The Liar's paradox

[Ahab]

"This sentence is true."

That looks to me like a form of nonsense. Just as "This sentence is false." is.

How could one determine the truth or falsity of either sentence?

 

[DBT]

What’s wrong with it not having a resolution? Logic is simply a way that we model reality and models aren’t perfectly accurate. This is just one instance where our logical models fails to accurately reflect reality.

A 'resolution' to the Liars Paradox seems to be impossible. But, who knows, maybe there is a Genius out there who has the brain power to tackle it.

 

[untermensche]

"This sentence is true."

That looks to me like a form of nonsense. Just as "This sentence is false." is.

How could one determine the truth or falsity of either sentence?

Absolutely correct.

No paradox.

Just nonsense.

A conclusion devoid of any context. Conclusions can only be made within a context.

Nonsense.

The stuff that fascinates small minds.

 

[fast]

What’s wrong with it not having a resolution? Logic is simply a way that we model reality and models aren’t perfectly accurate. This is just one instance where our logical models fails to accurately reflect reality.

A 'resolution' to the Liars Paradox seems to be impossible. But, who knows, maybe there is a Genius out there who has the brain power to tackle it.

Check out post #4

 

[Speakpigeon]

What’s wrong with it not having a resolution?

Ah, excellent. Thanks. Vox populi, vox dei.

So, no resolution.

Logic is simply a way that we model reality and models aren’t perfectly accurate. This is just one instance where our logical models fails to accurately reflect reality.

OK, so you think the sentence is unproblematic?

Would you say that is true?

Or do you mean by "logic" and "logical model" not mathematical logic but the way a human being reason logically?
EB

 

[Speakpigeon]

"This sentence is true."

That looks to me like a form of nonsense. Just as "This sentence is false." is.

How could one determine the truth or falsity of either sentence?

Sure, that why it's called a paradox to begin with. We can perfectly understand the sentence, including if we interpret it as referring to itself, namely that the sentence says of itself that it is false.

So, we understand the sentence and now you're saying it is nonsense. So, how come it's nonsense? We can understand it. It's not really nonsense. The problem seems to be we can't explain how it would possibly be true or false. Well, paradoxical...
EB

 

[steve_bank]

I take t paradox to mean a situation that can not be resolved logically and logic yields no solution.

Zeno's half step [redox.

 

[Speakpigeon]

I take t paradox to mean a situation that can not be resolved logically and logic yields no solution.

Zeno's half step [redox.

Wrong again.

Paradox
1. A statement that seems to contradict itself but may nonetheless be true: the paradox that standing is more tiring than walking.

This essentially means it can be solve logically.

Demonstration:

Proposed solutions
Diogenes the Cynic
According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see solvitur ambulando). To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

Aristotle
Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.[22][23] Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").[24] Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."[25]

Thomas Aquinas
Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."[26]

Archimedes
Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: Geometric series, 1/4 + 1/16 + 1/64 + 1/256 + · · ·, The Quadrature of the Parabola.) Modern calculus achieves the same result, using more rigorous methods (see convergent series, where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent). These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.[6][27]

Bertrand Russell
Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.[28][29]

Nick Huggett
Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest.[16]

Peter Lynds
Peter Lynds has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.[30][31][32] Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. For more about the inability to know both speed and location, see Heisenberg uncertainty principle.

Hermann Weyl
Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space, if not on time and space themselves. According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem".[33][34] According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.[6][35]

Can't you try to check a few facts before posting your inanities?!
EB

 

[untermensche]

Zeno's "paradoxes" merely show the absurdity of trying to apply a human invented concept (infinity) to a reality that does not include the concept.

Any attempt to apply infinity to anything real creates an absurd impossible situation.

 

[DBT]

What’s wrong with it not having a resolution? Logic is simply a way that we model reality and models aren’t perfectly accurate. This is just one instance where our logical models fails to accurately reflect reality.

A 'resolution' to the Liars Paradox seems to be impossible. But, who knows, maybe there is a Genius out there who has the brain power to tackle it.

Check out post #4

Maybe that is as close as it gets, who knows. EB didn't appear to understand the remark.

 

[Speakpigeon]

The liars paradox is neither true or false.

Easy to say. Please explain briefly why.

And do you think it should be possible to prove there is in fact no paradox.

This sentence is false. As the explanation of the paradox goes, if the sentence is false, then it is true since it says it is false. But if it is true, then it is false since it says it is false. Hence the paradox.

