What does paradox reveal about the nature of truth?

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A great challenge to philosophers and logicians has been in how to deal with paradox.

The classic case is the Liar's Paradox:

Quote:

This sentence is false.

If you presume the truth of the proposition, then it follows that it is actually false.
If you presume the proposition to be false, then it follows that it is actually true.
So what then IS the truth value of the proposition?

From about 600BC until the 19th century AD, this was able to be dismissed as perhaps just being an artifact or imperfection in human language. However a similar contradiction was discovered in Set Theory Mathematics by Bertrand Russell. In response attempts were made to modify mathematics to eliminate the possibility of self-reference. By 1931 however it was proven by Kurt Gödel that any system complex enough to be capable of at least arithmetic operations (simple number theory) either :
Must have true statements that are unprovably true,
or must be internally inconsistent (you can prove something to be both true and false).
The artifact or imperfection is not simply in human language, it is pervasive throughout formal logic.

What implications does this have for logical discourse?
What implications does this have for the nature of truth?

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[+3] - fresco - 02/09/2013You will find that w ehave discussed "truth" many times on this forum. Traditional logic (using binary truth values) is a [i]sub[/i]-aspect of general semantics, and in the sciences at...

@MattDavis,

u cant handle absolutes truth since u r meanin to b above it to get smthg from, that is why truth is always pointing the reverse being u, the else absolute meaning else absolute, cheap opportunist that fart first on what gives

why do u talk about truth when u mean it being false
is anyone or anything name is truth??? no, so why cant u b happy with everything existing which is never call true but rather choose to speak about what is never mentionned anywhere just to call it ****

do u think u can get away with ur means?? as if it wasnt clear that any can send u to hell

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@MattDavis,

You will find that w ehave discussed "truth" many times on this forum.

Traditional logic (using binary truth values) is a sub-aspect of general semantics, and in the sciences at least it has been partially displaced by the concept of mathematical coherence (e.g. in wave particle duality) which transcends the law of the excluded middle. It follows that that "truth"(small t) is a lay concept with "point of view" or "culpability overtones", "Truth" (capital T) is an absolutist religious claim, and possibly all we can say about truth is that it is an expression of "contextual social agreement as to what is the case".
To say more than this is to advocate a form of naive realism, involving observer-independent "things" in permanent "sets".

3 Replies

@fresco,

Wow! Thanks for the reply. Very Happy

That's a mouthful, and a head-full (at least for me). Wink

It will take me a bit to process each point.

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@MattDavis,

To me at least the paradox arises from definitively a stated assertion of an entity's falseness using a channel, declarative written out sentence in this case, which implicitly asserts its truth. Some see this as a never ending circular and mutually exclusive and exhaustive loop of true and false, but the problem in the statement itself is not circular but exist at the same time, in the same space.

And Kurt Gödel's Incompleteness Theorem reminds of a riddle I know. See if you can find the solution to the riddle.

This riddle have clearly stated facts: Three pal's register to a hotel and the desk clerk bills them $60 for the room, payable in advance. So, each man pays the clerk $20 and go to their room. Some minutes later, the clerk becomes aware of his blunder of overcharging the group by$5. So he tells one of the hotel attendant to return $5 to the 3 friends who checked in a moment ago. Seeing an opportunity to make $2, he keeps the $2 to himself thinking that the three friends would have some trouble evenly dividing $5 between themselves, and then decides to tell them that the clerk made an error of overcharging $3, giving a dollar back to each of the the three men. And the attendant goes home for the day, with the extra $2 stolen from the exchange. Now, each of the three friends gets a dollar back, thus they each paid $19 for the room which is a total of $57 for the night. We know the attendant pocketed $2 and adding that to the $57 leaves you at $59, and not $60 which was originally spent. Where did the one dollar disappear to?

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@fresco,

Interesting that you brought up observer-dependence/ observer independence. QM seems to literally find a phenomenon that mocks science with the paradox of the double slip experiment. The act of measurement (which some call observing) collapse the wave function. But then there is the Heisenburg Uncertainty Principle that says that regardless of our measurements both the exact position and velocity of a particle cannot be known at the same time. This is a property of nature regardless of our measuring devices. It disproves naive assumptions about what the universe at the quantum level is really like, and weirdly to put it, it is related to the discrete (vs continuous), a mathematical concept, but relatable to each other in the psychological perpetual torment of comprehending the paradox and of thinking how can it be like that?

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@Bennet,

Perhaps the phrase "mocks science" suggests a misunderstanding about what "science" is about. Contrary to the popular view, it is not about establishing "truth" per se, but about testing and modifying paradigms for "prediction and control". As Richard Feynman pointed out, nobody understands QM, and by that he meant that it was counter-intuitive or "illogical". It is nevertheless spectacularly successful in its predictive role, which raises the issue of what constitutes "scientific explanation". It would seem that mathematical elegance rather than binary logic is a major factor in that.

