Talk:Liar paradox/Archive 1

"Everything I say is a lie"

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)

That is correct, which is why I have again removed it. If any statement the speaker said was true, then this can be verifiably a lie. Nicholasink 00:44, 27 January 2007 (UTC)

yea but thats assuming - lets go extreme and believe for a second everything person A ever said in his life was a lie - then he proceeded to say "Everything I say is a lie" - which would officially be his first truth 68.202.136.112 (talk) 03:06, 25 October 2009 (UTC)

Greenough

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)

Literary versions

A version of this paradox appears in the Don Quixote (II, Chapter LI) and another in the letter of Paul to Titus; 1, 12:

"One of themselves, even a prophet of their own, said, the Cretians are alway liars, evil beasts, slow bellies." (KJV translation)

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)

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Why do we making such a big deal about the difference ? If someone says

• This statement is not true.

that's the liar paradox, right ? But if someone says

• Everything I say is a lie.

, doesn't that *include* the previous statement ?

-- DavidCary 04:52, 18 Jun 2004 (UTC)

No, not really. Wheras

• This statement is not true.

is neither true nor false, this statment may be false:

• Everything I say is a lie.

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)

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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)

"Cretans always lie"

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)

A. N. Prior

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)

xxx.lanl.gov reference

What the heck is it? It refuses my connection. RSpeer 04:35, Sep 22, 2004 (UTC)

xxx.lanl.gov is/was the "Physics ArXive" (or perhaps it went under some other title) at Los Alamos Nat'l Lab. Dunno if the url should be pointed somewhere else now. Wile E. Heresiarch 15:20, 25 Sep 2004 (UTC)

The link has already been fixed -- see article. Ropers 15:55, 25 Sep 2004 (UTC)

Another example?

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)

Yablo example deleted

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?

Stephen Yablo is a philosophy professor at MIT's deparment of linguistics and philosophy (one of the top US Philosophy programs). http://www.mit.edu/~yablo/home.html (his home page) has some papers of his, particularly the one in which he talks about the Liar "Circularity and Paradox." Posiduck 02:50, 5 Dec 2004 (UTC)

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.")

Why don't we believe there can be an infinite list of statements? At any rate, despite our personal beliefs on infinity, philosophers and mathematicians do in fact believe in it, and his paradox is a direct response to people's claims about self-reference and the liar. Posiduck 02:50, 5 Dec 2004 (UTC)

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)

Ok, let's examine two different issues: 1) Has Yablo proven that self reference is not at the heart of the liar? and 2) Is Yablo's attempt to prove such worthy of inclusion in this article?

We could debate 1 for quite some time without coming to agreement, however, as evidenced from the inclusion of the Graham Priest information; something need not be difinitively agreed upon by everyone in order to merit inclusion in the article. Some people think paraconsistent logics are ridiculous, others think paraconsistent logics are tenable. I daresay there is no consensus on the matter. So, I think 2 is the more interesting question for what should be addressed in this article. And, bearing that in mind, the relevant questions are, a) is Yablo's attitude part of the overall liar's paradox discussion in philosophy? b) Is Yablo/his idea influential enough to merit inclusion in the article, and c) How do we include his take on the paradox in an NPOV manner.

To answer the third question first; this is no problem; we attribute the claims he makes to him, and mention that it is not agreed upon by all that this paradox is in fact related. To answer the first question; Yablo wrote his paper about self-reference and the liar, so it seems like his paper on the subject is in some important way related to this discussion (rather than, perhaps, meriting its own entirely separate article). Thirdly, the question of whether Yablo is important enough to merit inclusion in the article. The Philosophical Gourmet Report (a reputational ranking survey of graduate philosophy programs in the english speaking world), ranks MIT's program as tied for 6th best. Since it is a reputational survey, the reputations of the professors in that department among those active in the field are the primary criteria by which such a ranking is generated. So, MIT's Philosphy program on the whole is certainly notable. Yablo is not only a professor in that department, but also the chair of the department. I think this ought to make him notable enough for inclusion in the article, and since he wrote a paper which proposes a fairly interesting revision of the way we examine the liar's paradox (i.e. that self reference is a bit of a red herring), I certainly don't see why there oughtn't be a section on the paradox he claims is related in the article, so long as that section makes it clear that this is merely one attitude that some take towards the paradox, and does not claim that this has demonstrated that the paradox is in fact not about self reference. Posiduck 16:50, 8 Dec 2004 (UTC)

Yablo's reputation is irrelevant. So is the reputation of his employer. The quality of a philosophical idea is measured by its content, not the reputation of it's author's employer. Many crappy papers have been written by even renowned philosophers or others on the faculty of schools with good reputations. I made a reasoned argument for why the Yablo stuff should not be included. Citing MIT's reputation doesn't refute that argument. An encyclopedia article cannot be an indepth study. It must limit itself only to the ideas that represent an expansion of our understanding of the topic. Because Yablo's "paradox" isn't really a paradox at all as I showed above, it doesn't really enhance our understanding of the Liar paradox. Priest's ideas are probably too weird for an encyclopedia article too. It would be better to have neither Priest nor Yablo rather than both. -- Finally, let me make an even stronger argument that Yablo's "paradox" isn't really a paradox, as distinct from merely a reductio of a dubious premise: Yablo's implied premise is not just that there is an actual infinite set of sentences, it is that there is an actual infinite sequence of sentences. There can be such a thing with integers because they are each different so there is a conceptual basis for ordering them. But all Yablo's sentences are identical. So the only way to order them is by their different positions relative to one another. This means that they must be ordered spatially or temporally. And that, in turn, means that they must be PHYSICAL entities (made of ink or chalk dust or some such). But no one, not even Yablo, believes there is an actual infinity of physical sentences. --Nate Ladd 19:28, Dec 9, 2004 (UTC)

It seems to me as though, to make judgements about which of the ideas currently in the field are viable, by our lights, borders on POV. Yablo is active in the philosophical community, he writes on this subject, and he has claimed some connection between the two cases. YOUR opinion on the merits of his argument are, in fact, completely irrelevant, because YOU are in no position to be judging whether or not Yablo is right that they are related. At the very least, the article ought to claim that some prominent Philosophers have believed that there are related paradoxes to the Liar, which do not contain self reference, and then link to an article on Yablo's paradox. However, to omit Yablo because you disagree with him is to bias the article towards what YOU think, and that's not what we should be doing. He even names his paradox after the liar, so clearly he is trying to claim some relationship. As for your problems with the concept of infinity; mathemeticians and philosophers do not have the objection you have to the concept, since the analysis of this semantic paradox is a debate in philosophy, we ought to present all of the opinions that people in the field have publicly asserted/defended. I think it's great that you have a response to Yablo's paradox. E-mail him with it, and maybe he'll recant his view. However, right now, as an encyclopedia meant to record what has been said about a particular paradox, we ought to record what he said. The reason I cited his credentials is to point out that he was not just some guy with a website who said, "hey, wouldn't it be cool if there were, like, infinite sentences," but instead that he is indeed a member of the philosophical community who has written about the topic of this article.

