd incompleteness theorem/the halting problem, my intuition tells me that it actually bears more resemblance to Tarski's undefinability theorem, which is discussed more in philosophy than mathematics and in my opinion is very underrated. Godel established his system of Godel numbering that encodes syntactic statements as natural numbers; Tarski proved that there's no "truth predicate" among the natural numbers that will always evaluate a natural number as true/false whenever its corresponding statement is true/false. He basically did this by constructing a mathematical liar paradox. I learned about this in AC Grayling's "An Introduction to Philosophical Logic", and it got me right into math. The breakdown in this book was essentially saying that the famous liar paradox, saying " this sentence is false ", is actually not a paradox, but a syntax error -- because of Tarski's theorem, a truth predicate can only ever refer to sentences in a different language. So the real sentence should be " 'this sentence' is a true sentence in English " (notice the extra quotes around 'this sentence'). ' This sentence ' would be the sentence in English, while the remaining ' is a true sentence in English ' would be a meta-language of English, separate from English itself. But ' this sentence ' is not a proper English sentence, hence the syntax error. Here it seems the Grelling-Nelson paradox assumes that there is some mapping between words as objects and words as predicates, i.e. if we have a word w then there is a corresponding word predicate W, and vice-versa. So if A is autological and H is heterological, then A is defined by A(w) iff W(w). Heterological then is H(w) iff ~W(w). But then we have H(h) iff ~H(h), which is the paradox (for those unfamiliar, ~ means NOT). Tarski's theorem is somewhat similar, where instead that you're assuming that there's a mapping between a truth evaluation function in a given language (for example, first order logic) and a truth evaluation function in an encoding of that language (eg. Godel numbering). The mathematical liar paradox he derives ends up being T(n) iff ~T(n), where T is a truth predicate assumed to exist and n is the Godel number of a specially crafted statement.

[@mistafizz5195]

10/10 youtube comment

[@mistafizz5195]

God bless your soul, do you know how long I have been tormented by people trying to argue that the Earth is flat under a video of the ISS? This is a breath of fresh air. Thank you, sir.

[@seriously3shade]

The third category should be one that is defined by its reference to describing other words traits. A descriptor that is described as being non-self describing because of its nature to describe the nature of words relation to themselves.

[@Ash-vi3gg]

The problem is that if you let any propositional function p(x) have a domain U, then the function p(x) itself can't be part of U, because it is of a higher order. p(p(x)) is simply a non predicative statement. It doesn't make any sense logically. Let p(x) be "x describes itself", then define heterological as ¬p(x), you can't say say whether heterological describes itself, or if p(px) is true, because it is just syntactically and semantically incorrect. It's the same problem with all of theparadoxes involving something in its own definition, like the set of sets that don't contain itselves. The everything of an entity (set, word, whatever) can't be a part of the definition of that entity. That was already solved by Rusell.

[@Chris.Davies]

It is my contention that ability with language is roughly equivalent to the ability to think. And also that human language is a product of a flawed process, which much like biological evolution, only has to keep an organism alive, and nothing more. So, our language is the least-worst communication method which allowed (most of) us to survive. :P And so the languages we have today control how and what we think, and I think our current language tends to inhibit our thinking and stops us advancing. And so I have a picture in my mind, in the future, where uploaded (and upgraded) humans who have hit English's limits, begin from scratch to formulate a language which analyzes itself, such that false statements make no sense at all to a listener. You have to learn all the words in a sentence before any sentence can make sense. This new human language would share some commonality with computer languages, and existing human languages, with the very great difference that no word is ever permitted to have two meanings, or definitions, and that each and every word is defined in detail, and the definitions never ever change. Words can be added at any time, to introduce new concepts, of course. In this language there could never be any paradoxes - because all paradoxes are fundamentally failures to express the ideas correctly. Maybe there's a special section of the language devoted to so-called paradoxes - and there will be rules about what category claimed paradoxes fall into, so they may be examined precisely to see what's wrong with the claim. I think it may be that beyond a certain point, the only possible way to scientifically advance is to use a different language both spoken, and thought, to allow the human mind to properly think on a subject, and to express ideas in a way which cannot possibly be misunderstood. It would, necessarily be a rather long-winded language, but suitable for virtual humans to use without undue difficulty. I'm convinced there are concepts, ideas, thoughts, and inventions, which can never be expressed, or even had with our current languages. And it seems to me that humans are currently separated by a common language, and that our language is a hindrance to our unity, because it is not sufficiently deterministic in nature. Nor can it ever be, unless we start again from scratch. It would be horrible to try to have conversations in such a language, and a simple one might take many hours of intense effort. But it would be by far the best language to negotiate deals with, and to advance science with. That's how the hyper-intelligent aliens in one of my stories discovered how to travel in Upper Space, anyway. :)

