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I removed the version Everything I say is a lie.
This isn't paradoxical when most people say it. It's simply false, assuming the speaker has said at least one true thing in his life. Evercat 18:40 21 Jun 2003 (UTC)
The section "Patrick Greenough—Free Assumptions and the Liar Paradox" is in dire need of editing/clarification. I am not familiar with his work, but the section makes no sense. Please fix it. Joshua.horton 03:32, 10 December 2006 (UTC)
A version of this paradox appears in the Don Quixote ( II, Chapter LI) and another in the letter of Paul to Titus; 1, 12:
The first (which I was unfamiliar with) seems to be a paradox of some sort, but on skimming the text, I don't think it's strictly the liar paradox.
The second is the Epimenides paradox, and both this article and that one make a big deal (correctly, I think) in asserting the difference between it and the liar paradox. Evercat 19:11 21 Jun 2003 (UTC)
Why do we making such a big deal about the difference ? If someone says
that's the liar paradox, right ? But if someone says
, doesn't that *include* the previous statement ?
-- DavidCary 04:52, 18 Jun 2004 (UTC)
No, not really. Wheras
is neither true nor false, this statment may be false:
for instance I told the truth yesterday, and when i said Everything I say is a lie., i was lying. So the statment is false. The difference is slight, but there is no reason not to be picky in an encyclopedia =)
Gkhan 17:00, Jul 17, 2004 (UTC)
To state why it's not a paradox another way: the statement Everything I say is a lie only implies This statement is not true if it is true. That would be a contradiction, so the statement must be false. The statement Everything I say is a lie being false does not imply This statement is true, because it could be some other statement that is true. So, as stated above, if the speaker has ever told the truth before, then Everything I say is a lie is a lie, and not a paradox. Rob Speer 17:12, Jul 17, 2004 (UTC)
Hello. I put a proposal to merge liar paradox and Epimenides paradox at talk:Epimenides paradox. Perhaps you'd like to respond there. Happy editing, Wile E. Heresiarch 19:16, 15 Aug 2004 (UTC)
Be careful, Ropers - in your temporary version, you said that "Cretans always lie", spoken by a Cretan, is a paradox. This is wrong; it's a lie, not a paradox, though it has been given the name of the " Epimenides paradox" because of how deceptively like a paradox it is. It's not a paradox for the same reason that "Everything I say is a lie" isn't. RSpeer 17:16, Aug 27, 2004 (UTC)
Wait... this makes no sense. He states (correctly) that: '2 + 2 = 4' is the same as: 'It is true that 2 + 2 = 4', so we can surmise that (It is true that) can be added to any sentence, without affecting the meaning. However later he states that: 'This statement is false' is the same as: 'This statement is true and this Statement is False,' but this does not follow the same theory. It should actually be: 'It is true that this statement is false.'
This, of course does remove the paradox, but the way it was written was terribly incorrect, thus I have been forced to change that. 58.175.169.47 ( talk) 07:28, 25 September 2008 (UTC)
What the heck is it? It refuses my connection. RSpeer 04:35, Sep 22, 2004 (UTC)
I think a good example to use would be All generalizations are false. Using This statement is not truedoesn't define what 'this' is, or so I feel that way. I'm wondering if anyone else is confused, or is it just me? -- KaiSeun 06:48, 2004 Nov 4 (UTC)
"All generalizations are false" is not paradoxical, because there is no contradiction in assuming that it is false.
I don't find the "this" confusing in "This statement is not true". The "This statement" has to refer to itself because there is no other statement that it could refer to. -- Nate Ladd 11:08, Nov 23, 2004 (UTC)
I deleted the material below for these reasons:
1. There is no reference to this "Yablo"s publications in the References section. Who is he/she? Is his/her work even published?
2. The Yablo paradox applies only to an infinite list of statements. But this is not genuinely a paradox at all. We don't believe there can be an infinite list of statements anyway, so the fact that the supposition of such an infinite list entails a contradiction is not disturbing. Yablo's argument is a disproof of the supposition, not an apparent counterexample to our notions of truth. (But people can actually say and write things like "This sentence is false.")
To Posiduck: Which philosophers/mathemmaticians believe in an ACTUAL infinity of sentences (as distinct from numbers)? More specifically, which ones believe that the particular infinite list that Yablo describes actually exists? Is such a list constructible and, if so, then how? Questions like these have answers when applied to, say, the infinite set of integers, but I can't see what the answer would be for Yablo's list of sentences. That's why I'm asking. For any integer, I know how to construct one that's one greater in size. Ultimately, my construction technique traces back to making a union of two sets (or, if you prefer an older theory of the foundations of math, to making a line one unit longer than it currently is using only a straight-edge and a compass.) The Liar paradox is important because it seems to show that our culture's cherished intuitons about truth lead to a contradiction. The cherished intuitions are
1. Every sentence s is either true or false. (Principle of Bivalence)
2. Sentence s is true iff and only if what s says is the case.
But Yablo's so-called paradox requires the additional assumption that there can be an actual infinity of sentences such as he describes. This is not a cherished intuition. Indeed, the typical member of our culture does not believe it is true at all. So Yablo's derivation of a contradiction is only an ordinary reduction ad absurdum argument of its premises. When two of the premises are cherished intuitions about truth and the third is a dubious claim about an actual infinity, then we simply take the argument as a disproof of the dubious premise. It is not, therefore, a counterexample to something at the heart of our culture or logic or mathematics. It is, thus, not what is meant by the word paradox. This means that Yablo has failed to show that self-reference (directly or indirect) is not at the heart of the Liar paradox. -- Nate Ladd 05:09, Dec 7, 2004 (UTC)
Here's what I suggest we should put back in, and unless there is some reason not to, beyond you disagreeing with Stephen Yablo as to whether or not this is related, I see no reason not to include it.
Related Paradoxes:
Stephen Yablo (2004) has published a paper "Circularity and Paradox" in which he claims that semantic paradoxes, such as the liar, can be generated even without direct or indirect self reference. He poses a paradox he calls the w-liar.
He asks us to consider a list of sentences which is infinitely long in both directions.
And so forth, so that each sentence N says, All sentences numbered N+1 or greater are false No statement in the sequence is consistently evaluable as true or false. Choose one arbitrarily. It is true if and only if all of the subsequent statements are false. But if all of the subsequent statements are false, then any of the following sentences also makes a true claim. If any one of the sentences is false, then that could only be because a sentence numbered higher than it is true. But we already know of any arbitrary sentence that it cannot be true. So, none of the sentences are consistently evaluable. Just as in the case of the standard liar's paradox, each sentence is true if false and false if true, yet, unlike most liar variants, none of the sentences predicate falsity of themselves. Yablo thinks that these sentences are suffering the same failure as the Liar's paradox, but without self reference. This claim is controversial.
Posiduck 22:27, 9 Dec 2004 (UTC)
HERE'S WHAT I DELETED:
Furthermore, there is Yablo's version of the paradox:
Consider a list of sentences which is infinitely long in both directions. The sentences all say the same thing: All of the subsequent statements are false. Pick one statement at random. It is true if all of the subsequent statements are false. But if all of the subsequent statements are false, then what they say is indeed the case: they say that all of the statements subsequent to them are false, and ex hypothesi they are false. That contradiction means that the picked statement should be false, but its selection was arbitrary, implying all the statements must be false; again this leads to their description of subsequent statements being true. So like the liar, they're true if they're false and false if they're true, yet no propositions predicate falsity of themselves. This is sufficient to suggest that the liar does not depend upon self reference.