1. A quick check of bibilographies of the Liar paradox on the web shows that there are over 100 philosophers who have published on the subject. We cannot include all of them. That is why it is not correct for you to say "we ought to present all of the opinions that people in the field have publicly asserted/defended". Given that an encyclopedia survey article cannot survey 100s of viewpoints, the mere fact that Yablo has written on the subject is not sufficient justification for including him.

2. Making judgements about whose ideas should be included is precisely what we are supposed to do. We are all collectively the writer/editors of the wikipedia. (Consider the idiocy of an encyclopedia article about Hitler that treated as equally plausible the views that (a) he was a killer and (b) he never did anything wrong.) I am in a position to judge whether Yablo is right. So are you and every other participant in the wikipedia. When we disagree, we hash it out with reasoned argument. I've made a reasoned argument for why Yablo's view should not be included. You have not attempted to refute it.

3. I don't have any problem with (or objection to) the concept of infinity. (I referred to the infinite set of integers in my argument.) There is, in fact, nothing non-standard at all in my views on infinity. Contrary to what you seem to think, neither philosophers no mathematicians believe that just any only kind of thing can come in infinite quantities. (Physical things cannot.) And only some of the things which are infinite, can be in a well-ordered sequence. As far as I know, every mathematician and philosopher would make a distinction between actual and potential infinities. I don't know of anyone who would claim that English has an actual infinity of sentences, distinct from a potential infinity of sentences.

4. See my suggested compromise below.--Nate Ladd 02:54, Dec 14, 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.

I've done more than disagree. I've given a reasoned argument. --Nate Ladd 02:54, Dec 14, 2004 (UTC)

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.

1. All sentences numbered 2 or greater are false.
2. All Sentences numbered 3 or greater are false.

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)

This is a better version because adding the numbers makes it possible for the sentences to be well-ordered and, hence, they need not be physical entites. Nevertheless, Yablo's derivation of a contradiction still has as a premise the claim that there is, in English, an actual infinity of sentences of the form "N. All sentences numbered N+1 or greater are false". If you believe this premise, Posiduck, you are entitle to your (unique, I think) opinion. But you are not entitled to pretend that this premise is just as plausible to the typical member of our culture as the premise that "This sentence is false" is a sentence of English. Here's my proposed compromise. There is an article just called "Paradox" in the widipedia, with a long list of paradoxes and links to wikipedia articles about them. Why don't you create an article on Yablo's work. Add a reference and link to it in the "Paradox" article. Then here in the Liar Paradox article have a very brief statement like "It has been alleged that a form of the Liar paradox can be created in which there is no self-reference. See 'name of Yablo article here'." Then link the title to your Yablo article. --Nate Ladd 02:54, Dec 14, 2004 (UTC)

It seems pretty clear that there can be infinitely many statements. "The number 1 is a number," "The number 2 is a number," "The number 3 is a number," and so on, for one obvious example. In fact, before reading the discussion here, I had no idea that some people believe there are only finitely many sentences. Nate Ladd, if you can find examples of notable philosophers who share that view, you might consider writing a Wikipedia article about it. I'm interested to know how many statements you think there can be. More than a million, presumably. But is it more than a googol? More than a googolplex? Can it be calculated at all? If the subject isn't just original research, it could probably be a useful article. So that's two new articles that would be useful: one about the finite statement theory, and one about Yablo's paradox. We can have those articles in Wikipedia and still mention Yablo's paradox in this article, since it's pretty relevant. By relevant, I mean that published papers by philosophers consider it to be relevant. We should try to accurately report on philosophical ideas, even if we don't like the concept of infinity. Factitious 06:14, Dec 14, 2004 (UTC)

Despite what you call yourself, I'll treat you with respect and reply. However, I'm going to resist the temptation to lecture you on the difference between potential and actual infinity, as well as on mathematical constructivism. Read up on these things and you'll have the examples you are looking for. As for how many sentences (not statements) I think there are in English: I think there is a potential infinity of them. To get back to what's at issue here: A mere "mention" of Yablo's work in this article is OK by me, as I suggested in my proposed compromise. "Considered relevant by published papers" is still to broad a criterion for inclusion. The professional philosophers are producing a literature for themselves, not an introductory encyclopedia article that includes mainly beginners in its readership. We, on the other hand, are producing just such an encyclopedia, hence we must use a more stringent criterion of inclusion. Also, consistently applied your criterion would require us to include over 100 writers views. (By the way, I like the concept of infinity.) --Nate Ladd 07:33, Dec 15, 2004 (UTC)

Quite plainly, I don't know anyone who doesn't agree that there is an (enumerable) infinity of sentences of a given language (generally defined recursively). The "potential" and "actual" infinity distinction is irrelevant, as is mathematical constructivism. The paradox still applies to an indefinite list of sentences. Sentences and statements are used interchangeably by most logicians, though some may make a distinction concerning closed formulas (i.e. sentences) and open formulas (i.e. statements). (Or sometimes, there is a distinction made between well-formed formulas (i.e. sentences) and simply formulas (not necessary well-formed -- i.e. statements).) At any rate, noting such a distinction is, again, irrelevant. It is still a legitmate paradox without self-reference, as far as I can tell. The solution is the same, however. There is no "global" truth predicate in any sufficiently strong theory (e.g. one that can talk about itself), so the statements are not actually statements at all because 'is false' is not a predicate of the language. In other words, you cannot form statements (or schemata) such as 'statements with number n>1 are false'. Nortexoid 06:51, 24 Dec 2004 (UTC)

1. Looks like I SHOULD have given Facetious that lecture! Aristotle and Aquinas did not believe there was an actual infinity of sentences (or anything else). In addition, mathematical finitists do not believe there is an actual infinity of anything. Nor can any kind of ontological constructivists, like Michael Dummett, allow that there is an an actual infinity of anything. Anyone who believes that meaningful truth-bearers are physical entities (sentences made of ink molecules, for example) does not believe that there is an actual infinity of truth-bearers. Likewise, anyone who thinks "meaning is in the head" (that is, truth bearers are thoughts in the heads of people), cannot believe in an actual infinity of sentences. Add ontological nominalists to the list as well (see below). I've never seen a poll of philosophers, but its quite possible that most philosophers fall into one or another of these various groups who do not believe there is an actual infinity of sentences.