[@joekirkup2624]

Someone has read GEB I see ;) Another complex concept very well explained @Dialect. Your content is some of the most intellectually stimulating and inspiring out there. Whenever I watch you, my desire to contribute to human knowledge is reinvigorated. Keep it up :)

[@karlbjorn1831]

What is GEB?

[@joekirkup2624]

@@karlbjorn1831 GEB is the book 'Gödel, Escher, Bach', written in the 1970s by a Douglas Hofstadter. It explores formal logic, recursion and meaning through exploring the work of those in the title and how it might interplay to produce consciousness. Infinity and circular/recursive reasoning paradoxes are at the centre of it, with 'autological/heterological' being a prime example of one. It's an intense book, but I would recommend it if you are interested in these kinds of questions :)

[@joekirkup2624]

@@karlbjorn1831 If you are going to read it, there is an excellent lecture series by MIT that you can watch alongside the book. It helped me enormously... kzbin.info/aero/PLBOgSgXfJ6B2nbZ_YREW_Nb-AX8FW9U9K

[@karlbjorn1831]

@@joekirkup2624 Often when I wander in the mind the problem of recursion comes up. Thank you! This seems very interesting, I will read it.

[@karlbjorn1831]

@@joekirkup2624 tyty!

[@timmo971]

This is how Sarah Conner should have defeated Skynet

[@angeloareniego3615]

The Only Term that is Self Contradictory by nature between Autological and Heterological is Heterological. He said It wrong between 9:28 to 9:31

[@ehsam8202]

i’ve never seen ur channel or had any interest in dialect but i jus wanted to say your thumbnail was really interest capturing. great work

[@livedandletdie]

This is all about if a set can contain itself or not. And is a pretty basic logical paradox.

[@Izurag]

Every system needs to have a "void", this simply means it exists and works. Like puzzles with images where you have to move the tiles to put them in order - you always need a free tile, so movement can occur.

[@CookieMasterRBLX]

logic doesn’t operate like that

[@Izurag]

@@CookieMasterRBLX but apparently, it does

[@JasonYu-bf3le]

feels like the Russell paradox itself is easier to understand

[@dialectphilosophy]

In a way Russell's paradox is easier to understand, at leasts with regards to its premise. What we like about this paradox however, is that the content-vs-form machinery behind the paradox is more easily made explicit than in the other paradoxes.

[@SloverOfTeuth]

Yep. As I was trying to follow all this, I was just thinking this is like the set of things that are not in that set, i.e. undecidable. I guess that doesn't make a long enough video, so you have to obfuscate and drag it out, dramatise with logic being "broken", then realise your example isn't very clear so give the examples the clearer thinkers gave in the first place to express their idea. At which point there's no time to explain what it's a result of, which is the obvious next step. So it's clickbait. But done in such a serious voice.

[@Matyanson]

Oh, I am so glad you highlighted the udiprod halting problem video there ^^. Great video btw.

