The Liar's Paradox, Contradiction, and Truth

Recently, I have been intellectually preoccupied with a consideration of the Liar's Paradox. Which for ease, and because I rather not discuss Cretans (those dirty islanders!), shall be rendered thus:

The following sentence is true: The previous sentence is false.

Now when considering this, I came to a revelation: Is not the liar's paradox simply an instance where one is testing the Law of Non-Contradiction by postulating a contradictory statement?

For consider the above, is it not saying that A = B when A is defined as -B and B as -A? Or to put it otherwise: -B = B or A = -A? Clearly, these statements are nonsensical (unless of course A and B are zero, in which case it isn't the liar's paradox and really could be cut down to 0) and this is precisely why we have -any- problem, whatsoever, with the Liar's Paradox. You are beginning with the impossible and ending with nonsense.

Yet what does this say about the Law of Non-Contradiction? That there can be sentences which take for granted contradiction and subsequently, even mathematical expressions (albeit it a simplistic one for sake of the conversation) would seem to indicate something wrong with the Law of Non-Contradiction at first glance, or rather, such would be our intuitions. But this is not so in the least. Rather, that a Contradictory Statement cannot be resolved to truth or falsehood, that it is litterally impossible to say that a statement which implies impossibility by the nonsense of its assumptions is true or false, demonstrates the infallibility of the Law of Non-Contradiction. How? If all truth is resolvable, and only non-contradicting statements are resolvable, then all truth is free of contradiction.

Also: Does this mean that all truth is tautological? For if we consider everything which is true as an equation, do not we have to affirm that in order for it to be truth, we must have truth on either side of the equal sign? And does not this resolve to simply one thing when further reduced?

 

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revelation indeed, i reconmend you ring someone important.

 

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Right on P_J, I do so love this stuff....

The real solution here is to recognize that this is a Category mistake. The problem linguistically presents iteself as a paradox because we, quite naturally, create a meta-argument out of an argument. The illegitimate move here is to allow self-referential statements regarding truth. Once we confine truth to just one argument, then there is no problem.

...


Exactly. This of course moves us to problematize the source of the Law of Non-Contradiction: the Law of Identity (from which, IMO, all nastiness arises..). Here, we take as axiomatic the particular nature of an entity (or the ability to identify it as such...) to be understood, even though there is no reputable process for doing so (thanks to the problem of induction...). Interestingly, this is why in Eastern logics, there is no Law of Identity.

You are correct. Admittedly, any good logician will assert this. This is why tautologies in logic are useless; they are void of content. More importantly, this also explains the downfall of the Rationalist movement (a la, Descartes, Spinoza, et. al.) and why the notion of truth is generally regarded as useless in contemporary logic, as opposed to validity.

 

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Nice post as usual, but can you explain this statement in light of Gödel's true but unprovable statement, i.e. his incompleteness proof - how Gödel's proof either agrees with your statement about the notion of truth or can be ignored or superceded.
Thanks.

 

Thanks cg,

It's important to remember that the ultimate result of Godel was not necessarily incompleteness rather it was undecidability. In brief then, for any closed system (our logical 'system' for defining truth) not only must it necessarily be incomplete (we cannot ever know all the laws) it must also therefore be undecidable (given the incomplete rule-architecture, at some point there must arise an explicit assumption). The notion 'truth' then, as it ultimately must be founded upon some sort of 'given', tautaulogy', or even (if one dares to go there..) 'a priori' will be at best, undecidable. The quandry then becomes this: of what use is an 'undecidable truth'? The obvious answer is: none.
And so most philosophers moved on to placing more importance on the notion of validity, while those philosophers who were loathe to give up their Platonic aspirations moved on to less rigid structures such as the Coherence Theory of Truth.

Ahhhh... epistemology.

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This, of course, is exactly the same situation with respect to the old "What if an unstoppable force met an immovable object" question. Clearly by defining one or the other you preclude the existence of the converse.