(all words in brackets are lies) hehe
I put a "mysteroius" tone flag on top mostly because of this section:
"If we assume that the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So assuming that it is true leads to the contradiction that it is true and false. OK, can we assume that it is false? No, that assumption also leads to contradiction: if the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either assumption, we end up concluding that the statement is both true and false. But it has to be either true or false (or so our common intuitions lead us to think), hence there seems to be a contradiction at the heart of our beliefs about truth and falsity."
Does nobody else think this can be avoided?
Nrbelex (
talk)
23:36, 15 Mar 2005 (UTC)
You are still not being explicit about what your complaint is. What is the "this" you want to avoid? Is it the use of "we"? Using an indefinite "we" is common in philosophy. -- Nate Ladd 02:12, Mar 18, 2005 (UTC)
I removed the following because the first sentence is an unjustified assertion, ex cathedra, and the second is so poorly punctuated that it makes no sense. Finally, it does not undercut the argument being made. If the anonymous Prior fan who wrote this wants to make changes in the discussion of Prior, he/she should make his case here on the Talk page. -- Nate Ladd 09:36, May 8, 2005 (UTC)
Such an assumption about clausal truth values can be done independently of sentential truth value only if the sentence itself does not make assertions about individual clauses. Of course, in this case undeniably the Prior assertion that the whole series of logical conjuctions of clauses is true is exactly identical with the whole series of assertions about the individual clauses.
Although this suggestion is somwhat amateur, we should have an article on the Two Guards and two doors logical problem. Where one guard always lies, one guard always tells the truth, and one door leads to death, and one door leads to life. You can only ask one question to ensure that you enter the door of life. Something along those lines. Colipon+( T) 21:49, 21 May 2005 (UTC)
The sentence
is not a type of liar paradox. It is a sentence that is always false whereever it appears, but it is not a sentence that is both truth and false. (And there is nothing unusual about sentences that are false wherever they appear, most false sentences are like that: "Ronald Reagan was a king of Egypt". Also, note that it is not a self-referring statement. It refers to the page on which it appears, but it does not refer to itself.
I'm not sure I understand the discussion below, but I think there is a much simpler argument against the current version of this page. When it says:
The last claim is wrong, I think. Here is a truth assignment that makes everything consistent:
In other words both sentences are false and Prior analysis works fine here too. Please correct either the page or me. F4810 16:36, 30 November 2005 (UTC)
The wiki author here made a fatal mistake. Applying the Prior prescription should invoke EVERY implicit assumption, and would look more like this
Just as with the previous reasoning this is the statement "(A and not B) and (B and A)" which can be reduced to "A and B and not B" which is obviously false. Therefore the statement is false and there is no paradox.
Now the last attack to this is the statement
I strongly disagree with this statement. The two clauses are coupled to each other. They are inherently not independent and therefore I see no reason why it should be possible to treat them independently.
I'm gonna fix this now. Please reply if I am in error ( CHF 09:57, 20 September 2005 (UTC))
The whole last paragraph of the section on Prior's argument should be removed unless someone can cite a reference that makes the arguments made in that paragraph (Note that I'm not claiming that arguments in that paragraph are necessarily wrong, but a citation is desperately needed) —Preceding unsigned comment added by 24.17.244.140 ( talk) 00:44, 4 September 2007 (UTC)
?? I guess I am a little confused. I entered a bit of info about the liar paradox and it was summarily removed with no explanation. I went back and looked at the editing guidelines and can't understand what I did wrong. I thought it was an interesting statement about the liar paradox, it eqivocates. It is one that my phil. prof, J.C. Beall, an expert in this area, thought was probably true. Any suggestions?
I see. I thought the encyclopedia was a means of presenting different views, ideas that did not need the Imprimatur of the academic establishment. My mistake-namaste
I don't know what that's supposed to mean, but this much I can tell you: The entire article is bogus. The statement "I am lying" has no truth value in and of itself, because it's not "about" anything. Go ahead, prove me and my Logic 101 professor wrong. Wahkeenah 04:42, 27 January 2006 (UTC)
To put it another way, you only think there's a paradox because you think the statement "I am laying" should have a truth value. Your initial assumption is incorrect. Once you realize that, you discover that there is no paradox, and that this article is based on a false premise. As my professor said, "When you start with incorrect assumptions, you are liable to get interesting results!" Wahkeenah 04:48, 27 January 2006 (UTC)
OK, I worded it a little better. Sorry to burst your bubble, Grasshopper. Wahkeenah 05:00, 27 January 2006 (UTC)
Note that in my writeup, I referred to something. I said "That assumption is false." That sentence has a truth value, because it refers to something else that has a truth value. A sentence such as "This assumption is false", by itself, has no truth value. Class dismissed! Wahkeenah 05:04, 27 January 2006 (UTC)
It would be a good idea to mention the name of that episode. Go for it Trekkers :)-- Manwe 17:49, 29 March 2006 (UTC)
Reply to Nate Ladd and anyone else who cares: There is not an infinite number of integers or lists or anything else. There is only a repeating (infinite) process to construct them, and the process is always used a finite number of times. His own words... "For any integer, I know how to construct one that's one greater in size." If a list has a beginning it can't be infinite. As for the statement "I am lying", I agree with Wahkeenah,it has no content that can be verified true or false. Phyti 01:19, 10 June 2006 (UTC)
195.153.45.54 14:23, 9 August 2007 (UTC)
Alright, I think I've come up with a solution to "This statement is false". It is false, as part of the statement is false and another is true. Saying "This statement is false" is obviously stating that the whole statement is false, which is false, because the statement as a whole is both true and false. With an easier-to-understand version that means the same thing, the part in italics is false, and the part in bold is true:
This whole statement is false
This whole statement is the false bit because it refers to the statement as a whole, whereas because only part of the statement is false, you cannot say that the whole thing is false, thus making only that bit false. Hugh Jass 23:27, 4 July 2006 (UTC)
Is the answer to this question "no"?
Does this self-referential contradiction fall under the liar paradox?
What about this:
If I ask you [question X], will the answer be the same as the answer to this question?
which forces an answer of "yes" to the question X.
If not, what topic should they go under? -- Spoon! 05:28, 30 July 2006 (UTC)
If [question X] is Can you make jam? and the answer is yes then I can answer no to your question.
195.153.45.54 14:31, 9 August 2007 (UTC)
I change a statements i found un clear to this, perhaps i was wrong. if that is so, please revert it i didn't mean no harm, and i may have bean right to change it. but i am having doubts
Would this qualify as an example of the Liar's Paradox? 65.12.114.98 14:09, 27 September 2006 (UTC)
Do statements have meaning if un-uttered? Do they not allways take place in time? If so then, "statement" in "This statement is false" contains a prediction.
In "This statement is false", the referent does not yet exist at the time "this statment" is written, spoken, or typed. The same is probably true when the statement is heard, and even read.
So what does "This statement" in "This statement is false," mean? As pointed out above "This statement" must be refering to the whole statement, that is to say the completed statement that does not exist at the time of writing "this statement". The future does not yet exist, so we can only make predictions about it. We say "we can talk about the future", but really we are predicting the repetion of a past event.