[@reegarou]

Logic isn't broken. What's broken is our understanding of logic. In this case we pose an either/or statement. Either/or are always inherently flawed, because of this this paradox. So, there must be another option that we haven't considered or discovered, an either/or/(both and neither) statement. For example, to bring this into computing, the flaw (and strength) of standard computing is that the states are either on or off. However, we now have another option with computing in the form of quantum computing where the states are on, off, both and neither (all bits are in all possible states at the same time). So, the flaw of this paradox is that it's based in bivalence, autological and heterological words demonstrate that the bivalence model is flawed, not logic. So, to overcome this, we need a new model to describe both bivalence and the flaw we observe. Don't ask me what that model is, I haven't the faintest clue. But, this is the beauty of logic, it's not a yes or no answer, it's a yes, no or your premise is wrong.

[@nruojos]

I am not an expert, simply a hobbyist in finding knowledge in all sorts of fields and the funny little links between them. Something in my heart wants to say this connects to graph theory. Surely these loops in logic can be correlated to some set of rules between nodes that makes these kinds of phenomena in logic generalizable

[@jfredett]

Indeed, you can find your way to graph theory from here by taking a trip through Category Theory and the work of Groethendieck, though the road is rocky and not easily traveled. CT is very much the generalization you're looking for (indeed, many of these theora come back to relatively simple statements about diagrams in category theory), and has it's own clever insights as well (Yoneda's Lemma is perhaps one of the most profound results of modern mathematics, and it is so simple, subtle, and powerful that you can learn it in a moment, and spend years trying to understand it).

[@johnculver9353]

Did set theory resolve this problem by prohibiting the definition of a set of n from containing the set n itself? I think it was Wittgenstein who suggested this (Russell's student)? I am in no position to confidently argue the merits of the veridicality of the solution as it (and virtually all of your great content) remains quite far outside of my formal schooling, but these fundamental cracks in logic always concerned me.

[@susulpone]

There is multiple resolutions, depending on which set theory you use. The wikipedia article on Russel's Paradox is quite good in describing resolutions en.wikipedia.org/wiki/Russell%27s_paradox

[@ABetterName22]

I mean I feel like the obvious problem here is that we are trying to figure out wether everything can be sorted into two distinct categories. And then they use the logic built around this presupposition (that everything is either one thing or another) and can’t prove it creating a paradox … it’s just circular logic. Introducing more options like “neither” will always create the same issue, but you are forgetting about superpositions or a state of being both. Why can’t the answer be both at the same time. Be

[@Houshalter]

The definitions of the bins exclude each other. The second bin is literally defined as objects which don't belong to the first bin. If you put an object into both bins, you are just ignoring the rules of the second bin which says it can't go there. It's as silly as putting "horse" into the bin of "two legged creatures", or "4" into the bin of "prime numbers".

[@ABetterName22]

@@Houshalter yeah but who is creating the definitions for these impossible to make machines. This is just a logical fallacy in the English language more than a real world paradox.

[@Atomchild]

There is an exception to every rule and the exception to this rule is the rule without exception: that there is an exception to every rule.

[@scientious]

I have to admit that I laughed when I heard your statement at the end about one tiny flaw putting a crack in the edifice. There's a major flaw in this type of reasoning since these claimed paradoxes can be worked through and resolved. That it shows the limitations of philosophy is not much of a surprise, but you are correct that it does relate to computational theory and the incompleteness theorem.

[@sapphie132]

Good old self-referential paradox. Love those.

[@vbcsalinasapologetics1242]

That's it. Bless you, Sartre. In _On Being and Nothingness,_ about page 12 if I recall, Sartre argues that technically Descartes' _Cogito Ergo Sum_ does not indicate that the individual exists, because the mind which observes the thought taking place cannot in fact be the same mind, at the same time, that is thinking the thought. Am I thinking a thought, or seeing myself think the thought? So, according to Sartre, we need a second _cogito,_ and almost an infinite regression of _Cogitos_ in order to justify _Cogito Ergo Sum._ But Sartre resolves this by saying that there is a duality in the mind or in the brain, which allows it to do both... Something like that; it's been 35 years since I read this... So applying it to the liar's paradox: The statement that "I am lying" does not apply to itself, but to the previous statement, whatever that may be. "I am lying" in that case is differentiated in time from the lie proper, thus it can be true even while I am (in the rolling present time) lying; that is, I have maintained a falsehood. But "I am lying" is not that falsehood which I have until now maintained; it is the terminal moment of that maintenance. The loose definition is of the word "am." Autological is not autological at the exact moment that it is describing itself; It is rather describing the concept which it would describe if it were autological. Wait, is that it? That's the edge of it; it just needs to be flipped over.... Let us think about this... Too late at night for deep thoughts...

