Logic at its Limit: The Grelling-Nelson Paradox 

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?"

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.

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.

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.

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

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

This is how Sarah Conner should have defeated Skynet

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

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!

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

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."

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!

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.

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

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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

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.

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

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?

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.

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

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

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.

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 🙏

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.

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

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

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.

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.

He managed to convert a debate into math and simple logics. This way, any subjectivity is voided. Can't get more objectively than this 👏

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

Probably one of the most underrated educational channels on RU-vid. I hope to one day see you gain many more subscribers. I love that you acknowledge past RU-vid 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.

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.

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.

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.

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.

Good old self-referential paradox. Love those.

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 RU-vid.

May I break it even further? "Self-referring" Looked at under the assumption it's autological, then it is self-referring, thus it's autological; now assume it's heterological, then it's not self-referring, thus it's heterological. And it gets even worse when you look at the word "paradoxical", which when you look at it in a normal way is heterological through and through. But lets assume for a moment it would be one of the paradoxical words that fit in no/both category... now suddenly it's autological. The inherent problem seems to be that logic is math based, so it dies when it encounters infinity, just as it does in those recursion problems.

The 2-part pattern in all these paradoxes is that they are 1) self referential, and 2) are negative (using the word "not"). These two notions appear to be incompatible together in our usual systems of logic.

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

The sentence below is true The sentence above is false

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.

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

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.

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

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.

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.

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.

When he asked, "Do you agree with this statement?" I let out a small "uh-oh" as it dawned on me

feels like the Russell paradox itself is easier to understand

Another fantastic video by dialect

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!

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?

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

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.

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!

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.

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.

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 :)

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

This seems simple to me. The assumption that all words are either autological or heterological is false. This resolves the paradox.

I've barely started the video but the moment he defined autological I knew where this was going... because I understand basic set theory!!!

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

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.

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

The literal description of putting words in a container labeled with either "heterological" or "autological" is the correct depiction. To put what you used to define a structure into an opposite definitional structure will be impossible. To be able to do so would be the actual paradox. The snake can eat its own tail, but its head will not come out the other end.

These aliens are amazing. They think we don't know. They gave us the transistor. They moved us along. And now they're teaching us everything. They're on TV every day. This is great.

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

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

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.

The way this started "Words are fun, aren't they?" felt like an accusation in an interrogation room lol

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?

“This sentence is a lie”

This all boils down to one of two things: 1 - _You can't hack from inside of Windows."_ 2 - (Rick and Morty) "How the hell are you gonna' fix time when you're *standing* in it?!" It's not a logic puzzle, per se, it's a philosophical puzzle. We view the world from our field-of-consciousness... but you can't define "consciousness", because you're using consciousness to do it. A knife doesn't cut itself. It's not a new concept.

This video was enough to know this channel is gold

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.

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.

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.

This isn't a paradox. It simply means that the property of "describes itself" is not sufficiently descriptive to admit a yes/no classification for every word. This is what happens when you play logic games with the English language. This is why mathematics is superior in all respects. Speaking of which, mathematicians had a similar "paradox" in set theory to deal with decades ago, but they fixed it.

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.

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

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.

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.

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

When the video started, I din´t expect to finally understand the struggle the struggle of the Turing problem fifteen minutes later

Sorting Machine: "Are you autological or hetrological?" Autological: "Yes." Hetrological: "No."

The problem isn't the words, it's the boxes. Just changing to "ALWAYS autological" fixes the sorting.

Reminds me of Wittgenstein - "Whereof one cannot speak, thereof one must be silent."

You can expect paradox everywhere where things are self-reflected