You are correct. it's not a contradiction, but rather, an unresolved paradox. Any self-referential statement represents the un-computable in nature. It stems from self-awareness/consciousness. It's also found in mathematics (Godels theorem).

Another example would be:

Socrates: What Plato is about to say is false.
Plato: Socrates has just spoken truly.

It does not make much sense to think and/or believe that a contradiction is even capable of being true. — creativesoul

How about the statement: 'I saw your brother today' and you reply my brother died yesterday. Wouldn't that be a true contradiction?

The liar paradox has multiple interpretations and resolutions. For instance, if 'this sentence' is interpreted syntactically as being recursive self-quotation, we end up with an infinitely deep nest of quotes ' ' ' ... ' ' ' that isn't even a sentence, let alone a well-formed sentence.

In my opinion, the Godel sentence used in the proof of the incompleteness theorem is best understood in an analogous fashion, since the proof is purely syntactical. Its semantic interpretation as a sentence asserting it's own lack of provability is a heuristic argument that isn't formally acceptable, because the Godel number supposedly referred to by the Godel sentence is infinitely long when the sentence is recursively unpacked by substituting the sentence into itself.

On the other hand, if 'this sentence is false' is interpreted semantically as being a pair of sentences, each sentence belonging to a different language whose meaning is the negation of the sentence in the other language, then we get the traditional semantic understanding of the sentence as a contradiction.

However, since the liar paradox is a paradox of natural language that is it's own meta-language, as opposed to being a paradox of formal language, my preferred resolution is to consider the liar paradox as being a meaningful sentence (since we can understanding the paradox), that isn't a contradiction, rather it is a self-negating sentence with alternating truth value. This interpretation best describes our use of the paradox. i.e. "It is true - hence it is false - hence it is true... etc"

How alternating? If it's true, it's false. If it's false, it's true. Same with Russell's paradox: if yes, then no (and vice versa).

If it's true, it's false. If it's false, it's true — frank

Put those together and you get:
If it's false it's true which makes it false which makes it true which makes it false........

I think that's what he meant

Nothing is alternating, though.

it goes from true to false to true to false to.....

Nothing is alternating, though. — frank

That depends on your notion of truth. Classically, you're right; for truth is not traditionally considered to be the property of a sentence or of it's construction, but of a timeless matter of fact referred to by the sentence that is existentially independent of, and external to, the sentence. From that perspective, the notion of 'alternating truth' i have sketched should be interpreted as referring to 'alternating belief' in the truth of a sentence, where a sentence is said to be 'true' merely if one accepts it and 'false' otherwise.

You are correct. it's not a contradiction, but rather, an unresolved paradox. Any self-referential statement represents the un-computable in nature. It stems from self-awareness/consciousness. It's also found in mathematics (Godels theorem). — 3017amen

I don't see the direct link between The Liar and Godel's proof that there is always an assumption buried somewhere within an inductive system of logic that cannot be proven by any means within that system. Nevertheless, I'll attend to the other example you've offered for consideration...

Socrates: What Plato is about to say is false.
Plato: Socrates has just spoken truly. — 3017amen

How is this a true contradiction? This reminds me of an amended Liar, or reinforced, or whatever they call it. It is a contradiction, but it is neither true nor false. The two claims are about one another, and are in direct conflict with one another. The same lack of truth conditions of the Liar, but with an improvement on the referents. There is nothing that can make either one true/false.

It's a confused way to speak. It's a contradiction, but contradictions aren't able to be true/false.

However, since the liar paradox is a paradox of natural language that is it's own meta-language, as opposed to being a paradox of formal language, my preferred resolution is to consider the liar paradox as being a meaningful sentence (since we can understanding the paradox), that isn't a contradiction, rather it is a self-negating sentence with alternating truth value. — sime

Could you continue on to an explanation of what counts as an alternating truth value? Is that what makes it self-negating? If it's true, it is false, etc...

How about the statement: 'I saw your brother today' and you reply my brother died yesterday. Wouldn't that be a true contradiction? — ovdtogt

No. It would be a false statement followed by a true one about the same referent. It is a contradiction, but only the latter half of it is true(assuming the brother died). Or, if the brother did not die, and was seen by the speaker, it would be a true statement followed by a false about the same referent, with the first half of the contradiction true.

'alternating belief' in the truth of a sentence, — sime

Or, to simplify matters, alternating assertion and denial of a sentence.

Or, to simplify further, production or selection of sentence tokens (utterances/inscriptions) that successively contradict each other.

"This sentence is false" is utterly meaningless.

Without a frame of reference (context) most sentences are meaningless.

Practically every statement is referential.
That would be like a math formula without an =.

If a sentence being true is for it to correspond to a fact and being false is for it to not correspond to a fact then we can rewrite the liar sentence as "this sentence does not correspond to a fact."

We can then consider each of these:

"this sentence does not correspond to a fact" corresponds to a fact
"this sentence does not correspond to a fact" does not correspond to a fact

I'm unsure about the first, but there's a way to interpret the second as reasserting the liar sentence rather than contradicting it.

'I am a liar'.

Is this sentence/statement true or false?

Neither of them are contradictions. I said they were paradox's; not contradictions.

Godel's theorem speaks to the un-computable, undecidable, middle ground or gray area in logical deduction. Another axiom in life that is illogical, if you will.

https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Truth_of_the Gödel_sentence

Otherwise, a contradiction is A and-A. This ball is red and not red. But there are also conscious phenomena that breaks that rule, including the limitations of language/descriptions about a thing, thus:

https://en.wikipedia.org/wiki/Principle_of_bivalence

Read 'Vagueness" halfway down.

