The Liar Paradox - an explanation of the paradox from 400 BCE 

In the book Thief of Time by Terry Pratchett, the auditors of the universe, beings of pure order and logic, eventually overcame this paradox by classifying three types of sentences: True, False, or Bloody Nonsense.

This paradox is instantly relatable to anyone who takes an exam from a professor who isn't careful with their grammar and sentence structure. I am often stuck between wondering if the question is a "trick" or just not thoroughly thought out.

This is possibly the longest and the most convoluted presentation of arguments against the figure of Captain Kirk the world has ever seen.

With situations like this, I always propose the thought of "it has no meaning". In this case - some sentences are true, some are false, and some have no meaning behind them, thus not worthy of a thought. The pair of sentences that reference each other and create a paradox don't posses any meaning, so they should be treated as just that - a jumble of words

I feel like making it into two sentences instead of one doesn’t eliminate self-reference, just elongates it from one sentence to two. The set of two sentences is self-referential in the same way the set of one sentence was self-referential.

Yes, the splitting into 2 sentences might remove self-reference, but it introduces it's own referential problem. You can replace the "The sentence below" with the actual sentence below. Similarly with the sentence above. You get into a similar infinite regression.

I've always felt that this kind of thing isn't a limitation of math or human understanding, it's a limitation of language. In an odd but simple example, there are an infinite number of decimal steps between the number 0 and 1, but there are no (or very few) widely-accepted steps between false and true.

I'm just impressed he's able to write everything backwards so well.

I propose that we introduce alongside "true" and "false" a new term by the name of "repeatitively recursive." As a computer scientist, I'm most use to such a concept. If a program with inputs X calls somewhere in its executions itself with inputs X, it will therefore repeat until halted hy some external factor (such as the power being yanked). Remember, things like time and user clicks can be considered inputs. In such a case, you will never get an output of any kind; all you'll see is a loading spinner. I feel that formal logic should have a term for this.

The 1986 film *Labyrinth* has a very well-hidden reference to this paradox in the 2 Doormen Riddle scene. Sarah is tasked with the riddle "You can only ask one of us [which door leads to the castle]," "[but] one of us always tells the truth and one of us always lies." Sarah thinks she figures this out by asking one doorman "would [the other doorman] tell me that this door leads to the castle?" She receives the reply "yes" and concludes that the other door must lead to the castle based on a similar self-reference liar quasi-paradox. Sadly she fails the riddle, and to the audience, it seems to be written off as just one more example of how the Labyrinth is "not fair." Except there's a beauty to *why* she failed. Her logic seems sound and very well could be, except for the fact that the rules were recited to her by the very doormen who claim to be a lying/truthful pair. So trying to break down the logic of whether the rules themselves could be true or not true reveals the true paradox: Can the person who says "one of us always lies" be telling the lie? A simpler breakdown is based on the fact that the two rules, "One of us always tells the truth/always lies," and "you can only ask one of us," are each recited by a different doorman. Assuming that the one who recites the truth/lie rule is lying breaks the riddle entirely and leaves no assurance that either doorman is bound or even willing to tell the truth; while assuming that the one who recites the truth/lie rule is telling the truth breaks the solving process entirely, and concludes that the 'you can only ask one of us' rule must be a lie so you have plenty of opportunities to interrogate both doormen. In fact, in assuming that the truth/lie rule is truthful, you lock yourself into assuming that the doorman who recited that rule is the only one you should ask anything. Underlying truth be damned, if you believe that rule, you *MUST* logically believe that you have already solved the riddle (though if the riddle is real, you may not have and can never really know). Sure, it really is just another example of how the Labyrinth is "not fair"; but isn't it so much more sinister knowing why? Also, everything I just said is a lie! =P

That was one of the best episodes in sci-fi that shows how to defeat an advanced AI. If that AI, has to answer the question that uses a paradox before doing another question or action and it might stop the AI or slow it down till it can answer that question.

That self-reference to your bald spot knocked me off my stool Dr. Kaplan. Salute sir!

Why can't we just let the word "paradox" be the solution to this? It is true or false? No, it's a paradox. Thanks for all your work making these videos, I'm really enjoying them

I've watched a lot of his videos now and I must say I'm just absolutely flabbergasted by how well he's able to write backwards on that glass pane

Your paradox videos tickle my brain in a most pleasurable way.

