Talk:Liar paradox

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)

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And I removed:

Contents

1 Literary versions 2 "Cretans always lie" 3 xxx.lanl.gov reference 4 Another example? 5 Yablo example deleted 6 Tone 7 Reversions in Discussion of Prior 8 Two Guards and two doors

Literary versions

A version of this paradox appears in the Don Quixote (II, Chapter LI (http://www.panzaconsulting.com/quixote/dq_108.html)) and another in the letter of Paul to Titus; 1, 12 (http://www.blueletterbible.org/tmp_dir/popup/1054442648-7471.html#11):

"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)

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.

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(all words in brackets are lies) hehe

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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)Missing image
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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)Missing image
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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)