The Liar's paradox

What is the basis for reason? And mathematics?

Moderators: AMod, iMod

User avatar
Speakpigeon
Posts: 963
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

The Liar's paradox

Post by Speakpigeon » Wed Feb 06, 2019 3:39 pm

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

User avatar
Eodnhoj7
Posts: 4982
Joined: Mon Mar 13, 2017 3:18 am

Re: The Liar's paradox

Post by Eodnhoj7 » Wed Feb 06, 2019 6:53 pm

Speakpigeon wrote:
Wed Feb 06, 2019 3:39 pm
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
The answer is that the sentence is true/false/neutral all at the same time.

Paradox leads to never ending paradox, thus necessitating all paradoxes as the meaning to further paradoxes as a solution in themselves.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: The Liar's paradox

Post by Logik » Thu Feb 07, 2019 7:38 am

Bertrand Russel himself agreed that the problem with the paradox is self-reference.
Tarski's undefindability theorem tells us that truth-predicate satisfying convention-T for the sentences of a given language cannot be defined within that language. The problem cannot be resolved without a meta-language.

Meta-languages lead to recursion and Chomsky ultimately formalised the problem: https://en.wikipedia.org/wiki/Chomsky_hierarchy

Self-reference is recursion. Recursion is computation. T

In the computational realm we default to Type Theory so the sentence "This sentence is false" is a proposition. Lets call it X.

Every proposition can be determined as decidable, undecidable or unknown (Turing's halting problem).

For you to determine on the truth-value of X first you need to define the meta-language's grammar/semantics in which you evaluate the truth-value of propositions.

If you want an intro to type theory see this: https://www.heidelberg-laureate-forum.o ... TkzR_qDckM

surreptitious57
Posts: 3517
Joined: Fri Oct 25, 2013 6:09 am

Re: The Liar's paradox

Post by surreptitious57 » Thu Feb 07, 2019 7:59 am

The sentence is neither true or false because while it is gramatically correct it makes no sense logically
The reason is because it lacks a referent which would determine what it was true or false in relation to

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: The Liar's paradox

Post by Logik » Thu Feb 07, 2019 8:01 am

surreptitious57 wrote:
Thu Feb 07, 2019 7:59 am
The sentence ... is gramatically correct
In which grammar?

surreptitious57
Posts: 3517
Joined: Fri Oct 25, 2013 6:09 am

Re: The Liar's paradox

Post by surreptitious57 » Thu Feb 07, 2019 8:18 am


Despite having no referent it is still true according to the rules of descriptive grammar

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: The Liar's paradox

Post by Logik » Thu Feb 07, 2019 11:41 am

surreptitious57 wrote:
Thu Feb 07, 2019 8:18 am

Despite having no referent it is still true according to the rules of descriptive grammar
I am not sure how to parse this.

Rules describe the grammar of any particular language.
We say that the set of rules R determine the grammar for language L.
Or maybe even language L is determined by the rules outlined in R.

The rules are not prescriptive, but it begs a question. If you aren't adhering to the rules in R are you speaking language L?

Second (back to Tarski's undefinability theorem) a self-descriptive grammar is a paradoxical notion.

In order to be able to describe, then you are necessarily being prescriptive about semantics.

Example: The sky is green.

Grammatically valid. Semantically false.
Last edited by Logik on Thu Feb 07, 2019 1:16 pm, edited 4 times in total.

User avatar
Speakpigeon
Posts: 963
Joined: Sat Nov 11, 2017 3:20 pm
Location: Paris, France, EU

Re: The Liar's paradox

Post by Speakpigeon » Thu Feb 07, 2019 12:42 pm

surreptitious57 wrote:
Thu Feb 07, 2019 7:59 am
The sentence is neither true or false because while it is gramatically correct it makes no sense logically
OK, sounds right to me.
surreptitious57 wrote:
Thu Feb 07, 2019 7:59 am
The reason is because it lacks a referent which would determine what it was true or false in relation to
As I understand the idea of the paradox, although the grammar of it isn't enough of itself to suggest as much, the sentence is referring to itself. If so, it doesn't lack a referent.
EB

surreptitious57
Posts: 3517
Joined: Fri Oct 25, 2013 6:09 am

Re: The Liar's paradox

Post by surreptitious57 » Thu Feb 07, 2019 2:51 pm

Logic wrote:
Rules describe the grammar of any particular language
Are unsound syllogisms grammatically wrong because they are factually wrong as well
No they are not as the validity of a sentence does not extend to how true or false it is
So the sky is green is a grammatically correct sentence even though it is actually false

