Why is the Liar's Paradox, a paradox?

[Graeme M]

I was reading today and came across the Liar's Paradox. I read it a couple of times and couldn't quite see what was being claimed. I went off to read some internet explanations (eg Stanford's philosophy encyclopedia) but these were all explained in rather unfathomable language so I was left none the wiser. Can someone quickly explain how it is considered a paradox? I asked my wife what she thought and her immediate response was exactly the same as mine - there's no obvious paradox here.

Take "This sentence is false". The argument seems to be that if we take it at face value, we end up in a predicament. If the sentence claims it is false it must be true yet it says it is false. or something like that.

However, I think this is just an error in thinking. "Sentence" in this case is a noun naming an object, the sentence itself. We are therefore just saying "This object is false". It is the same as saying "This car is false". An object cannot be true or false - such states can only be assigned to a proposition, such as "This car is black" or "This sentence has ten words in it". I can say "Sentence A's proposition is false" where A refers to a sentence containing a proposition, but to say "This sentence is false" is meaningless.

Similarly the sentence "I am lying". To be lying is to be in the act of presenting an untruth. An untruth is as noted above - a proposition that is not true such as "I am really a Martian". There is no intrinsic act of lying in the sentence "I am lying". Lying about what? For the sentence to have a valid claim, there must have been some proposition I have made (or will make) to which I now refer. Otherwise all I have is a statement which is neither true nor untrue on its own. A valid construction might be "I am lying when I say I am lying" but this really takes us no further. That is, there is still no actual proposition at the heart of the claim.

[alpha]

I was reading today and came across the Liar's Paradox. I read it a couple of times and couldn't quite see what was being claimed. I went off to read some internet explanations (eg Stanford's philosophy encyclopedia) but these were all explained in rather unfathomable language so I was left none the wiser. Can someone quickly explain how it is considered a paradox? I asked my wife what she thought and her immediate response was exactly the same as mine - there's no obvious paradox here.

Take "This sentence is false". The argument seems to be that if we take it at face value, we end up in a predicament. If the sentence claims it is false it must be true yet it says it is false. or something like that.

However, I think this is just an error in thinking. "Sentence" in this case is a noun naming an object, the sentence itself. We are therefore just saying "This object is false". It is the same as saying "This car is false". An object cannot be true or false - such states can only be assigned to a proposition, such as "This car is black" or "This sentence has ten words in it". I can say "Sentence A's proposition is false" where A refers to a sentence containing a proposition, but to say "This sentence is false" is meaningless.

Similarly the sentence "I am lying". To be lying is to be in the act of presenting an untruth. An untruth is as noted above - a proposition that is not true such as "I am really a Martian". There is no intrinsic act of lying in the sentence "I am lying". Lying about what? For the sentence to have a valid claim, there must have been some proposition I have made (or will make) to which I now refer. Otherwise all I have is a statement which is neither true nor untrue on its own. A valid construction might be "I am lying when I say I am lying" but this really takes us no further. That is, there is still no actual proposition at the heart of the claim.

"this sentence is false" or "i am lying" is a paradox when it's referring to another statement. e.g. if someone who lies 100% of the time says "i live in london", then says "this/that statement is false", both statements can't possibly be lies (false). either he does actually live in london, or "that statement is false" is true. whichever is true, it would contradict the fact/premise/supposition that nothing he ever says is true (100% of his statements are lies), hence the paradox.

[Graeme M]

Ahhh... so it's not the statements themselves alone! I can see I misunderstood the way it was being presented. Thanks for that, I felt I was just completely missing something obvious. Which of course, I was!

[alpha]

i could've actually misinterpreted it, lol. some sentences can contradict themselves, such as "everything i say is a lie" can't be true or false, because if everything i say is a lie, it would include this sentence, making it true, which makes it false, because then i wouldn't be lying, so everything i say isn't a lie.

[Graeme M]

alpha, I thought about this again and agree completely that the sentence has to be considered in relation to another sentence, which was my original point. I went back and reread a couple of explanations (eg Stanfords: http://plato.stanford.edu/entries/liar-paradox/ ) and as far as I can tell, they definitely do refer to the sentence in isolation. That is, the paradox is entirely contained within the statement "This sentence is false". Go figure.

I think I am missing some notion of representation. That is, the statement itself shouldn't be taken as literal, it's more representing the idea of a particular logical dilemma, as that reference shows when they apply the idea in different contexts. For example the Max and Agnes statements at 1.3 on that referred web page do what we have agreed here, ie they refer to another claim rather than to themselves.

