The Liar Paradox - an explanation of the paradox from 400 BCE
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- Опубліковано 14 бер 2023
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The liar paradox goes back at least to Eubulides, the Ancient Greek philosopher and student of Euclid in the 4th century BCE. That’s the year negative 400. In this lecture video I explain what the liar paradox is and why it just won't go away. I also try to make some jokes about Captain Kirk of Star Trek.
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.
Para = beyond/aside from - let's say separated from. Dox = opinion/idea/belief - let's say sense. So a paradox is separated from sense - i.e. in the modern vernacular of Ankh-Morpork, bloody nonsense.
And this demonstrates clearly the genius of Pratchett.
he was a genius
Thief of Time, one of the most delicious books ever writen!
Reminds me of the caption in my Statistics 101 textbook which proudly stated that there are , "Three kinds of lies, lies, damned lies and statistics."
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 kind of reminds me of how my teachers used to teach the lessons, and told us to write everything down because it was going to be on the exam. But when you took the exam, they worded the questions so unintentionally, that you had a does not compute situation, and your brain started to fry itself trying to figure out what the hell the answer supposed to be, even though you knew the answer. Because the teacher said this was going to be on the test.
Ha. Exactly.
Definitely relate, grammar can be a bitch, but so can philosophy. Combine them and even the most experienced and knowledgeable tutor can be 'strung up'!
"Choose the answer that's least wrong"
Been there. It sucks.
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
Yes, similar to how the correct answer to dividing by zero is that it is ‘undefined’. Or to put it another way: garbage in, garbage out.
@@philcorrigan5641 I just had this same thought when watching the video!
That's just a less eloquent, and potentially incorrect way of saying "neither true nor false".
@@Squant em, no. You just refuse to play the game because it makes no sense, that's all
@@Squant Take a sentence that doesn't contain a claim, like a command: "Go fetch me some milk." Or a question: "Who are you?". You could technically say that the sentence is "neither true nor false" and therefore "not true", but in doing so you're stretching the definition of "untrue" to the point where it simply makes no sense to use it linguistically. "That command is untrue" or "That question is untrue" is wordplay at best and gibberish at worst.
The logic you're using operates under the false dichotomy that something can either be true or untrue and there's nothing in between or outside that. But sometimes something IS in between, half true, or (Most importantly) simply irrelevant or meaningless.
This is possibly the longest and the most convoluted presentation of arguments against the figure of Captain Kirk the world has ever seen.
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.
Really enjoying these videos. Yes so to my mind (with no formal philosophy training) the paradox arises due to the existence of a circular reference. Remove these from any set of sentences. Problem solved! I'm sure I'm missing something here....
It's more recursive than self reference. You could say a recursive system can't make reliable statements about itself.
I was also kind of thinking of dual self-reference, but I wasn't convinced I was right.
@@jeffwells641 Good point.
Circular logic be circular.
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
Fuck.
Hi, this was done earlier in Dr Who with Tom Baker - Pyramids of Mars
It was well explained
If he asked the "truth" guardian where would the other guardian point to as correct exit the truth guardian would point to the false door - death
If he asked the "lying/false" guardian which door would the other one point to he would lie and also point to the death door
The analysis was that both answers would highlight the lie every time so the other exit was the correct one
@@georgemaragos2378 i think the original comment takes it one step further by stating: if these rules are told by the brothers, (of which one is lying) that must mean that the rules are a lie. Which creates a new set of problems like: maybe they are both always lying. But you will never know for sure, and that makes it unsolvable
Right, just like a picture that looks “real”, the structure of language seems like “truth” because it’s the structured symbols that we base all learning and communication. Those brothers were both correct in what they said. It wasn’t until she put meaning to it that she was wrong. Like Schrödinger’s cat.
I like how they handled this riddle in episode 5 of Journey Quest. Where the guy presented with the riddle kills one of the door keepers(the one that speaks truth) the asks the other which way is correct. Then tells him that if he is lying he would kill him and asks if he understands. The door keepers says no, so the guy asks if he wants to die so the door keepers says yes, and the guy throws him through the door that is thought to lead to doom and when the door keepers doesn't die the guy seduces that that is the safe pass. The lying doorkeeper eventually becomes the guy's squire by always flattering him and being his "yes" man.
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.
It isn't a limitation of language. It is an intentional choice that "true" means "completely true".
An equivalent is true means 1 and false means 0. If the value is 0.5, then it is either true (1), not true (not 1), false (0), or not false (not 0). In the 0.5 example, it is both "not 0" and "not 1", so it it satisfies both "not true" and "not false".
Typically when people say something is false, they mean "not true". Or rather that the statement is not completely true.