 

Louisos:

Thank you.

glaucon:

"Right on P_J, I do so love this stuff...."

I am most certainly glad to be of service, Glaucon. I was waiting for your input! I always appreciate our conversations.

"The real solution here is to recognize that this is a Category mistake. The problem linguistically presents iteself as a paradox because we, quite naturally, create a meta-argument out of an argument. The illegitimate move here is to allow self-referential statements regarding truth. Once we confine truth to just one argument, then there is no problem."

I would agree with you completely. This is a perfectly acceptable way of dealing with the argument in almost all ways. However, why I chose an alternative route was to illustrate what seems to be a crucial difference in true statements, false statements, and unresolvable and therefore contradictory statements. This was also to consider that all true statements are resolvable and it seems that all truths are fundamentally tautological. I also was attempting to show that one can basically show that Godel's Incompleteness Theorem does not come into play into the Liar's Paradox if you address the matter as I have. That if truth can only be found in resolvability, and that all non-resolvable things are contradictions, and contradictions essentially prove their own untruth, then one does not have in this instance an "undeciable truth", but a "certain falsehood" as certain as regular falsehoods.

"Exactly. This of course moves us to problematize the source of the Law of Non-Contradiction: the Law of Identity (from which, IMO, all nastiness arises..). Here, we take as axiomatic the particular nature of an entity (or the ability to identify it as such...) to be understood, even though there is no reputable process for doing so (thanks to the problem of induction...). Interestingly, this is why in Eastern logics, there is no Law of Identity."

Actually, I do not believe the Problem of Induction is really an issue here. For two reasons:

1. In at least in Hume's sense, I believe I have an answer. His argument basically rests on the premise of: Chaos is not irrational, thus the presumption of order is false. I affirm the opposite: Chaos is irrational, thus the presumption of order is true. For if we construe chaos as "the absolute opposite of order and exemplifying pure randomness" then we immediatly see that the concept can hold no water. For if something is completely random, it could not itself contain within it a capacity to persist, for all persistance is an example of order. Indeed, chaos could not be coherent enough to even manifest for a moment, for in order for it to do as such it must be, for a period, non-random, and therefore in violation of itself through not being chaotic. Moreover, even a less extreme form of chaos would suffer from the same aspect, as its randomness would have to be ordered. Contra-Kant, I thus affirm that metaphysics is possible not through relation to the pseudo-metaphysics of the Transcendental Aesthetic, but through the impossibility of chaos. Thus causality I construe as legitimized on this level, too, as well as certain aspects of causal-events and emergent properties, which imply that causality does in fact exist, and that we can infact determine how certain things can act upon other things, contra-Leibniz.

2. Identity does not depend upon the problem of induction. The problem of induction itself is properly a matter of causality. The Law of Identity rests basically on a tautology and doesn't even need to take into consideration such properties as change, movement, et cetera. It is basically affirming that anything is itself at any given time and to be non-self would imply that it did not exist, even if its existence is just illusionary or ephemeral.

However, I do agree with you that the Law of Identity and Non-Contradiction are ultimately linked.

"You are correct. Admittedly, any good logician will assert this. This is why tautologies in logic are useless; they are void of content. More importantly, this also explains the downfall of the Rationalist movement (a la, Descartes, Spinoza, et. al.) and why the notion of truth is generally regarded as useless in contemporary logic, as opposed to validity."

Yet the question is: Can validity offer us anything else either?

However, I am loathe to affirm that truth is impoverished.

superluminal:

"This, of course, is exactly the same situation with respect to the old "What if an unstoppable force met an immovable object" question. Clearly by defining one or the other you preclude the existence of the converse. "

Also the "Can God create a rock that not even he can lift?"

 

My understanding is that Gödel's proof consists of a statement which is BOTH true and undecidable within the system. If he just came up with an undecidable statement, well that wouldn't have been a big deal at all.