Who thinks that truth is in the head? I would've thought that after Hilary Putnam's attack on it, internalism had long been abandoned.

Hardly. (There are rebuttals to Putnam, you know.) And suppose there WAS no one left who believes that meaning is in the head? Are you saying that there used to be a significant number of people who would reject Yablo's assumption of an actual infinity of sentences, but now there isn't, so we should include his "paradox"?

Second, most philosophers, logicians, and mathematicians are not constructivists.

Says who? Did you take a poll? Besides, you missed the point. I was asked to provide examples of people who don't believe there is an actual infinity of sentences and I did. If even a significant minority of people don't believe in an actual infinity of sentences, that means that Yablo's argument depends on a controversial premise. So his contradiction is a reductio of that premise, not a paradox for our culture's cherished beliefs about truth. Hence, it is not a variation of the Liar paradox.

Most invoke an infinity of sentences or quantify over infinite domains or allow for nonenumerable languages, or etc. Pick up any accessible mathematical logic text and find out for yourself.

I have several on my shelf. None presuppose that there are actual (distinct from potential) infinities. Quantification generally presupposes only indefinitely sized domains. It does not presuppose domains with an actual infinity of objects. Which is not to say that it presupposes that there are no such domains either. Quantification, per se, is neutral on whether or not there are actual infinities.

(Many set theorists also accept the axiom of choice.) Reasoning about the infinite does not commit one to any ontology consisting of infinities.

Exactly! I think that Factitious and Posiduck may have interpreted every reference to infinity in the literature as an indication that the writer believed in actual infinities. I'm glad you agree with me that this is not the case.

I can define a domain consisting of infinitely many objects and not be committed to an ontology of infinitely many objects.

Well, that depends on exactly what you do with the domain you define. If you reach only conclusions of the form "If there were an actual infinity, then P" you're OK. But if you try to conclude "P" unconditionally, then you are, indeed, committed to the claim that there is an actual infinity. In particular, if P is a contradiction, then all you've got is a reductio of the claim that your defined infinite domain could actually exist. You do not have a contradiction that derives solely from uncontroversial premises. You don't have a paradox.

Your arguments miss the point since they assume that Yablo's paradox is making some sort of ontological claim.

I'm not assuming it. I concluded it. His argument (at least as it was presented in the material I deleted -- see below) presupposes the claim that there is an actual infinity of sentences in his list. (Whether you call this an "ontological claim" or not doesn't matter to me.) His derivation of a contradiction does not work without this assumption. The list has to be actually infinite. (None of his other defenders in this discussion deny that.) I concluded this from the fact that if the list is finite, then there is a consistent assignment of truth values

2. To have a pray of a chance of believing in an actual infinity of truth-bearers, you have to believe that truth-bearers are propositions -- timeless, locationless, immaterial entities, which already existed "prior" to humans (and human languages). (So you can add all ontological nominalists to the list of people who do not believe that there is an actual infinity of truth bearers because they do not believe that there are propositions.) But there are two reasons why this won't justify including Yablo's "paradox" in this article: (1) Reformulating Yablo's argument for propostions (not sentences) adds as a premise that propositions exist and that they are the locus of meaning (the truth bearers). But this puts Yablo's argument in the same situation as I described above: he doesn't derive a contradiction from premises that our culture holds near and dear. Rather, he uses a premise that is highly controversial anyway, so what he really has is just a run-of-the-mill reductio of a dubious premise. (2) Since propositions are language-transcendent (e.g. "Snow is white" and "La neige est blanc" are the same proposition), they do not have any features that are not universal to all languages, thus they do not have truth predicates. So the Liar paradox cannot be formulated for propositions.

A theory of truth-bearers is irrelevant when the paradox is formulated in a theory of logic -- as it should be. The semantics is defined for the theory by an assignment (e.g. of denotations to constants) under a given interpretation.

You could not be more wrong. The paradox is not "formulated in a theory of logic". It is formulated in natural language using premises that embody cherished pre-philosophical intuitions of truth.

Truth-bearers play no role in defining that semantics (i.e. the truth conditions of the formulas of the theory), nor do platonic propositions. Also, the argument is not making any ontological claims.

Most versions of the Liar paradox are not making ontological claims(unless you count as ontological something like "The sentence 'This sentence is false' is a well-formed formula of the language" as ontological.) But Yablo's "paradox" does make an ontological claim. It claims that there is an actual infinity of a certain class of sentences.

It is a logical paradox which happens to be (here) phrased in natural langauge. You're giving all sorts of bizarre arguments concerning natural languages, theories of meaning, and ontologies.

The Liar is not a logical paradox. A logical paradox would be a contradiction that seems to follow from uncontroversial topic-neutral purely logical axioms and/or rules of inference alone. (Arguably, the Surprise Quiz is a logical paradox.) Russell's paradox is a paradox of set theory. The Liar paradox is what has been called a "semantic paradox" but even that name is more a side-effect of Tarski's writings about it. The best name for it would "truth paradox", because it is a contradiction generated from premises that are (mostly) about truth, not purely logical axioms/rules.

3. The distinction between actual and potential infinity matters because almost no one disputes that there is a potential infinity of sentences, but many (maybe most) people do not believe there is an actual infinity of sentences and Yablo's argument presupposes that there is an actual infinity. (I mentioned mathematical constructivism because Facetious asked for examples of people who don't believe in an actual infinity of sentences.)