As for non-duality (observer-observed interdependency), its role in modern science has been examined by such authors as F. Capra (The Tao of Physics) and the cognitve scientist F. Varela (articles on "Embodied Cognition") .

For an interesting paper involving some of these issues I suggest...
http://www.fdavidpeat.com/bibliography/essays/nat-cog.htm

1 Reply

Fresco wrote:

Quote:

It follows that that "truth"(small t) is a lay concept with "point of view" or "culpability overtones", "Truth" (capital T) is an absolutist religious claim, and possibly all we can say about truth is that it is an expression of "contextual social agreement as to what is the case".

I consider that blather.

TRUTH (with all capital letters) is an absolute...or at least, there is no way language can express that it is not.

To suggest that truth "...is that it is an expression of "contextual social agreement as to what is the case" is to assert that there IS NO OBJECTIVE TRUTH.

That simply cannot be the case.

Whatever actually IS...IS.

The fact that Fresco's religion is not willing to accept that...and in fact denies it out of hand, notwithstanding.

And your paradox example does not impact on that, Matt.

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@fresco,

If you want my proper context, I was relating to the analogy of how Gödel's incompleteness theorem applied to physics or for that matter any sufficiently complex logical systems, and that is by reducing it to down the base level of formal systems and run afoul of it. In sufficient frameworks, there will always be statements which can be constructed which you can't tell are true or false using just the laws of the defined system. The ridicule aspect is the lack of success in certain area which can be identified with an inconsistency in the mathematical model as formulated as it is. Not only that, can the math we use model all true facts about physics? Well of course not, and we know this without Godel. I think that being able to self-referentially talk about truth implies inconsistency. (As usual, only for the appropriate class of theories). Is it meaningful to talk about a physical statement being a prediction of a given model if it is absolutely unmeasurable? I would contend that it isn't and therefore, as such statements don't count as fruition of modifying paradigms to the ultimate level of completeness, and Gödel's incompleteness theorems still applies any theory of physics would, just as any logical systems, contain undecidable statements. To think as such is self defeating, definitely, or what's the expression: seeing the cup half empty.
In some sense, Gödel's Incomplete theorems seems intuitively obvious.

2 Replies

@Bennet,

Bennet wrote:

To me at least the paradox arises from definitively a stated assertion of an entity's falseness using a channel, declarative written out sentence in this case, which implicitly asserts its truth. Some see this as a never ending circular and mutually exclusive and exhaustive loop of true and false, but the problem in the statement itself is not circular but exist at the same time, in the same space.

Are you suggesting there may be in error in assuming that something's truth value is persistent (through "time")?
In the Liar's paradox, does the proposition alternate between between true and false in succession as it is evaluated (having no persistent truth value)?
Or are you saying that there is an error in assuming simultaneity?
If we reject simultaneity, I think that their are going to be too few logical operations left to derive anything meaningful. We can no longer use any transitive operations. I can no longer assert that if "A" is true and "B" is true, then "A and B" are true, because I have no right to presume that they are true simultaneously. All propositions then become logically isolated.
Or are you saying that the Liar's Paradox is simply an artifact of language not necessarily reflected in pure logic?
Pure logic (in the form of number theory) has an analogous paradox as proven by Gödel.

Bennet wrote:

And Kurt Gödel's Incompleteness Theorem reminds of a riddle I know. See if you can find the solution to the riddle.

This riddle have clearly stated facts: Three pal's register to a hotel and the desk clerk bills them $60 for the room, payable in advance. So, each man pays the clerk $20 and go to their room. Some minutes later, the clerk becomes aware of his blunder of overcharging the group by$5. So he tells one of the hotel attendant to return $5 to the 3 friends who checked in a moment ago. Seeing an opportunity to make $2, he keeps the $2 to himself thinking that the three friends would have some trouble evenly dividing $5 between themselves, and then decides to tell them that the clerk made an error of overcharging $3, giving a dollar back to each of the the three men. And the attendant goes home for the day, with the extra $2 stolen from the exchange. Now, each of the three friends gets a dollar back, thus they each paid $19 for the room which is a total of $57 for the night. We know the attendant pocketed $2 and adding that to the $57 leaves you at $59, and not $60 which was originally spent. Where did the one dollar disappear to?