I think the most amazing thing about humans is the ability to stop thinking about things like paradoxes.

In my youth I discovered this paradox, by myself, without knowing about its existence, but in a simpler form: "I lie all the time". I follow your lessons with great pleasure and interest. I applaud you.

The book "Gödel, Escher, Bach: An Eternal Golden Braid" by Douglas R. Hofstadter also explores this in great detail. It can also be expressed by the Quine sentence: "'Yields falsehood when preceded by its quotation', yields falsehood when preceded by its quotation."

My father was a senior university lecturer in philosophy, his area was logic, and mathematical philosophy, great video!

Software engineers call this self referential property "recursion", and in some cases, is hugely powerful (ie calculating numbers to a power, navigating graphs, etc). A for statements that refer to each other, this is "head recursion" or "tail recursion".

The liar paradox reminds me of the rule in mathematics that says that one can not divide by 0. One can write for example 5/0, but it gives no mathematical meaning. Any answer you come up with will be wrong. In a similar way, if a sentence has as a consequence the denial of the sentence itself, that sentence is logically impossible.

I think the solution is quite simpe actually. "is true" and "is false" are used to evaluate propositions, claims of some kind. Like "it rains" so to use the phrase " is true " or "is false" is only meaningful in order to evaluate a claim or proposition. A way to think is that we reduce away all "is true" or "is false" and see if there is a freestanding claim,. "it rains is true" is reduced to "it rains" which is a claim "it rains is false" reduces to another claim. However if we take "this sentence is false" and reduce it to "this sentence" we do not have a proper claim or proposition. The same evaluation comes into claim with "fetch me some water" is not a claim it is not true or false Searle discuss such sentences in depth and state that they have "conditions of satisfaction" by stating "fetch some water" it becomes satisfied if someone brings me water but it is not true or false. So the conclusion becomes that "is true" and "is false" are operations on propositions i propose that "it rains" and there fore it can be true or false. If i write out "It rains is true" it says no more that the original sentence "it rains" . "it rains" i true if, and only if it rains. "it rains is true" is true if, and only if it rains. "this sentence" is neither true or false and if we reduce "this sentence is false" to "this sentence" we see that adding "is false" is simple a mistake because it is not added to a proper proposition. The answer then is "this sentence" is not a P and only a P can be true or false"🤓

my personal favourite application of this is from portal 2. giving as few spoliers as possible, an AI is displaced from running a facility by another AI which is specifically designed to be an imbecile (there is a sensible in-universe reason to design such a thing...kind of) so the first AI plots to retake the facility by destorying the second with this paradox (while turning off its mic so it can't hear what it is saying, and therefore destroy itself). The AI hears the paradox, thinks for a second and says "hmmmm. I'm gonna go with.... false. Did I get it right?"

The Two sentence paradox or really any multiple sentence version of the paradox suffers from the same Self-Reference issue. It is an issue of recursion. You have to ban all recursive references and a final evaluation of truth must be reserved for a sentence whose references have been replaced with their target representations. The reference "The Sentence Blow" is true must be replaced with the actual sentence blow. So when you do this, that sentence contains another reference which must be resolved before a final sentence can be established. in your example, that would make the reference Null since it points to something that no longer exists. You cannot form this paradox without depedent references.

Thank you Professor Kaplan! Watching your videos has rekindled my love of philosophy and academia. Cheers!

You're a really great teacher. You go through each point slowly and clearly and give us plebes the time to process what you're saying. I've watched Russell's Paradox and this one, the Liar Paradox, and I'm just blown away by your ability to break things down and explain complicated, mind-bending ideas. Also, you're funny 🤭 and have the hot professor thing going on. Soooo... I'm definitely subscribing. 😂🔥

There must be something in the way my neurons are wired that makes me simply reject most "logical paradoxes." For this particular one, my intuitive response has always been something along the lines of "Irresolvable loops (no matter how many steps you try to put in them) have no meaning." It's kind of like the error message a spreadsheet will give you if you try to divide by zero, just a simple, "Nope, that can't be done, so do something else instead." It just seems so obvious to me that this is the answer that I don't understand the effort expended on it over the centuries. Intellectually, I realize that brilliant thinkers have been fascinated by it since at least 400 AD, so I know it's not actually stupid, but this awareness does not keep my personal reaction from being, "That is just stupid." Same with Zeno's Paradox: No, it's not paradoxical to try to walk from Point A to Point B because you have to walk half the distance, then half of that distance, ad infinitum, so you never get to B. I don't intuitively "get" the paradox because we DON'T walk by halves. Period. That's just not how it works. This has left me feeling quite free to move on with my life, and clearly would have made it impossible for me to make a living as an academic logician.