1.3 Liar cycles
Consider a very concise (viz., one-sentence-each) dialog between siblings Max and Agnes.

Max: Agnes' claim is true.
Agnes: Max's claim is not true.
What Max said is true if and only if what Agnes said is true. But what Agnes said (viz., ‘Max's claim is not true’) is true if and only if what Max said is not true. Hence, what Max said is true if and only if what Max said is not true. But, now, if what Max said is true or not true, then it is both true and not true. And this, as in the FLiar and ULiar cases, is a contradiction, implying, according to many logical theories, absurdity.

The trouble with this though is that in the original case, the sentence refers to itself. In the Max and Agnes cases, they refer to something else. if we use the same self-referential form, the Max and Agnes cases should read:

Max: My claim is true.
Agnes: My claim is not true.

Which is meaningless. So clearly it must be that the original form is meant to be representational rather than literal.

I think...

[Blueswing]

A paradox is self-contradictory, those sentences are self-contradictory, so they are paradoxical.

They're not very interesting though, are they?

[alpha]

some sentences can contradict themselves, such as "everything i say is a lie" can't be true or false, because if everything i say is a lie, it would include this sentence, making it true, which makes it false, because then i wouldn't be lying, so everything i say isn't a lie.

"everything i say is a lie" is an example of a self-contradicting statement.

[SpheresOfBalance]

I was reading today and came across the Liar's Paradox. I read it a couple of times and couldn't quite see what was being claimed. I went off to read some internet explanations (eg Stanford's philosophy encyclopedia) but these were all explained in rather unfathomable language so I was left none the wiser. Can someone quickly explain how it is considered a paradox? I asked my wife what she thought and her immediate response was exactly the same as mine - there's no obvious paradox here.

Take "This sentence is false". The argument seems to be that if we take it at face value, we end up in a predicament. If the sentence claims it is false it must be true yet it says it is false. or something like that.
As one reads the sentence, "this sentence is false," it's to be taken as though it's true that the sentence is false, but then how could it be true if in fact it's false. If the sentence says that it's false, then it's false that it's false, which means that it's true, yet it says that it's false. In other words, it seemingly creates a paradox, because it can be neither true nor false; It defeats itself.


However, I think this is just an error in thinking. "Sentence" in this case is a noun naming an object, the sentence itself. We are therefore just saying "This object is false". It is the same as saying "This car is false". An object cannot be true or false - such states can only be assigned to a proposition, such as "This car is black" or "This sentence has ten words in it". I can say "Sentence A's proposition is false" where A refers to a sentence containing a proposition, but to say "This sentence is false" is meaningless.

Similarly the sentence "I am lying". To be lying is to be in the act of presenting an untruth. An untruth is as noted above - a proposition that is not true such as "I am really a Martian". There is no intrinsic act of lying in the sentence "I am lying". Lying about what? For the sentence to have a valid claim, there must have been some proposition I have made (or will make) to which I now refer. Otherwise all I have is a statement which is neither true nor untrue on its own. A valid construction might be "I am lying when I say I am lying" but this really takes us no further. That is, there is still no actual proposition at the heart of the claim.

[Graeme M]

some sentences can contradict themselves, such as "everything i say is a lie" can't be true or false, because if everything i say is a lie, it would include this sentence, making it true, which makes it false, because then i wouldn't be lying, so everything i say isn't a lie.

"everything i say is a lie" is an example of a self-contradicting statement.

Hmmm... I am no philosopher so I don't know the proper form for evaluating logic. But to me there is no contradiction here.

If Bob says "Everything I say is a lie" then what he just said is a lie. If it's a lie, then at least one of his statements must be true. I must therefore evaluate his statements using my knowledge to ascertain the truth or otherwise of his statements, which is no different to what I face when talking to anyone. The statement "Everything I say is a lie" therefore provides no useful clue and can be considered meaningless, or more exactly, redundant.

[alpha]

Hmmm... I am no philosopher so I don't know the proper form for evaluating logic. But to me there is no contradiction here.

If Bob says "Everything I say is a lie" then what he just said is a lie. If it's a lie, then at least one of his statements must be true. I must therefore evaluate his statements using my knowledge to ascertain the truth or otherwise of his statements, which is no different to what I face when talking to anyone. The statement "Everything I say is a lie" therefore provides no useful clue and can be considered meaningless, or more exactly, redundant.

it might be redundant, but it's self-contradicting, nonetheless.