Paradox and truth

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01 Mar 2010, 2:08 am

"A man says that he is lying. Is what he says true or false?" -Eulibides of Miletus

The liar's paradox is a well-known paradox, but the issue is a matter of truth. If the man saying that he is lying is making a true claim, then the claim is false. If the man saying that he is lying is making a false claim, then it is a lie, and thus it is a true claim.

Similar issues come up with Russell's paradox. First, let's define a "set". A set is a comprehensive grouping of objects according to a specific description. So, if I talked about a set of leaves, then I would mean the logical idea of a grouping containing all things that are considered to be "leaves". Well, an issue is that sets can contain other sets. So, the set of leaves contains the set of tree leaves, for instance. Well, the issue comes in that if sets can logically exist, and if sets containing sets logically exist, what about a set of sets that do not contain themselves? If this set did not contain itself, then it would contain itself. If the set contained itself, then it would not contain itself. The set is a paradox because neither possible state of affairs can be true. This paradox is known as Russell's paradox.

The issue is what paradoxes mean. It is easy to point to a paradox, but if our naive claims about reality are true, then how is the paradox dissolved? If the paradox cannot be dissolved, then what can we think about reality?

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01 Mar 2010, 2:20 am

The liers paradox is an easy one. A lie is an only a lie in a given context. Outside of the given context it is both a truth and a lie, because all truths are a lie at some level and all lies are true on some level..

As for the other one, I have never heard of it before now, and I am too tired to think about sets within sets within the first set. Maybe tomorrow..

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01 Mar 2010, 6:14 am

I think it might be nonsensical. True / non-true and false / non-false are not quite the same distinction. It is possible for statements to be simultaneously non-true and non-false, without being either true or false.

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01 Mar 2010, 11:39 am

Hmm...

The issue is why do we consider the statement of the paradox neither true nor false as a category rather than a paradox? It seems that such a move is driven more by the utility and less by the validity of such a move.

I also don't think that the liars paradox is so easily broken down. Let's just make the claim "This statement is false". The issue is that the statement refers to itself, meaning that if it is true, then it cannot be true. If it is taken as false, then it must be true. I mean, the issue isn't context so much as the idea of a self-referential statement that claims to be false.

Even if the liar's paradox is soluble, what about Russell's?

Part of the question is really what paradoxes say about truth, not just "can you solve these paradoxes".

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01 Mar 2010, 1:56 pm

I should quickly mention that I do not understand you well.

With that out of the way, I can continue.

Quote:

The issue is why do we consider the statement of the paradox neither true nor false as a category rather than a paradox?

Does this quoted sentence say that we do not consider the statement a paradox?

Quote:

Even if the liar's paradox is soluble, what about Russell's?

Is the liar's paradox more soluble than Russell's sets paradox?

Quote:

Part of the question is really what paradoxes say about truth, not just "can you solve these paradoxes".

The only thing I can draw about truth from these paradoxes is that truth isn't readily applicable to their statements.
What do you see?

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01 Mar 2010, 2:15 pm

PLA wrote:

I should quickly mention that I do not understand you well.

With that out of the way, I can continue.

Quote:

The issue is why do we consider the statement of the paradox neither true nor false as a category rather than a paradox?

Does this quoted sentence say that we do not consider the statement a paradox?

The quoted sentence is saying that your solution makes the paradox solved. The issue is that paradoxes are paradoxes due to the problems in their solubility but the necessity of them being maintained.

Quote:

Even if the liar's paradox is soluble, what about Russell's?

Is the liar's paradox more soluble than Russell's sets paradox?

I'd think it is, but I am not sure. I am not even sure if there is a general category of "solubility".

Quote:

The only thing I can draw about truth from these paradoxes is that truth isn't readily applicable to their statements.
What do you see?

Well, right, but I see issues with the nature of truth as well, at least in as much as we understand logic as related to truth.