"This statement is false", begins as a prediction about something that does not exist until the last "e" is written. Hence the "statement," which starts as a prediction, can perhaps be re-written as
"I predict that the statement that will exist when I finish writing this will at that time be false." or
"It is predicted that the statement that will exist will be at that time false."
It is only because the statement starts as a prediction that it can refer to itself. It needs time to loop back on itself, but admixture of time prevents the looping back from being complete.
As a prediction/statement it relates to two times, the time of predicting, and the time being predicted. The truth value of the same sentence may be different at each of these two times. Just as "Tim is alive" is true now, but will not be true in considerably less than 100 years time.
I think that there is a case for saying that the prediction came true: the statement is false. The sentence as statement does not say anything true. But as a predicition, in its unfolding prior to its completion, it was a good, i.e. veracious prediction. The mistake that everyone is making are
1) To mistake a prediction for a statement just because it calls itself a statement.
2) To think that there is such a thing as a living word that sits on the page and just means, without a reader or writer. But as far as I know, language only gets meaning in use, i.e. in time, and it is in time that the liars "paradox" fails to be paradoxical as it unfolds. When you take the fact that the statement says nothing true, it does not allow use to travel in time and take that false-ness back to the time of predition since the prediction was accurate - as it surely was.
3) Perhaps again you could say that "the statement" always remains a prediction "This will be false," and it always will be. The claim that this is a "paradox" is also akin to claiming that it is impossible to predict that "Tomorrow is Sunday," because when Sunday comes, the next day will be Monday, and the preditiction "Tomorrow is Sunday" will no longer be true. But if the next day was indeed Sunday then at the time of prediction, "Tomorrow is Sunday" was true and as a predition it remains true. The sentence in question always has the same referent, and is as false as a statement and as true as a prediction today as it every will be.
"I pledge that the statement that will exist when I finish writing this will at that time be false." -- Timtak 10:21, 1 October 2006 (UTC)
The following is not really a paradox, as the person is only guessing because no outcome has yet been established, therefore they have not identified them self. If the person is decapitated, then like after being married you are married, but before you are single, so you must be decapitated first to know if indeed it happened, but then you are now a different person after decapitation. So you are one person before, and a different one after, but the question was asked who they were before they entered not what they will be after. They may hang her next week, not the next day, or decapitate her after she dies from hanging, so she fufills both conditions, but first they may torture her with drugs, like Zyprexia, so she is even stupider than before. (That is the modern method used by the USA today for those who express their first amendment right of freedom of religion. We should have an amendment that protects us after our expression as well. That is a better paradox than the current one shown here.) Therefore it is not a paradox since no outcome can yet be established on the current information. However, it does clearly show how a typical Jew thinks they are clever, when they are only fooling them self, where they should be hung, and decapitated for their attempt at humor. Here follows the so called paradox.
In a Jewish folktale an anti-semitic king makes an edict that any Jew who enters the capital city will be asked to identify himself. If he tells the truth he is to be hanged, but if he lies he will be decapitated. One Jewish woman comes to the gates of the city. She tells the guard she is a woman who is going to be decapitated that day. If they do that she will be telling the truth, in which case she will have to be hanged. But then she would be lying, meaning she will have to be decapitated. And the cycle of logic repeats ad infinitum.
Danross 03:42, 11 December 2006 (UTC)Dan Ross
1: The sentence below is true. 2: The sentence above is false.
Here's how I tried to solve his:
Let's call sentence 1's truth vale A, and sentence 2's truth value B. Then it's the same as:
A = B B = 1-A
Wich gives:
A = 1-A 2A = 1 A = 0.5
This means A is both true and false/neiter true or false. However, say you do not allow anything but 1 or 0. Concider the truth values boolean (true, false) instead of real (-2.7, -1, 0, 1.2, 5). Then:
A = B B = !A
Wich gives:
A = !A
If A can only be 1 or 0, then the equation is never true.
My conclusion:
The second sentence is as false as the first sentence is true. If we allow them to be "half true/false", then it works. Otherwise, the sentences are incompatitible.
It seems the execution paradox has a weak point in this story. It can be avoided by the person who makes the paradoxical statement. In this story the man said "I'm going to be executed today", the inquisitor could have let him live one day longer and have him executed the next day for making a false statement, since the inquisitor did not set any time limit. A better thing to say would have been: "I will die by execution".
--Anonymous —Preceding unsigned comment added by 62.177.253.214 ( talk) 13:28, 30 July 2009 (UTC)
Hi, I'm kinda new (or at least I've never contributed) so I'm kinda worried about editing the page and messing something up or anything (hehe)
Anyway, I noticed that under "In popular culture" it states that
"If the first statement was "Everybody lies all the time", then it by itself would constitute a liar paradox."
However, that's not a liar paradox, after all, it can only not-be true (then it'd be contradictory) but it can be false (I think) If somebody tells the truth then not everybody lies all the time, but it doesn't mean that person didn't lie that time. —The preceding unsigned comment was added by 80.126.65.34 ( talk) 22:20, 5 March 2007 (UTC).
seems to me this is just a false dichotomy. The sentence is neither true nor false.
I changed "false conclusion" to "solution", as it is clearly correct. —The preceding unsigned comment was added by 82.93.92.62 ( talk) 09:50, 13 April 2007 (UTC).
I added some lay examples, including why the examples were self-contradictory, but was reverted saying they weren't examples of the liar paradox and simply false statements. These examples are valid, accessible liar paradox examples:
Paradox: ...including those that contradict this sentence.
Paradox: ...including those that declare this statement false.
Paradox: ...including those asserting the opposite of this statement.
Paradox: ... except this statement.
Paradox: ... except the existence of dichotomies.
-- Loodog 18:08, 9 July 2007 (UTC)
I have deleted the following sentence from just below the first appearence of the 2-sentence version: "However, it is arguable that this reformulation is little more than a syntactic expansion. The idea is that neither sentence accomplishes the paradox without precisely its counterpart. "
My reason is that "syntactic expansion" is not defined and, more importantly, there doesn't seem to be any implication of the remark. So what if "neither accomplishes the paradox without precisely its counterpart"? That's not a criticism of the 2-sentence version, its just a description of it. Its still a paradox and still needs resolving. —Preceding unsigned comment added by 24.16.98.193 ( talk) 00:51, 4 November 2007 (UTC)
wouldnt that paradox automaticly resolve itself because you can ignore the 1st line it currently says The statement below is false The statement above it true
but to make it a paradox it should be The statement below is true The statement above is false -- Tjayh913 03:49, 10 November 2007 (UTC)
I was about to reference someone to all the references in pop culture that were in this article, but they're not there anymore. Can anyone put them back or were they erased for a good reason? 159.90.9.83 17:27, 15 November 2007 (UTC)
What we really need is a summary of its use in humor that cites popular culture, rather than a list of occurrences. I understand the arguments for and against the inclusion of Trivia sections, and am torn on the issue myself. What is less controversial is when an article summarizes and provides context for these examples, often not simply as "popular culture," but as some particular aspect of it that serves to expound upon the importance of the topic, instead of trivializing it into a list of particular instances. I'm gonna try to look through the history pages and see if I can find a unifying theme. I have no desire to get into a serious edit war, so please feel to discuss this here or on my talk page. J Riddy ( Talk || Contributions) 21:58, 17 February 2008 (UTC)
If we can derive this statement is false from This statement is true and this statement is false, then the paradox is back. And if we are not allowed to make such a derivation, then Prior has, in effect, invented a new kind of conjunction whose truth value characteristics are so mysterious, we cannot really say with any confidence that the paradox has been dissolved.
