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The Liar Paradox of antiquity goes something like this:

A man says, "Everything I say is a lie."

(I find the modern variation -- "This statement is false" -- to be less interesting. It seems to me to be nothing more than a simple self-contradiction.)

Define 3 logical predicates:

$S(x)$ means $x$ is a sentence

$T(x)$ means $x$ is true

$M(x)$ means the man says $x$

EDITED:

  1. $S(x)\land M(x)\land \forall y(S(y)\rightarrow (M(y)\rightarrow \neg T(y)))$ (Premise)

  2. $S(x)$ (Splitting premise, 1)

  3. $M(x)$

  4. $\forall y(S(y)\rightarrow (M(y)\rightarrow \neg T(y)))$

  5. $S(x)\rightarrow (M(x)\rightarrow\neg T(x))$ (Universal Specification, 4)

  6. $M(x)\rightarrow\neg T(x)$ (Detachment, 2, 5)

  7. $\neg T(x)$ (Detachment, 3, 6)

  8. $\forall a (S(a)\land M(a)\land \forall y(S(y)\rightarrow (M(y)\rightarrow \neg T(y)))\rightarrow \neg T(a))$ (Conclusion, 1)

My question is, does this proof resolve the Liar Paradox?

FOLLOW-UP:

By definition, everything a constant liar says is false. A contradiction arises only, it would seem, when he says something like, "Everything I say is a lie," that is, when he claims:

$$\forall x (S(x)\rightarrow (M(x) \rightarrow \neg T(x))$$

If, as required, this is false, then

$$\exists x (S(x) \land M(x) \land T(x))$$

This contradicts the requirement that everything he says is false. If the constant liar can refrain from making such an admission, no paradoxical situation should arise. No such contradiction would arise, for example, from his saying, "Everything I say is true."

EDITED:

To answer my own question then, the above theorem does not resolve the original Liar Paradox of antiquity. It doesn't "prove" much at all, I'm afraid, but I now feel I have indeed resolved the paradox: It arises from the liar himself claiming that everything he says is false, and my (not necessarily well-founded) assumption that everything he says is indeed false. In hindsight, I think the "prize" must go to Alex Becker for his insightful comment. See my formal proof (Corollary starting on line 19) at http://www.dcproof.com/LiarParadox.htm

FOLLOW-UP TWO YEARS LATER

In the time since I first posted this question, I have come to realize that "This sentence is false" is nothing more or less than meaningless nonsense. Somehow, it is easier to see that "This sentence is TRUE" is meaningless nonsense. Simply changing "true" to "false" should not somehow suddenly imbue this sentence with meaning. I don't really think you can formalize the notion of meaningless nonsense.

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    What is your question? (Also, do you mean to have a $\wedge$ in 4?)2012-08-21
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    The old liar paradox has never made sense to me. Clearly the man is lying now, but does not always lie.2012-08-21
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    @CameronBuie My question is in the heading. (I have deleted the $\land$ in 4. Thanks.)2012-08-21
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    (1) Your version reduces to *This statement is false* if the man makes only that one statement. (2) What makes *This statement is false* interesting is that unlike a simple self-contradictory $\varphi\land\lnot\varphi$, it cannot be assigned **any** truth value.2012-08-21
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    There are three sentences in this comment. Exactly two of the sentences are false. You owe me one million dollars.2012-08-21
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    @AlexBecker I guess that's why they usually say the man **always** lies or **always** tells the truth. But even in that case, a sentence of the form "Everything I say is a lie" must be false.2012-08-21
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    @BrianM.Scott True, but again, a sentence of the form "Everything I say is a lie" is false whether he makes any number of statements.2012-08-21
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    @Dan: Not if it’s the only statement that he ever makes. Then it is precisely equivalent to *This statement is a lie*.2012-08-21
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    @BrianM.Scott Agreed. See my previous comment.2012-08-21
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    @BrianM.Scott I think I see your point now: The modern variation is just a special case. Thanks.2012-08-21
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    You may also be interested in George Spencer-Brown's construction of "imaginary" truth values in his book Laws of Form, which I'm currently sort of obsessed with. He creates a coherent way of dealing with such paradoxes.2012-08-22
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    @JBeardz: would you recommend any particular reference for that?2012-08-22
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    As I state the "paradox" above, there is nothing to suggest the man always lies. I didn't realize the importance of this assumption at the time. From Wiki: "Cretan philosopher Epimenides of Knossos (alive circa 600 BC) who is credited with the original statement." http://en.wikipedia.org/wiki/Epimenides_paradox A Cretan himself, he famously said, "All Cretans are liars." Somewhere along the line, perhaps to make it more interesting and "paradoxical", writers began to assume that this meant that *everything* Cretans say is a lie.2012-08-22
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    @NieldeBeaudrap, yeah, just the book I mentioned.2012-08-24

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