If you think this is a paradox, please explain briefly how you solve the paradox, if you think you do.

Second, if you think it is not a paradox, please explain briefly why.

Finally, do you think it should be possible to prove there is in fact no paradox.

Thank you to stick to the point and refrain from personal attacks.

EB

 

[Tom Sawyer]

What’s wrong with it not having a resolution?

Ah, excellent. Thanks. Vox populi, vox dei.

So, no resolution.

Logic is simply a way that we model reality and models aren’t perfectly accurate. This is just one instance where our logical models fails to accurately reflect reality.

OK, so you think the sentence is unproblematic?

Would you say that is true?

Or do you mean by "logic" and "logical model" not mathematical logic but the way a human being reason logically?
EB

I mean the actual logical model. Allow me to expand on what I was saying. Logic is designed to help us understand the way that the universe works and we create these models to allow us to apply rules to a fixed subset of data in order to see if the premises we're assuming about the universe reach valid conclusions which we can apply to the universe. This doesn't always work. One of the ways which it doesn't work is when you get a model which is internally consistent but can't actually be applied to anything outside of the model, such as "This sentence is false". That can basically be summed up as saying:

"A = true" = "A = false"

Internal to a model, that's actually fine, since it's a theoretical construct and you can define whatever you want within it. However, when the rules you set up for your theoretical constructs are different than the rules of actual reality, you can't apply any conclusions from that construct to actual reality.

It's no different than if you posit a theoretical universe where pi = 4. All the math you do within this construct you created is perfectly valid math and it all yields correct results. However, if you try to apply that construct to the real universe and stumble upon the confusing result that all the diameters of the circles you're measuring have different values than this perfectly internally consistent model you're testing, you haven't discovered some sort of paradox, you've discovered a situation where the rules of your model don't map to the rules of reality.

Similarly, the rule "A = true" = "A = false" isn't a rule with an equivalent in reality. Therefore, any logical model which uses it as a rule isn't one which can be applied to reality. It can be talked about hypothetically and you can build valid sentence structures off of the model, but those sentences are meaningless outside of the model since they cna't be applied to anything.

 

[untermensche]

It is less than [A = false]

It is ["the undefined" = false]

An absurdity on it's face.

Just because there is a way to say "the undefined" through omission using human language that is no reason not to clearly see it is what is being said.

 

[Tom Sawyer]

It is less than [A = false]

It is ["the undefined" = false]

An absurdity on it's face.

Just because there is a way to say "the undefined" through omission using human language that is no reason not to clearly see it is what is being said.

No, any statement has the implicit "It is true that ..." attached to it. So "Jack is a dog" is the equivalent of "It is true that Jack is a dog". So, the statement "This sentence is false" is similarly the equivalent of "It is true that this sentence is false", which works out to "A = true" = "A = false".

Of course it's absurd. That's just another way of saying that it can't be applied to reality. That is wholly different, however, from it not being internally consistent with the rules defined from the model.

 

[untermensche]

The phrase "This sentence" and nothing else is equivalent to saying "the undefined"

It is equivalent to handing somebody a brick and saying "This brick is true".

Nonsense.

 

[Speakpigeon]

I mean the actual logical model. Allow me to expand on what I was saying. Logic is designed to help us understand the way that the universe works and we create these models to allow us to apply rules to a fixed subset of data in order to see if the premises we're assuming about the universe reach valid conclusions which we can apply to the universe. This doesn't always work. One of the ways which it doesn't work is when you get a model which is internally consistent but can't actually be applied to anything outside of the model, such as "This sentence is false". That can basically be summed up as saying:

"A = true" = "A = false"

Internal to a model, that's actually fine, since it's a theoretical construct and you can define whatever you want within it. However, when the rules you set up for your theoretical constructs are different than the rules of actual reality, you can't apply any conclusions from that construct to actual reality.

It's no different than if you posit a theoretical universe where pi = 4. All the math you do within this construct you created is perfectly valid math and it all yields correct results. However, if you try to apply that construct to the real universe and stumble upon the confusing result that all the diameters of the circles you're measuring have different values than this perfectly internally consistent model you're testing, you haven't discovered some sort of paradox, you've discovered a situation where the rules of your model don't map to the rules of reality.

Similarly, the rule "A = true" = "A = false" isn't a rule with an equivalent in reality. Therefore, any logical model which uses it as a rule isn't one which can be applied to reality. It can be talked about hypothetically and you can build valid sentence structures off of the model, but those sentences are meaningless outside of the model since they cna't be applied to anything.