But the second italic proposition is (A=)A and not A, from which logically we can derive any sentence. I don't see why this property of the conjunction would be mysterious, given the Principle of explosion.
Take "classical" logic: to determine if a sentence is true, then either (1) we derive it from the axioms, in which case it is true, or (2) we assume it's true and try to derive a contradiction, in which case we call it false. (That's not really correct as far as I can tell; in the resulting system all propositions are true, given the PoE, and I'm not sure I see how that makes the sentence false in the original axiomatic sentence.)
In this case, the sentence _is_ contradictory. It contravenes the law of non contraction (according to Prior, a sentence claims that its content is true, and this one claims that true is false). So in a system where we use the law of non-contradiction, the sentence is false. This can be nicely expressed by it's negation, "The sentence 'This sentence is false' is false". If you don't allow the law of non-contradiction, then truth isn't well defined anymore, anyway. —Preceding unsigned comment added by Bogdanb ( talk • contribs) 18:54, 5 March 2008 (UTC)
"This is to be distinguished from the common colloquial expression "I tell a lie." when the speaker has realized that he has just accidentally told an untruth."
I don't, uh, what? I've never in my life heard someone say "I tell a lie" after they realized they just 'accidentally' lied. That isn't even proper English. -- Dbutler1986 ( talk) 06:48, 18 June 2008 (UTC)
"Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed" by which he meant a language in which it is possible for one sentence to predicate truth (or falsity) of another sentence in the same language (or even of itself)."
I don't see how this resolves the paradox. Tarski is just saying that such a statement can't even be formulated in a semantically closed language, whereas he's offered no resolution for assigning a non-contradictory truth value to "This is a lie." in semantically open languages (like colloquial English).-- Loodog ( talk) 15:58, 18 June 2008 (UTC)
Removed from the article, to discuss what parts of it might be saved:
My specific criticisms:
That's it for now. It would be reasonable to include some discussion of Goedel here, but there was not enough correct stuff in the text as it stood to make it worth trying to fix piecemeal, in my judgment. Any thoughts on how it could be done (especially with sources) would be welcome. -- Trovatore ( talk) 00:10, 17 August 2008 (UTC)
(outdent) I think this gets rather far afield from the Liar. The two things really are not that closely analogous (largely because provability is not very much like truth). Something the two arguments have in common is that they are both diagonal arguments, but there are an awful lot of those.
Just for fun I'll try and write down something that I think barely might be defensible here, if it could be sourced.
Now it's true that this version doesn't get into showing that A also does not prove ¬GA, but that part would require bringing in either ω-consistency, or Rosser's modification (which no longer looks like the liar), and is less important anyway since we've already shot down A as a way to generate all arithmetical truths (since it fails to prove GA, which is a true statement of arithmetic).
But I'm not seriously proposing this addition. As I say, it's not really all that closely related; there's a nice formal similarity in the formulation of the sentence itself, but the analogy doesn't go too much further. -- Trovatore ( talk) 07:35, 19 August 2008 (UTC)
We begin with the paradox:
Therefore it is false.
But if it is false, then what it says is true.
But true is not false.
Therefore it is also true to say
Therefore the conjunction is true:
the negation of which is
The first part of the disjunction is just the original paradox and so cannot be true, which implies the latter part of the disjunction must be true:
In other words,
which is obviously true -- which resolves the paradox.
-- Vibritannia ( talk) 17:47, 20 November 2008 (UTC)
Begin with
Therefore
which negated is
which is
which is just
The paradox is resolved.
-- Vibritannia ( talk) 11:37, 22 November 2008 (UTC)
The sentence called the Liar paradox is absurd.
It is potentially absurd because the definition of the sentence (the words) refers to the thing being defined (the sentence). And it is actually absurd because the definition of the sentence (the words) asserts the negation of the thing being defined (the sentence).
An assumption of the paradox is that it begins from a valid definition (the words of the liar sentence), but the definition is not valid -- because it is absurd. The sentence is grammatically correct, but that is not the same as saying that the definition of the sentence is logically valid.
The starting premise of the paradox, that the grammatical definition of a valid sentence and the logical definition of a valid sentence are equivalent, is false.
Vibritannia ( talk) 15:48, 4 April 2009 (UTC)
The section "Non-paradoxes" seems misguided. Though the statement in question may not be an example of the liar paradox, it is paradoxical.
Consider the following: The statement "I always lie" is either true or false (this ignores the problem that the use of the indexical "I" introduces). If we suppose that the statement is true, then it follows that the statement is false because we have supposed I am lying. Alternatively, if we suppose that the statement is false then it is, of course, false. Since the statement cannot possibly be true, the statement is necessarily false. It seems then that this is a case where a statement about an apparently contingent state of affairs (my lying habits) turns out to be necessarily false. The idea that a logically indeterminate statement could be necessarily false is paradoxical.
It seems wrong to call this statement non-paradoxical. Anyone have a proposed solution? —Preceding unsigned comment added by 208.89.36.58 ( talk) 06:48, 29 August 2009 (UTC)
1. Self-reference statements are meaningless.
2. There is one or more true statements.
If 1 is true then 2 can not be true. If 2 were true then it would be self-reference. If 2 can not be true then 1 can not be true. If 1 were true then there would be a true statement. That would mean 2 is true. If 1 is true then 2 can not be true.
1 CAN NOT BE TRUE.
David L Davidsstorm ( talk) 02:00, 25 November 2009 (UTC)
You guys............ All this and nobody asks "What's the difference between "execution" and "hanging"?
After deleting the "Spanish Inquisition" story, everything under
"Explanation of the paradox", ending with:
"lf (C) is both true and false then it must be true. This means that (C) is only false, since that is what it says, but then it cannot be true, creating another paradox."
should be deleted. lf someone has the wherewithal to create links to the various Schools of Thought (using the term VERY loosely), then go for it. Otherwise, it ALL should be considered POV or argumentative.
I waded thru the various discourses above and, personally, I think that the ONLY on-topic "talk" is by "Vibritannia" (see "Explanation of the paradox" a few lines [31] above) and the subject... LIARS PARADOX... is covered quite competently in the first 9-10 paragraphs of the article. —Preceding unsigned comment added by Paleocon44 ( talk • contribs) 06:46, 26 December 2009 (UTC)
Please feel free to add and strike out done items. Paradoctor ( talk) 15:06, 15 December 2009 (UTC)
"The next statement is true. The previous statement is false." A sentence has to assert something verifiable or unverfiable in order to be true or false. For example, the sentences "Go to the store for me, will you?" and "Ring up my groceries" are both neither true nor false. Just as the sentences "The next statement is true. Go to the store for me, will you?" are both neither true nor false (because the first sentence hinges on the verity of the second sentence for it to assert anything verifiable or unverifiable), the sentences are both neither true nor false. The same goes for "This sentence is false."- it asserts nothing verifiable or unverifiable, and is no more true or false than the sentence "Hell yes!". Perhaps the sentence can be noted for being the only type of sentence that makes a claim that is unverifiable- whereas "Hell yes!" doesn't make any claim, "This sentence is false" does make a claim, though it is neither true nor false. —Preceding unsigned comment added by 24.30.56.142 ( talk) 07:24, 4 February 2010 (UTC)
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I removed the version Everything I say is a lie.