[@lukedowneslukedownes5900]

This is an extremely summarized video, thank you for sharing. Semantics is so tricky and requires a great deal of philosophical thinking as well

[@mosubekore78]

The sentence below is true The sentence above is false

[@chrimony]

This statement is false.

[@theburritokids6151]

I'd like feedback on this, but every word except for those two have a definition, and whether or not it describes itself... "autological" and "heterological" are the words we use to define whether or not a word applies to them. They each have a definition which is used to define OTHER words and not themselves. Its easer if you think of those words as boxes, you can't put a box in itself so there's no point in trying. I hope this makes sense and if anyone sees a flaw in this, feel free to reply abut it.

[@medexamtoolscom]

Yeah I knew that was coming. It's just like the "paradox" of the town where every man must shave himself or be shaved by the barber, or the claim that all natural numbers are interesting because being the first uninteresting natural number would be an interesting property even though obviously there isn't something interesting about every one of them, which actually just proves that it can't be partitioned into 2 sets. Or the words "I am lying". Or wishing with a genie that he not grant your wish. I don't see it as "breaking logic" for something that involves cases of self-reference to have no solution for being partitioned into one set or another.

[@medexamtoolscom]

The omnipotence paradox, there's another one. The impossibility of omnipotence because god can't create a rock too heavy for himself to lift. It's all a result of the rules for a case being conditional upon the outcome of the case.

[@tedsheridan8725]

Similar example: The statement "This statement is false" can be neither true nor false. But the statement "This Statement is true" can be either true or false.

[@ddacoe0]

thing is, there's nothing to deem as false or true in this "statement"...because your statement isn't really positing a complete thought; it's just using the phrase "this statement" itself as a stand in for an ACTUAL statement. so, yeah...your "statement" is lacking a statement.

[@dialectphilosophy]

Sure does seem a lot like a disguised version of the Liar's Paradox 👀 Would be awfully coincidental if the other paradoxes also turned out to be disguised version's of the Liar's Paradox as well... 🤔

[@hugoballroom5510]

@@dialectphilosophy i hope this is going to lead to/through Lawvere.

[@applimu7992]

Diagonal arguments (the 4 mentioned in the video, along with Cantor's diagonal argument and a few others) are a very interesting result of self-referential theories :D

[@dialectphilosophy]

Yes indeed. This paradox and diagonal argument used in the other paradoxes are very closely related

[@lawrencedoliveiro9104]

I have a problem with the diagonal argument. It has to do with the distinction between the computable numbers and the real numbers, and the fact that any number you can write down as part of a list must necessarily be computable.

[@yerimkone4835]

Around the 4th minute when you show how sorting auto/heterological into categories results in contradiction, it seems you use definitions and negations. However in both instances, it seems like there is some tricky business happening with double negatives. Is there an error with the proof somewhere or am I misunderstanding something?

[@noirox4891]

Your comment is 11 days old already, but I am going to answer anyway: There is no tricky business going on here. Here is a walkthrough without any double negatives: Heterological is defined as a word that does not describe itself, autological is defined as a word that does describe itself. Now consider the cases: Assume the word "heterological" is an autological word. Thus, "heterological" falls into the category of words that do describe themselves. So let's apply the description: "Heterological" is a word that does not describe itself. So it seems to be heterological after all. Assume now that "heterological" is a heterological word. Thus, "heterological" falls into the category of words that do not describe themselves. But wait! If "heterological" means a word that does not describe itself and it falls into the category of words that do not describe themselves, then these are equivalent, it is describing itself! And thus, "heterological" seems to be autological after all. If you are interested, look up the problems at the end (That is, Russel's Paradox and Gödel's Incompleteness Theorem). They are much harder to understand, especially without a heavy mathematical or formal logical background, but give a stronger, more general statement that can be proven with mathematical methods.