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: The Liar's paradox

Post by Logik » Thu Feb 07, 2019 3:10 pm

surreptitious57 wrote:
Thu Feb 07, 2019 2:51 pm
Are unsound syllogisms grammatically wrong because they are factually wrong as well
We've missed each other.

The 'Sky is green' is not factually wrong. It's semantically 'wrong'.
I mean the same thing when I say 'The sky is green' as you mean when you say 'The sky is blue'.

So re-writing your question: Are unsound syllogisms grammatically wrong because they are semantically wrong as well?

And you will find the answer in the definition of decidability.

https://en.wikipedia.org/wiki/Decidability_(logic)

Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines the notion of logical validity.

Semantics determine validity.
Grammar determines provability e.g soundness.

You can then ask the question: Can an invalid argument be sound?

Yes! The sky is green.

Classical logicians got EVERYTHING backwards.
Last edited by Logik on Thu Feb 07, 2019 3:12 pm, edited 1 time in total.

Walker
Posts: 6845
Joined: Thu Nov 05, 2015 12:00 am

Re: The Liar's paradox

Post by Walker » Thu Feb 07, 2019 3:11 pm

surreptitious57 wrote:
Thu Feb 07, 2019 2:51 pm
So the sky is green is a grammatically correct sentence even though it is actually false
A green sky is false to both perception and inference.

Sky is Sky
Blue is Blue
Sky is not Blue

After eliminating the inference of equality that tags along with “to be,” playing golf under a blue sky means that when I look up I see blue, when I look down I see green, when I look all around I see an organized kaleidoscope of kolor that familiarity arranges into order.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: The Liar's paradox

Post by Logik » Thu Feb 07, 2019 3:12 pm

Walker wrote:
Thu Feb 07, 2019 3:11 pm
surreptitious57 wrote:
Thu Feb 07, 2019 2:51 pm
So the sky is green is a grammatically correct sentence even though it is actually false
A green sky is false to both perception and inference.

Sky is Sky
Blue is Blue
Sky is not Blue

After eliminating the inference of equality that tags along with “to be,” playing golf under a blue sky means that when I look up I see blue, when I look down I see green, when I look all around I see an organized kaleidoscope of kolor that familiarity arranges into order.
Bullshit.

That which you call 'blue' I have DECIDED to call 'green'.

Semantic difference.
Last edited by Logik on Thu Feb 07, 2019 3:13 pm, edited 1 time in total.

Walker
Posts: 6845
Joined: Thu Nov 05, 2015 12:00 am

Re: The Liar's paradox

Post by Walker » Thu Feb 07, 2019 3:13 pm

different word is different word

Furthermore, x is x.
Last edited by Walker on Thu Feb 07, 2019 3:14 pm, edited 1 time in total.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: The Liar's paradox

Post by Logik » Thu Feb 07, 2019 3:13 pm

Walker wrote:
Thu Feb 07, 2019 3:13 pm
different word is different word
Same meaning is same meaning. It's called a synonym.
Walker wrote:
Thu Feb 07, 2019 3:13 pm
Furthermore, x is x.
That's not a grammatically coorrect proposition in type theory.
Last edited by Logik on Thu Feb 07, 2019 3:17 pm, edited 2 times in total.

Walker
Posts: 6845
Joined: Thu Nov 05, 2015 12:00 am

Re: The Liar's paradox

Post by Walker » Thu Feb 07, 2019 3:15 pm

x is x

y is y

x is not y

Post Reply

Who is online

Users browsing this forum: No registered users and 1 guest