 

Cole Grey:

I don't think that Godel ever showed a "true but undeciable statement". If it is undeciable, it cannot be true (otherwise it would be decidedly true!). Rather, it is like the Liar's Paradox, which would simulteneously be true and false.

 

Though we're tending to move into ontology here and away from epistemology....

I still think, given your excellent point, that the Problem of Induction obtains here.

2 points:

Firstly, don't you think that your presumption of order leads your argument to end up as circular to some extent? In granting order, doesn't the feasibilty of a deductive analysis become possible? In assuming an order, it must be possible therefore to observe an event and from there incorrigibly deduce the result (assuming of course one knows all the relevant rules and has the time for the calculations). The assumption of order here then, for all intents and purposes, ends up guaranteeing one's prediction; the truth then, becomes a priori.

Seconly, doesn't a coherence necessitate an 'ordering agent'? If ordered reality is to be granted (as opposed to an incoherent chaos) then either this order is apparent, or endemic to the system. If it's apparent, then we have to determine if it's an effect of the nature of the system, or an effect of the observer (a la Kant's a priori synthetic...). The former possibility leads us into metaphysics, and all sorts of empiricist nastiness. The latter leads us into some sort of phenomenalism. On the other hand, if the coherence is supposed to be endemic to the system, then we're right back to straight up physics (and eventually back to the fatalist position).

I agree with you on Identity's reliance on tautology, but I think nevertheless that it's inexorably tied up with the induction problem. If one object is to be said to be unique, then this object must always remain as it is (unique to itself.. though this sounds odd..). The problem here is that persistence requires induction. Perhaps not ontologically speaking, but at least epistemologically, for anyone to be able to say "This is A.", it must be possible for the speaker to always be able to say that. Ultimately, either a thing cannot change, or we must have a valid means of making a number of observations of that thing, and still be able to determine that it is still that same thing.

See the point i just made. I think in the long run, validity enables us to be able to make fairly reliable predictions. What's more, validity, as a criterion of 'truth', is flexible; over time we can emend our definitions and systems, accomodating new information quite readily (for the most part).
[Unfortunately, this opens up the door to the notion of 'flexible truth'.

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]

As am I. But I think a more serious danger is to be found in less rigid application of 'truth-hood'.

 

that is what you would think, but the fact that his statement is both true and undecidable is why his work is valuable. If his statement is false it is decidable, and means little.

 

Glaucon:

"Though we're tending to move into ontology here and away from epistemology...."

You are right. We ought to steer it back to epistemology for the most part. However, it always seems that as soon as one begins to speak of epistemology, it is almost certain that one will move towards ontology in the course of the discussion. They do so seem to be linked! But yes. Let's try to keep this on course as much as we can, with asides when needed otherwise.

"Firstly, don't you think that your presumption of order leads your argument to end up as circular to some extent? In granting order, doesn't the feasibilty of a deductive analysis become possible? In assuming an order, it must be possible therefore to observe an event and from there incorrigibly deduce the result (assuming of course one knows all the relevant rules and has the time for the calculations). The assumption of order here then, for all intents and purposes, ends up guaranteeing one's prediction; the truth then, becomes a priori."

I do not think it is so much a presumption of order, only a consideration from what chaos entails. That is to say, my conclusion "chaos is irrational, thus order is true" stems from whether or not chaos is a rational consideration, rather than any consideration of order as superior until the above was proven one way or another. Thus the argument begins without such pre-declaration of order, but instead goes:

If chaos is to exhibit persistance - which would be a given if chaos were to be able to exist, yes? - and persistance is not random, yet chaos is "pure randomness", then chaos must have qualities not in accordance with itself, and therefore, would not be able to exist. That is to say, if one were to put forth an idea of a "persistant chaos", one is putting forth an idea of "chaos which is ordered", which would imply no chaos at all.

Thus it was merely stylistic to begin with the affirmation of my conclusion, for I have written this argument before and so was just simply stating that such woudl be my conclusion based on the argument I gave immediatly after.

