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If "This is a lie" were a true statement, its fulfilled claim of being a lie implies it can't be true, leading to a contradiction. If it were false, it could not be a lie and thus had to be true, again leading to a contradiction. So with this argumentation the statement "This is a lie" can be neither true nor false and therefore binary logic is not enough to treat logic.

Is this argumentation correct?

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If you are talking about human speech, you might be correct. If you are talking about mathematical logic, see below. Basically, there is no way for a statement of mathematical logic to be (directly) about the statement itself. –  Thomas Andrews Mar 13 '12 at 11:58
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If you want to find something which implies the insufficiency of binary logic, sorites paradoxes would serve you better. –  Doug Spoonwood Mar 13 '12 at 13:21
    
I love the sorites paradox! Hrbacek uses it to explain nonstandard analysis. –  Robert Haraway Mar 13 '12 at 19:12

5 Answers 5

up vote 32 down vote accepted

The problem lies in your interpretation of the sentence. If you want to apply logic to it, you need first reformulate it into the language of logic. There are many different ways to do that, but in majority of "usual" methods the curious thing happens: for a sentence to talk about it self you need a set that contains the sentence which in turn uses the set again -- you are unable to find a formal representation of it. This is closely related to Russel's paradox and his proof that there's no set of all sets, e.g. there is no set of all sentences (if they can refer to this set).

Of course, there are some "unusual" structures, e.g. you could take sets defined using the bisimilarity relation where x = {1, x} would be a proper definition, although, this is far beyond binary logic.

Concluding: the simple binary logic is not contradictory, but it is too weak to express this sentence.

On a happier note: this kind of thing happens all the time, and even innocent fairy-tales fall for that, e.g. consider what would happen if Pinocchio had said "my nose will grow"... (Have anyone tried ever this on their's native language teacher?)

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ah, nice relation to Russel's paradox, thanks! So... if a sentence cannot be formulated in the language of logic, doesn't this mean the latter is incomplete? –  Tobias Kienzler Mar 13 '12 at 13:46
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It depends what do you mean by formulated. I can create a grammar that will generate such sentence, but the semantics is missing. But, if you would want semantics that somehow reflects the "natural understanding", then such system would be self-contradictory (the "natural understanding" is the key part here). Of course you could derive other semantic systems, but I guess every time you would feel cheated, because the formalization of the sentence will not represent what you had in mind (i.e. the "natural understanding"). –  dtldarek Mar 13 '12 at 14:20
    
"is a string from which a false sentence can be formed by writing a quote, the string, a quote, and again the string." is a string from which a false sentence can be formed by writing a quote, the string, a quote, and again the string. This sentence talks about itself without needing the kind of sets you describe. But no meaning can be ascribed to it. –  Marc van Leeuwen Mar 13 '12 at 16:14
    
@MarcvanLeeuwen I have no idea how to comment upon your comment. It this a further example or you are asking about something? –  dtldarek Mar 13 '12 at 17:56
    
@dtldarek: My comment was questioning your "for a sentence to talk about itself you need a set that contains the sentence". I don't think that is accurate, nor indeed do I think it even describes the Russell paradox: that is a set whose definition makes an implicit reference to the universe of all sets. And the paradox doesn't prove there is no set of all sets, but it shows that naive set theory (which allows a formula for the set of all sets) is inconsistent. And by the way, by any reasonable definition of "sentence", there is a (countable) set of all sentences. –  Marc van Leeuwen Mar 14 '12 at 8:06

No, this just shows that logic should not allow arbitrary sentences and ascribe a truth value to all of them. Completely nonsensical utterances clearly fall under the forbidden category. But sentences like this classical liar paradox which mix levels cannot be allowed either. If you want to know more interesting forms of the liar paradox and applications of them, I advise you to read Hofstadters "Gödel, Escher, Bach", where such matters are discussed at great lengths.

