## The Liar Paradox - Is it even a valid statement?

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The liar paradox is that the statement: ‘This statement is false’ appears neither true nor false. If the statement is true, then it is false. If the statement is false, then it is true (and so on):

I’m going to argue that the paradox is due to the statement not being a valid statement. The definition of a statement:

https://en.wikipedia.org/wiki/Statement_(logic)

In our case, we take the first definition: ‘a meaningful declarative sentence that is true or false’.

I think the word ‘declarative’ is important; a statement declares a fact; it does not in addition instantiate that fact to a given truth value.

So a statement is associated with but distinct from a truth value. For example: ‘all cats are black’ is distinct from ‘false’.

Contrast this with ‘this statement is false’; the truth value is embedded in the ‘statement’. It is sufficiently different in structure to a normal statement as to be classed as not a statement. The truth value is also instantiated to false making the 'non-statement' both contradictory and redundant.

It should IMO be handled differently in logic than a normal statement as it has an in-built truth value.

So if statements including their own truth value where excluded from logic, then the liar paradox would not be a paradox. This would make logic self consistent again (I believe Godel's objections would go away though I need to look at that conjecture further).
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You can also interpret 'this statement as false' as:

1. this is a statement
2. and it is false

So 2 says 1 is false. IE it is not a statement.
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So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'. If you can’t prove its a statement then you can even start to prove it.
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We need to compare a statement with something to determine its truth value. When we say "This statement is false", it can be given a truth value when "this statement" refers to a statement that can be given a truth value. But here, "this statement" refers to "This statement is false", which can be given a truth value if "this statement" refers to a statement that can be given a truth value, and so on and so forth, so we have an infinite regress because we're never specifying what "this statement" refers to.

Not all statements in a given language can be given a truth value, in that they don't refer to anything that allows to determine a truth value. Language is used to refer to things we observe, but in "This statement is false" we're never saying what we're referring to.
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These self-denying statements are acceptable according to formal logic and they lead to Godel's Incompleteness Theorems.

Not all statements in a given language can be given a truth value, in that they don't refer to anything that allows to determine a truth valueleo

If it can't be given a truth value, its not a statement would be a simple rule to adopt.
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If it can't be given a truth value, its not a statement would be a simple rule to adopt.

Yes I'm not disagreeing, I was just giving my point of view on the 'paradox'.

We can say things such as "This sound is purple" or "This smell is true", but they don't refer to anything in the range of what we experience, or at least I can't form a mental image of whatever these statements may refer to. Same goes with "This statement is false", not all statements that can be uttered in a language are meaningful, and I agree it's not much use to spend much time pondering about them or trying to fit them into some grand framework.

Like the paradoxes in the theory of relativity, they are a consequence of the postulates at the basis of the theory, we can choose to ignore them and just "shut up and calculate" and make predictions that fit somewhat with observations, or we can change the framework (change the theory, pick different postulates) so that the paradoxes disappear while making similar observable predictions, in the end it depends whether we're looking for mathematical 'elegance' with symmetries and so on or if we're looking for intuitive simplicity. I'm a bit like you on this, I prefer intuitive simplicity that can be grasped by many over mathematical elegance that leads to complexity, paradoxes and confusion.
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We can say things such as "This sound is purple" or "This smell is true", but they don't refer to anything in the range of what we experience, or at least I can't form a mental image of whatever these statements may refer toleo

I think that anything that is self contradictory or nonsensical should be classed as a non-statement - such statements disprove themselves if you see what I mean.

Like the paradoxes in the theory of relativity, they are a consequence of the postulates at the basis of the theory, we can choose to ignore them and just "shut up and calculate" and make predictions that fit somewhat with observations, or we can change the framework (change the theory, pick different postulates) so that the paradoxes disappear while making similar observable predictions, in the end it depends whether we're looking for mathematical 'elegance' with symmetries and so on or if we're looking for intuitive simplicity. I'm a bit like you on this, I prefer intuitive simplicity that can be grasped by many over mathematical elegance that leads to complexity, paradoxes and confusionleo

Yes, just like infinity clouds and complicates set theory, the loose definition we have of statement greatly complicates logic.

It is complicated stuff that I admit I don't fully understand:

'The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.'

https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems

These theorems place profound epistemological limitations on formal logic and mathematics. If we remove this class of malformed, contradictory statements then these limitations do not apply any more.
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Yeah, it makes sense to say that P is true or false, but P has to be a proposition. It needs to claim something. And then it's the claim that's true or false. "This statement" doesn't claim anything.
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So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'. If you can’t prove its a statement then you can even start to prove it.

No, no, no, no, no. It is a statement. It says, roughly, "this statement is not provable," which means, this statement is not provable. "Roughly" because Godel's sentence is carefully crafted and we here have not done it justice, but only approximated it.
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'This statement is not provable' means:

1) this is a statement
2) this is not provable

IE it is not provable if it is a statement. Also if you read the OP, it is syntactically different from a statement.
• 2.2k
in "This statement is false" we're never saying what we're referring to.leo

Right. The so-called "Liar's paradox" seems quite silly, akin to something a third grader thought up at recess.

I agree it's not much use to spend much time pondering about themleo

Me too. :up:
• 3.2k
The so-called "Liar's paradox" seems quite silly

Knowing something about logic and the context helps to understand why the liar paradox is of interest.

I agree it's not much use to spend much time pondering about them
— leo

Me too.