This isn't paradoxical when most people say it. It's simply false, assuming the speaker has said at least one true thing in his life. Evercat 18:40 21 Jun 2003 (UTC)
The section "Patrick Greenough—Free Assumptions and the Liar Paradox" is in dire need of editing/clarification. I am not familiar with his work, but the section makes no sense. Please fix it. Joshua.horton 03:32, 10 December 2006 (UTC)
A version of this paradox appears in the Don Quixote ( II, Chapter LI) and another in the letter of Paul to Titus; 1, 12:
The first (which I was unfamiliar with) seems to be a paradox of some sort, but on skimming the text, I don't think it's strictly the liar paradox.
The second is the Epimenides paradox, and both this article and that one make a big deal (correctly, I think) in asserting the difference between it and the liar paradox. Evercat 19:11 21 Jun 2003 (UTC)
Why do we making such a big deal about the difference ? If someone says
that's the liar paradox, right ? But if someone says
, doesn't that *include* the previous statement ?
-- DavidCary 04:52, 18 Jun 2004 (UTC)
No, not really. Wheras
is neither true nor false, this statment may be false:
for instance I told the truth yesterday, and when i said Everything I say is a lie., i was lying. So the statment is false. The difference is slight, but there is no reason not to be picky in an encyclopedia =)
Gkhan 17:00, Jul 17, 2004 (UTC)
To state why it's not a paradox another way: the statement Everything I say is a lie only implies This statement is not true if it is true. That would be a contradiction, so the statement must be false. The statement Everything I say is a lie being false does not imply This statement is true, because it could be some other statement that is true. So, as stated above, if the speaker has ever told the truth before, then Everything I say is a lie is a lie, and not a paradox. Rob Speer 17:12, Jul 17, 2004 (UTC)
Hello. I put a proposal to merge liar paradox and Epimenides paradox at talk:Epimenides paradox. Perhaps you'd like to respond there. Happy editing, Wile E. Heresiarch 19:16, 15 Aug 2004 (UTC)
Be careful, Ropers - in your temporary version, you said that "Cretans always lie", spoken by a Cretan, is a paradox. This is wrong; it's a lie, not a paradox, though it has been given the name of the " Epimenides paradox" because of how deceptively like a paradox it is. It's not a paradox for the same reason that "Everything I say is a lie" isn't. RSpeer 17:16, Aug 27, 2004 (UTC)
Wait... this makes no sense. He states (correctly) that: '2 + 2 = 4' is the same as: 'It is true that 2 + 2 = 4', so we can surmise that (It is true that) can be added to any sentence, without affecting the meaning. However later he states that: 'This statement is false' is the same as: 'This statement is true and this Statement is False,' but this does not follow the same theory. It should actually be: 'It is true that this statement is false.'
This, of course does remove the paradox, but the way it was written was terribly incorrect, thus I have been forced to change that. 58.175.169.47 ( talk) 07:28, 25 September 2008 (UTC)
What the heck is it? It refuses my connection. RSpeer 04:35, Sep 22, 2004 (UTC)
I think a good example to use would be All generalizations are false. Using This statement is not truedoesn't define what 'this' is, or so I feel that way. I'm wondering if anyone else is confused, or is it just me? -- KaiSeun 06:48, 2004 Nov 4 (UTC)
"All generalizations are false" is not paradoxical, because there is no contradiction in assuming that it is false.
I don't find the "this" confusing in "This statement is not true". The "This statement" has to refer to itself because there is no other statement that it could refer to. -- Nate Ladd 11:08, Nov 23, 2004 (UTC)
I deleted the material below for these reasons:
1. There is no reference to this "Yablo"s publications in the References section. Who is he/she? Is his/her work even published?
2. The Yablo paradox applies only to an infinite list of statements. But this is not genuinely a paradox at all. We don't believe there can be an infinite list of statements anyway, so the fact that the supposition of such an infinite list entails a contradiction is not disturbing. Yablo's argument is a disproof of the supposition, not an apparent counterexample to our notions of truth. (But people can actually say and write things like "This sentence is false.")
To Posiduck: Which philosophers/mathemmaticians believe in an ACTUAL infinity of sentences (as distinct from numbers)? More specifically, which ones believe that the particular infinite list that Yablo describes actually exists? Is such a list constructible and, if so, then how? Questions like these have answers when applied to, say, the infinite set of integers, but I can't see what the answer would be for Yablo's list of sentences. That's why I'm asking. For any integer, I know how to construct one that's one greater in size. Ultimately, my construction technique traces back to making a union of two sets (or, if you prefer an older theory of the foundations of math, to making a line one unit longer than it currently is using only a straight-edge and a compass.) The Liar paradox is important because it seems to show that our culture's cherished intuitons about truth lead to a contradiction. The cherished intuitions are
1. Every sentence s is either true or false. (Principle of Bivalence)
2. Sentence s is true iff and only if what s says is the case.
But Yablo's so-called paradox requires the additional assumption that there can be an actual infinity of sentences such as he describes. This is not a cherished intuition. Indeed, the typical member of our culture does not believe it is true at all. So Yablo's derivation of a contradiction is only an ordinary reduction ad absurdum argument of its premises. When two of the premises are cherished intuitions about truth and the third is a dubious claim about an actual infinity, then we simply take the argument as a disproof of the dubious premise. It is not, therefore, a counterexample to something at the heart of our culture or logic or mathematics. It is, thus, not what is meant by the word paradox. This means that Yablo has failed to show that self-reference (directly or indirect) is not at the heart of the Liar paradox. -- Nate Ladd 05:09, Dec 7, 2004 (UTC)
Here's what I suggest we should put back in, and unless there is some reason not to, beyond you disagreeing with Stephen Yablo as to whether or not this is related, I see no reason not to include it.
Related Paradoxes:
Stephen Yablo (2004) has published a paper "Circularity and Paradox" in which he claims that semantic paradoxes, such as the liar, can be generated even without direct or indirect self reference. He poses a paradox he calls the w-liar.
He asks us to consider a list of sentences which is infinitely long in both directions.
And so forth, so that each sentence N says, All sentences numbered N+1 or greater are false No statement in the sequence is consistently evaluable as true or false. Choose one arbitrarily. It is true if and only if all of the subsequent statements are false. But if all of the subsequent statements are false, then any of the following sentences also makes a true claim. If any one of the sentences is false, then that could only be because a sentence numbered higher than it is true. But we already know of any arbitrary sentence that it cannot be true. So, none of the sentences are consistently evaluable. Just as in the case of the standard liar's paradox, each sentence is true if false and false if true, yet, unlike most liar variants, none of the sentences predicate falsity of themselves. Yablo thinks that these sentences are suffering the same failure as the Liar's paradox, but without self reference. This claim is controversial.
Posiduck 22:27, 9 Dec 2004 (UTC)
HERE'S WHAT I DELETED:
Furthermore, there is Yablo's version of the paradox:
Consider a list of sentences which is infinitely long in both directions. The sentences all say the same thing: All of the subsequent statements are false. Pick one statement at random. It is true if all of the subsequent statements are false. But if all of the subsequent statements are false, then what they say is indeed the case: they say that all of the statements subsequent to them are false, and ex hypothesi they are false. That contradiction means that the picked statement should be false, but its selection was arbitrary, implying all the statements must be false; again this leads to their description of subsequent statements being true. So like the liar, they're true if they're false and false if they're true, yet no propositions predicate falsity of themselves. This is sufficient to suggest that the liar does not depend upon self reference.