Even constructivists believe that there are infinitely many sentences of any classical theory of logic in which the language is defined recursively, even if they insist only on finite domains and even if they insist on potential infinities. There is a big difference, however, in considering infinitely many (potential or actual) sentences all at once, and considering any particular arbitrary sentence. The paradox could be interpreted not to appeal to a consideration of an infinity of sentences all at once but to appeal to a consideration of any given sentence of a (potentially) infinite sequence.

There is no paradox if you interpret it the latter way. A "(potentially) infinite sequence" is a FINITE sequence. And for a finite sequence there is a consistent assignment of truth values that does not lead to a contradiction.

If you find the word 'all' abhorrent, then use 'any'. (1) Any sentence Pn>1 is false.

Now I think you are pulling my leg. "Any given X is P" is synonymous with "All X are P", so substituting "any" doesn't help Yablo at all. He's still presupposing an actual infinity. By the way, your use of the phrase "(potentially) infinite sequence" makes me think that you think it refers to a sequence that may or may not be infinite, we just don't know. That is not what "potential infinity" has meant in the history of philosophy dating to Aristotle. A "potentially infinite sequence" is a finite sequence. The phrase "potentially infinite" conveys only that for whatever is currently its greatest member, it is always possible to construct another that is even greater. But at any moment, it has a largest member.

4. If you can formulate Yablo's paradox for an "indefinite list of sentences", please do so. When I try it, I find that I can consistently assume that all sentences before the next-to-last one are false, the next-to-last one is true, and the last one is false. (And there IS a "last" one when the list is "indefinite" as opposed to infinite.)

Let us imagine, for sake of faulty argument, that there is a last in an indefinite n-tuple of sentences (which one that happens to be remains a mystery). The first claims that all the rest are false including the next-to-last. If we suppose the first is true, then the next-to-last is not true, contrary to what you are able to imagine. On the other hand, if we suppose that the next-to-last is true, as you imagine, then the first sentence must be false -- i.e. not all of subsequent sentences, e.g. the next to last, are false. Pick the next subsequent sentence after the first that is not false.

What do you mean the "next" one that is not false? There is only one that is not false. It is the next-to-last one. I said there is a consistent assignment of truth values to a finite list: The next-to-last is true and all of the others are false. I did not say that there was more than one consistent assignment of truth values. In particular, I did not claim that there is a consistent assignment in which more than one of the sentences is true.

If it is the second and the second is not the next-to-last

You are describing a situation in which there are two sentences that are true. I never claimed there was such a situation in which there is no contradiction. I claimed only that there is ONE assignment of truth values which does not lead to a contradiction. But one is all I need. To get a Liar paradox it must be the case that there is NO assignment of truth values which doesn't imply a contradiction.

(why would it be in an indefnite sequence?), then the next-to-last is false according to the second, contrary to our supposition. Therefore the next-after-first that is true must be n>2 in the sequence. If it is the third and the third is not the next-to-last (why would it be in an indefnite sequence?), then the next-to-last is false according to the third, contrary to our supposition. Therefore the next-after-first that is true must be n>3 in the sequence, and so on.

None of this refutes my point that if Yablo's sequence of sentences is finite, there is a consistent assignment of truth values. Hence, there is no contradiction. Hence, no paradox. Hence, Yablo's "paradox" presupposes that there is an actual infinity of the sentences. To prove me wrong, you have to take the assignment of truth values I proposed (the next-to-last is true, and all others false) and derive a contradiction from it.

5. No doubt there are times when this or that philospher uses "sentence" and "statement" interchangeably, but in discussions of truth and the Liar, that is not generally the case. --Nate Ladd 18:59, Dec 25, 2004 (UTC)

That's news to me. Provide some credible or reputable sources which raise the distinction in any meaningful way with respect to the liar paradox. Nortexoid 04:40, 28 Dec 2004 (UTC)

Barwise and Etchemendy, for example, talk about the nature of the truth bearer. But you've missed my point anyway. I was trying to bend over backward to be fair to Yablo. I was, speculatively, wondering if he could rescue his argument by claiming that the truth bearers are propositions. (I concluded that he could not.)--Nate Ladd 11:43, Dec 28, 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

---

Tone

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)

Common or not, it doesn't seem to fit the general tone of Wikipedia. I've never read any other Wikipedia pages with this style of writing. Also the response to questions ("OK, can we assume that it is false? No, that assumption also leads to contradiction...") strikes me as a little odd for an encyclopedia aimed at describing the topic, not teaching it. I guess if this is common in philosophy then it should stay but I've just never heard of questions and answers in an encyclopedia. Nrbelex ^(talk) 01:16, 2 Apr 2005 (UTC)

OK. I'm convinced that the question/answer is not appropriate, but I still think we should keep the indefinite "we". --Nate Ladd 09:41, Apr 7, 2005 (UTC)

Reversions in Discussion of Prior

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.

Two Guards and two doors

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)

This Page Intentionally Left Blank

The sentence

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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.

• How is "I am lying now" both true and false??? It is a meta-reference, however, as I stated later in the article. Not all of that sentence was incorrect, so it was obviously wrong of you to delete the whole sentence. -- BRIAN0918  23:45, 31 July 2005 (UTC)

The only sentence I deleted from the article was "This page intentionally left blank". What part of this whole sentence do you think I should have left in? Also, the article itself explains the reasoning that leads to the conclusion that the Liar sentence is both true and false. If you find that that is not clear, I'll try to restate it here for you. --Nate Ladd 23:57, July 31, 2005 (UTC)

In defense of Prior and against wiki

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:

On Prior's analysis these would be equivalent to:

This whole sentence is true and the next sentence is false.

This whole sentence is true and the preceding sentence is true.

Neither of these is by itself contradictory, but there is no way to assign truth values to them consistently, so we still have a paradox.

The last claim is wrong, I think. Here is a truth assignment that makes everything consistent:

This whole sentence is true and the next sentence is false. → FALSE and TRUE = FALSE

This whole sentence is true and the preceding sentence is true. → FALSE and FALSE = FALSE

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)

You might be right, but now that I think about it, Prior didn't attempt to apply his response to the the paradox to the two-sentence version; so it is "original research" to speculated on what he would have said if he did. We should delete the whole passage about applying Prior to the 2-sentence version. --Nate Ladd 03:01, 1 December 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

It is true that

It is true that the next clause is false

and

It is true that the preceding clause is true

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

it should be possible to consistently assign truth values to each one of those clauses

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))

Actually, the whole passage on Prior has long since passed over the line into "original research" which is not allowed in wikipedia (something I didn't realize myself until recently). I am as guilty of that as the people who have been defending Prior. I have now gone back to the last version that reflects what is actually in published philsophical literature.