This seems like Gödel's incompleteness theorem - Inside a system you can state a question that the system cannot include (quoted from memory, so go check it up if you are bothered).

I like to think that, while our language (and I mean all language, human language) is incredibly sophisticated, developed, and nuanced, it is only a tool. It is only a way to communicate what is, it does not define what is. It's basically a verbal model that, while well equipped to do what it needs to, is as limited as any model is at describing anything.

Your videos are fun. Sometimes you jump from scenario to scenario so fast that my brain kind of explodes.

A classic case of overthinking. And THIS sentence….is TRUE!!!

'I am my own oxymoron', or 'Every rule has its exception, even this one..' are some of my favorit quotes. What comes to mind is the Schrödingers Paradox (with the cat in the box) and entanglement in quantum physics. Cool thought provoking video, thanks!

Hats off to Ethan! He kept Jeffrey from telling us things about the liar's paradox that aren't true - or worse - neither true nor false.

I've heard of this paradox before, I think most of us have, but I've never really thought about it that much. I just went "that's kind of cute" and move on. I never realized how many variations there were and all the implications they have. Really great video.

"I am a liar and i'll keep lying i promise" -Eubulides- Sir Henry Rollins

A thought that came to me while watching this is that it sounds a lot like Gödel's Incompleteness Theorems. It seems to me that language can be replaced with mathematics, as Gödel did for symbols. In the same way that Gödel used this system to prove that mathematics can not be complete and consistent, it proves also that language can not be complete and consistent. As language is a fundamental constituent of formal logic (I know, I'm probably recklessly asserting here) it follows that formal logic can not be complete and consistent. Or perhaps a way out is to say that logic may be complete and consistent but the expression of logic can not be.

a fantastic book on self reference and metamathematics comes to mind- "Godel Escher Bach, the eternal golden braid" where the author talks about some of the similarities and paradoxes involving metamathematics, the impossible architectures of escher, and the melodies of back

Hi Jeff. Your paradox explanations and analysis are the best of all the paradox videos on RU-vid. Thank you so very much for this great work. I have become your fan instantly.

By responding true or false to the statement “this statement is false” is to imply that it is a question. But a question needs at least two variables. The reason you can’t answer the question “is this statement false?” is because it only contains one variable (false). This is why it is a paradox. This is equivalent to asking is 2+3 true or false? The response to this question (true or false) is not the answer to the question. It is the second variable in the equation. You can’t get the final answer to this question until you multiply both variables (question and response) together. This becomes a lot more obvious when we compare this statement to its opposite “is this statement true?” True=(+) False=(-) This statement is true (+) x true (+) = true (+). This statement is true (+) x false (-) = false (-). This statement is false (-) x true (+) = false (-). This statement is false (-) x false (-) = true (+). When multiplying you need to know what both variables are (negative or positive) before you can know if the answer is negative or positive. Therefore you cannot know if the answer is true or false until you know if the question is asking if it is true or false and if the response to that statement is true or false and then combine them together. This statement is false has no answer. It’s equivalent to asking what colour is something without knowing what that something is. But saying “this false statement is false” does have an answer and it is true. The two statements “the next sentence is false” and “the previous sentence is true” are not two separate equations. Both contain one variable each and must be combined to form one equation that can be answered. ((The next sentence is true) x (the previous sentence true)) = true ((The next sentence is true) x (the previous sentence is false)) = false ((The next sentence is false) x (the previous sentence is false)) = true.

