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

the liar Paradox is very simple and very old it goes back at least to the year 2265. [Music] you say you are lying but if everything you say is a lie then you are telling the truth illogical illogical Boy the original Star Trek from the 60s was wild huh come on spark I know that look okay the liar Paradox in the real real back at least you book philosopher and student of Euclid in the 4th Century BCE that's the year negative 400. this sentence is false what do we make of a sentence like this is it true well when a sentence is true you look at what the sentence says suppose that there's a sentence in here and we know that this sentence whatever it is is true well we look at what that sentence says the ship is on fire so if the sentence is true then the ship is on fire back to the liar sentence if this sentence is true well then we look at what the sentence says and what it says about itself is that it's false so if it's true then it's false and that's a contradiction so this option is out so then the sentence is false right well the sentence says that it's false and so if this claim here is itself false then it's false that it's false which means it's true and that's a contradiction or another way to put this is just that the claim of the sentence the claim that the sentence is false well that claim is false so it's false that it's false okay well if it's false that it's false then it's not false so this sentence is not true it's not false what if we say that it's both true and false unfortunately that's not going to work because in order to be both of two things you have to be each of them individually to make an analogy if a character is not well written and it's not well acted then it definitely can't be both well written and well acted as we all know the greatest Captain was Jean-Luc Picard William Shatner is old and Rich enough that he's not going to care he's not going to sue me that's why it can't be both we've already eliminated for example true so it can't be both true and anything else so it definitely can't be both true and false what about neither true nor false this one might work but it's controversial you might have thought that all sentences or claims could only be either true or false logicians call this the principle of bivalence the idea that there's only two options true and false but since it's clear that in this case neither of those two options works it's tempting to introduce a third option a sentence or a claim can be true it can be false or it can be neither true nor false and this works this sentence is not true or false it's neither true nor false no Paradox end of video wait a minute what if we replace the word false with not true logicians have a name for this sentence they call it the strengthened liar sentence remember the Paradox arose originally because there were only two options and neither of those two options worked so we introduced a third option but now now we're talking about the category of not true and notice false is one way of being not true a sentence that's false is not true but being neither true nor false that's also one way of being not true by changing the word false to not true what we've done is we've regenerated the situation in which there were only two options there's the option of true and the option of not true this whole area encompasses the larger category of not true false is not true and neither true nor false that's another way of being not true so we're back to having just two options and if it turns out that neither of those two options work we're going to regenerate the Paradox is this sentence true well the sentence just says that it's not true so if it's true then you check whatever it says oh it says it's not true so if it's true then it's not true okay this option is out is it false well if a sentence is false then you just have to reverse whatever the sentence says the sentence says that it's not true so if it's false we do the opposite of that which is that it's true so if this sentence is false then the sentence is true contradiction this option's out is it both true and false no because in order to be both of these you need to be each of them individually we've already eliminated both of them individually so it's definitely not both true and false finally is it neither true nor false being neither true nor false is one version of being not true and the sentence says that it's not true so if the sentence is neither true nor false well then this is accurate what the sentence says about itself is accurate well if a sentence is accurate then it's true so if this sentence is neither true nor false then it's true and it says that it's not true so it ends up being true and not true that's a contradiction this option is out as well what's going on here well if you've ever heard of this other Paradox Russell's Paradox which I have a video about Link in the description then you might have noticed that these two paradoxes though different have something very important in common they both seem to result from self-reference Russell's Paradox results from the fact that a set can contain itself and the liar paradox seems to result from the fact that a sentence can be about itself so maybe what we should do in order to wriggle out of this Paradox is call into question the possibility of self-reference like look at this phrase this sentence how do we even know that this sentence is talking about this sentence ah but this won't work because we can generate a liar sentence without any phrase anything like this sentence and we do this by naming the sentence we can name things we can name people places things well a sentence is a thing so let's name it let's call it uh fribble these are called fribbles no they're not they're called Tribbles there was an episode about them oh can CBS sue me now the sentence reads fribble is not true fribble is just a name for the sentence so this sentence says of itself that it's not true and this is gonna