The error is in representing the $57 paid as a positive number while also representing the $2 stolen as a positive number (the assertion to "We know the attendant pocketed $2 and adding that to the $57 leaves you at $59").
Here is a table representing the situation:
Guests Desk Clerk
-----------------------------------------
Starting | 60 0 0 |
Paid for Room | 0 60 0 |
Rebate to clerk | 0 55 5 |
Guests shorted | 3 55 2 |
-----------------------------------------
Net Change -57 +55 +2

What is finally being asked of us is to know is an accounting of where all the money is. The "Guests Shorted" row shows this information.
The attempt is to mislead us into using the "Net Change" information to derive this accounting, which is possible only so long is care is taken to pay attention to the (+/-) signs of this information.
[-57 + 2 = -55] (net change for guest and clerk combined)
[+55] (net change for desk)
[-57 + 2 + 55 = 0] (net change for guest and clerk and desk combined)

1 Reply

@Bennet,

Thanks Bennet!
Your latest response is very much along the lines of what I was trying to explore with my question.

Bennet wrote:

... and Gödel's incompleteness theorems still applies any theory of physics would, just as any logical systems, contain undecidable statements. ......
In some sense, Gödel's Incomplete theorems seems intuitively obvious.

I wonder if some of the "strangeness at the bottom" effects seen in quantum mechanics, could have been predicted as purely logical consequences of any sufficiently complex axiomatic system.
Similarities exist between how paradoxes arise in artificial axiomatic systems (like number theory, like the MU system, like set theory) and how paradoxes manifest themselves in our understanding of physics. Is this evidence that our reality is an axiomatic system?

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@fresco,

fresco wrote:

Traditional logic (using binary truth values) is a sub-aspect of general semantics, and in the sciences at least it has been partially displaced by the concept of mathematical coherence (e.g. in wave particle duality) which transcends the law of the excluded middle. It follows that that "truth"(small t) is a lay concept with "point of view" or "culpability overtones", "Truth" (capital T) is an absolutist religious claim, and possibly all we can say about truth is that it is an expression of "contextual social agreement as to what is the case".
To say more than this is to advocate a form of naive realism, involving observer-independent "things" in permanent "sets".

Still trying to parse most of this.
I think it may assumes/requires a familiarity with terms that I do not have. Sad

I think you are saying that simple binary logic (only true or only false), is not an adequate method of understanding (at least physical) reality.
I wasn't intending for an extension to physical reality in my OP, but I do appreciate your inclusion of it. Smile

As for 'truth' (small t) requiring a reference frame ( "point of view").
In the case of the Godel Incompleteness Theorem, that reference frame is from within the axiomatic system.

If we confine this discussion to properly strict axiomatic systems (maybe excluding physical reality as not being in this class), can we then demonstrate (per Godel's Theorem) the need for non-binary logic, even within a binary logical system?

1 Reply

@MattDavis,

MattDavis wrote:

Are you suggesting there may be in error in assuming that something's truth value is persistent (through "time")?

No, I never brought up the discussion of the expiration date of a true statement. That's a whole different topic, a tangent we can discuss later if you want. And I was talking about paradoxes and how opposites exists at the same time, not through time or in alternative manners. This may be redundant but for the sake of limiting confusion let's take the Liar's paradox for example: the idea that if its telling the truth then its lying, and if its lying then its telling the truth, exists at the same time, making it a paradox. Now in trying to understand the paradox, one can isolate the ideas and evaluate separately these two statements and compare alternatively with each other in ultimately trying to find a possible solution.

MattDavis wrote:

Or are you saying that there is an error in assuming simultaneity?

MattDavis wrote:

Or are you saying that the Liar's Paradox is simply an artifact of language not necessarily reflected in pure logic?

Yes, it is simply an artifact of language, if it were reflected in pure logic, then we could formalize the liar's paradox. But we can't, now can we?
There is no formula P (x ) which defines PA. That is, there is no
formula P (x ) such that for every formula ρ:
P(#ρ) ↔ ρ

Regarding the puzzle: There is an initial $60 charge. It should have been $55, so $5 must be returned and accounted for. $3 is given to the 3 friends, $2 is kept by the attendant: there you have the $5. The trick to this riddle is that the addition and subtraction are done at the wrong times to misdirect your thinking - and quite successfully fools many for the most part. Each of the 3 friends did indeed pay $19, not $20, and as far as the friends are concerned, they paid $57 for the night. But we know that the clerk will tell us that they were charged only $55 and when you add the $3 returned with the $2 kept by the bellhop, you come up with $60.

From the puzzle I was trying to make an analogy that it is not an exact scientific system, or theorems, or mathematical equation out there that tells the complete truth. To attain complete absolute truth is impossible and you can only get pieces of ideas at a time and try to put it together. As mice we are just following the bread crumbs in an endless journey.

1 Reply

@Bennet,

Bennet wrote:

...if it were reflected in pure logic, then we could formalize the liar's paradox. But we can't, now can we?