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+1 for "Gödel, Escher, Bach" –  draks ... Mar 13 '12 at 11:42
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+1 for "great lengths"! :-) –  Asaf Karagila Mar 13 '12 at 13:44
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For those unfamiliar, Gödel, Escher, Bach is a good (soft) introduction to Gödel’s incompleteness theorems, but it also includes a lot of philosophy and speculation on the nature of these so-called strange loops. It’s a bit of work to get through, but well worth it. –  Jon Purdy Mar 13 '12 at 13:52
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While this is a good overview, the situation is much more subtle than "mixing levels is not allowed." Consider even Gödel's Incompleteness Theorem: there you construct a sentence in the formal language of number theory which is interpreted as "This sentence cannot be proven." Surely this is at the very least a kind of mixing of layers (a sentence of the formal language talking about the formal theory itself). –  Arthur Fischer Mar 13 '12 at 14:31
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@ArthurFischer: The great thing about the Gödel sentence is that it is itself clearly not mixing levels. As you say it is a sentence in the formal language of number theory, and its intended meaning is determined by interpreting that as a statement about natural numbers: the nonexistence of a number with a certain property. The way it is constructed we can see that it is equivalent to a statement about logic (the nonexistence of a derivation of a certain formula in a certain formal system) but that is not at all its direct interpretation. It is our interpretation that mixes levels. –  Marc van Leeuwen Mar 13 '12 at 16:03

You're right,

binary logic is not able to express the sentence This is a lie. The binary logic is the simplest logic and works with logical operators and True/False values. There are other more powerful logic systems that can work better with meaning (semantics) of natural language.

Here, we don't know what insufficiency means. What should we prove here? Insufficiency for what? But it's possible to say that we can't express This is a lie in binary logic.

To be able to express it we have to use more powerful logic systems like intensional logic that are able to express propositional attitudes and other pretty constructs of natural language. Though, I think that This is a lie is unexpressable in intensional logic too.

We have to analyze the sentence meaning (semantics) first. Usually the word this has a semantical pragmatic function (depending on context where the sentence is spoken or written). I would first think about the concept lie.

You can see that lie is a predicate on proposition. Proposition in intensional logic is a function depending on possible world and time. The concept of possible world is quite complex so assume you have only time function of type (time -> boolean).

example: At this moment (7.7.2012 07:07) these propositions hold

Peter asserts that Charles is slim.

Charles is fat

We can easily derive (reason) that "Charles is slim." is a lie. You can see that the meaning of lie is a predicate on tuple (proposition, time moment).

This is a lie can't be applied to this time moment because in this predicate is missing an argument (proposition).

You can see that sentence this is a lie is pure nonsense because the sentence is "functionally incorrect", you can't express it unambiguously in type theory.

The meaning (our understanding) would look like 1 or 2:

  1. lie(this, 7.7.2012 07:07). But this has only pragmatic function.
  2. lie(lie(lie(...(...(...never ending story))), 7.7.2012 07:07), 7.7.2012 07:07).
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Very interesting point about the incompleteness of "this is a lie". From that perspective, the trueness of "this is a lie" basically oscillates between true and false - if it has been true, it must be false now, and therefore it will be true, leading to falsehood, ... –  Tobias Kienzler Mar 14 '12 at 12:25
    
Here we are unable to determine the meaning of "this is a lie". The meaning is an application of predicate "lie" on argument which is the meaning we're trying to determine (the meaning that does not exist yet). We are unable to construct the meaning in our mind. We will never have a chance to ask on a truth value of this expression. –  xralf Mar 14 '12 at 20:53
    
when you put it like that this sound almost like an axiom :-7 –  Tobias Kienzler Mar 14 '12 at 22:09
    
My thoughts are from the little I know about intensional logic especially transparent IL (TIL) which is very advanced logical system. –  xralf Mar 15 '12 at 8:40

This argument is know as the Sevilla Barber paradox or the all set argument! The problem is when you can't attribute a logical value in the sentence. In the other hand the sentence "This sentence is true" can have both logical value without leading us to any contradiction!

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In philosophy, there is a logic system known as "protothetic" logic that is set up to treat these sorts of statements. It uses Polish notation, but is binary. There was a book or paper by Timothy McGrew on the subject, but I can't seem to locate the title.

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sounds interesting, is it among these? –  Tobias Kienzler Mar 13 '12 at 13:50
    
I don't think any of those. I e-mailed Professor McGrew. When I know I'll put down a comment here. –  Joe Johnson 126 Mar 14 '12 at 1:13
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Professor McGrew said it's in the appendix to his book "The Foundations of Knowledge" (Littlefield Adams, 1995). He also mentioned a good introductory paper is J. L. Mackie's paper "Self-Refutation: A Formal Analysis," in his book "Logic and Knowledge". –  Joe Johnson 126 Mar 14 '12 at 9:45
    
Thanks a lot for checking. I found said paper: jstor.org/stable/2955461 –  Tobias Kienzler Mar 14 '12 at 12:20

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