Although this page is a discussion of the article and not a general discussion of the paradox, I will say that your notion of a clause that is "inherently not independent" is original with you and makes no sense to me.

I was refering to the implicit AND

A clause is a clause is a clause and any clause has a truth value. If Prior needs to resort to some exotic new notion of "clause" to defend himself, then that in itself is a pretty powerful objection to his analysis of the paradox.

Write it both ways in symbolic form and then tell me what the difference is.

What is the "it" and what is the "both"? --Nate Ladd 17:10, 24 September 2005 (UTC)

Also, what you say above following "this is the statement" is false. Your "A" and "B" business is a different statement, not a symbolic representation of the the English sentence that precedes it.

--Nate Ladd 17:12, 21 September 2005 (UTC)

Are you saying that the original statement is irreducable? I disagree strongly. Otherwise I don't follow you. Please write them both in symbolic form and show me the difference.

I'm not saying that the statement cannot be formalized in any formal logic (if that's what you mean by "reducible"). I'm saying that your "(A and not B) and (B and A)" is not a symbolic representation of

It is true that

It is true that the next clause is false

and

It is true that the preceding clause is true

To formalize the preceding, we would need a logic with the sentential (really, clausal) operator "It is true that" and some way of expressing the indexicals "next" and "preceding". Let's see if what would happen if we waived the latter requirement: Let the sentence as a whole be S. It says of itself that its true, so S = T(S) where T is the sentential operator. But S has two clauses so it is T(A & B). Each clause says of itself that it is true, so now we have T[T(A) & T(B)]. If we interpret "is false" as equivalent to a negation operator (as you seem to want to do), then "the next clause is false" is "not-B" and "the preceding clause is true" is just "A". So the complete formalization would be T[T(not-B) & T(A)]. But wait, since B is just T(A) and A is just "T(not-B)", we really have T[T{not-T(A)) & T(T(not-B))]. I think you can see that we are on an infinite regress. Since B and A refer to each other, we can always substitute one more time. We never really complete the formalization without the indexicals.

But all of this misses the point anyway. It doesn't matter if the sentence as a whole can be consistently assumed to be "False". There are lots of sentences in the world that can be consistently given a truth value. (Most of them!) What matters is that there is also a set of clauses here that CANNOT be assigned any set of truth values. You are not in any way refutting the latter claim by pointing to some OTHER sentence/clause and saying "well this one CAN be given a truth value". To refute the claim, you have to state an assignment of truth values to the clauses that does not lead to a contradiction.

Also, I what is the "both" you are talking about? --Nate Ladd 17:10, 24 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)

Equivocates

?? 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 think I was the one who deleted that. I did explain why in an edit annotation. One problem was that the remark was so brief it made no sense. It wasn't clear what word you thought was being given two meanings. Moreover, the view wasn't attributed to any published philosopher, so it appeared to be what is called "original research" in wiki terms. --Nate Ladd 23:28, 28 September 2005 (UTC)

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

Information on wikipedia does not need the imprimateur of academia; but it needs some verification that it's not just something you came up with. You know best whether what you're writing is something verifiable elsewhere or just your own personal ideas... but you may have to convince other people by using some kind of citation. wasserperson 05:22, 14 December 2006 (UTC)

"Bad faith" edit?

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)

The wikipedia does not allow original research, which means that your own (or your teacher's) response to the paradox does not belong here. You can cite a published work in philosophy that you agree with if you want. As it happens, your view (that the troublesome sentences have no truth value) is already represented in the article. Kripke, for example, has this same view. By the way, when you say that the article is based on a false premise, you are confusing paradoxes with antinomies. A paradox is an "apparent antinomy" or "alleged antinomy." If a paradox is completely unsolved, then it is an actual antinomy. If it has a solution, then it is "merely" a paradox. So the fact that you think this paradox has a solution does not mean that it is not a paradox. It is an apparent antinomy on its face and that is enough to justify an article about it, especially since, although virtually everyone believes it is a mere paradox (that is, virtually everyone thinks there is a solution to it), there is disagreement about what the solution is. The point of the article is to introduce readers to those various solutions. There is nothing bogus about this. --Nate Ladd 02:38, 28 January 2006 (UTC)

I say again that the alleged paradox does not exist. And if "common sense" is considered "original research", then neither this article nor yourself are worth wasting my time on. Wahkeenah 03:16, 28 January 2006 (UTC)

The article already covers your viewpoint. Also, please refrain from making personal attacks. rspeer / ɹəədsɹ 18:14, 28 January 2006 (UTC)

The whole "personal attacks" thing on Wikipedia is ultimately annoying. It's like the admins here have a hall-monitor complex. Any negative evaluation of a person doesn't qualify as a "personal attack" unless you're on Wikipedia —The preceding unsigned comment was added by 68.38.196.212 (talk) 08:22, 11 January 2007 (UTC).

Perhaps you are right. But it is called an ad hominem attack, and, as a logical fallacy, it is completely useless in a civil forum. The only thing that saying something negative about another person (while debating something else) accomplishes is making that person mad, possibly leading to more personal attacks. So why shouldn't Wikipedia outlaw them? -- trlkly 04:59, 28 July 2007 (UTC)

So let's get back to what Wahkeenah wrote. Is Wiki's official viewpoint that unless information is published elsewhere then it must be original information and therefore removed? That leaves us in all kinds of a mess when I submit the phrase: "This information is not yet published".

It also leads to the situation where Wiki can remove a comment stating that they couldn't verify the information even although it's not something which requires verification. If it is a commonly held opinon amongst laypersons then how do we verify that? In this case it seems that it might be a commonly held viewpoint amongs philosophy lecturers.

It also provokes me into commenting that it's probably just as well Einstein didn't publish his paper on General Relativity as an edit on a Wikipedia article about Newtonian physics. He was only a filing clerk after all.