An example of this is "Hobbits are small humans". It is true that Hobbits are small, but it is false that they are humans. People will say that the statement is "false" as a shorthand for "not completely true".
a sentence as such is not binary maths and can be not only "true" or "false", it can also be "an incorrect set of words", so there is a third option: "the sentence is rubbish"
That's one of the (not so great) options but you can formally express the paradox too. Language is what conveys you the paradox but not the paradox itself
I very much a novice in this area but I was wondering about that very thing as I listened. Then I wondered if this problem arises in the real world , or just a puzzle…and if that’s a measure of it being a language thing…
I'm just impressed he's able to write everything backwards so well.
Can’t tell if this is a joke or not but in case anyone doesn’t know, you just write normally and then mirror the video in the edit.
@@GlutenEruption Well that would make a lot more sense!
You my friend should watch some "shapes and colors" type of video.
Might to early for paradoxes for you hehe
When he wrote the sentence "fribble is not true." But wrote the sentence in backwards order from true to fribble, to much lol
@@GlutenEruption I didn't know either
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 essential duality of the Universe is how Edwin Schrodinger came up with his classic theory where he named his two cats Anny and Sheila after his wife and mistress.
We do, it's called "endless loop".
We should call it Schrodinger's Sentence.
In logic circuits, we often allow such things, in fact we even design them. The most basic example is the ring oscillator - just an odd number of inverters (n>1) connected in a loop. Many oscillator designs have some form of that at their core - self contradictory logic. They might have a bunch of analog trickery to get better timing, but at their heart, they are just some form of logic that can't make up it's mind which state it should be in.
@@ccoder4953 always saw that more akin to a pendulum. The potential is transferred and thus gone, but it comes back around again. Not necessarily paradoxical because you can trace it through time, even if we’re talking about speeds akin to the speed of sound through nanometers of copper. It becomes a ridiculously fast NASCAR race, full of nothing but left-hand turns.
In the same way, languages that aren't based on stack frames are generally more amenable to running recursively in perpetuity, whether singularly recursive, or mutually recursive.
Most Lisp languages (that aren't built on top of Java) for instance could happily oscillate back and forth to the heat death of the universe (or random traces on the motherboard, whichever comes first).
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.
That's where I was at. Which means you can ban all logic based on any loop of truth, and we already do that by calling out circular logic.
It's like an indirect self-reference
in short, the two sentences don't exist in a vacuum by definition they are linked by referencing one another,so saying that you can do liar paradoxes by making liar PERIODS instead of liar SENTENCES is just... more recursion.
That's exactly right. Splitting into two sentences doesn't remove the self-reference, it's just a higher order of self-reference.
Indeed! 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 in 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 it’s 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 reference 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.
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?"
I always loved this part of the game. GlaDos doesn't seem to know that even her knowing about the liar's paradox is, itself, a paradox. Just merely knowing and understanding the paradox and what it apparently does to a being of pure logic should destroy her as well based on her understanding of it (which means it shouldn't matter if she hears it or not.) But it doesn't destroy Wheatley, just like it doesn't destroy her. Man, there are just layers upon layers to that whole exchange.
It's not necessarily a paradox for her to know that she shouldn't use it on herself. All she really needs to know is that it'll break her circuits if she thinks about it.
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.
Computer, compute to the last digit the square root of 2.
Unfortunately, I don't think the AIs we're about to replace ourselves with will fall for it.
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
He is writing normally and the camera is inverting it. Look at his wedding ring. Some new U-Tubers have not figured out how to fix that camera glitch. He is simply using it.
@@tomboyd7109 Yup, the ring, and his watch, and his shirt buttons...
Ahh new here?
lol they just flip the video so its readable lady
Interesting..I simply assumed he was writing on a mirror, not a glass window, and the black background was to mask the camera; I never imagined anyone would see it any other way! 😂
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
Probably because they always try to force the sentence into a classical logic-ish interpretation. Like he did with the "this sentence is not true" example. Then used axioms and inference rules of classical logic on it to generate a contradiction. But in classical logic... a sentence that is true and false or neither is not a sentence at all. (well not a "proposition").
I concur. Our universe should contain, must contain, needs to contain mysteries.
Because you would have to consider the statement ; "This sentence is not a paradox"? In essence "paradox" == "Neither True nor False". Introducing a new term doesn't solve the problem.
Like the definition of x/0?
False, paradoxes do not exist
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."
GEB:EGB mention nice
That self-reference to your bald spot knocked me off my stool Dr. Kaplan. Salute sir!
I think the most amazing thing about humans is the ability to stop thinking about things like paradoxes.
That’s because language was made to save energy instead of spending more of it.
Seems I don't have that ability. Like it.or not, I keep seeing them.
I see dead theories.
I see “Mostly” dead theories!!!
Eh, because alot of people instantly see this as invalid. Regardless true or false, it is invalid as it violates the given ruleset it proposes. More paradoxes than you might think end up being invalid because humans are just masters at breaking things. With some being closer to untested Exploitive things and lack the deeper knowledge to either resolve the truth of it or make it clear it isn't possible outside imagination & fiction.
Paradoxes often happen because of a VERY simplistic take on a system with much deeper understandings required, and past that are invalidated because it was crafted to be that way.