work like we don't have any weird uncertain ambiguous phrases like this sentence it's pretty clear that this sentence is referring to itself or is it the name fribble is just shorthand for the whole sentence right so if we want to know for sure what exactly this name stands for then we should just swap it out for the thing that it stands for okay so we're saying that this sentence fribble is not true we're saying that that sentence is not true so now we know what we're talking about oh but the name fribble is right there right and that's just shorthand for the sentence so in order to be really sure what exactly we're talking about we should replace that for the thing that it stands for okay now we're done the sentence fribble is not true is not true and that sentence is not true oh but the name fribble is still in there so we have to replace it again this has been an argument against the possibility of self-reference when we have a term in a sentence that refers to the sentence itself we end up in this regress this endless cycle of replacing that name with the thing that that name is supposed to stand for we keep trying to clarify what exactly this sentence is about but we never reach Clarity so maybe self-reference is impossible after all if we're trying to resolve the liar Paradox this is good it means that self-reference is impossible the sentence can never fully or successfully refer to itself and so the Paradox can't arise this is where most discussions of the liar Paradox in public discourse end but the professional logicians know that there's more we can generate the liar Paradox or something almost exactly like it without any sentence that even tries to refer to itself first notice that a sentence can easily and unproblematically refer to another sentence this sentence says the sentence below is false and then the sentence below says Kirk was a good captain now this is false Kirk was not a good Captain a captain has to be responsible they are entrusted with the lives of their crew they can't just go around trying to hook up with every alien babe they come across anyway there's no problem here this sentence is false and this sentence which is about this one says that it's false and so this sentence is true like no problem but if this is possible then we can generate a pair of sentences that work together to do the same thing that the liar sentence Did in the first place I'll go through this one real quick let's just focus on the top sentence is this top sentence true well if it's true we have to check what it says the sentence below is false so if this sentence is true then this one is false but this sentence just says that this first sentence is true so if this one's false then this one's false so if the top sentence is true then the top sentence is false contradiction okay what if the top sentence is false well it says that the sentence below is false so if it's false then the sentence below is true well if this sentence below is true then this sentence is true because this one down here just says that this one up here is true so if this sentence is false then this sentence is true contradiction and then the same thing for both true and false and then neither true nor false we could run the same thing where we replace this with not true and we're exactly back to where we were with the strengthened liar sentence and we did all of this without any kind of self-reference these sentences aren't referring to themselves they are referring to each other and they make this kind of circle the problem of course comes from this kind of circular thing but it's not really a problem for self-reference any attempt to ban self-reference from our language in order to prevent liar sentences from arising that's doomed to failure because we can generate liar sentences with multiple sentences in the end the liar Paradox shows that there is a deep problem with one of two things first our notion of Truth itself the way that logicians formalize the notion of Truth within a logical language is with what are called t schemas T stands for truth and the letter P stands for proposition and so we're just talking about any sentence or any proposition you can think of these as inference rules Like rules for how you can think logically if you can say something P like that Jean-Luc was the better Captain if you can say that then you can also say that P is true that Jean-Luc was the better Captain is true and then you can go the other direction if you can say that something is true Jean-Luc was the better Captain is true well then you can just drop these true and you can also say Jean-Luc was the better Captain you don't have to remember any of this you just have to know that this is a really intuitive way of capturing the notion of Truth and one way to get out of the liar Paradox is just to drop this but that seems crazy the other way out might be worse and that's to abandon classical logic which goes all the way back to Aristotle and then gets developed by Fraga you replace it with dialethianism or Gap theories I'm not going to go into what those are but they're less intuitive they're weird and they seem less logical and that's bad if you're developing Logic the point is that this is an area of live and active debate today in philosophy and logic I should thank two things for help with this video the first one is Tim maudlin's 2004 book truth and Paradox I've never met Maudlin but I read his book back in graduate school and then years later before recording this video I skimmed it again it was very helpful thank you Tim and the other is Ethan jerzak who is in my cohort in graduate school and now he is the renowned assistant professor of philosophy at the National University of Singapore and we had a zoom conversation where he fixed a few things that were wrong in my understanding of the liar paradox

Channel: Jeffrey Kaplan

Views: 961,776

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Id: in4u2i9v4vg

Length: 14min 16sec (856 seconds)

Published: Wed Mar 15 2023