What is the basis for reason? And mathematics?

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Logik
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More from the realm of Constructive mathematics.

Here is a consistent logical system in which both A = B AND A != B are true.

This is SUPPOSED to be a logical error?!?!?

https://repl.it/repls/FantasticTenseDividend

#ShatteringTheAristotelianDream
surreptitious57
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Joined: Fri Oct 25, 2013 6:09 am

As I have already said before this is only true for a computer programme / mathematical algorithm
It is not however a universal truth otherwise it would have disproven the Law Of Non Contradiction
Logik
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surreptitious57 wrote: Sun Feb 24, 2019 2:24 pm As I have already said before this is only true for a computer programme / mathematical algorithm
ONLY?

What other kind of logical system do you have in mind that is a super-set of Lambda calculus?

What can your system DO that this system can't?
surreptitious57 wrote: Sun Feb 24, 2019 2:24 pm It is not however a universal truth otherwise it would have disproven the Law Of Non Contradiction
Well what do you expect a "universal truth" to look like? Glitter, fireworks and rainbows?
Big bright LED signs that say "Universal Truth" ?

The contradiction of non-contradiction is right before your eyes and you can't recognize it for what it is.

It is apparently "not possible" for A = B and A != B to be true at the same time.
So how come I was able to write a computer program to demonstrate that it is, in fact possible?

If I have made an error, then surely you have an explanation for what you see?
If you can't find an error - surely you should believe your eyes?
surreptitious57
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Are you absolutely sure that Lamda calculus is the most rigorous logic system there is ?
Can it be improved upon or will it be replaced by more rigorous systems in the future ?

There are known unknowns and unknown unknowns so how does Lamda calculus address these ?
As it presumably cannot know / solve them does this not mean that it is not absolutely rigorous ?

What does it have to say for example about a solution for Godels Incompleteness Theorem or for Russells Set Paradox ?
Are these not examples of logical contradictions that cannot be solved by any logic no matter how rigorous it may be ?
Logik
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surreptitious57 wrote: Sun Feb 24, 2019 4:13 pm Are you absolutely sure that Lamda calculus is the most rigorous logic system there is ?
I didn't say rigorous. I said complete.
surreptitious57 wrote: Sun Feb 24, 2019 4:13 pm Can it be improved upon or will it be replaced by more rigorous systems in the future ?
I can't say. until we find problems with it.
surreptitious57 wrote: Sun Feb 24, 2019 4:13 pm There are known unknowns and unknown unknowns so how does Lamda calculus address these ?
By starting top-down rather than bottom up.

You start with an empty canvas then you populate the canvas with all the context/details/facts that are relevant to your "argument".
You are explicit rather than implicit. This way there's little room for (mis)interpretation.

And IF you find out that you left something out the first time - you correct-and-retry.
surreptitious57 wrote: Sun Feb 24, 2019 4:13 pm As it presumably cannot know / solve them does this not mean that it is not absolutely rigorous ?
It knows anything you can define. If you can't define it - well. It's stuck in your head until you can express it!

surreptitious57 wrote: Sun Feb 24, 2019 4:13 pm What does it have to say for example about a solution for Godels Incompleteness Theorem or for Russells Set Paradox ?
It solves them
surreptitious57 wrote: Sun Feb 24, 2019 4:13 pm Are these not examples of logical contradictions that cannot be solved by any logic no matter how rigorous it may be ?
No. All those discoveries were about a logical world made up entirely of numbers and sets. Different paradigm.
There are no such things as "Cats" in set theory. Or rather both "Humans" and "Cats" were expressed as sets.
Sets and sets of sets is ALL you have to express yourself. Which is fine for a mathematician, but the world is faaaaar more colourful than that!

And perhaps this is an important point to draw.

Lambda calculus (Type theory) is an alternative foundation for Mathematics to set-theory.
You don't have the problems of set theory because you aren't using set theory.

https://en.wikipedia.org/wiki/Type_theory
Speakpigeon
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Location: Paris, France, EU

Logik wrote: Sun Feb 24, 2019 2:11 pm More from the realm of Constructive mathematics.

Here is a consistent logical system in which both A = B AND A != B are true.