About the Star Trek trivia

It would be a good idea to mention the name of that episode. Go for it Trekkers :)--Manwe 17:49, 29 March 2006 (UTC)

abc

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)

It is easy to verify that an infinite list can have a boundary (or beginning).

Consider a hotel with an infinite number of rooms, they can be numbered from 1 upwards and the hotelier will never run out of numbers. (If you need a mathematical example then consider the set of non-negative integers. There is a boundary at zero, no non-negative integer can be less than this).

I believe that the previous example also fatally damages your theory about a process being repeated an infinite number of times. You would need to repeat the process an infinite number of times just to reach the smallest positive number you can think of. That number does actually exist, as does any other positive number you care to mention. Repeating the process is unneccessary - you already know how to decide if a number is a member of the set of positive numbers.

Obviously no-one is going to be able to write out an actual infinite list in order to prove you wrong. An actual infinite list can never be written down (your life is too short for one thing!)

There isn't even an infinite number of particles in the universe but that doesn't mean diddly squat. Consider the set of all the places in the universe that one atom can have been since the start of time. Now consider the set of all places (and combinations) that each and every particle can have been. The new set is bigger than the largest possible counting device - the number of particles in the universe. It's still possible to concieve of though. It's even possible to describe without presenting you with a process or formula...

If you want another example of something infinite then consider differentiation. There are an infinite number of changes in gradient under a certain curve. That wouldn't stop you enumerating the area under the curve. (Using other types of geometry, the gradient could be expressed as a straight line and the area under it would be infinite).

We can't easily see the infinite because that's the way our brains have been wired in prehistory.

195.153.45.54 14:23, 9 August 2007 (UTC)

A solution I thought of

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)

Only statements with subject and predicate can be true or false. The noun this whole statement' is neither true nor false. Neither is is false true or false. The object which can be assigned a truth value is the connection between the two.--Loodog 23:57, 5 July 2007 (UTC)

is the answer to this question "no"?

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 I ask you [question X], will the answer be the same as the answer to this question?

If [question X] is Can you make jam? and the answer is yes then I can answer no to your question.

If [question x] is Can you make jam?, I can give a logically consistent answer by saying No but Katie can make jam and still truthfully answer your combined question with a no. The answers were actually different answers even although both questions were answered in the negative.

195.153.45.54 14:31, 9 August 2007 (UTC)

All enlish men are liers, i am english

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

Well the thing is, I don't think that the "Englishmen" example is really a paradox, because the speaker could be telling the truth about all Englishmen being liars, but lying about being English. I think we should go back to the example that was used on this page before June 29; namely, "I am lying now." (On June 29 someone modified this to "Everything I say is a lie. I am lying now," but the additional sentence was completely unnecessary.) --The Lazar 17:50, 17 August 2006 (UTC)

Thank God I'm an Atheist

Would this qualify as an example of the Liar's Paradox? 65.12.114.98 14:09, 27 September 2006 (UTC)

No, it's merely nonsensical. If you take "Thank God" literally (rather than just as a way to say "It's good that...", as it is used most of the time), "Thank God" essentially means nothing to an Atheist. If there is no god, then "Thank God" means the same thing as, say, "Thank The Non-Existent Thing Which Does Not Exist", or simpler: "Thank nothing".

Being an atheist renders the semantic meaning of "God" (as in, The God; not A God, as a Christian would talk about Ra, Thor, or any other god he does not believe in) void. Well, save for the meaning of "that idea of God which others have but is invalid".

It's not a statement about its own truthiness, so it can't be a paradox. It can be reduced to "I am an atheist" and, if you go by the common usage of the phrase "Thank God", "I am happy about this circumstance". It's witty nevertheless. — Ashmodai ^{(talk · contribs)} 21:04, 10 November 2006 (UTC)

The Liar Paradox in Time

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.

Upon further reflection, the "prediction" is less a prediction, than a promise. The Liar "Paradox" is a speech act. In a sense every utterance is a speech act since it creates an utterance. The utterance "My keyboard is grey" is not only a statement about my keyboard but it results in a quotable event. It is in effect, "My keyboard is grey, and I said so." But usually this is of no relevance. However in the case of the liar "paradox" the 'thing that is done with words' is relevant because it is also making the subject of the statement. Thus, "This statement is false" is in a sense a promise assuming that there is such a thing as an essentially binding promise, akin to "I promise I have spoken." But this is not the usual meaning of a "promise." So the LP may considered perhaps a declaration, or pledge:

"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)

Execution Paradox

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)

Everybody lies all the time?

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).

You're right. That's basically another example of the Epimenides paradox, which is not equivalent to the liar's paradox even though most people who have heard of it think it is. rspeer / ɹəədsɹ 06:49, 28 July 2007 (UTC)

Paradox?

seems to me this is just a false dichotomy. The sentence is neither true nor false.

It's because it's not a statement, it's an opinion. It was uttered by a human who could not cite any references to back up the information.195.153.45.54 14:35, 9 August 2007 (UTC)

Paradox in the movie Labyrinth

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).

Provided examples aren't examples, but false?

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:

• All beliefs are true.'

Paradox: ...including those that contradict this sentence.

• All moral systems are good.

Paradox: ...including those that declare this statement false.

• All realities are true.

Paradox: ...including those asserting the opposite of this statement.

• Absolutes don't exist.

Paradox: ... except this statement.

• Dichotomies don't exist.

Paradox: ... except the existence of dichotomies.

--Loodog 18:08, 9 July 2007 (UTC)

The reverter was right. Those are simply false statements. As you showed, you can tell they're false because they defeat themselves if they're true; but their negations do not defeat themselves at all. The reason for their falsehood may be "paradoxical" in a loose sense, but they aren't examples of the liar paradox, because the entire point of the liar's paradox is that it can't be true and it can't be false.

rspeer / ɹəədsɹ 22:58, 9 July 2007 (UTC)

I see. Is there a term for a statement that contradicts itself like the above examples?--Loodog 23:31, 9 July 2007 (UTC)

Deleting mysterious sentence

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)

"the following statement is false/the previous statement is true"

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)

references in pop culture... gone?