Just like we view those who think flying is proof of either a paradox or conspiracy as idiots... in time we will view those who honestly believe such paradoxes in earnest as fools as well. The issue is 'we do not fully understand what we need to' for alot of these paradoxes, and some are just people being douches and breaking a system because they can.
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.
Chuck Norris can divide by 0!
@@irgendwieanders2121 lol Chuck Norris's beard can divide by zero.
@@randomrandomizer You can also define 0/0=1
Depends on your choice of axioms...
@@randomrandomizer That doesn't sound too wrong. Until you try it the other way around and multiply infinity with zero. How long you continue the row 0+0+0+0+...., you'll never make it to 5. Or to put it the other order: Zero infinities is not 5.
So your answer doesn't work, even though your suggestion is a tempting conclusion.
If Zero Mostel had a high school diploma than he understood basic maths. At one point he must have proved he understood division. Therefore at one point something was successfully divided by 'Zero'.
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.
Language is a man made construct. Assuming there is no fault in a manmade construct is false.
@@CodeguruX That statement is true.
Actually, we could potentially “walk by halves” depending on your definition. The only reason the paradox is false is because infinite sums don’t produce infinite values.
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. 😂🔥
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.
hey me too.
@@kennarajora6532 Nice. :)
For me it was the episode of SpongeBob where he proposes a "no rules" rule.
@@WWLinkMasterX
Oh, that's so cool and funny. :)
Yep me too 😁
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.
I hate how he said the sentence "this sentence is not true" can't be neither true not false because it can like this sentence isn't true ok then it's false no then it's true no ok so it's neither true not false
And then bro pulled the if it's Accurate it's true and I was like that's an ass pull it pissed me off cuz like yes technically you could do that but when the "this sentence is false" it doesn't apply then it wouldn't with not true as well or it would apply to both the same way
The fundamental problem isn’t recursion. Recursion is a symptom. The fundamental problem is the inability to resolve any kind of mutual exclusivity when both options are EQUAL. And this goes back to a fundamental and flawed assumption in logic itself:
That there is always a better/more accurate/superior option. And baked into that assumption is a second assumption: That logic itself is always the superior option to resolve X. Without even realizing it, every one of us who relies on logic has a secret belief we may not be aware of:
Logic is always the superior option to resolve X. If X cannot be resolved by [current understanding of logic], [current understanding of logic] is flawed. Ergo: A perfect, pure form of logic must exist that can resolve all variables of X.
At no point is the first premise ever countered. Deep down in our souls, we believe there is a pure, omega logic that perfectly resolves everything. Every possible paradox; Every possible scenario; takes into account every possible variable; and with perfect results every time.
This is an unfalsifiable assumption, and for reasons that baffle me - completely rejects the Null Hypothesis without even attempting to prove the premise is true.
My premise: Logic may not always be the superior option to resolve ALL instances of X, and by extension - no perfect, pure, omega logic exists.
For example - to the best of my understanding, logic cannot be used to solve the following:
A has a value of 2
B has a value of 2
Choose the option with greater numerical value. A or B. (And the question provides a radio button with which to select your response).
Since you cannot enter both/neither, or an another answer of your choosing (and are limited to choosing A or B as your only answers), this question cannot be resolved without relying on something other than logic (personal bias, random selection, etc).
@@kalanivernon7273 The problem is not (and cannot be) logic, since logic is rigourosly defined to be correct.
The problem is the presumption that language must be logically correct.
This and other so called paradoxes merely show that language lacks rigid (logically correct) semantic rules.
@@kalanivernon7273 Ohh, and to answer your question: B in virtually all cases...
While A and B might have the same value, B has a greater name.
In fact, B is the correct answer regardless of the values of A and B since your question only asks for the greater option, not the greater value of the option.
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".
Welcome to "deadlock"; a known potential problem with recursion. Ye olde GW-BASIC had a "deadlock" error message built into it, but I've never seen it come up. I assume it would only occur when using shared files which have locked records and GW-BASIC is the only version of BASIC I've seen which had the ability to use shared files and locked records, and I've never seen a program which needed to use them (although I did write a prototype "chat" program which used them, just for fun.)
Any computer programmer knows to be careful with recursion that you exit the loop. This seems like a case of recursive sentences that don't exit their loop. As soon as you refer to yourself, even indirectly via another function, you have to have a plan for when that loop ends
Speaking of computers/software .... which I am only vaguely familiar through game modding software , so pardon my ignorance.
But is that why "AND / OR / NOT" are used ? ( logic gates I guess you call them )
Giving 3 options instead of 2 ... and pretty much the idea of truth is somewhat singular in that it is the journeys destination . So false is in a sense taking no journey ?
IDK ... but maybe that is how the liar paradox can be solved ... embrace the journey ?
@@Mk101T it can be solved by a simple inversion truth table. Just take the sentence at face value then invert it.