"Seconly, doesn't a coherence necessitate an 'ordering agent'? If ordered reality is to be granted (as opposed to an incoherent chaos) then either this order is apparent, or endemic to the system. If it's apparent, then we have to determine if it's an effect of the nature of the system, or an effect of the observer (a la Kant's a priori synthetic...). The former possibility leads us into metaphysics, and all sorts of empiricist nastiness. The latter leads us into some sort of phenomenalism. On the other hand, if the coherence is supposed to be endemic to the system, then we're right back to straight up physics (and eventually back to the fatalist position)."

I am in fact implying a total and complete order, so one could indeed charge me as rightfully a determinist/fatalist. Whereas I reject the pessimissim and defeatism of moral fatalism, I do recognize that this absence of chaos as implying a perfection of order. So I shall go with the "endemic order" of what you speak.

I'll respond to the rest of your post in about an hour. Yours too, Cole Grey.

 

Glaucon:

"I agree with you on Identity's reliance on tautology, but I think nevertheless that it's inexorably tied up with the induction problem. If one object is to be said to be unique, then this object must always remain as it is (unique to itself.. though this sounds odd..). The problem here is that persistence requires induction. Perhaps not ontologically speaking, but at least epistemologically, for anyone to be able to say "This is A.", it must be possible for the speaker to always be able to say that. Ultimately, either a thing cannot change, or we must have a valid means of making a number of observations of that thing, and still be able to determine that it is still that same thing."

A very interesting way to look at it, but I think you're a little off base here. Let us consider things by making recourse to numbers (as they are so damn conveinent!). What does it mean when we say "1" or "22"? Well we are speaking of a numerical entity whose value is "1" or "22". What does this mean? That Numerical Entity = 1. That 1 is its identity. Now in order for it to be 1, it must not be any other number besides 1, lest it not be 1 to start off with. And in order to be one, it must be 1, that is, 1 must = 1. Yet what would happen if we added 1 and 22 together? Well, we'd have 23. And what would it mean to have 23? 23 = 23, 23 != any other number...Thus even if we change the numbers in question, the new number corresponds to the Law of Identity.

Yet does this require induction in order to distinguish?

No, we only need to analyze the thing in question and go straight for the core of what is implied in the numbers. We could indeed make a general, inductive rule from this likely, but we needn't. That is to say, at any time can the Law of Identity can be validated by simply speaking of the object and then analyzing what is implied through referencing it. The Law of Identity thus remains tautological without recourse to induction, although clearly, one can formulate an inductive law based on "the Law of Identity always holds true".

"See the point i just made. I think in the long run, validity enables us to be able to make fairly reliable predictions. What's more, validity, as a criterion of 'truth', is flexible; over time we can emend our definitions and systems, accomodating new information quite readily (for the most part).
[Unfortunately, this opens up the door to the notion of 'flexible truth'.

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]"

How far would you suggest we take "validity over truth"? It would seem useful in empirical analysis, where we aren't dealing in general with things we can make wide-spread inductive statements about (although we can say one or two things at least about them in advance if induction is valid as I say). But as to using it for logic, I have always found it a bit lacking. For analytic statements seem to always imply necessity (and thus truth and not mere 'validity') and if grounded in association with reality (as opposed to being of the type of "if dogs are cats, and cats are horses...") then we seem to find absolutely true things.

"As am I. But I think a more serious danger is to be found in less rigid application of 'truth-hood'. "

I agree. The ultimate in sophistry results from such considerations.

cole grey:

"that is what you would think, but the fact that his statement is both true and undecidable is why his work is valuable. If his statement is false it is decidable, and means little. "

Can you give me an example of one of these true but undecidable statements of his? For in all the readings on the subject, it basically all seems to resolve to issues like the Liar's Paradox. Also, we should take into consideration that Godel's Theorem only applies to systems of a certain complication. It doesn't hold true in Euclidean Geometry, for instance.

 

You've given it yourself.
The Liar's Paradox.