[@bcataiji]

@@noirox4891 , the problem is the word is being used as both a definition and a category and used too loosely. For instance, lets look at noun. Noun is autological because noun is a word that describes itself, being a person, place, or thing (noun is a thing). However, a cow is a thing. But cow is not autological. You can't just substitute definitions and categories and expect it all to work out. The whole thing is framed improperly. This shows it.

[@lrrobock]

there is a trick. he changed the rule of the game as we were playing it. the initial rule was that if sentence 1 is True, it goes in box Left. And that if Sentence 2 is True, it goes in box Right. When it came to Heterological. He showed that sentence 1 was True and that Sentence 2 was False. Therefore, heterological, by the initial rules of the game clearly goes in box Left and not in box Right. Sure, the sentence 1 itself seem to point that the word belong to the other box, but the rule is to simply evaluate whether the sentence was True/False, not to act on what the sentence spells out, but only on if it was True/False. Same with the sentence 2, that spells out that it goes in box Left; but the only thing that matter is if the sentence was True/False and from that it shows that it did not belong to the box Right.

[@noirox4891]

@@bcataiji I disagree, but i can see how you arrive at your conclusion. I think you are trying too hard to make language rigorous, which it is not. I am with you in that I agree that in the video, definitions are loose and there can arise confusion. Yet the video is entirely correct, and it seems like the "paradox" is still working on you. If you care enough to investigate further, read into Gödel's Incompleteness Theorem and Russel's Paradox, which are formal mathematical formulations of this exact problem. The advantage of math is that it does not need to deal with confusing language, and I think if you take the tme and make the effort to understand those properly, your confusion will be resolved.

[@DesignateVoid]

Every Computer Scientist knows that things can be true, false, or undefined.

[@spectralglory6920]

The word "brown" is a word that "is the color of brown." The word brown may only sometimes be autological.

[@kanabhprates2103]

I'm a logician, so I'm super cool 😎 please value me, I literally graduate seeking this goal.

[@dialectphilosophy]

We support and value you!

[@30hp]

“This sentence is a lie”

[@mysticalword8364]

You know, people have it really easy these days in a lot of different ways... but damn, back then you could academically immortalize your name with a philosoraptor meme.

[@NotSomeJustinWithoutAMoustache]

You still can, philosophy is still a thing

[@StridersBored]

@@NotSomeJustinWithoutAMoustache Modern philosophy is ripe for discovery. We live in a unique age with an abundance of everything that’s never been known in any time in human history and we’ve made incredible strides in this field in the past century. You can definitely slap your name on a discovery

[@NavnikBHSilver]

Effectively it's a circular logic, which can be iterated upon but never perfected, and due to the logic being binary, it will simply alternate back and forth. It's really cool and reminds me of how I recently found out that some spreadsheet applications can now in fact iterate upon circular logic to come to answers that whilst flawed, can be excellent approximations.

[@lorenzodiambra5210]

if a tree falls in the forest and no one hears it, has it made a noise?

[@loulasher]

Prior to placing this into the context of set theory or any other related paradox, this sounds to clueless me like it could be chalked up as an error of nominalism. It also sounds like there should be a distinction between words and categories of words such that the words describing categories of words are either outside the words being placed in them, sort of the way a homonym is multiple words or how a word has multiple definitions. By analogy maybe this is like a category is a different dimension than the things placed in the category, such that the word being placed in a category is not the word labeling the category somehow-- and that "somehow" can remain a mystery to me, at least for the foreseeable future. It might be analogous to creation and a creator that is outside creation: words speculating about categories of words are like words speaking above their paygrade.

[@jonathandawson3091]

I'm not sure we can call Incompleteness Theorem or Halting Problem paradoxes. They are theorems. Also this is probably the most convoluted presentation of Russell's-like self referential paradoxes. In fact Russell's paradox is a lot easier to understand because it doesn't need you to define or create ambiguous things like words. Another good presentation of the same thing is the library formulation about books that do or don't refer to themselves.