(all words in brackets are lies) hehe
I put a "mysteroius" tone flag on top mostly because of this section:
"If we assume that the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So assuming that it is true leads to the contradiction that it is true and false. OK, can we assume that it is false? No, that assumption also leads to contradiction: if the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either assumption, we end up concluding that the statement is both true and false. But it has to be either true or false (or so our common intuitions lead us to think), hence there seems to be a contradiction at the heart of our beliefs about truth and falsity."
Does nobody else think this can be avoided?
Nrbelex (
talk)
23:36, 15 Mar 2005 (UTC)
You are still not being explicit about what your complaint is. What is the "this" you want to avoid? Is it the use of "we"? Using an indefinite "we" is common in philosophy. -- Nate Ladd 02:12, Mar 18, 2005 (UTC)
I removed the following because the first sentence is an unjustified assertion, ex cathedra, and the second is so poorly punctuated that it makes no sense. Finally, it does not undercut the argument being made. If the anonymous Prior fan who wrote this wants to make changes in the discussion of Prior, he/she should make his case here on the Talk page. -- Nate Ladd 09:36, May 8, 2005 (UTC)
Such an assumption about clausal truth values can be done independently of sentential truth value only if the sentence itself does not make assertions about individual clauses. Of course, in this case undeniably the Prior assertion that the whole series of logical conjuctions of clauses is true is exactly identical with the whole series of assertions about the individual clauses.
Although this suggestion is somwhat amateur, we should have an article on the Two Guards and two doors logical problem. Where one guard always lies, one guard always tells the truth, and one door leads to death, and one door leads to life. You can only ask one question to ensure that you enter the door of life. Something along those lines. Colipon+( T) 21:49, 21 May 2005 (UTC)
The sentence
is not a type of liar paradox. It is a sentence that is always false whereever it appears, but it is not a sentence that is both truth and false. (And there is nothing unusual about sentences that are false wherever they appear, most false sentences are like that: "Ronald Reagan was a king of Egypt". Also, note that it is not a self-referring statement. It refers to the page on which it appears, but it does not refer to itself.
I'm not sure I understand the discussion below, but I think there is a much simpler argument against the current version of this page. When it says:
The last claim is wrong, I think. Here is a truth assignment that makes everything consistent:
In other words both sentences are false and Prior analysis works fine here too. Please correct either the page or me. F4810 16:36, 30 November 2005 (UTC)
The wiki author here made a fatal mistake. Applying the Prior prescription should invoke EVERY implicit assumption, and would look more like this
Just as with the previous reasoning this is the statement "(A and not B) and (B and A)" which can be reduced to "A and B and not B" which is obviously false. Therefore the statement is false and there is no paradox.
Now the last attack to this is the statement
I strongly disagree with this statement. The two clauses are coupled to each other. They are inherently not independent and therefore I see no reason why it should be possible to treat them independently.
I'm gonna fix this now. Please reply if I am in error ( CHF 09:57, 20 September 2005 (UTC))
The whole last paragraph of the section on Prior's argument should be removed unless someone can cite a reference that makes the arguments made in that paragraph (Note that I'm not claiming that arguments in that paragraph are necessarily wrong, but a citation is desperately needed) —Preceding unsigned comment added by 24.17.244.140 ( talk) 00:44, 4 September 2007 (UTC)
?? I guess I am a little confused. I entered a bit of info about the liar paradox and it was summarily removed with no explanation. I went back and looked at the editing guidelines and can't understand what I did wrong. I thought it was an interesting statement about the liar paradox, it eqivocates. It is one that my phil. prof, J.C. Beall, an expert in this area, thought was probably true. Any suggestions?
I see. I thought the encyclopedia was a means of presenting different views, ideas that did not need the Imprimatur of the academic establishment. My mistake-namaste
I don't know what that's supposed to mean, but this much I can tell you: The entire article is bogus. The statement "I am lying" has no truth value in and of itself, because it's not "about" anything. Go ahead, prove me and my Logic 101 professor wrong. Wahkeenah 04:42, 27 January 2006 (UTC)
To put it another way, you only think there's a paradox because you think the statement "I am laying" should have a truth value. Your initial assumption is incorrect. Once you realize that, you discover that there is no paradox, and that this article is based on a false premise. As my professor said, "When you start with incorrect assumptions, you are liable to get interesting results!" Wahkeenah 04:48, 27 January 2006 (UTC)
OK, I worded it a little better. Sorry to burst your bubble, Grasshopper. Wahkeenah 05:00, 27 January 2006 (UTC)
Note that in my writeup, I referred to something. I said "That assumption is false." That sentence has a truth value, because it refers to something else that has a truth value. A sentence such as "This assumption is false", by itself, has no truth value. Class dismissed! Wahkeenah 05:04, 27 January 2006 (UTC)
It would be a good idea to mention the name of that episode. Go for it Trekkers :)-- Manwe 17:49, 29 March 2006 (UTC)
Reply to Nate Ladd and anyone else who cares: There is not an infinite number of integers or lists or anything else. There is only a repeating (infinite) process to construct them, and the process is always used a finite number of times. His own words... "For any integer, I know how to construct one that's one greater in size." If a list has a beginning it can't be infinite. As for the statement "I am lying", I agree with Wahkeenah,it has no content that can be verified true or false. Phyti 01:19, 10 June 2006 (UTC)
195.153.45.54 14:23, 9 August 2007 (UTC)
Alright, I think I've come up with a solution to "This statement is false". It is false, as part of the statement is false and another is true. Saying "This statement is false" is obviously stating that the whole statement is false, which is false, because the statement as a whole is both true and false. With an easier-to-understand version that means the same thing, the part in italics is false, and the part in bold is true:
This whole statement is false
This whole statement is the false bit because it refers to the statement as a whole, whereas because only part of the statement is false, you cannot say that the whole thing is false, thus making only that bit false. Hugh Jass 23:27, 4 July 2006 (UTC)
Is the answer to this question "no"?
Does this self-referential contradiction fall under the liar paradox?
What about this:
If I ask you [question X], will the answer be the same as the answer to this question?
which forces an answer of "yes" to the question X.
If not, what topic should they go under? -- Spoon! 05:28, 30 July 2006 (UTC)
If [question X] is Can you make jam? and the answer is yes then I can answer no to your question.
195.153.45.54 14:31, 9 August 2007 (UTC)
I change a statements i found un clear to this, perhaps i was wrong. if that is so, please revert it i didn't mean no harm, and i may have bean right to change it. but i am having doubts
Would this qualify as an example of the Liar's Paradox? 65.12.114.98 14:09, 27 September 2006 (UTC)
Do statements have meaning if un-uttered? Do they not allways take place in time? If so then, "statement" in "This statement is false" contains a prediction.
In "This statement is false", the referent does not yet exist at the time "this statment" is written, spoken, or typed. The same is probably true when the statement is heard, and even read.
So what does "This statement" in "This statement is false," mean? As pointed out above "This statement" must be refering to the whole statement, that is to say the completed statement that does not exist at the time of writing "this statement". The future does not yet exist, so we can only make predictions about it. We say "we can talk about the future", but really we are predicting the repetion of a past event.