0 = 1
1 = 0
@@Mk101T AND / OR/ NOT, and others, are used because computers operate on pure logic. Computers cannot perform mathematical calculations directly. They simulate them using logic operations on true and false values. I would give a better explanation, but it's been decades since I studied how CPU's work.
Here is an image from Wikipedia that shows the logic gates within a certain CPU used to perform basic arithmetic. upload.wikimedia.org/wikipedia/commons/thumb/c/c0/74181aluschematic.png/600px-74181aluschematic.png
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".
It's interesting, because your explanation actually bears a lot of similarities to an explanation for the liar paradox made all the way back in the 5th Century AD, by a linguist/philosopher called Bhartrhari.
a computer is sequential tests and operations, but the"this sentence is false" problem implies at the current point in time. like division by zero; always meaningless. or if a result is demanded for some reason, then "undefined"
@@jgunther3398 the brain is just sequential tests. Things don't make sense if you don't process one word at a time. It might seem instantaneous, but it's not.
@Kenna Rajora just looked into him. Yeah it does seem very similar to what I was thinking.
Right! It’s like “GNU is not Unix”!
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"🤓
Your paradox videos tickle my brain in a most pleasurable way.
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).
First thing I thought of.
Me too - I thought that Dr. Kaplan would mention, or explain more, linking this to Godels Theory
The last work should be theory (damn spelling correction!).
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.
That’s the whole point of the paradox. Our language is just how we communicate reality, its not reality itself. And because of that fact there are inherent quirks and ambiguities in our language which create logical paradoxes.
“This sentence is false” isn’t actually a paradox. It’s neither true or false. It’s not real… Which is kind of the whole point of the paradox…
For Chomsky it wasnt that thinking was a result of language, but that language taught us how to think.
I think it's both, and anyone who knows 2 or more languages pretty well will testify to that. The better your grasp of language the deeper you can develop thought but deep thought can also reveal a lack of words in a language (which can end up affecting whole societies and cultures unaware of the existence of such words known in other languages. @@ilovepavement1
Exactly. Our shared understanding of the "sentence", a construct in itself, presents a mobius strip of a kind, a flawed vehicle we use to navigate our universe: if-this-then-this in perpetuity. @@Joe-nh8eq
Your videos are fun. Sometimes you jump from scenario to scenario so fast that my brain kind of explodes.
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 would prefer "indeterminate". Self referential statements and the like can be indeterminate. You can't give it a label of either true nor false as it cannot be evaluated.
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.
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 am a liar and i'll keep lying i promise"
-Eubulides-
Sir Henry Rollins
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.
'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!
The Shrodinger's cat thought experiment is not a paradox. The cat really is neither alive nor dead until it is "observed".
@@N.i.c.k.H Thanks for correcting the terminology. I'm no mathematician, so: as the rules in creation seem valid, until 'observed' (investigated) and then revealing the paradox,- it still strongly reminds me of the Schrödingers two simultaneously valid states.
My favorite is "All sweeping statements are wrong."
It's not directly self referential, which I like.
@@miriam-english Haha, yes! Or similar: 47.5% of all statistics are made up on the spot.
@@miriam-english This is not a paradox, it is wrong. It is not a paradox because being wrong does NOT imply that "All sweeping statements are right" only that "SOME sweeping statements are right" and there is no reason to assume that this is one of those. If you believed it was true THEN you would have a paradox.
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.
To me, the resolution to the paradox is simple; the world is under no obligation to make every possible statement self consistent or even meaningful. The fact that a statement is neither true nor false is interesting (mostly in what it says about the language the statement is made in), but not at all surprising. In fact, the non-existence of such statements would be the surprising thing.
@@benjaminshropshire2900 Agreed, and I would go even further. A statement can NEVER equate to absolute truth or absolute falsehood, since words have no precise meaning, being sounds or markings that trigger memories of experiences, not having any fixed meaning. This even includes mathematical statements such as 1+1=2, because numbers have no actual meaning until assigned to represent something (e.g. 1 apple + 1 apple = 2 apples).
@@Real_Lion_of_Judah I wouldn't go that far. Words have precise meaning, at least to the extent the people using them agree on what they mean. And societal discourse depends on having wide agreement on that.
Where things get strange is when one side of a debate tries to win by trying to choose the meaning of terms the other side already used and then say that is what the other side meant. The problem there is that changing the meaning of words doesn't change the meanings people are expressing. At best it just pisses off the people trying to talk about something. At worst it tricks the other side into thinking they succeeded making it impossible to resolve anything in a civil way.
@@benjaminshropshire2900 What I meant is words never have *absolutely* precise meaning (to speak of relative precision in this context doesn't make sense). A statement can never be absolutely true or absolutely false. One could argue a mathematical statement can, but I would again point out that math has no actual meaning when the numbers are isolated (i.e. when they aren't representing anything "real"), and that always includes concepts with no absolutely precise meaning. What is the average airspeed of an unladen swallow? What do you mean? African or European swallow?
@@Real_Lion_of_Judah your clarification is saying what I already understood your original statement to say. I still disagree with both.
