Logic at its Limit: The Grelling-Nelson Paradox
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- Опубліковано 5 чер 2024
- Whoever said logic was infallible? Certainly not the German logicians who, in 1908, discovered this mind-bending paradox. Learn how they beat reason at its own game, all by asking a single question about two simple words. Don't skimp out or you'll miss the surprise twist at the end! Plus: a very brief, totally not copyright-infringing appearance by The Sorting Hat.
Enjoy the video and please consider helping support us on Patreon -- / dialect_philosophy
Contents:
00:00 - A Tale of Two Words
02:04 - Semantic Sorting
03:30 - Paradox in a Nutshell
04:41 - Grammatical Formulas
06:53 - Breaking Logic
10:12 - The Naive Objection
11:18 - This was never about words, was it? - Наука та технологія
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?"
Who cares
@@akshay4107 Trevor Cronath cares, hence why he pointed it out
@@akshay4107 11:55 We care 😁
@@akshay4107 I care
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.
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.
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.
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.
I can smell contradiction
agreed. this paradox is constructed; not natural
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.
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
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.
> too unnatural to be described as fully human, but too unique to be generated
this is the perfect description for bill wurtz
You have just described the uncanny valley
He just says everything slowly and clearly, that's hardly enough to call inhuman.
dear grido. what has been declared here?
becose there are paradoxes, there is no logic.
not here no;)
LMFAO isn’t that a paradox?
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
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)
@@dialectphilosophy also you can do "What the Tortoise Said to Achilles"
@@dialectphilosophy I hope you bring up the catuskoti or tetralemma.
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...
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.
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.
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.
you did it, you solved it
but what if you make a bin to put bins in 🤔
@@daneo1952 this is basically the solution that ZFC proposes by forcing sets to be unable to refer to themselves.
@@quorryraphael9980 what if you want to throw that bin away
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!
Rather than like the NP complete set, they are exactly the undecidable set of problems.
I loved how you pulled together Russell's, Godel's and Turning's paradoxes!
Sod Turning, what about Turing?
@@SigFigNewton even more than that. they are essentially the same paradox taking on different "disguises".
@@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.
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."
@@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 :)
@@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
You ignore consciousness, that's why you get paradoxes.
"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.
@@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.
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!
That's not a word.
@@SoulDelSol Word.
A word is a word but a sentence is not a sentence
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.
Adopting Russell's system in "On Denoting" solves the paradox. Basically, the thing is words have no intrinsic meaning, only full propositions do. No words can denote by themselves, so the whole idea of those two words is inconsistent, they don't exist the way they appear to. The propositions "there is X such that X is a word and X describes itself" (or doesn't) are false propositions, there can be no such words. Logic is flawless, we are flawed.
I feel like you can easily sort autological into the autological category. There is only a problem when you try heterological.
Yeah, I got confused by that. I was waiting for the explanation why autological can't be categorized as autological.
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.
@@vanlepthien6768 Thanks, I didn't think about that.
@@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 ] ❌
@@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.
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 🙏
Oh, I am so glad you highlighted the udiprod halting problem video there ^^.
Great video btw.
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
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.
There are only two kinds of words, paradoxoids, which create a self-referential paradox when describing themselves, and all other words.
@@mikicerise6250 Then how about a word like "non-paradoxoid" ?
@@faran_iqbal That would be heterological.
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.
Keep up the good work. Your videos are truly amazing and inspirational.
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!
You just earned yourself a subscriber. Awesome stuff ! Great explanation !
Thanks for watching!!
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.
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 UA-cam.
This is a great channel. I like how ti focuses on logic and math in a simple visual manner.
i have been personally struggling with this since I was young and I didn't eloquently phrase the concept/issue in my mind, but after watching your wonderfully explained video I finally somewhat understand what I was thinking about/struggling with. :)
I knew where this was going from the beginning but still loved every second! Amazing vid!
So far this is one of the most interesting and accurate channels I've ever seen in youtube.
Is it?
@@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) UA-cam 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.
@@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.
@@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).
@@TheOnlyGeggles Yes, that would be -- but where is the demonstration?
Great video. Very stimulating and relaxing. I love your channel.
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
yeah, that's the lack of so called "common sense logic", categorizing objects or anything as if they're captured phenomenon like numbers or letters in our mind; ...nominalism problem...
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.
@@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
It’s similar to a categorical error. The contradiction exists when you attempt to treat the word as the category itself.