"This statement is false", begins as a prediction about something that does not exist until the last "e" is written. Hence the "statement," which starts as a prediction, can perhaps be re-written as
"I predict that the statement that will exist when I finish writing this will at that time be false." or
"It is predicted that the statement that will exist will be at that time false."
It is only because the statement starts as a prediction that it can refer to itself. It needs time to loop back on itself, but admixture of time prevents the looping back from being complete.
As a prediction/statement it relates to two times, the time of predicting, and the time being predicted. The truth value of the same sentence may be different at each of these two times. Just as "Tim is alive" is true now, but will not be true in considerably less than 100 years time.
I think that there is a case for saying that the prediction came true: the statement is false. The sentence as statement does not say anything true. But as a predicition, in its unfolding prior to its completion, it was a good, i.e. veracious prediction. The mistake that everyone is making are
1) To mistake a prediction for a statement just because it calls itself a statement.
2) To think that there is such a thing as a living word that sits on the page and just means, without a reader or writer. But as far as I know, language only gets meaning in use, i.e. in time, and it is in time that the liars "paradox" fails to be paradoxical as it unfolds. When you take the fact that the statement says nothing true, it does not allow use to travel in time and take that false-ness back to the time of predition since the prediction was accurate - as it surely was.
3) Perhaps again you could say that "the statement" always remains a prediction "This will be false," and it always will be. The claim that this is a "paradox" is also akin to claiming that it is impossible to predict that "Tomorrow is Sunday," because when Sunday comes, the next day will be Monday, and the preditiction "Tomorrow is Sunday" will no longer be true. But if the next day was indeed Sunday then at the time of prediction, "Tomorrow is Sunday" was true and as a predition it remains true. The sentence in question always has the same referent, and is as false as a statement and as true as a prediction today as it every will be.
"I pledge that the statement that will exist when I finish writing this will at that time be false." -- Timtak 10:21, 1 October 2006 (UTC)
The following is not really a paradox, as the person is only guessing because no outcome has yet been established, therefore they have not identified them self. If the person is decapitated, then like after being married you are married, but before you are single, so you must be decapitated first to know if indeed it happened, but then you are now a different person after decapitation. So you are one person before, and a different one after, but the question was asked who they were before they entered not what they will be after. They may hang her next week, not the next day, or decapitate her after she dies from hanging, so she fufills both conditions, but first they may torture her with drugs, like Zyprexia, so she is even stupider than before. (That is the modern method used by the USA today for those who express their first amendment right of freedom of religion. We should have an amendment that protects us after our expression as well. That is a better paradox than the current one shown here.) Therefore it is not a paradox since no outcome can yet be established on the current information. However, it does clearly show how a typical Jew thinks they are clever, when they are only fooling them self, where they should be hung, and decapitated for their attempt at humor. Here follows the so called paradox.
In a Jewish folktale an anti-semitic king makes an edict that any Jew who enters the capital city will be asked to identify himself. If he tells the truth he is to be hanged, but if he lies he will be decapitated. One Jewish woman comes to the gates of the city. She tells the guard she is a woman who is going to be decapitated that day. If they do that she will be telling the truth, in which case she will have to be hanged. But then she would be lying, meaning she will have to be decapitated. And the cycle of logic repeats ad infinitum.
Danross 03:42, 11 December 2006 (UTC)Dan Ross
1: The sentence below is true. 2: The sentence above is false.
Here's how I tried to solve his:
Let's call sentence 1's truth vale A, and sentence 2's truth value B. Then it's the same as:
A = B B = 1-A
Wich gives:
A = 1-A 2A = 1 A = 0.5
This means A is both true and false/neiter true or false. However, say you do not allow anything but 1 or 0. Concider the truth values boolean (true, false) instead of real (-2.7, -1, 0, 1.2, 5). Then:
A = B B = !A
Wich gives:
A = !A
If A can only be 1 or 0, then the equation is never true.
My conclusion:
The second sentence is as false as the first sentence is true. If we allow them to be "half true/false", then it works. Otherwise, the sentences are incompatitible.
It seems the execution paradox has a weak point in this story. It can be avoided by the person who makes the paradoxical statement. In this story the man said "I'm going to be executed today", the inquisitor could have let him live one day longer and have him executed the next day for making a false statement, since the inquisitor did not set any time limit. A better thing to say would have been: "I will die by execution".
--Anonymous —Preceding unsigned comment added by 62.177.253.214 ( talk) 13:28, 30 July 2009 (UTC)
Hi, I'm kinda new (or at least I've never contributed) so I'm kinda worried about editing the page and messing something up or anything (hehe)
Anyway, I noticed that under "In popular culture" it states that
"If the first statement was "Everybody lies all the time", then it by itself would constitute a liar paradox."
However, that's not a liar paradox, after all, it can only not-be true (then it'd be contradictory) but it can be false (I think) If somebody tells the truth then not everybody lies all the time, but it doesn't mean that person didn't lie that time. —The preceding unsigned comment was added by 80.126.65.34 ( talk) 22:20, 5 March 2007 (UTC).
seems to me this is just a false dichotomy. The sentence is neither true nor false.
I changed "false conclusion" to "solution", as it is clearly correct. —The preceding unsigned comment was added by 82.93.92.62 ( talk) 09:50, 13 April 2007 (UTC).
I added some lay examples, including why the examples were self-contradictory, but was reverted saying they weren't examples of the liar paradox and simply false statements. These examples are valid, accessible liar paradox examples:
Paradox: ...including those that contradict this sentence.
Paradox: ...including those that declare this statement false.
Paradox: ...including those asserting the opposite of this statement.
Paradox: ... except this statement.
Paradox: ... except the existence of dichotomies.
-- Loodog 18:08, 9 July 2007 (UTC)
I have deleted the following sentence from just below the first appearence of the 2-sentence version: "However, it is arguable that this reformulation is little more than a syntactic expansion. The idea is that neither sentence accomplishes the paradox without precisely its counterpart. "
My reason is that "syntactic expansion" is not defined and, more importantly, there doesn't seem to be any implication of the remark. So what if "neither accomplishes the paradox without precisely its counterpart"? That's not a criticism of the 2-sentence version, its just a description of it. Its still a paradox and still needs resolving. —Preceding unsigned comment added by 24.16.98.193 ( talk) 00:51, 4 November 2007 (UTC)
wouldnt that paradox automaticly resolve itself because you can ignore the 1st line it currently says The statement below is false The statement above it true
but to make it a paradox it should be The statement below is true The statement above is false -- Tjayh913 03:49, 10 November 2007 (UTC)
I was about to reference someone to all the references in pop culture that were in this article, but they're not there anymore. Can anyone put them back or were they erased for a good reason? 159.90.9.83 17:27, 15 November 2007 (UTC)
What we really need is a summary of its use in humor that cites popular culture, rather than a list of occurrences. I understand the arguments for and against the inclusion of Trivia sections, and am torn on the issue myself. What is less controversial is when an article summarizes and provides context for these examples, often not simply as "popular culture," but as some particular aspect of it that serves to expound upon the importance of the topic, instead of trivializing it into a list of particular instances. I'm gonna try to look through the history pages and see if I can find a unifying theme. I have no desire to get into a serious edit war, so please feel to discuss this here or on my talk page. J Riddy ( Talk || Contributions) 21:58, 17 February 2008 (UTC)
If we can derive this statement is false from This statement is true and this statement is false, then the paradox is back. And if we are not allowed to make such a derivation, then Prior has, in effect, invented a new kind of conjunction whose truth value characteristics are so mysterious, we cannot really say with any confidence that the paradox has been dissolved.