Good then that no one is forcing you to spend time on it. But meanwhile it is worth time to people who study logic.
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a statement declares a fact; it does not in addition instantiate that fact to a given truth value.

Some statements do mention truth values.

I believe Godel's objections would go away though I need to look at that conjecture further

The incompleteness theorem is not at all disqualified by the liar sentence. And it's not a conjecture.
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1. this is a statement
2. and it is false

So 2 says 1 is false. IE it is not a statement.

We can formulate the liar paradox without saying "is a statement".
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So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'.

That is not the Godel sentence.

If you can’t prove its a statement then you can even start to prove it.

We do prove it is a sentence in the language of the theory at hand. And we don't the sentence in the theory.
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Not all statements in a given language can be given a truth valueleo

For a formal language, per a given interpretation, every sentence has a truth value.
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These self-denying statements are acceptable according to formal logic and they lead to Godel's Incompleteness Theorems.

The incompleteness theorem does not rely on the liar sentence.
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they are a consequence of the postulatesleo

Regarding the liar sentence, what postulates?
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If we remove this class of malformed, contradictory statements then these limitations do not apply any more.

The incompleteness theorem does not rely on any sentences that can't be formed in the language of arithmetic.
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'This statement is not provable' means:

1) this is a statement
2) this is not provable

That is not the Godel sentence.
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The only difference is that its circumscribed to a given system, though, right?
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The differences between the Godel sentence and "this is a statement and this is not provable"?

The Godel sentence is a sentence in the language of arithmetic. Given the standard interpretation of the language of arithmetic, the Godel sentence says something about natural numbers. But also, the Godel sentence is true in the standard interpretation if and only if it is not provable in the system (whichever system the incompleteness theorem is being proven about). In that sense, the Godel sentence says "I am not provable", but keep in mind that "I" is only our informal description; the language doesn't have such pronouns. And, the Godel sentence does not say "I am a sentence" nor mention "sentence". Rather, it is in the meta-theory that we show that the Godel sentence is indeed a sentence in the language of arithmetic and that, if the system is consistent, then the Godel sentence is not provable in the system, and the negation of the Godel sentence is not provable in the object theory, and that the Godel sentence is true in the standard model for the language. (Note that Godel did not specify formal models, as formal models were not explicated until later, and he proved for a different kind of system. Instead, he simply worked in ordinary mathematics regarding natural numbers without putting a fine point on that in terms of models and a standard interpretation.)
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How is an unprovable proposition different from a postulate?
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There are two different things:

(1) The incompleteness theorem. It's not a conjecture. It is proven. It is a theorem about certain kinds of object theories.

(2) The Godel sentence. In proving the incompleteness theorem, we prove that the Godel sentence is a sentence in the language of the object theory. And we prove that, if the object theory is consistent, then the Godel sentence is not provable in the object theory.
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But to my point, if Godel proved there are limitless unprovable propositions and if these propositions can be used to make theorems then they are no more than postulates and we already knew that math had unprovable postulates on which the whole structure was built (you can't prove everything). So what exactly did Godel add to our body of knowledge?
• 486
So what exactly did Godel add to our body of knowledge?

You can perfectly know the construction logic of a system but that does still not allow you to know its complete truth. So, even if we manage to figure out the perfect theory of the physical universe, we will still not be able to predict most of its facts.
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My point was that the incompleteness theorem is not a conjecture.

/

I take it that 'postulates' means axioms.

For every sentence, there is a system of which the sentence is an axiom.

A sentence is provable or not relative to a given system.

Every axiom is trivially provable in a system in which it is an axiom (by the trivial proof of putting the axiom itself as the only line in a proof).

For every sentence, trivially, there is a system in which the sentence is provable (by making the sentence an axiom).

So, the incompleteness theorem is not about what is provable simpliciter, but what is provable in certain kinds of systems.

The incompleteness theorem is: If a theory is formal, sufficient for a certain amount of arithmetic and consistent, then the theory is incomplete. That is highly informative: It tells us that there is no axiomatization of arithmetic such that every sentence of arithmetic is a theorem or its negation is a theorem. It tells us that there is no axiomatization that proves all the true sentences of arithmetic. It tells us that there is no algorithm to determine whether any given sentence of arithmetic is true. And the methods of the proof lead to profoundly informative results such as the unsolvability of the halting theorem and that there is no algorithm to determine whether a given Diophantine equation is solvable.

You seem to not be distinguishing between (1) For any given system, the axioms are not proven from previous theorems and (2) Given any consistent set of axioms sufficient for a certain amount of arithmetic, there are sentences of arithmetic such that neither the sentence not its negation is provable from the axioms, thus there are true sentences of arithmetic not provable from the axioms.

(1) is a trivial given. (2) is a remarkable result.
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You can perfectly know the construction logic of a system but that does still not allow you to know its complete truth. So, even if we manage to figure out the perfect theory of the physical universe, we will still not be able to predict most of its facts.

That is ridiculously overbroad and vague.
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That is ridiculously overbroad and vague.

That is what Hawking has said on the matter:

https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

Isn't it rather your own criticism that is ridiculous?
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Unlike your post, that quote seems at least fairly clear and doesn't make an overbroad mischaracterization of incompleteness.
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Unlike your post, that quote seems at least fairly clear and doesn't make an overbroad mischaracterization of incompleteness.

Noson Yanofsky writes:

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

There are more true but unprovable statements than we can possibly imagine.

We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.

What I wrote, is the combination of what Hawking and Yanofsky wrote on the matter. Why would that be an "overbroad mischaracterization"?
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