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.
good reply
I don't agree
to category theory?
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."
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
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.
Is it Sylvan's box?
@@tomholroyd7519 no, but I see what you did there :)
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.
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).
@@khajiithadwares2263 interesting, though doesn't address my question.
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.
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
@@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
@@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.
@@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.
@@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.
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.
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.
We miss PBS Infinite Series 😢
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.
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, “⊥”.
@@lawrencedoliveiro9104 That sounds a lot like creating that "neither" category the video mentioned
@@SporeMystify The “neither” category represents a different answer returned after a finite number of steps.
Another fantastic video by dialect
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.
amazing video! just one suggestion: i wish you had shown the case for why "autological" could apply to either category
The issue with Autological is actually that it can fit into both categories, not that it can't fit into either.
@@JosephVozzo Prove it
@@allstar4065
Autological is a word that describes it self (autological)
Autogical is a word that does not describe itself (heterological)
@@vilmernyberg193 So, in reality, does Autological describe itself or not?
@@TiagoCavalcanti-ji6hu Yes
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!
Popped his YT cherry! Masseltov bro.
Welcome. It's an interesting road, to say the least.
Probably one of the most underrated educational channels on UA-cam. I hope to one day see you gain many more subscribers.
I love that you acknowledge past UA-cam 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.
Appreciate the support and thank you for watching! We will be returning to Relativity soon...
Thank you for producing this thought-provoking video.
Those where some of the most worth 13 minutes of my logical life. What a work of genius brother.
Gödel incompleteness in action?!
Thank you it is a really great work!
I think this fits more into Russell's paradox (at least intuitively). Of course all of them suffer from being self referential.
@@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.
Nice.
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.
@@argfasdfgadfgasdfgsdfgsdfg6351 lol. You are not conscious of consciousness.
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.
This comment is as interesting as the video lol
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!! 😄
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.
I'm a computer science student and I love to see how this is strictly connected to the Halting problems and the fact that math is cannot be consistent and complete by Incompleteness theorems of Gödel, I'm waiting for the next video.
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.
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.
@@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.
@@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).
@@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!
@@dialectphilosophy The essential mystery is that you don't take consciousness into account.
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.
i caught some sleepless nights too after dialekt's GR videos
We think you'll be resting easier after our next couple videos...
I have a feeling this is just a digression.
@@dialectphilosophy this gives me a lof of hope. No doubt you are the best
69thliker
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.
10/10 youtube comment
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.
This video was enough to know this channel is gold
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
Physical world doesn't exist. "Physical world" is just an idea in consciousness. See my paper "Meaning and Context: A Brief Introduction".
@@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.
@@GamingBlake2002 Give 1 single example of something outside consciousness.
@@GamingBlake2002 Also, why are you angry ? Lack of girlfriend ? So many vargins on za internet.
@@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.
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.
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.
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?
@@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
@@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?
@@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)
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".
I see this happening on UA-cam newer smaller channel forming and providing great academic content these days!!! Keep it up!
This paradox is like the machine that simulates it's self to see if it works and then gives out the opposite Boolean value
That's precisely what the solution to the halting problem he mentioned is..
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
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 ...
@@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.
@@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
@@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.
???
@@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
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.
just commentating to say the editing is fantastic
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?
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?
A second barber? 😅
@@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
@@feynstein1004 there is only one barber
@@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...
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.
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.
Okay, I'm gonna take a swing at it.
There's a third category, but the third category is "the first two categories." We were looking for a third thing but the third thing was the first two things as a whole.
I was thinking this sounds like that halting (or "stopping" as I remembered it) problem Turing had. Glad you ended up mentioning it.
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.
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 :)
What is GEB?
@@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 :)
@@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... ua-cam.com/play/PLBOgSgXfJ6B2nbZ_YREW_Nb-AX8FW9U9K.html
@@joekirkup2624 Often when I wander in the mind the problem of recursion comes up. Thank you! This seems very interesting, I will read it.
@@joekirkup2624 tyty!
Editing is marvelous
thank you for your time and persistence
Another great video Dialect. You have a special talent in distilling complex topics into easy to comprehend segments.
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.
Indeed, this is called "impredicativity". Having an infinite (cumulative) hierarchy of (type) universes avoids the issue entirely
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.
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.
@@TomKaitchuck Except most languages allow recursive definitions--definitions which refer (directly or indirectly) to themselves.
@@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.
I instantly thought of the Halting problem when this was presented. Really interesting stuff.
