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

• 2.5k
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.
• 830
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.
• 830
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.
• 2.5k
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.
• 13.8k
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.
• 4k
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.
• 2.5k
'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.
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