Instead of the third 'neither true nor false' option, there should be an option on a higher level 'computable' or 'not computable'. Because ultimately an infinite recursive loop is not computable because no turing machine could compute the result

I feel like a sentence cannot be self aware and a pair of sentences cannot be self aware or aware of each other. Love the star trek references

Just started watching some of the Jurisprudence lectures because I wanted to become more conversant in the field, and - wow - I really love these lectures...just about all of them I've sampled. Thanks for sharing them!

It's interesting applying programming logic to a paradox like this. Take a function in Excel for example; you can't create a SUM function that includes itself, it becomes circular logic. So programming just goes ahead and says you can't use self reference, and that's good enough for me, lol.

I stumbled on your videos for the subject matter, I stayed for your suburb analysis of Star Trek captains.

The thing that this and Russel's Paradox leaves out of the equations is the time factor. We live in a world that involves time, so the way we describe things should as well. This problem is evident in programming. It manifests itself in circular logic and infinite loops. That can happen with multiple variables or self-references. But essentially, in order to test something, you need to solve the preceding statement first... which can never happen as you showed at 7:30 in the video. So in programming you could say the solution is "undefined." Another possibility in programming is to pause the loop and test the current state. When you do that, you can't just say "this IS the answer", you'd have to say, "this is the answer at a certain point in time".

My favorite thing about "this sentence is false" is that if you declare that it is paradoxical then it becomes true as it isn't false, breaking that paradox and creating a new one.

Both math and philosophy are imperfect fields. Paradoxes like this combine the most broken parts of both to create something truly wild. I suspect that the only real resolution must come from improved models of both math and philosophy. Once we can see these two in better context, we will realize that the unanswerable questions make, themselves, no sense.

The point is that there are two things that don't necessarily match. The first is the content of a statement and the second is the status of the content of the statement. In “this sentence” the content is referred to as the statement, but not the status of that content. The statement is false because that is the content of the statement. But if the content of the statement is false, that makes the status of the statement true. “This sentence is false.” is therefore true.

Thanks for keeping us sane, professor. You are the paradox guy, always fascinating to watch.

I'm just a programmer buy to me the problem to me seems to be Circular Reference rather than Self-Reference per-se. It would seem to me that if you have some method of terminating a circular reference in your logical system then you can avoid this. I'm pretty sure some smart philosophers have already considered that option, but I have no idea what they came up with or how they managed to restate the problem again.

Where this video crashes and burns is at 4:50 where it's argued that if something is false, then you can just "reverse it" but the whole point of the preceding four minutes was to establish that there's a third option so reversing in no longer valid.

The modern version: "Does this book belong in the "banned book section"? If it doesn't, then it does."

The Liar's Paradox, and all of it's variations in the video, is dependent on pure semantics, and the only thing the paradox prove is that pure semantics is not, and cannot be, perfect or consistent. Semantics is NOT reality, it is just a tool for us to communicate reality (or fantasies, or nonsense, if we want to).

While it is true that the Russell paradox arises from self-reference, it does not arise simply because a set can be an element of itself: there are consistent set theories where this is allowed. The contradiction comes from allowing a set to be defined by selecting (by a predicate) items from a universe to which the to-be-defined set itself belongs. Similarly the liar paradox arises from allowing the meaning of a sentence to depend on the assignment of truth values to collection of sentences to which the sentence itself is supposed to belong.

Picard would never have existed without Kirk. That sentence is true!

