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How would one go about proving that the Liar sentence is paradoxical?

I ask because to me it obviously isn't one so I'd need a decent proof to change my mind. I've always been baffled as to why it's considered paradoxical.

It seems rather obvious to me that in the sentence "This sentence is false", the sub-phrase 'This sentence" is not a sentence and has no truth-value. So how does one create a paradox out of this?

I might as well say "This elephant is false", or "This word is false".

Where is the paradox? How would one go about proving there is one?

I've been bothered by this for years and never seen a convincing argument that there is a paradox here. It appears to be a basic and simple error that creates the paradox, but this would be odd when so many people make it. So, it must be me. But what am I missing?

EDIT: I just found this article and it gives my view. http://steve-patterson.com/resolving-the-liars-paradox/

What is Steve Patterson missing?

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The paradox comes from the act of negational self-reference. Self-reference carries identity, and identity statements are tautologically true; things fall out of whack when a statement explicitly negates the identity that self-reference is carrying.

This is easier to see if we take the original form of the liar's paradox, where a man stands up and says: "Every statement I make is a lie." The self-reference here is easy; it's a reference to the man himself, who has identified himself as a liar (i.e., someone whose every statement is a lie). But clearly the statement itself is a statement he has made, so it too should be a lie, as a matter of identity. So...

  • If we take that statement to be true, (meaning that all his statements are factually false) it directly contradicts the identity the self-reference is carrying
  • If we take that statement to be false (meaning that some of his statements are true), then the statement itself conforms to the identity being carried, but it implicitly falsifies that identity by asserting that not all the speaker's statements are lies

It is impossible to assign a truth-value to this particular statement because one way or another the statement contradicts the identity being conveyed. When we make this more abstract by removing the person of the liar — the 'This statement is false' construction you've used — then we have to keep in mind that the 'identity' being conveyed is in the meaning of the statement. 'This statement is false' is problematic because the statement is identifying itself as false. If we assert that the statement is true it directly contradicts the identity it is trying to convey; if we assert that the statement is false it indirectly contradicts the identity it is trying to convey.

The Patterson piece you linked tries to get around this paradox using the common tactic of denying that there is a self-reference. In the first case he isolates the fragment 'this statement' as non-referential: a word combination that does not rise to the level of a proposition or refer to any proposition, and thus has no truth-value. In the second case he takes a more sophisticated approach — following Wittgenstein and several others — in which he allows 'this statement' to be a reference to a proposition, but asserts that the proposition being referred to is different from the proposition doing the referring. Since this reference isn't a self-reference, Patterson can replace the circular contradiction with an infinite regress; still problematic, of course, but not a paradox. But it's questionable as a solution to the problem, because it runs against that intuitive sense of identity.

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  • Thanks. Nice summary.
    – user20253
    May 3 '20 at 10:57
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I had the same puzzlement for a long time.

The important thing is a distinction between sentences and propositions. I define a sentence as simply a string of characters in some language. Propositions are the "meanings" that sentences convey. Propositions are what we usually take as true or false.

So sentences "convey" propositions which in turn are true or false... or they convey no proposition in which case they are meaningless. For example:

Sentence A: "sdfsdfsdlsdf"

Sentence A is meaningless in the english language.

We need to "define" what it means for a sentence to be true, and it's slightly different from truth for a proposition.

I call a sentence true if it conveys a true proposition. In all other cases I call it false... so if the conveys a false proposition I call it false... if the sentence conveys no proposition (meaningless) I also call it false. So by this definition every sentence (string of characters) is true or false.

Now look at the liar paradox using the fact that a sentence is simply a string of characters in a particular language, and all sentences are true or false. The same paradox pops up.

Sentence A: Sentence A is false.

This seems perfectly meaningful. 'Sentence A' is just a string of characters and it exists. It can be true or false using our new definition of true and false. So the meaning seems clear, but we can't decide on true or false.

EDIT:

We can reformulate the liar as simply referring to properties of "strings of characters" itself. Say a string of characters translates to a true proposition in english. I'll define that as "brue" and in all other cases the string of characters is "balse". So all strings of characters are either "brue" or "balse". String A: String A is balse. We get the same liar paradox now. If String A is brue then it is balse. If it is balse it is brue.

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    If sentences conveying no proposition are false then their negations are also presumably false, so logical negation would not behave as we expect. I also do not see how the sentence/proposition distinction helps. We'll just have to be more verbose with "the proposition expressed by this sentence is false", but this is what is meant anyway. Words/sentences are normally treated as transparent labels that redirect to their senses/propositions when pointed at, unless quotes are used to indicate otherwise.
    – Conifold
    May 2 '20 at 10:29
  • You haven't defined negation for a sentence. In my definition a sentence is simply a string of characters. If 'sentence' is problematic we can use 'formula' or 'string' instead. In any case the entity being referred to is a string of characters. May 2 '20 at 10:34
  • In that case you equivocate. This is called "uninterpreted sentence", and not how the Liar uses the word. If the topic is the Liar I do not see what this does for it. You should also think about the Strengthened Liar:"This sentence is either false or meaningless".
    – Conifold
    May 2 '20 at 10:37
  • I'm doing the same thing Godel does in his incompleteness proof. He refers to the Godel sentence as a formula... a string of characters. That way the referrant is perfectly clear. It is the string of characters itself, not the proposition which may or may not exist. I haven't seen the Liar Paradox formulated in formal logic, but I'd be really surprised if it wasn't done similarly. In any case, the liar paradox crops up the same way. My intent was to show the OP that the existence of the referrant is not the issue in the liar paradox. May 2 '20 at 10:50
  • It is not quite the same. Godel'd sentence is "I am unprovable", not "I am false", and it has quite a different meaning even intuitively. In particular, it does not lead to an intuitive contradiction, nor to a formal one. Formalization of the Liar by Tarski in terms of a definable truth predicate of natural languages does lead to a contradiction, as expected. This is why he had to dissociate natural language into a hierarchy of meta-languages.
    – Conifold
    May 2 '20 at 11:00

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