Nothing is alternating, though.
— frank

That depends on your notion of truth. Classically, you're right; for truth is not traditionally considered to be the property of a sentence or of it's construction, but of a timeless matter of fact referred to by the sentence that is existentially independent of, and external to, the sentence. From that perspective, the notion of 'alternating truth' i have sketched should be interpreted as referring to 'alternating belief' in the truth of a sentence, where a sentence is said to be 'true' merely if one accepts it and 'false' otherwise. — sime

So let's say the sentence is A:

If I believe A is true, then I believe A is false.

It's still a paradox. If/then isn't describing a sequence of events.

'I am a liar'.

Is this sentence/statement true or false? — ovdtogt

The syntax is correct, but the true meaning is indeterminant. You could say it's 50% true. It wouldn't make it a contradiction or paradox though.

(But, Godel' would say: It is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone. )

To make it a paradox you would re-word to say: This statement is a lie.

To make it a paradox you would re-word to say: This statement is a lie. — 3017amen

'I am a liar.'

is already a paradox. Tell me if you know I am a liar or not?

'I am a liar'
is already a paradox. Tell me if you know I am a liar or not? — ovdtogt

Godel' would say: It is not 'syntactically complete', since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone.

'I am a liar' then is considered incomplete or indeterminant.

Kind of like : The girl ran. The girl is a liar. (Or otherwise a simple sentence structure of subject-verb, which is actually syntactically correct, but not according to Godel's incompleteness rules.)

In either case, one does not know whether the girl ran, or whether she or you are a liar. It's indeterminant or incomplete. It cannot be proved or disproved.

Again, to make it a paradox you would have to change it to: This statement is a lie. Because if it's true, it's a lie. But if it's a lie, it's not true.

I think the more practical implication here is the fact that deductive logic (formal logic/mathematics) has inherent limitations in the world of experience. To that end, one could argue that 'I am a liar' is a synthetic proposition. It would require empirical evidence/experience to determine it's truth value.

Again, to make it a paradox you would have to change it to: This statement is a lie. Because if it's true, it's a lie. But if it's a lie, it's not true. — 3017amen

All we can say about these sentences is that they contain no information.
This statement is false.
Everything I say is a lie.
'I am a liar'
Don't believe anything I say..... etc....
I am crazy
These all contain the same paradox.

Ever read Catch 22? That novel is all about being caught up in a paradoxical situation you can't escape from.

'I am a liar.'

is already a paradox. Tell me if you know I am a liar or not? — ovdtogt

The strengthened liar paradox – “this very sentence is false” - is an abstraction obtained from either “I am a liar” or “I am lying”.

A liar: a) someone who tells lies (not “someone who never tells truths”), b) someone with a propensity to lie, such that they are attracted to lying, or c) someone whose propensity to lie is greater than average.

A lie: an intentionally (or, less commonly, unintentionally) told statement known to the speaker to be untrue whose contents are intended to be believed true by those to whom the statement is told. (If the speaker believes her statement to be true, the statement would not be a lie.)

“I am a liar” (or “all people are liars”, etc.) cannot then feasibly be a contradiction, for it intends to correlate to the fact that the individual (or that all people, etc.) has told lies or, else, has some propensity to lie in certain contexts. Or, in some contexts, it would be the momentarily honest expression that one or all of one’s cohort has a greater than average propensity to lie. In all such cases the statement would be unequivocally true.

It is not feasible that a human never utters a true sentence in the entirety of their lives. If for no other reason, no such person could tell successful lies, for no trust would be imparted upon such person and, so, none of their false statements would be believed true by others. Hence, the equivalence of a liar to someone that never tells truths – something that appears required for the liar paradox to obtain - is a product of mistaken reasoning.

“I am lying,” on the other hand, is in real world application made in reference to sentences that have already been spoken or, less commonly, that have yet to be spoken. This, again, is not a contradiction. The statement of “I am lying” would itself be unequivocally true in these cases.

The liar paradox is then always abstracted from mistaken reasoning applied to the significance of real world cases in which “I am a liar” or “I am lying” is spoken. Given the verity of this, the liar paradox – both strengthened and non-strengthened – is the product of faulty reasoning. And, if the product of faulty reasoning, then the contradiction it presents is itself as specious as would be any other contradictory outcome of reasoning.

I still don't know what your answer is when I say, 'I am a liar'. This statement is either true or false. Which one is it?

It's true.

The only way it's truth-value would be problematic is *if* you have never told a lie in your life *and* you know this to be true *and* you speak what you know to be untrue. You find this scenario feasible in real-world applications?

X = This sentence is false.

Assume X is true, then X is false.
Assume X is false, then X is true.

Y = This sentence is false and this sentence is true.

Assume Y is true, then Y is false (and true, but we knew that already)
Assume Y is false, then Y is true (and false, but we knew that already)

X and Y are equivalent.

But Y evaluates as:

Y is True and False.

True and False evaluates as False.

Y is false.

Y and X are equivalent.

X is false.

X is just false. No contradiction here.

Problems?

Ever read Catch 22? — ovdtogt

...thanks. No, but I remember it was popular even a few decades after its release, both in movie and book form.

Back to the topic, generally speaking, all anyone has to remember about liar's paradox is the common theme of self-reference and negation of same... . I think there is only one exception outside of self-reference where paradox appears, but can't remember right now....maybe fdrake knows

Neither X nor Y are capable of being true or false.

True and False evaluates as False. — fdrake

Why?

Why? — frank

That's how conjunction works. Look at its truth table.

This sentence is short.
This sentence is false.

Why is the first truth apt but not the second?