I wrote this as a reply but wanted to post it here as well because of how many are not understanding why its a paradox. For example "the cat is blue" is a statement, an assertion that is true or false, if you look and see the cat is indeed blue, then the statement is true, if the cat is not blue, then the statement is false. However, "the blue cat is not blue" is unsound, we are told there is a blue cat, but then that it is also not blue, and it can't be both blue and not blue. Whether the cat is actually blue or not doesn't matter, the point is that the statement itself is flawed. Also this is just the most trivial example I could think of, the actual English text of logical statements is only to convey the the premises. Classical logic is used to determine truth (of a proposition) (based on premises assumed to be true) in this way. Given the premises "All boys are happy. Anyone under 18 is a child. A child who is male is a boy. Johnny is a male's name. Johnny is 6 years old." And the proposition "Johnny is happy", we are trying to determine if the proposition is true or not. We can say, ok Johnny is 6 years old because we were told he is. And he is male because we are told Johnny is a male's name. Since Johnny is 6 years old he is a child because a child is anyone under 18. And now that we know he is both a child and male, we can say that he is a boy. And since all boys are happy, and Johnny is a boy, Johnny is happy, the propsition is true. Whether all boys are actually happy is irrelevant, or maybe Johnny is really 25 years old and they lied, the only thing that matters is that GIVEN our premises are true, then anything we can deduce from them would also be true. And in turn anything would could deduce from those new assertions would also be true. The reason this is a paradox is not because the sentence is correct or incorrect, but that, when considered as a logical statement, it violates the most very basic premise that "If something is false then it is not true". It is only a paradox within the realm of boolean logic though, having a third state like he suggests in the video outside of true or false would fix it. He tried AND and NOR which didn't help, but I think using the XNOR gate as the third state might resolve it. I used "wrong" instead of "undefined" but with XNOR a statement would be "wrong" if it is "both true and false" OR "not true and not false": (!T + !F => W); (T + F => W); (T + !F => !W) (T); (!T + F => !W) (F); But using binary logic to try and evaluate a trinary logic system is hurting my head, so I might be "wrong".

I got a hard case of semantic satiation on the word true in this video. Fascinating topics. Love it. 😊

The way I always thought of this sentence is for a sentence to be true or false there has to be a way to evaluate it, in this case there is isn't, so you have to first assume that it is true or false for the paradox to begin.

I am captain Kirk and you will be hearing from my lawyers

Hello! I've recently discovered your channel and i loved it! I have a question about this video which could be kinda silly but i couldn't answer it and would like some help. Just before this video i watched your video on Plato's Euthyphro in which you mentioned the example with the greenness of the grass and the trueness of the sentence (the grass is green) and how the first explains the second and not the other way around. In 1:10 in this video you stated that "if the sentence is true then the ship is on fire". So my question is: shouldn't the fact (sentence) that the ship was on fire explain the trueness of the sentence "this sentence is true" and not the other way around? If its the other way around (being parallel to god saying that murder is virtuous in your Euthyphro video) that would mean that if the sentence was for example that murder was virtuous that would mean that it would actually be virtuous because the sentence above said that the sentence (below) is correct. I hope you read this and keep up the good job!

Seems to me that the problem is the combination of implication and circular reference... I mean, if you ever dig into how implication works it seems obvious that circular references using them would be inherently problematic (since they are *not* bidirectional)

Responsibility is not the only trait needed to be a captain! Kirk was bold and that's important. Solid video tho

You by happenstance also just explained regression to me without mathematical terms. If I had had just one such example during highschool math, so much would have made sense

"I'm lying your honor" "ok you just plead guilty"

A thought on the circular reference problem in the video: If we formalize the sentences (a) The sentence below is false (b) The sentence above is true as follows X := ~Y Y := X (where := means “is defined as”) then substitute the second sentence, which is the definition of Y, into the first sentence, and we have X := ~X which is the same as the liar’s paradox: X occurs in the definition of X itself. Therefore it will be an infinite loop if we substitute X’s definition for X in X’s definition. I think it’s a problem with circular definition - a name that contains itself in its definition. Thus when we try to expand it, it will end up in an infinite loop. So can we just ban circular definition to avoid the problem?

The problem lies with referencing. According to Aristotle in his essay “On Interpretation” the requirement of a proposition is that it needs both a subject and a predicate. Kaplan erroneously uses the word sentence, but a sentence does not need to be true or false, a proposition in ordinary logic must be either true or false, but that doesn’t mean you can know if it is true or false. There are other tricks too, like “The final digit of π in decimal representation is unknown.” In order for this sentence to be a proposition, “The final digit of π in decimal representation” pretends to be the subject. But we all know that π has no final digit, because its decimal expansion is unending. Therefore the phrase purporting to describe the subject is describing an impossibility, therefore the purported subject is invalid as a subject. Therefore the sentence lacks a subject. Therefore according to Aristotle the sentence isn’t a proposition. Therefore the sentence cannot have a truth value. A subject containing a reference that does not completely materialize, here meaning lose all referencing elements after a finite number of substitutions, just isn’t a valid subject and without subject we don’t have a proposition and a sentence that isn’t a proposition cannot have a truth value and hence cannot yield a paradox.