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)

It was deleted for being uncyclopaedic trivia; there was some discussion here. --McGeddon 18:01, 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)

This whole discussion seems to have been bypassed, as there is a new "pop culture" section now there. --190.74.102.246 (talk) 22:32, 24 July 2008 (UTC)

I don't mind IPC sections when they list genuinely important references to a topic in popular culture, but the ones there now seem just too minor to be worth including. skeptical scientist (talk) 07:08, 19 August 2008 (UTC)

The A.N. Prior part

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)

I agree, as far as his use of conjunction is concerned. As axiomatic logic and the truth tables both show, a-->(b&~b) is equivalent to saying that a is false. So, since no sources were given to show that this invalid argument was even in the philosophical literature anyway, I'm going to delete the relevant comment - assuming no one has any objections. 67.240.34.78 (talk) 00:43, 20 June 2008 (UTC)

Uh, hrm?

"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)

It's a perfectly valid expression (try Googling it), but as a minor coincidence of language, it doesn't belong in this article. I've removed it. --McGeddon (talk) 09:26, 18 June 2008 (UTC)

Tarski?

"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)

What to do with Goedel section

Removed from the article, to discuss what parts of it might be saved:

The proof of Gödel's incompleteness theorem uses self-referential statements that are similar to the statements at work in the Liar paradox.

In the context of a sufficiently strong axiomatic system A of arithmetic:

This statement is not provable in A. (1)

The statement (1) does not mention truth at all (only provability) but the parallel is clear. Suppose (1) is provable, then what it says of itself, that it is not provable, is not true. But this conclusion is contrary to our supposition, so our supposition that (1) is provable must be false. Suppose the contrary that (1) is not provable, then what it says of itself is true, although we cannot prove it. Therefore, there is no proof that (1) is provable, and there is also no proof that its negation is provable (i.e., there is no proof that it is also unprovable). Whence, A is incomplete because it cannot prove all truths, namely, (1) and its negation. Statements like (1) are called undecidable. We take for granted that all the provable statements of logic and arithmetic are true; Gödel showed that the converse, that all the true statements of a system are provable in that system, is not the case. (This does not mean that all true statements are not provable in some system or other. Additionally, there are systems, such as first-order logic, in which all true statements of the system are provable.)

Tarski's indefinability theorem, closely related to Gödel's Theorem, is a more direct application of the Liar Paradox, though there is no actual paradox involved; instead, the "paradox" simply demonstrates that all the true sentences of arithmetic are not arithmetically definable (or that arithmetic cannot define its own truth predicate; or that arithmetic is not "semantically closed").

My specific criticisms:

1. While the informal rendition of the Goedel sentence for A is fine, the argument as to why it's independent of A is not. Suppose (1) is provable, then what it says of itself ... is not true. But this conclusion is contrary to our supposition. Which supposition? We didn't suppose that what (1) says of itself is true, only that (1) is provable.
2. There is no mention of the requirement that A be consistent, which is necessary for the conclusion that (1) cannot be proved in A.
3. The argument in general speaks of what can be proved, rather than "proved in A", which sort of misses a lot of the point.
4. "True statements of a system" makes no sense, if "system" means "axiomatic system"; truth is semantic, not syntactic. Again, this is a big part of the whole point.
5. (This one I could have just fixed, but there were too many other problems to save the section): The link to decision problem is not on point. We want the other sort of undecidability, being independent of a certain given formal system.
6. Additionally, there are systems, such as first-order logic, in which all true statements of the system are provable. Hard to tell what the author was thinking of here, but this is completely wrong. Possibly he meant propositional logic; in that case the statement would be imprecise, but defensible.
7. The bit about Tarski actually looks OK, except that it should say "the set of all true sentences..." rather than "all true sentences".

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)

Hmm, I never noticed this section before. Here's a possible revision, maintaining some of the previously quoted section but revised to deal with your criticisms. I tried to keep it in as plain language as possible while mentioning a couple of the technicalities (and providing relevant wiki-links) and keeping it true in spirit to Godel's original proof.

Gödel's incompleteness theorem in mathematical logic says that any consistent axiomatic system of sufficient strength is incomplete; that is, there are certain statements that cannot be proven or disproven in the system. Gödel's proof of this theorem uses self-referential statements similar to the Liar paradox.

In the context of a sufficiently strong axiomatic system A of arithmetic, one can use a technique known as arithmetization of syntax to produce a statement which means:

This statement is not provable in A. (1)

The statement (1) does not mention truth at all, only provability, but is otherwise identical to the liar paradox. Taking for granted the possibility of writing the statement (1) in the language of A, the argument proceeds as follows: Suppose (1) is provable in A. Since (1) asserts that it is not provable, one can encode this proof, again using arithmetization of syntax, to provide a proof of the negation of (1) in the system A. Since A can prove both (1) and its negation, A is inconsistent. Suppose now that just the negation of (1) is provable, and not (1) itself. Then A can prove the existence of a proof for (1), as that is what the negation of (1) asserts. However, since (1) is not provable, the system A is inconsistent. (There is a technicality here, due to the technique of arithmetization of syntax, where proofs are actually coded by natural numbers. More precisely, A can prove that there is a natural number which codes a proof of (1), but it can also prove that 1 does not code such a proof, nor does 2, nor 3, etc. This is not a direct inconsistency, which would be a pair of provable statements S and ~S which assert the exact opposite, but rather something called an ω-inconsistency.) Therefore, if A is consistent (or rather, ω-consistent), then A is incomplete, since neither (1) nor its negation can be proved in A.

The fact that the sentence (1) refers to provability rather than truth is not merely a choice, but has to do with the restrictions imposed by the technique of arithmetization of syntax. It is possible to encode the notion of provability in arithmetic, and therefore create a statement in the language of number theory which in some sense talks about its own provability, but the same thing cannot be done with the notion of truth. In fact, this is known as Tarski's indefinability theorem, and is closely related to Gödel's incompleteness theorem.

I think I covered most/all of the issues you raised. I removed the link to decision problem and the mention of the existence of complete consistent logical systems entirely - those aren't strictly relevant. I think it's reasonable to be fairly brief on this page - if people want to know more they can check out the main article on the incompleteness theorems. I used "the incompleteness theorem" rather than "the first incompleteness theorem" since I don't think confusion is likely to arise, and this is the theorem people usually mean; again, interested readers can easily navigate to the main article.