But the second italic proposition is (A=)A and not A, from which logically we can derive any sentence. I don't see why this property of the conjunction would be mysterious, given the Principle of explosion.
Take "classical" logic: to determine if a sentence is true, then either (1) we derive it from the axioms, in which case it is true, or (2) we assume it's true and try to derive a contradiction, in which case we call it false. (That's not really correct as far as I can tell; in the resulting system all propositions are true, given the PoE, and I'm not sure I see how that makes the sentence false in the original axiomatic sentence.)
In this case, the sentence _is_ contradictory. It contravenes the law of non contraction (according to Prior, a sentence claims that its content is true, and this one claims that true is false). So in a system where we use the law of non-contradiction, the sentence is false. This can be nicely expressed by it's negation, "The sentence 'This sentence is false' is false". If you don't allow the law of non-contradiction, then truth isn't well defined anymore, anyway. —Preceding unsigned comment added by Bogdanb ( talk • contribs) 18:54, 5 March 2008 (UTC)
"This is to be distinguished from the common colloquial expression "I tell a lie." when the speaker has realized that he has just accidentally told an untruth."
I don't, uh, what? I've never in my life heard someone say "I tell a lie" after they realized they just 'accidentally' lied. That isn't even proper English. -- Dbutler1986 ( talk) 06:48, 18 June 2008 (UTC)
"Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed" by which he meant a language in which it is possible for one sentence to predicate truth (or falsity) of another sentence in the same language (or even of itself)."
I don't see how this resolves the paradox. Tarski is just saying that such a statement can't even be formulated in a semantically closed language, whereas he's offered no resolution for assigning a non-contradictory truth value to "This is a lie." in semantically open languages (like colloquial English).-- Loodog ( talk) 15:58, 18 June 2008 (UTC)
Removed from the article, to discuss what parts of it might be saved:
My specific criticisms:
That's it for now. It would be reasonable to include some discussion of Goedel here, but there was not enough correct stuff in the text as it stood to make it worth trying to fix piecemeal, in my judgment. Any thoughts on how it could be done (especially with sources) would be welcome. -- Trovatore ( talk) 00:10, 17 August 2008 (UTC)
(outdent) I think this gets rather far afield from the Liar. The two things really are not that closely analogous (largely because provability is not very much like truth). Something the two arguments have in common is that they are both diagonal arguments, but there are an awful lot of those.
Just for fun I'll try and write down something that I think barely might be defensible here, if it could be sourced.
Now it's true that this version doesn't get into showing that A also does not prove ¬GA, but that part would require bringing in either ω-consistency, or Rosser's modification (which no longer looks like the liar), and is less important anyway since we've already shot down A as a way to generate all arithmetical truths (since it fails to prove GA, which is a true statement of arithmetic).
But I'm not seriously proposing this addition. As I say, it's not really all that closely related; there's a nice formal similarity in the formulation of the sentence itself, but the analogy doesn't go too much further. -- Trovatore ( talk) 07:35, 19 August 2008 (UTC)
We begin with the paradox:
Therefore it is false.
But if it is false, then what it says is true.
But true is not false.
Therefore it is also true to say
Therefore the conjunction is true:
the negation of which is
The first part of the disjunction is just the original paradox and so cannot be true, which implies the latter part of the disjunction must be true:
In other words,
which is obviously true -- which resolves the paradox.
-- Vibritannia ( talk) 17:47, 20 November 2008 (UTC)
Begin with
Therefore
which negated is
which is
which is just
The paradox is resolved.
-- Vibritannia ( talk) 11:37, 22 November 2008 (UTC)
The sentence called the Liar paradox is absurd.
It is potentially absurd because the definition of the sentence (the words) refers to the thing being defined (the sentence). And it is actually absurd because the definition of the sentence (the words) asserts the negation of the thing being defined (the sentence).
An assumption of the paradox is that it begins from a valid definition (the words of the liar sentence), but the definition is not valid -- because it is absurd. The sentence is grammatically correct, but that is not the same as saying that the definition of the sentence is logically valid.
The starting premise of the paradox, that the grammatical definition of a valid sentence and the logical definition of a valid sentence are equivalent, is false.
Vibritannia ( talk) 15:48, 4 April 2009 (UTC)
The section "Non-paradoxes" seems misguided. Though the statement in question may not be an example of the liar paradox, it is paradoxical.
Consider the following: The statement "I always lie" is either true or false (this ignores the problem that the use of the indexical "I" introduces). If we suppose that the statement is true, then it follows that the statement is false because we have supposed I am lying. Alternatively, if we suppose that the statement is false then it is, of course, false. Since the statement cannot possibly be true, the statement is necessarily false. It seems then that this is a case where a statement about an apparently contingent state of affairs (my lying habits) turns out to be necessarily false. The idea that a logically indeterminate statement could be necessarily false is paradoxical.
It seems wrong to call this statement non-paradoxical. Anyone have a proposed solution? —Preceding unsigned comment added by 208.89.36.58 ( talk) 06:48, 29 August 2009 (UTC)
1. Self-reference statements are meaningless.
2. There is one or more true statements.
If 1 is true then 2 can not be true. If 2 were true then it would be self-reference. If 2 can not be true then 1 can not be true. If 1 were true then there would be a true statement. That would mean 2 is true. If 1 is true then 2 can not be true.
1 CAN NOT BE TRUE.
David L Davidsstorm ( talk) 02:00, 25 November 2009 (UTC)
You guys............ All this and nobody asks "What's the difference between "execution" and "hanging"?
After deleting the "Spanish Inquisition" story, everything under
"Explanation of the paradox", ending with:
"lf (C) is both true and false then it must be true. This means that (C) is only false, since that is what it says, but then it cannot be true, creating another paradox."
should be deleted. lf someone has the wherewithal to create links to the various Schools of Thought (using the term VERY loosely), then go for it. Otherwise, it ALL should be considered POV or argumentative.
I waded thru the various discourses above and, personally, I think that the ONLY on-topic "talk" is by "Vibritannia" (see "Explanation of the paradox" a few lines [31] above) and the subject... LIARS PARADOX... is covered quite competently in the first 9-10 paragraphs of the article. —Preceding unsigned comment added by Paleocon44 ( talk • contribs) 06:46, 26 December 2009 (UTC)
Please feel free to add and strike out done items. Paradoctor ( talk) 15:06, 15 December 2009 (UTC)
"The next statement is true. The previous statement is false." A sentence has to assert something verifiable or unverfiable in order to be true or false. For example, the sentences "Go to the store for me, will you?" and "Ring up my groceries" are both neither true nor false. Just as the sentences "The next statement is true. Go to the store for me, will you?" are both neither true nor false (because the first sentence hinges on the verity of the second sentence for it to assert anything verifiable or unverifiable), the sentences are both neither true nor false. The same goes for "This sentence is false."- it asserts nothing verifiable or unverifiable, and is no more true or false than the sentence "Hell yes!". Perhaps the sentence can be noted for being the only type of sentence that makes a claim that is unverifiable- whereas "Hell yes!" doesn't make any claim, "This sentence is false" does make a claim, though it is neither true nor false. —Preceding unsigned comment added by 24.30.56.142 ( talk) 07:24, 4 February 2010 (UTC)