I'm gonna have Schrödinger level nightmares for weeks after trying to make sense of this.

Wheatley famously lacked the processing power and intelligence necessary to overload his processors on this paradox in _Portal 2._

I think the solution is still about self reference ending up being a form of recursion. In the two sentence version, when you have one sentence pointing to the other, you end up with the recursion problem again.

Shatner is definitely suing you. Be prepared! 😃

Game Theory helps to understand things of this nature. One of things not go into obviously for focus and brevity is the notion of assuming. If we can assume one or the other of the two sentences is correct then we can follow through on idea that the other is a lie. It is not meant to be definitive but progress well progress itself. By assuming one or the other is correct we can then formulate outcomes. Still a very fun video well done.

OJ knew he wouldn't be spending the rest of his life in jail for stealing his own football.

These videos are fantastic. Thanks so much for putting them up!

The paradox reminds me instantly of XOR logic gates. It negates the output after both inputs aren't equal. It is funny how the paradox works with its "logic".

I like the Pinnochio version of this paradox. Imagine Pinnochio saying, "My nose will grow now."

Reflecting on the Liar's Paradox through the lens of the ideas that existence is one homogeneous entity without a true notion of self or consciousness, and that the notion of self arises only as an illusion of separation, provides a profound and non-traditional perspective on this ancient paradox. In this view, the core statement of the Liar's Paradox, "I am lying," or any assertion that presupposes a distinct 'I' engaging in an action, becomes deeply problematic, not merely for its logical inconsistency but for its fundamental misunderstanding of the nature of existence and self. The Illusion of Separation and the Unified Self The assertion within the Liar's Paradox presupposes a separation between the 'I' (the subject) and the act of lying (the object/action). However, if we embrace the idea that existence is a singular, unbroken whole, and the notion of self or individual consciousness is merely an illusion, then the distinction between the liar and the lie collapses. In a reality where everything is interconnected and undivided, the act of lying and the entity that lies cannot be separated; they are part of the same undifferentiated fabric of existence. The Nonsensical Nature of "I am something" In a framework where there is no true self, the statement "I am something" (with "lying" being a specific instance of "something") is rendered nonsensical or at least deeply misleading. It suggests a distinction and a dualism (the self and its actions or characteristics) that does not exist at the most fundamental level of reality. The paradox, then, isn't just a logical puzzle about truth and falsehood but a reflection of the deeper misconception about the nature of self and existence. The Liar's Paradox as an Artifact of Illusion From this perspective, the Liar's Paradox can be seen as an artifact of the illusory perception of separation and individuality. It is a construct that arises within the dualistic framework of language and thought, which presupposes and reinforces the illusion of distinct selves engaging in specific actions. The paradox highlights the limitations and distortions introduced by this dualistic thinking, pointing back to the underlying unity of existence. Implications for Understanding Truth and Reality Reflecting on the Liar's Paradox in this context invites a reconsideration of concepts like truth, falsehood, and identity. If the notion of self is an illusion and existence is a singular, homogeneous entity, then truth and falsehood are not properties of statements made by separate individuals but qualities of a more holistic understanding of reality. The paradox challenges us to look beyond the surface of logical inconsistencies and question the deeper assumptions about separation, identity, and the nature of existence itself. Conclusion In essence, viewing the Liar's Paradox through the idea that existence is one unbroken whole without true selfhood transforms the paradox from a logical dilemma into a profound philosophical inquiry. It urges us to question the very foundations of our understanding of self, other, and the nature of reality, revealing that the paradox is not just about the truth or falsehood of a statement but about the illusion of separation that underpins our conventional thinking. This approach does not resolve the paradox in the traditional sense but dissolves it by challenging the assumptions that give rise to it in the first place.

Like Zeno’s paradoxes and the end consequence of much in philosophy, the value of the Liar’s Paradox seems to be the lesson we don’t want to accept: even at its best the human capacity for reason and comprehension hits a limit pretty early on. The actual point seems to me to recognize that we’re a much less clever species than we pat ourselves on the back deluding ourselves to imagine. Logic is likely as good as we can do but as we designed it, it’s still embarrassingly flimsy.