--skeptical scientist (talk) 22:10, 18 August 2008 (UTC)

P.S. I haven't mentioned any sources because my main sources are classes and exercises rather than published material. There's a long list of sources on the incompleteness theorem article which could be used. —Preceding unsigned comment added by Skeptical scientist (talk • contribs) 22:13, 18 August 2008 (UTC)

Well, you've certainly addressed many of the inaccuracies. But now it's very hard to follow, which I don't think is your fault; I think the narrative just sort of degenerates once made precise. It seems to me this exercise suggests that nothing along the lines of what I removed is salvageable (which, to be fair, was what I already thought, so possibly this is not an objective test).

My current feeling is that we should just leave this out. The analogy is not really terribly close anyway. A completely rewritten, and much shorter, allusion to the Gödel theorems and to their rough analogy with the Liar, might be OK, but ought to be sourced. --Trovatore (talk) 04:16, 19 August 2008 (UTC)

I think you're right about it being hard to follow once made precise, but I think that something should be kept. There really is a close analogy between the liar paradox and Godel's sentence, I think. What about something like the following, which forgoes most of the details in order to discuss the analogy a bit more?

Gödel's incompleteness theorem in mathematical logic says that certain logical systems are necessarily incomplete; that is, there are statements that cannot be proven or disproven in these systems. Gödel's proof of this uses self-referential statements similar to the liar paradox.

In the context of a sufficiently strong axiomatic system A of arithmetic, one can use a technique known as arithmetization of syntax to produce a statement about natural numbers which can nevertheless be interpreted as saying, "This sentence is not provable in A."

This statement is known as "the Gödel sentence" for A. Gödel's sentence replaces "true" with "provable", but is otherwise identical to the liar paradox. (The fact that it refers to provability rather than truth is not merely a choice, but has to do with the restrictions imposed by the technique of arithmetization of syntax. It is possible to encode the notion of provability in arithmetic, and therefore create a statement in the language of number theory which in some sense talks about its own provability, but the same thing cannot be done with the notion of truth.)

Just as the liar paradox is contradictory, a contradiction results if Gödel's sentence is either provable or disprovable in A, so if A is consistent, Gödel's sentence must be neither provable nor refutable.

--skeptical scientist (talk) 06:59, 19 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.

In some sense, the proof of Gödel's first incompleteness theorem, which demonstrates that a consistent axiomatic system satisfying certain technical stipulations cannot prove all truths about the natural numbers, may be viewed as analogous to the liar paradox. Given a consistent axiomatic system A, the proof formalizes the sentence

This sentence cannot be proved from A

into a claim about the natural numbers. This formalization is known as the Gödel sentence of A, denoted G_A. If G_A could be proved in A, then A would also be able to prove that G_A cannot be proved in A (since that is the content of G_A) and therefore A would prove a contradiction, which is impossible because A is consistent. However the analogy breaks down in that there is no paradox derivable from the assumption that A cannot prove G_A, and this is in fact the case; the Gödel sentence of a consistent system is therefore true.

Now it's true that this version doesn't get into showing that A also does not prove ¬G_A, 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 G_A, 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)

Meh. Imo the completeness theorem shows that provability is a lot like truth, but I guess I'm not that attached to having a section on Godel in this article. skeptical scientist (talk) 08:40, 19 August 2008 (UTC)

There is an important link between Goedel's incompleteness theorem, the liar's paradox and the statement "This sentence is true" - they all involve a hidden referential failure which renders them semantically defective. Take the third of these first - "This sentence is true". In order to try to pin down its meaning, you have to insert the entire sentence in place of the words "this sentence", so you then get "This sentence is true is true". In order to pin down the full meaning of this new sentence, you have to do the same thing again (inserting the whole original sentence again, not the new one, because "this sentence" always refers to the original sentence), so it becomes "This sentence is true is true is true". The task of trying to pin down the root meaning of the sentence is actually an infinite process which never terminates, which means the sentence is not fully meaningful, even though it looks at first sight as if it is true. It is neither true nor false, but simply defective.

Goedel's sentence suffers from the same defect (as does "This statement is false"), so none of them have a complete meaning. This has implications for Goedel's incompleteness theorem itself, because it is based on an error: "This statement is not provable" is semantically deficient and therefore cannot be taken as true in any sense.Djvyd (talk) 06:06, 29 January 2010 (UTC)

Solution to the Liar Paradox

We begin with the paradox:

This sentence is not true.

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

This sentence is not this sentence.

Therefore the conjunction is true:

This sentence is not true, and this sentence is not this sentence.

the negation of which is

This sentence is not true is false, or this sentence is not this sentence is false.

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:

This sentence is not this sentence is false.

In other words,

This sentence is this sentence.

which is obviously true -- which resolves the paradox.

--Vibritannia (talk) 17:47, 20 November 2008 (UTC)

Quicker solution

Begin with

This sentence is not true.

Therefore

This sentence is this sentence is not true.

which negated is

This sentence is this sentence is not true is false.

which is

This sentence is this sentence is true.

which is just

This sentence is true.

The paradox is resolved.

--Vibritannia (talk) 11:37, 22 November 2008 (UTC)

Where do you get off claiming a sentence and its negation are equivalent? --Loodog (talk) 23:34, 9 March 2009 (UTC)

Explanation of the paradox

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)

Non-paradoxes

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)

self-reference

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 LDavidsstorm (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)

More applications

• Halting problem
• Rice's theorem

Please feel free to add and strike out done items. Paradoctor (talk) 15:06, 15 December 2009 (UTC)

Neither True Nor False

"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)

Agreed, it is Null. Only propositions can be true or false. "This sentence is false" is not a proposition. Just what is being evaluated when someone tries to judge "this sentence is false"? The truth or falsity of a proposition isn't part of the proposition itself. If I make a proposition "London is a city in France", that's a false proposition, but we wouldn't say "London is a city in France is false". So, if we treat "this sentence is false" like a proposition, the proposition becomes "this sentence", and we can see clearly that "this sentence" has no content, and is not a proposition, so "this sentence is false" is meaningless and null.--RLent (talk) 17:46, 29 March 2010 (UTC)

Please remember WP:NOTFORUM. What you propose is a truth gap solution. To see how it fails, consider this: A sentence L that is "meaningless and null" is clearly not true. Therefore, "L is not true" is true, and the liar paradox follows. Paradoctor (talk) 20:04, 29 March 2010 (UTC)