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.
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.
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!
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).
This is a macroscale example of the standard models superposition. At the quantum scale answers that are in superposition are accepted as such, but bring them up to the macro world and suddenly they are paradoxes.
I am captain Kirk and you will be hearing from my lawyers
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
Thanks for keeping us sane, professor. You are the paradox guy, always fascinating to watch.
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.
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
Thinkin' Things? Never forget to go to fripple town
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.
@@CorwinSTP No, we just learned what constitutes a logically consistent entity, and we refined our definitions to call those sets. Clearly, derivations of properties of inconsistent entities that are set-like fall apart, so it is important to make the distinction. There's no cherry picking involved.
This is why we created or invented the "paradox."
@@CorwinSTP it seems to me like most people are unwilling to accept that this whole thing represents a limitation of our present brain power; or our modus operandi to which we perceive/create the universe.
@@CorwinSTP The Russell paradox arises in a _formalization_ of naïve set theory. As an informal theory, naïve set theory only informally describes what sets are, how they behave, and what one can do with them (like take intersections of them, form sets of sets). For instance, while Georg Cantor definitely meant to study infinite (as well as finite) sets, I doubt whether he actually stated a clear rule that some infinite set exists (and without such a rule, one cannot _prove_ that any infinite sets exist; indeed Greek philosophers held that nothing can possess actual infinity, as an infinite set would, and this is a logically consistent point of view). So if one wants to do rigorous mathematical reasoning about sets, one needs to fix the rules, i.e., formulate axioms of set theory. Gottlob Frege undertook such a formalization (maybe not exactly of Cantor's set theory, but something essentially equivalent).
The set of rules must be proposed; there is no (cherry-)picking from a pre-existent set of rules involved. The main things that formalized naive set theory proposes about sets involves equality of sets (two sets are equal if anything is member of the first if and only it is member of the second) and an axiom about sets that (must) exist; for the latter it states that any well formed predicate (in the language of set theory) defines a set (of everything that satisfies the predicate). This is elegant and economical (for instance one does not need to state explicitly, as starting point, that an empty set exists, since the predicate that is always false "creates" the empty set; similarly, the predicate that is always true creates a universal set, as set of which everything is member. Unfortunately this elegant axiom also make the theory inconsistent, as Bertrand Russell pointed out to Frege. (Incidentally, the inconsistency arises by applying Cantor's theorem, stating that every set has strictly smaller cardinal than the set of all its subsets, to the universal set; no great originality on the part of Russell was required.)
So nowadays we use a formalization of set theory (usually the ZFC axioms), in which the naive rule for "creating" sets is replaced by several axioms that state that certain sets exist. One for instance states explicitly that an empty set exists, another that some infinite set exists, and ZFC has some other existence axioms. Most relevant here is that instead of "naïve comprehension" mentioned above it has "restricted comprehension", stating that for any set X and predicate P one can form the set (subset of X) of all x in X for which P(x) holds. The fact that an explicit (already existing, in a sense) set X must be supplied, instead of implicitly selecting from the universal set as in the naïve theory, avoids Russell's paradox in ZFC. This is what I meant by "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".
Excellent! Much appreciated, thanks!
Thank you Professor Kaplan! Watching your videos has rekindled my love of philosophy and academia. Cheers!
“This sentence is partially 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".
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.
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
That book rules
it rules so much
I got a hard case of semantic satiation on the word true in this video. Fascinating topics. Love it. 😊
Hi Jeff. Your paradox explanations and analysis are the best of all the paradox videos on UA-cam. Thank you so very much for this great work. I have become your fan instantly.
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
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.
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.
I was going to comment on this as well. It seems obvious enough to me that I'm certain it's already been brought up. I'd be curious how logicians respond to it.
A programmer has self reference, its called recursion. Recursion that has no action is disposed of by the compiler, because it is indeed nothing.
The beauty of being a programmer when it comes to philosophy is that you work with logic on a regular basis. Programming is applied logic in the same sense that engineering is applied physics.
The usual way to deal with circular references is to either ban or ignore them. Given the references are really numbers (i.e. memory pointers or database index ids) you could also wilfully misinterpret the circular references as numbers (change the context) for the sake of something like a serialisation task. I wonder if there's anything from all that we can transfer to this philosophical debate?
I am an amateur programmer thus probably I can understand what you try to say. But I can tell you that the problem is not Circular Reference or Self-Reference (i.e. recursive reference). If you read my separate comment I think you will understand where the "cheating" is. 😀
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 agree here. Processing the statement "This sentence is false" also requires a parser of sorts. I don't see a paradox in the statement, rather a misinterpretation (and perhaps blind trust) that the parser of the sentence behaves in a way that it does not.
A trivial example: a 5 year old kid is running around the room saying "This statement is false lol". Obviously a "language parser" for this context would be a little wiser than simply being flung into an infinite recursion and deadlock. It would discard the sentence as nonsense and invalid.