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.
if a tree falls in the forest and no one hears it, has it made a noise?
This reminds me of sentences that can be pronounced the same but have completely different meanings
This is all about if a set can contain itself or not. And is a pretty basic logical paradox.
I came up with a paradox:
If one wants to become different one can either think or not think about it.
If one thinks that one wants to be unique one cannot because at least one person in the world is thinking the same.
If one doesn't think about being unique there's also at least one person in the world not thinking about it.
And if one is unique without thinking there's also at least another person like you.
So uniqueness is not able to be achieved by anyone at least when there are two people in the world.
Really enjoyed the graphics/animation
Good old self-referential paradox. Love those.
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...
This is how Sarah Conner should have defeated Skynet
Great great GREAT video, congrats!
Excellent explanation, I already love this kind of thing and you managed to make me even more curious! Please continue making videos in such vein, and congratulations for your channel, Mr. Dialect!
feels like the Russell paradox itself is easier to understand
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.
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.
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.
logic doesn’t operate like that
@@CookieMasterRBLX but apparently, it does
He managed to convert a debate into math and simple logics. This way, any subjectivity is voided. Can't get more objectively than this 👏
But, in the end, he didn't manage to see his own genius
When the video started, I din´t expect to finally understand the struggle the struggle of the Turing problem fifteen minutes later
This is an extremely summarized video, thank you for sharing. Semantics is so tricky and requires a great deal of philosophical thinking as well
Hahaha “if you were able to follow that, you’re a natural logician,” nooooo I just suffered through Logic I at MIT this semester
amazing work
I've barely started the video but the moment he defined autological I knew where this was going... because I understand basic set theory!!!
Amazing analysis. Looks one could argue that there are 3 groups here.
1. Heterological and NOT Autological. Largest group.
2. Autological and NOT Heterological. This group includes the word Autological.
3. The word heterological.
Alternatively the definition of heterological could be redefined to include itself.
This would be similar to the way mathematics defines factorial zero as 1 using a different definition than the usual formulaic one which appears to be 0.
I assume that you mean
1. Heterological and not a synonym (including the word itself) of the word "autological"
2. Autological and not a synonym (including the word itself) of the word "heterological"
3. The word "heterological"
This does not solve the problem, though, since the word that describes words that fall into the first group (for which English doesn't have a word right now, but let's invent it: "overlogical"), would recover the paradox:
If "overlogical" fit into the first bin, then it would be heterological, meaning (according to the definition of "heterological"), that "overlogical" is not overlogical, meaning (according to the definition of "overlogical"), that "overlogical" is either not heterological (which would be a contradiction), or it is a synonym of "autological". But if it has the same meaning as "autological", then it would have to be in the second bin.
If "overlogical" fit into the second bin, then it would be autological and not a synonym of "heterological". Being autological, though, would mean that "overlogical" is overlogical, so, by the definition of "overlogical", the word "overlogical" is both heterological (which already gives us a contradiction), and a synonym of "autological".
You cannot avoid the paradox by adding categories. Also redefining words makes no difference, since the labels you use to refer to these definitions is immaterial.
Also, why would zero factorial be equal to zero? Zero factorial denotes the amount of permutations of the empty set, of which there is exactly one, namely the empty permutation.
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
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).
This video reminds me that when I was at university studying computing and linguistics, every time we got to look at logic, I would get a headache.
The way this started "Words are fun, aren't they?" felt like an accusation in an interrogation room lol
The sentence below is true
The sentence above is false
This statement is false.
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
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".
@@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.
You've just pointed out something that has been recognized for quite a long time - the fact that self-referential statements often lead to paradoxes.
This looks like s great channel. U hardly know where to begin. Most of its going to be over my head Im sure
i can already tell this is going to be a great video
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
Yes indeed. This paradox and diagonal argument used in the other paradoxes are very closely related
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.
Wow, right. Russell’s Paradox, the Incompleteness Theorem, the Halting Problem, and the Grelling-Nelson Paradox are really the same problem worded differently. Never thought about that.
This is a great description of the halting problem in computer science.
Well, I had this problem programming using a 1-bit flag and could not get the logic to work consistently. When changing the code, the other way a bug would re-appear. Then, changing it back, the same problem would re-appear. Now I know why it's not a clean function. LOL. Thanks.
wtf are you talking about.
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.
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
Every Computer Scientist knows that things can be true, false, or undefined.
Well done.