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01 Mar 2010, 4:28 pm

Awesomelyglorious wrote:

Language. The trouble lies in the design and structure of language. It's great for telling people ideas, but only to a certain extent. There are so many paradoxes that are just not paradoxes to me, because I think kinesthetically. There is no way to solve some paradoxes in any known language, but I guarantee, if someone had enough time and wits, they could design a language meant for communicating paradoxes in a way that is logically sound, but when translated, loses all of it's sense.

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01 Mar 2010, 4:49 pm

fidelis wrote:

So, basically it is just this:
1) Language is a system of expressing truth
2) Language as a system of expressing truth is limited.
3) The limits of language are expressed as paradox.
4) The truth itself though is not hampered by real paradoxes though.

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01 Mar 2010, 5:20 pm

I also think the issue is language. We can talk about a square circle, but that does not mean that one can necessarily exist in reality as anything other than a linguistic construct.

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01 Mar 2010, 5:50 pm

pandd wrote:

I also think the issue is language. We can talk about a square circle, but that does not mean that one can necessarily exist in reality as anything other than a linguistic construct.

Hmm... ok. Just a small question for proponents of this view: how does logic relate to mathematics? What is mathematical truth?

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01 Mar 2010, 6:05 pm

Ha!

Ask a mathmatician.

I do not claim to understand mathmatics all that well. I understand it entails a set of axioms that at least a sub-set of logic is applied to. So I suppose it might not be wrong to describe the relationship as reliant (mathmatics relies on [at least a sub set of] logic). But I have to say I have no firm opinion on this as I do not feel particularly well acquainted with the theoretical aspects of maths (nor with many of the practical aspects either for that matter).

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01 Mar 2010, 6:16 pm

Awesomelyglorious wrote:

pandd wrote:

I also think the issue is language. We can talk about a square circle, but that does not mean that one can necessarily exist in reality as anything other than a linguistic construct.

Hmm... ok. Just a small question for proponents of this view: how does logic relate to mathematics? What is mathematical truth?

From my limited precalculus understanding, mathematical truth can be proven. Axioms are based off logic, and used chiefly for proofs.

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01 Mar 2010, 6:30 pm

pandd wrote:

fidelis wrote:

From my limited precalculus understanding, mathematical truth can be proven. Axioms are based off logic, and used chiefly for proofs.

Well, the reason I asked is because if logic works, and math is a form of logic, then math has to work. The issue is that math has major issues with it working, as noted in debates in the early 20th century. In fact, it was proven that not all math could be proven from a given set of mathematical axioms, which severely undercut projects relating to the foundations of mathematical knowledge.

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01 Mar 2010, 6:38 pm

Awesomelyglorious wrote:

Language->logic
Logic->axioms
Axioms->proofs

If language is limited, then it's only logical that so should math to some to degree. If we change language we could build new logic for new axioms for new branches of mathematics.

Is that what you were thinking?

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01 Mar 2010, 6:56 pm

fidelis wrote:

Awesomelyglorious wrote:

Are you assuming that language creates logic?

At first I thought you held that language approximated underlying logic.

But if we hold that language creates logic, then this upholds the Sapphir-Whorf hypothesis, which basically says that the language taught creates the mental structure for organizing reality. And that then creates a lot of interesting epistemic issues.

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01 Mar 2010, 7:08 pm

I believe that both language and logic are created from thought, but I also believe that if language can't express an idea then that idea can only stay in an individual at kinesthetic form (which sadly people aren't too used to dealing with, and are very tempted to just forget it.)
Because I believe this, I also believe that language limits logic, although it does it indirectly.

I guess this would have been a bit better:

thought->logic------->axioms->proofs
thought->language->axioms->proofs

They way I wrote it was for convenience. Sorry about that, it's just one of those habits that I can't seem to shake.

So, if language changed, it would allow expressing a broader range of logic, and therefore broader ranges of mathematics.

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