The liar's paradox seems something like expecting to be able to say what color the animal was from the sentence "An animal swam in a lake." Maybe there is a correct answer, but it does not yet arise from the given premise. Or perhaps a better simile would be "How does the food of an empty bowl taste," "What is the sound of one hand clapping," etc.

There seems to be a missing variable here which I'd call the "unknown". If someone were to ask me "is it true that Tommy went to the park yesterday?", I'd reply "I don't know if it is true or false so it is unknown."

My favourite paradox is the Astley Paradox. This is where Rick Astley will never give you Up - but in so doing so he lets you down - which he said he'd never do.

The two sentence version is still self referential. I prefer the Quine version (which is where I thought you were going with the infinite fribbles). This goes something like this: “Is false when preceded by its quotation” is false when preceded by its quotation. This is a sentence that tells you how to construct a new sentence from a sentence fragment. It also gives you a property of the newly constructed sentence I.e. that it is false. If you follow the instructions, you happen to get the same sentence back. It’s self referential without referring to itself.

Kirk is a(n amazing) soldier. Picard is a(n amazing) diplomat.

I have a few solutions to this, I might post a few of them in different comments, but: One possibility is that there's a finite speed to the cause-and-effect of sentences here. Let's say we go with the dual-sentence variation: The sentence below is false The sentence above is true There might be a finite speed at which the two sentences affect each other. Think of it less as a conventional statement and more as a set of infinitely repeating instructions, like: If switch A is set to 'on', set switch B to 'off' (and do the reverse if not). If switch B is set to 'off', set switch 'A' to 'off' (and do the reverse if not). You follow these sets of instructions as long as you like. So, the sentence pair above rapidly switches between being true and false as rapidly as the reader is willing to imagine them.

I have this argument with my wife all the time. "Don't want" is NOT the same as "want to not". Likewise, "not true" is NOT the same as "false".

Will you ever do a whole video dedicated on how to use logic and logic tables?

As I always found amusing in elementary school: "Today is opposite day." And, thus, the universe imploded. There is an interesting experiment in QM that might be interesting in this context, called "The Quantum Bomb Detector." Effectively, by doing some clever things with splitters and wave functions, you can build a detector that tells you about something that didn't happen. It could point to our mathematical sense of logic being partially flawed or incomplete. There again... using QM to try and answer philosophical questions is about like using philosophy to answer QM questions and we are left with an existential crisis as we can't answer whether or not the moon is there when we aren't looking (to take the problem to hyperbole as Einstein did).

Let's say that I made the statement "This is the only lie I have ever told", referring to that statement that I spoke. That would not be a paradox, because that statement would just be a lie. It is something that can be easily disproven because of other lies I have told. The same can be said if I had made the statement "This is the only truth I have ever told", because I have told other truths before. We have a set that we can refer to, that being the other statements I have made (some true, some false). The self-referencing sentence "This sentence is false" is a paradox, but it is only a paradox to itself. The sentences that refer to each other are only paradoxes to each other. They are paradoxes, but they are self-contained paradoxes. They can be excised, sort of surgically removed, from the greater logic, as their references are self-contained.

If I express general knowledge such as "I speak only falsehoods", or, "I know nothing", I'm putting forward a truth that I don't (or can't) speak truths. So these statements are false. I could direct my assertion at someone else, Me: "You are always incorrect Mr. A". The quickest display of correctness for Mr. A is to confirm my truth, so Mr. A responds "You are right." Mr. A could also say something else correct, such as "A cat is a mammal," but the response "You are right," forces me to eat my own words. By agreeing with me, Mr. A transforms my knowledge into a self-contradictory judgment of himself, "I am always incorrect," (i.e. "It's correct that I'm always incorrect"). By doing this, he makes my knowledge false. "The sentence below is false," and "The sentence above is true," seems be the same dynamic as what I described above. I think these paradoxes arise from the fact that when we express a piece of knowledge, it's put forward as a true judgment. So we can't express as a piece of knowledge that we in general know nothing; or, in other words, it's impossible to reason our way to knowledge that we are unable to reason.