[@javierflores09]

This is just another intetpretation of the paradox, while it does not quite do it for you, it may do for others. I find this easier to understand than Rusell's paradox

[@jonathandawson3091]

@@javierflores09 Ahh I see. Interesting, then I stand corrected - I could be wrong in thinking this would be more convoluted to most people. Edit: Actually, I think even if you liked the linguistic approach, this is not the best way to present the paradox. You could instead pose it as a job of classifying "Sentences that describe themselves" and "Sentences that do not describe themselves". You'd have exactly the same paradox, except you'd cut all that crap about defining two new words being mixed up and somehow diluting the effect of the paradox, like a drop of black ink in a cup of water. Those words, were were needless (except attaching the names who coined them - honestly another useless fact given Russell was the one whose name is already attached to the set-theoretic description), and could be avoided for a cleaner presentation.

[@freshbakedclips4659]

When you realized that "Wave-particle duality" in Quantum Mechanics is also part of this logic breaking insanity.

[@bubblegumgun3292]

that one is easy, modern scientist are just stupid, "wave particle " is a coherent as 3 gods in 1 light is a wave not a particle

[@langdonlycroft704]

Wow, I got an A in my logic class in college because I made basically the same argument to say why I thought what he was teaching was inherently illogical. (I was never a fan of how logic is taught in schools) we used to sit after class all the time and debate. Thought he was going to fail me but at the end of the semester he told me that he enjoyed our debates a lot. I did too. But I still think this is a weird way to look at logic. I only took a 101 class so I'm guessing there's an amount of mathematics background you're supposed to apply to this way of thinking but this "paradox" is kinda stupid to me because you're comparing two things that are so arbitrary and intangible, any conclusion you come to is sort of an opinion anyway. A word that describes itself? We never really even layed down clear ground rules for this. Like how is dog a word that doesn't describe itself? Do we have to examine the etymology because dog in English has multiple meanings? What about the word "Get" can anyone come up with a good argument for why that word does or doesn't describe itself? The whole premise of this seems pretty illogical to me not to mention the method for testing the idea. Maybe these German logicians came up with this paradox to prove a point about how silly all of this really is.

[@Luizfernando-dm2rf]

You're problematizing the wrong thing, of course if you change the meanings you won't arrive at any paradoxes. The point is that if you get those words with those definitions and tried to categorize them, the paradox is inevitable, breaking logic reasoning.

[@blizzard4231]

Maybe try to understand the meaning of „a word that describes itself“ better, because this is not at all undefined. Also, „dog“ does not describe itself, since the word „dog“ is not a dog. The word „noun“ is a noun though, and the word „pronounceable“ is pronounceable.

[@8stormy5]

I wish you would have explained why the word "Autological" also cannot be sorted. It's not nearly as clear why that produces a contradiction.

[@mahabubrahaman2729]

You just earned yourself a subscriber. Awesome stuff ! Great explanation !

[@dialectphilosophy]

Thanks for watching!!

[@drew8347]

The topic of this video is alright, but everything about the video could be made in higher quality by publicly available ai given the same premise

[@Leibniz_28]

This video was enough to know this channel is gold

[@carlosnr4434]

At first glance, the problem of the autological word cames as a confusion of the OR logic table, the XOR logic table, and the word "or" in english. By the OR logic table, if both (being autological and heterological) are truth, the table returns true. But for the XOR logic table, if both are true, it returns false. By that, we could think of "autological" as being an autological word.

[@dahatmuhseen1610]

Can someone explain why this paradox is different from the other paradoxes? Don't all paradoxes break logic?

[@_RedRightHand_]

Wouldn't it work like this? Heterological is autological because it's the word heterological, meaning it must be classified as autological, meaning it isn't classified as heterological. You need to separate the word itself from its meaning. Autological is autological because of the same reason, which is that it's the word you're defining it as.