Don't have much time here, but this is one of my favorite topics. I'll answer this question with a quote from ago:

"This well-formed formula is unprovable in the system."

So, the sentence can be proven to be neither true nor false. Yet, let's consider if it were false. This would mean that it is provable in the system. Which implies that it is true. Which means that it is unprovable. This is a blatant contradiction. Therefore, the sentence is a.) true in a consistent system yet unprovably so, or b.) false in an inconsistent system.
In this way, it is unprovable yet able to be seen as true through informal methods.​

If Godel's theorem were simply about undecidability, then it would be of little interest and would never have stirred up the crowd as it has (outside of mathematics, that is.)
It is about, as Cole has said, things that are undecidable within a system and yet still true. Just not provable within the system.

Human cognition is not a formal system. It is subject to inconsistencies, and rarely subject to Turing's Halting Problem.
But, despite its inconsistency, it has far more power than any formal system will ever possess.

GEB is a mandatory read at this point, of course. As is The Emperor's New Mind by Penrose. Deutsch's Fabric of Reality also plays with this idea. I suspect that more than one of you have read at least one of these....

Another interesting read on this subject is available on the web here:
Lucas versus Mechanism.

(Disclaimer: The quoted section above is from my earliest discovery of GEB. I've restated this proposition in much better forms elsewhere, but I don't recall where. I knew exactly where this one was to be found, hence its inclusion. I believe it suffices to answer your question.)

 

Invert_Nexus:

"You've given it yourself.
The Liar's Paradox."

When you have the time to give more of a response, what do you think of my argument to show that the Liar's Paradox is basically an example of postulating a contradiction and that its unresolvability reveals that all truthful statements must be resolvable and that all unresolvable statements are in fact not truthful?

"So, the sentence can be proven to be neither true nor false. Yet, let's consider if it were false. This would mean that it is provable in the system. Which implies that it is true. Which means that it is unprovable. This is a blatant contradiction. Therefore, the sentence is a.) true in a consistent system yet unprovably so, or b.) false in an inconsistent system.
In this way, it is unprovable yet able to be seen as true through informal methods."

I'd like to see how you'd reconcile this idea with my argument when you have the time. Do you think that if we add to truth my purposed "resolvability" and keep in mind that the statement is affirming a contradiction as true (therefore postulating that A = -A in essence) that we can say that the statment is not true with certainty? And would not this allow for consistancy if we made sure to consider this in all theorems?

Funny, though, that you should have mentioned "Godel, Escher, and Bach". I am currently reading that book for a course I am taking as I pursue my Ph.D. in philsophy. This is part (but not all) of the reason why I have put my mental efforts to considering this issue.

Thanks for the link, too. I shall read it.

 

Reading GEB was beautiful.
I will read it again soon.
Taking it in a class sounds amazing.

 

Cole Grey:

GEB is a pretty good read, yes. I am not particularly fond of the Achilles and the Tortoise dialogues, though, because I rather prefer Zeno's originals to the C.S. Lewis style, which I believe invalid.

 

What's GEB stand for?

 

If you remember "the taoist trap", I think it's fundamental to the issues at hand with tautology and such.

Hmm.

Let me make a few noob comments cuz I'm noobish like that.

The two sentences link to each other circularly.

a points to b which points to a, ad infinitum.

IMO, this is an interesting opportunity to introduce the notion of perspective.

As it is when A and B and independently viewed, they are valid.

In other words, if you maintain the perspective of A and stop there, it's valid.

Likewise with B.

But if you follow the circle, you formulate a conceptually geometric vortex of circularity, wherein the observer (while continuing to contemplate both statements rather than focusing on the perspective of either) is forced into the circle eternally.

This is the trap, and it's much more prevalent in human pyschology that ego generally allows to be considered.

Dang ole' fascinating.

I don't know what's wrong with me, but IMO, there is much to be learned about perspective from these type of conceptual oddities.

*shrug*

 

"Godel, Escher, and Bach"