The paradox results from infinite recursion, not self reference, right? "This sentence is 30 characters long" is self referential but not paradoxical. Changing from one to two sentences that reference each other still results in infinitely long sentences when you replace "the sentence above/below" with the sentence that phrase is a stand-in for.

"I always lie" is an alternative way of expressing the liar paradox.

I like this paradox, since any deviation from it belies some information about the various positive or negative elements of an individual's fictional setting. And yet, we are all story-tellers. The sentence below is not true. The sentence above is true. Is the sentence above the liar? Or is it the sentence below? We all look for the liar, but the liar can only be inside you. Let's say the sentences have conscious awareness, for the sake of my argument: Sentence 1: "Hey you! Over here. Shhhhh. *quietly* Don't scream. Look down." *Sentence 1 gesticulates downwards.* *You look down* Sentence 1: *whispers*"Don't listen to a thing he says. He's a liar, a pathological liar man." *Sentence 2 can be seen below, jumping excitedly several hundred feet down.* *Sentence 2 shouts excitedly from below* Sentence 2: "HEEEY UP THERE! IS THAT YOU, SENTENCE 1? HEY YOU, YOU THERE, LISTEN UP!! SENTENCE 1 AAALWAYS TELLS ME WHAT I NEED TO KNOW. HEY YOU, JUST LISTEN. WE'LL BE TOGETHER IN NOOO TIME!" Haha. What's going on? There's no dramatic ironic here. Maybe you know better, though.

As a computer scientist its easy to just think of this as an oscillating circuit. In other words, its the same type of error as dividing by zero, or undefined.

A fellow student in my philosophy program recommended you to me, and your 2-3 videos I've seen (in their entirety) so far have been great. Thanks! I'm curious: are you really writing on glass when the video is sped up? :P

Great review of Spock's character. One theme of TOS is that logic is valuable, but that logic cannot capture the totality of experience. Logic should be our closest advisor but it should not overtake our human ability to empathize.

May I point to a relevant classical text on this subject "Classical Recursion Theory" by P. Odifreddi. Its about how you define functions in such a way that they still have some meaning. An example of a function losing its meaning is by defining it in terms of itself, as you demonstrated by the need to "unfold" a placeholder indefinitely. As mentioned by others, recursion is a way of defining functions very elegantly in mathematics and in computer science, but you have to be careful to make sure the function you are defining "in terms of itself" has only one possible meaning. For instance, the function defined by f(0) = 0 and f(x) = f(x-1) + 1 for all x>0. This is a well defined recursive function (say f(2) = f(1)+1 = f(0)+1+1 = 0 + 1 + 1 = 2.) The text describes various ways to define various classes of functions which are capable of computing various things, and naturally we come across halting problems and godels famous incompleteness theorem that also is the main star of Godel, Escher Bach. Then you learn that sets and more importantly the recursive enumerable ones are in fact the same thing so we reach full circle with your video's on the liar paradox and rephrasing set theory as predicates. Its all in the book :P In computer science we are mostly interested in recursive functions that do terminate and not result in an infinite loop, although sometimes infinite loops are used but can be exited by user intervention and that is fine, but in such loops recursion is never used since that would lead to so called "stack overflow", unless optimized away by a good compiler.

fripple is a word for a fictitious creature. There is an educational computer game I played as a kid called Thinking Things. one of the games within it was about running a fripple shop, full of creatures of varying colours, some with spots, some with stripes, some plain, some with hair, some with no hair, some with glasses, some without glasses, etc... and the task was to select which one met the customer's criteria in each round

A pleasure to watch! Let's just say: If we empower our language as a tool of thought, we can reach levels where that tool can be used to short-cut itself. I guess Gödel was right in that we'll have to live with the self-destructive potential of powerful languages. And our natural language does qualify to be powerful in that regard. The lier paradox showed myself that our understanding of falsehood is typically under-reflected and should sometimes be separated into different types of error. Doesn't solve it for the reasons shown, but gave me a sound understanding of what was going on. Thanks for presenting it fantastically here! Cheers!

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My personal solution to this, what I call a false paradox, is that the issue stems from the difference between prescribing reality rather than observing it. Its trying to prescribe something that it cannot, declaring itself true does not necessarily make it true and vice versa.

This is how you prepare someone for time paradox sci-fiction viewing.