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.
I relate this to a different concept. In math, we can define an infinite series, which is the sum of infinitely many terms. Some of these series converge and have a well-defined sum, like 0+0+0+… or 1/2+1/4+1/8+1/16+…. Others don’t converge, like series that increase endlessly (1+1+1+…) or series that alternate endlessly (1-1+1-1+…).
Likewise, some self-referential statements “converge” by being well-defined (this sentence is true) or “diverge” by not being well-defined (this sentence is not true). The diverging series or sentence is real because we can write it down and describe it, but we cannot assign it a value.
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)
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.
Glad to see someone called that out. Referencing something which references you is still self-referential, it's just a 3 step loop instead of a two step one.
These videos are so great! I have a thought:
At 4:55, you say “if a sentence is false, then you just have to reverse whatever the sentence says”. Is that even accurate anymore? We’ve just introduced a third option other than just true and false, so can we still think of false as just “the opposite of true”? That still feels like bivalent thinking.
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.
These videos are fantastic. Thanks so much for putting them up!
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.
But what's the premise which is insufficient to get to that "correct answer"?! Also, in your "how does non-existent food taste" example, i think any answer would be valid, since in logic, false premises can imply anything.
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.
imagine this is the question in your test papers
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.
Precisely. There are sentences for which no truth value can be assigned. 👌
Responsibility is not the only trait needed to be a captain! Kirk was bold and that's important. Solid video tho
William Shatner the bed.
Frankly I think Kirk should have been the chief of security and Spock the captain. Since Starfleet always insists on not being a military organization, it'd make much more sense to have a more diplomacy/science oriented leader in that position.
But I don't know why I'm discussing that in a video about logic. ^^
He was also more likely to sacrifice himself than Jean-Luc.
Jean Luc was bald too.
@@TheSwiftCreek2 It was quite hard for Kirk to sacrifice Jean-Luc given that they lived in different eras! ;-) I know they overlapped a few times, but still I had to make this important point. Heh heh.
I stumbled on your videos for the subject matter, I stayed for your suburb analysis of Star Trek captains.
Seems to me, comparing Star Trek captains just within the context of their particular shows without also considering the popular culture of the time these shows were produced is an offshoot of the 'self-reference' problem. There is a 30 year lag between TOS and TNG. Cultural attitudes regarding women changed somewhat over that time.
I remember the first time I figured out this paradox. I was a kid and I saw a trailer on TV for some movie or TV show in which a character says "everything I say is a lie. In fact I'm lying right now". I thought it over and was like wait a minute......
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.
Exactly, but "there will be a surprise test tomorrow" has absolutely no reference and still gets a similar self contradicationary cycle (there cannot be an announced surprise, but once the sentence is considered absurd it is true because it will be a surprise)
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".
that makes it a valid third option
Another starship computer story I read in a short story was about two men in a spaceship one gave the ship’s computer some logic problem, or perhaps a benchmark which used 100% of CPU and could not operate the life support systems and begun to get colder and colder…
Someone else already mentioned that in programming, this is called infinite recursion.
One important thing you did not mention: this is still a problem even when there's no contradiction. For example, the biggest logical fallacy in all religions is circular reasoning. "This holy book proves my beliefs are true" and "My beliefs show this holy book to be true" are the two sentences equivalent
(Apologies to anyone religious reading this comment. You could be 100% right about your beliefs, but it would be an accident. It is still circular reasoning that would have led you to the correct conclusion, and the reasoning is flawed even if you reached the right conclusion)
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?
I’m thinking of a macro-language compiler that substitutes every name in a sentence with its definition (except for the primitives). In order for the compiler to finish in finite time, at any point of expansion, it should not be the case that a name occurs in its own definition (which is a thing we can test for, say, write a program for it).
Banning circular definition is, essentially, removing self-reference. Kaplan sort of says you can avoid self-reference, but it's really just hiding it. The issue is with series of statements whose _entire_ definition relies on other terms. Statistically, you can say that the system of circular definitions you give have zero degrees of freedom, which means the error cannot be measured. Ergo, the truth cannot be determined in such a system.
Interesting idea. ChatGPT could probably help you expand on it if you wanted to test it quickly. Good luck!
Well, the problem is that you cannot ban something from existence if it exists :) So we cannot ban circular definitions "just because".
Also, there are practical real-life situations when we have to deal with circular references - it's when serializing a parent-child data model where they both reference each other. The developers of serializers implemented different tricks to deal with this, but they could not "ban" it.
I don't think circular definition in and of itself is the problem i.e.
X:=~Y
Y:=~X
is circular, but is always true and without paradox.
I think it's self-referential denial (whether immediate or mediated), which is what you presented, that is the failure mode of the liar's paradox.
Shatner is definitely suing you. Be prepared! 😃
Except that Shatner is objectively a worse actor than Stewart.
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.
I handled one of your other *_alleged_* paradoxes before, so here's the solution to this one as well.