OK, sounds at least plausible, but then, what about the statement "It is true that the conjunction A and B implies B". We all say it's true. It's a rule, so what's the equivalent in reality?

How do we get to know that the conjunction A and B implies B?
EB

 

[Tom Sawyer]

OK, sounds at least plausible, but then, what about the statement "It is true that the conjunction A and B implies B". We all say it's true. It's a rule, so what's the equivalent in reality?

How do we get to know that the conjunction A and B implies B?
EB

I assume that you're asking if we do something more complex than plug values into A and B and see if the result is either absurd or not absurd? Because I don't know of anything more complex than that which is required.

A standard logical conjunction would be along the lines of "It is true that they got married and had a child and therefore this implies that they had a child". That's a non-absurd statement and therefore there's nothing to invalidate the rule as being something which has an equivalent in reality.

Similarly, you can plug values into "A = true" = "A = false" and get "This sentence is true is the same as this sentence is false". That's an absurd statement which invalidates the rule as something which has an equivalent in reality. That doesn't make it a paradox, it just makes it a model which can't be applied outside of itself.

 

[Speakpigeon]

I assume that you're asking if we do something more complex than plug values into A and B and see if the result is either absurd or not absurd? Because I don't know of anything more complex than that which is required.

A standard logical conjunction would be along the lines of "It is true that they got married and had a child and therefore this implies that they had a child". That's a non-absurd statement and therefore there's nothing to invalidate the rule as being something which has an equivalent in reality.

So, A and B implies B is true just because it's not absurd?!

How does being absurd equates to having no equivalent in reality?

Similarly, you can plug values into "A = true" = "A = false" and get "This sentence is true is the same as this sentence is false". That's an absurd statement which invalidates the rule as something which has an equivalent in reality. That doesn't make it a paradox, it just makes it a model which can't be applied outside of itself.

"That doesn't make it a paradox"?!

Paradox
A paradox involves two or more facts or qualities that seem to contradict each other.

Isn't that a description of the sentence "The sentence is false"?
EB

 

[Tom Sawyer]

"That doesn't make it a paradox"?!

For something to ACTUALLY be a paradox, it would need to exist in reality. Theoretical paradoxes are just word games. That's all that this is - a word game. For instance, you can talk about a married bachelor all you want but until you actually have someone who's both married and not married at the same time, no actual contradiction exists because there is no "thing" that the words refer to. You can talk about God creating a rock too big for him to lift all you want because until someone real waves his hand and makes this rock, no actual rules of reality are ever broken by it.

The phrase "this sentence is false" has no reference to anything outside of that one sentence, so there is no "thing" in the real world which actually holds contradictory properties.

 

[Speakpigeon]

"That doesn't make it a paradox"?!

For something to ACTUALLY be a paradox, it would need to exist in reality.

But that's not the definition of paradox.

Paradox
A paradox involves two or more facts or qualities that seem to contradict each other.

W. V. Quine (1962) distinguished between three classes of paradoxes:
A veridical paradox produces a result that appears absurd but is demonstrated to be true nonetheless.

A falsidical paradox establishes a result that not only appears false but actually is false, due to a fallacy in the demonstration.

A paradox that is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling–Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.
https://en.wikipedia.org/wiki/Paradox

You might as well re-define the notion of reality whilst you're at it.

Theoretical paradoxes are just word games. That's all that this is - a word game.

Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. "This sentence is false" is an example of the well-known liar paradox: it is a sentence which cannot be consistently interpreted as either true or false, because if it is known to be false, then it is known that it must be true, and if it is known to be true, then it is known that it must be false.
https://en.wikipedia.org/wiki/Paradox

For instance, you can talk about a married bachelor all you want but until you actually have someone who's both married and not married at the same time, no actual contradiction exists because there is no "thing" that the words refer to. You can talk about God creating a rock too big for him to lift all you want because until someone real waves his hand and makes this rock, no actual rules of reality are ever broken by it.

The phrase "this sentence is false" has no reference to anything outside of that one sentence, so there is no "thing" in the real world which actually holds contradictory properties.

A sentence is as real as anything else.

Suppose an evil man in the White House asks you, using Twitter, to prove the sentence "This sentence is false" either true or false under threat of the electric chair. What do you do? You try to argue it's not possible or you try resolve the paradox?
EB