Well it has been formalized:
For any sufficiently complex axiomatic system that is internally consistent (number theory for example),
it has been proven that there exist propositions (within the system) that are both true and unprovable (within the system). It has also been proven that there exist propositions (within the system) that are both false and unprovable (within the system). Since their are unprovable propositions within the system, and an unprovable proposition could be either true or false, when the system attempts to evaluate such a proposition there is no way for it to do so. From the the system's perspective such propositions have no truth value.

This basically amounts to proving that within formal logic there exist propositions that have a truth value that is undecidable within formal logic. While at the same time proving that those propositions actually have a truth value (true or false), but to evaluate their truth value can only be done by "stepping outside" of the system.

One might then simply suggest having as your logical system (formal logic + whatever needed to step outside of the system). This successfully handles the paradoxes within formal logic, but now within this larger system there will be a different set of paradoxes. To resolve these new paradoxes we have to then step outside of (formal logic + additional system) to have (formal logic + additional system A + additional system B).... on and on to an infinite regress.
Creating an axiomatic system with an infinite number of axioms in order to handle all cases of paradox.

1 Reply

@MattDavis,

http://www.utm.edu/research/iep-wp/wp-content/media/tarski.jpg

Click the image.
"Tarski has proved that truth is not definable in a classical language–thus the name “Undefinability Theorem.” Tarski’s theorem establishes that classically interpreted languages capable of expressing arithmetic cannot contain a global truth predicate. A language containing its own global truth predicate is said to be semantically closed, so his theorem implies that classical formal languages with the power to express arithmetic cannot be semantically closed, and consequently cannot have a Liar sentence."

I should have been more clear when I said it hasn't been formalized.
According to the opinion of many philosophers the shorshortcomings existing in logical paradoxes, could be the problem with formal logic or problem with our language (may be something else?). For these reasons the solution of logical paradoxes was transferred from the frame of Formal Logic to the frame of mathematical logic (for example by Russell) and to the frame of classical formal languages (for example by Starski).

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@Bennet,

I agree that Godel's incompleteness theorem is intuitively obvious (e.g. from a consideration of the infinite regress of words defining words , etc). However we need to distinguish between "truth" and "validity". The first involves metaphysical issues about ontology (existence) and epistemology (knowledge). The second involves formal consistency of argument from axioms. Godel's incompleteness theorem is applicable to "validity"rather than "truth", despite the fact that we tend to use second order language (...it is true that argument A is valid...) which tends to blur the boundaries.

In general I tend towards Richard Rorty's pragmatism regarding the word "truth"...
http://www.youtube.com/watch?v=CzynRPP9XkY

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@Bennet,

Bennet wrote:

Click the image.
"Tarski has proved that truth is not definable in a classical language–thus the name “Undefinability Theorem.” Tarski’s theorem establishes that classically interpreted languages capable of expressing arithmetic cannot contain a global truth predicate. A language containing its own global truth predicate is said to be semantically closed, so his theorem implies that classical formal languages with the power to express arithmetic cannot be semantically closed, and consequently cannot have a Liar sentence."

Thanks Bennet :^)
the article at that link is great
It lead me to a related article at that resource about "inconsistent mathematics" which is fascinating.

1 Reply

@MattDavis,

Does your "inconsistent mathematics" paper deal with Paul Cohen's "proofs" that there both IS and IS NOT an infinite set lying between between Cantor's first two infinities? I seem to remember he received the Field's Medal for this, and that Godel was one of the referees.

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@MattDavis,

Quote:

Quote:
Re: fresco (Post 5248155)
fresco wrote:

Traditional logic (using binary truth values) is a sub-aspect of general semantics, and in the sciences at least it has been partially displaced by the concept of mathematical coherence (e.g. in wave particle duality) which transcends the law of the excluded middle. It follows that that "truth"(small t) is a lay concept with "point of view" or "culpability overtones", "Truth" (capital T) is an absolutist religious claim, and possibly all we can say about truth is that it is an expression of "contextual social agreement as to what is the case".
To say more than this is to advocate a form of naive realism, involving observer-independent "things" in permanent "sets".

Still trying to parse most of this.
I think it may assumes/requires a familiarity with terms that I do not have.

There is no way to say what I have to say here without seeming petty and disrespectful to Fresco in this comment. I truly do not mean it that way...but the only way to communicate what I have to say is to be direct.

Don't bother, Matt. The comment was not posted to impart information or to actually deal with the issue at hand. It was posted, as Fresco does and has done for over a decade, to write the way some stupid people think smart people write. It was an attempt to say, "See how smart I am."

You are not stupid, Matt. Don't fool for a device meant to fool the foolish and stupid people.

If Fresco actually wanted to say something reasonable, he seems capable of doing so.

Apparently he doesn't because he is more interested in "See how smart I am" than in "This can help."

2 Replies

@Frank Apisa,

What's the matter with you today Frank. Has the snow ruined your golf fix ?

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What does paradox reveal about the nature of truth?

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