"Is the sentence true or false" is the problem, that's not the only options.
The opposite of "True"... is *not* "false"... and vice-versa. The opposite is of True... is "Not True". The opposite of "False" is "Not False".
The Sentence is "Not true" *_And_* "Not False"
Just like the sentence:
"Train happy dog blue smelly"... is *ALSO* "Not True" and *Not False".
There is no paradox
Well put. The video creator didn't expressly pose the problem as 2 clauses and just inferred the 2nd clause. Are we trying to prove validity, truth condition or what. Depending on the context specious arguments could occur. It could be said that there is no paradox because the claim and the request are distinct and in this case ambiguous.
"I always lie"
is an alternative way of expressing the liar paradox.
And a better one I think. It feels like the other examples could be argued from a grammatical stand point.
@@ShredPile Thank you!
Its not really.
The sentence is false ->
I always lie is false
This doesnt mean "I never lie", it means "I dont always lie" so you can still sometimes lie, you just dont do it always
@@ShredPile how is it better if it's incorrect? :-)
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.
Or the liars paradox is a collection of words that have been arranged according to grammatical rules, but which actually have no meaning. Such as: The invisible pink unicorn. Just because concepts can be shoved together doesn't mean that they will have meaning once they are put into the same arbitrary container.
On the contrary - the problem is inherent to logic, and it's only human cleverness that lets us see outside the logic box and actually solve the problem (Godel's Incompleteness Theorem).
In order to be true or false, a sentence has to convey information. The problem is not self-reference: the sentence "This sentence contains five words" is self-referential, but it conveys information that allows us to determine its truth value. "This sentence is false" conveys no information, and therefore cannot be either true or false. In fact, I would call it a "non-well formed formula",
Wheatley famously lacked the processing power and intelligence necessary to overload his processors on this paradox in _Portal 2._
Will you ever do a whole video dedicated on how to use logic and logic tables?
you might want to try out the following playlists if your interested in logic and logic tables:
ua-cam.com/play/PLqEJ_rxb3Xf1l1KbR33vNyjAqwg8Adq8K.html
ua-cam.com/play/PLKI1h_nAkaQq5MDWlKXu0jeZmLDt-51on.html
The first one is more philosophy oriented and is taught by David Agler (a phil assistant professor) and the second playlist is more general and taught by William spaniel, an assistant professor from a political science department (doesn't use standard philosophy of logic syntax, which can make it a bit easier for beginners, but this is not so useful later on at an advanced level)
Kirk is a(n amazing) soldier. Picard is a(n amazing) diplomat.
At last a decent answer to the only real debate in this debate... xxx ;-)
If I were making up math, I would consider each assertion in a list of assertions to be a node in a directed graph. If an assertion references another assertion, it would get an edge directed from one corresponding node to the other. If there is a cycle in the graph, I would consider the whole system to be self-referential, and therefore useless in standard logic. I would also consider the individual nodes in the cycle to be effectively self-referential. More advanced logic and graph theory would probably have lots to say about nodes in the graph that aren’t part of cycles.
This video does not exist
This comment does not exist
This comment is false
Your comment is false
Meaning my comment is true
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).
That led me to an amusing question - are were "here" when we are not aware of ourselves (e.g. sleeping)?
@@camelCased It's entirely possible the answer could be 'no', if we're actually a type of simulation. It would resolve many fundamental problems with reality.
@OP I would argue that, similarly to QM, there are more than two logically achievable states. Specifically: true, false, and recursive (which is, ultimately, just another word for 'paradox' - but it just might be the only method of addressing the issue).
@@DarkVeghetta Yeah, it feels like Schrodinger's cat - it's both true and false, until you open the box (stop recursing) and do the measurement.
There are great videos on the quantum bomb experiment by Sabine Hossenfelder and PBS Spacetime.
It's a completely different weird thing than this, but they are both weird!
@@DarkVeghetta
Is there really a third state, or is it simply our inability to perceive reality which creates the appearance of two distinct states?
For example, asking whether something is a particle or a wave results in the conclusion it is somehow both... which is weird only because we are using our perception of reality and mathematical tools of prediction to describe the behaviors of something which does not necessarily have to conform to either.
There is a very interesting work, a manga, called "Dead Dead Demon's Dededededestruction" - it's a sort of modern Gulliver's Travels in a sense and mostly a social critique - but there is an interesting sci-fi premise as it pertains to the perception of reality.
It also falls into the "deathworld" and "humans are space orcs" category.
A sort of fusion of the anthropic principle wherein we don't necessarily create reality, but our imperfect perception/grasp of what is truly occurring has a consequence for how that reality is experienced.
Consider how easy it is for us to talk about a subject in the abstract. If I come up and start describing a set of directions for how to get somewhere, you can couple the use of sounds and symbols to the abstract ideas of things that are not within your perception at the moment. I can describe a building or set of landscape features that you can understand as something not currently present.
Now consider the perspective of a dog. You come up to it and start saying things. It might recognize the word grass... or tree... but clearly, you're being silly, as neither such thing is present. Or maybe you're telling it you intend to go outside?
The capacity of a dog to formulate abstract concepts is extremely small relative to people. They are obviously not without intelligence, but the multiple layers of reasoning to create language, that are second nature to us, are completely alien to anything a dog understands.
Likewise, perhaps our inability to resolve these challenges is a similar mental block and with more capacity to process information, it would become obvious to us.
I kind of suspect we are smart enough that, given enough time, we can reason through any problem - but there again, the chasm between 140 IQ children and 110 IQ children would indicate that there may be hurdles that can only be cleared through improved baseline performance.
..... of course, behavioral disorders and the like among high IQ examples kind of draw into question whether or not that would hold true..... i am rambling at this point.
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
Yeah. My question too. How is written on the glass
This and the Russell paradox are essentially similar in my mind. The part I missed in both videos is where you resolved it?
If they were resolved, they wouldn't be paradoxes, would they?
A classic case of overthinking. And THIS sentence….is TRUE!!!
I love it. I would have asked for more in the application area (bring up how it was used by Godel and Turing), but this is great.
Would you consider covering the Curry-Howard Correspondence? It might be a bit more on the practical side, but it shows great application of many different areas of logic to computer science beyond the obvious.
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.
It actually reminds me a bit of Thomson's lamp.
*35. Spaces are characters. ;-)
But even if we grant you that it's not "any kind of" self-reference that is the issue, only infinitely recursing ones; how can we decide whether some given self-ref is infinitely recursing or not? The halting problem isn't solvable... :-|
PS You're correct that the 2-sentence alternative goes into the same infinitely-recursive box, but it was presented as a way to bypass the SELF-referential nature of the paradox, not the infinitely-circular nature.
Any sentence that is scrutinized on its own merit without any context is an exercise in futility. You might get the true meaning, as in when someone made this statement, they were conveying meaning, or you might get stuck in some logic loop, forever lost.
If I make a road that loops in upon itself like an infinity symbol, tell you to drive to the end of the road, you really only have 2 options at this point. Either keep driving or get out of the car and tell me that I've made a mistake in my road design.
So, just because we can play with words doesn't mean that logic is broken. It means that the person who constructed the sentence made a mistake, or that CONTEXT HAS BEEN FORGONE.
Thank you.
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.
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!
Bruh, you were doing so well. This is the second video of yours I've seen after Russel's Paradox and was loving this one as well. Right up to the instant where you wrote "The sentence below is false" and claimed you had avoided self reference. Are you kidding me? I'm dumber than the gum on your shoe but even I can see that you didn't do that because "The sentence below is false" is clearly a shorter way to say "The sentence below THIS ONE is false". The sentence below what is false? The sentence below THIS ONE.
So it becomes "The sentence below 'The sentence below is false' is false" recursive to infinity. Not sure how you could miss that.
it could have easily referred to the space occupied by "this one"
the sentence on line one says _"the sentence on line 2 is false"_
the sentence on line two says _" __-this was a good video and not just a steaming pile of crap intended to harvest view time-__ the sentence on line one is true"_
@@Mitchell_is_smart._You2bs_dumb I agree. It *could* have. Which makes it all the more surprising he didn't do it that way. It's a glaring error with an easy fix. But I also agree with your redaction so I guess I'll give him another chance.
One can still self reference across multiple sentences. Plus the assumption that a double negative is the same as a postive, yet thus creates one is a distinction that splays this into 4 options.
Not sure whether this could be possible, but it seems if we include an option "Does not make sense" such that this option not only is "neither true not false" but it also invalidates whatever the sentence is saying. In this case "This sentence is not true" is the option "does not make sense", which while indeed being neither true nor false, but also conveys the message "the sentence is invalid", so there is no further conclusion that the sentence is not true. This means we need to introduce a concept "valid vs. invalid" whose priority is above "true and false" but then the paradox can happen with "This sentence is invalid." So in order to resolve this once and for all we need an infinite hiararchy of validness, let's call them valid, invalid, valid2, invalid2, valid3, invalid3, ... , such that valid2 is of higher priority than valid and valid3 is of higher priority than valid2 and so on. Then we define the "does not make sense" option for a sentence "This sentence is invalid(x)" to always be "neither valid(x) nor invalid(x), plus the message that it is even not valid(x+1)". Notice that we can completely get rid of the notion "valid" and use "true(x)" instead, but I think the word valid(x) makes more intuitive sense. I think this method in general makes intuitive sense as well, because when we say a sentence "does not make sense", basically we are ignoring its content and overwrite it with a higher-priority "more absolute" validness.
I realized there is still the problem of universal reference "for all x in Z, this sentence is invalid(x)". I think the solution could be to redefine the hierarchy to be not the size of integers, but define it as "for any sentence, there always exists a higher-priority invalidness" as an axiom and then make the "does not make sense" option invalidate the sentence using that higher-priority invalidness.