This is SUPPOSED to be a logical error?!?!?
It depends trivially on the formal system and how "A", "B", "true" and "=" are defined.
Here nothing has been defined so your post is shit.
The real issue is what it is such formal systems could possibly represent of the real world.
Definitely not human logic.
And calling something "logic" means shit.
Calling "logic" a formal system that doesn't give the same results as human logical reasoning is just a fraud.
EB
Logik
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Speakpigeon wrote: Sun Feb 24, 2019 6:50 pm It depends trivially on the formal system and how "A", "B", "true" and "=" are defined.
Well, not quite. What actually matters in formal systems is that "=" is INTERPRETED in EXACTLY the same way as you INTEND it to be interpreted.
Or to put it another way, what matters is that "=" is NOT misinterpreted.

How do you define and interpret "=" in classical logic? However you want to, I guess.
Scott Mayers wrote: Sun Feb 24, 2019 6:28 pm Here nothing has been defined so your post is shit.
That is a total fucking lie. Unlike Classical logic every single operator in Python is defined and documented!
And it is all PUBLIC KNOWLEDGE! Much unlike the contents of your mind.

You can examine the definition of ANY function/operator you like over here: https://github.com/python/cpython
And you can read the documentation over here: https://docs.python.org/3/

The cherry on top is that it's a RATIONAL language ( https://en.wikipedia.org/wiki/Regular_language ) .
You really need to read and understand Kleene's theorem: https://en.wikipedia.org/wiki/Kleene%27 ... on_theorem
Speakpigeon wrote: Sun Feb 24, 2019 6:50 pm The real issue is what it is such formal systems could possibly represent of the real world.
Definitely not human logic.
And calling something "logic" means shit.
Oh it sure is human logic. Like all logic - humans invented and perfected it!

You just haven't learned how to speak Mathematics.
Speakpigeon wrote: Sun Feb 24, 2019 6:50 pm Calling "logic" a formal system that doesn't give the same results as human logical reasoning is just a fraud.
EB

You DO know that another name for constructive mathematics is intuitionistic logic, right. RIGHT???
Python is a CONSTRUCTIVE logic and so it is far closer to the flexibility of human intuition than the rigid and archaic first-order logic.
Secondly - you can't model complex reality in first-order models! It is too complex! So you need a complex logic for the job.

Classical logic is a sledge hammer.
Constructive logic is a scalpel.

Not only does it give the same results as human logical reasoning, in some cases it performs even BETTER than humans!
That's why Artificial intelligence is a thing you know...

You know who the best chess player in the world is, right? It's this program: https://github.com/official-stockfish/Stockfish
Speakpigeon
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Location: Paris, France, EU

Logik wrote: Sun Feb 24, 2019 7:45 pm How do you define and interpret "=" in classical logic? However you want to, I guess.
Classical logic doesn't need to define equality because equality is not a logical connective.
That I need to explain this to you shows how ignorant of logic you are.
This is also why an argument using the symbol "=" has to contain premises giving the interpretation of it in logical terms. For example, you could have a premise saying something like (A = B) ⇔ (A ⇔ B). I'm sure you're going to misunderstand this but, still, that's what's needed.
The notion of equality is not used at all in logic because it's not a logical function and it's not a logical function because it's not a truth-functional function. So, all methods that use the notion of equality are mathematical theories, not methods of logic.
So, how is "=" defined in Python?
Remember, we need a logical definition, all based on logical connectives, the conjunction the disjunction etc.
Logik wrote: Sun Feb 24, 2019 7:45 pm That is a total fucking lie. Unlike Classical logic every single operator in Python is defined and documented!
Ok, good, so you'll have no difficulty producing the definition of the equal sign "=" in Python. Yes?
Logik wrote: Sun Feb 24, 2019 7:45 pm And it is all PUBLIC KNOWLEDGE! Much unlike the contents of your mind.
At least I'm here to explain and justify what I say. You, on the other hand, have proved yourself pathologically unable to explain anything or justify your position. All you can do is provide links that never have the information needed.
Logik wrote: Sun Feb 24, 2019 7:45 pm You can examine the definition of ANY function/operator you like over here: https://github.com/python/cpython
I couldn't find the definition of "=". It's not even mentioned.
Logik wrote: Sun Feb 24, 2019 7:45 pm And you can read the documentation over here: https://docs.python.org/3/
Yeah and if you read the Bible you'll find the proof God exists.
Logik wrote: Sun Feb 24, 2019 7:45 pm Oh it sure is human logic. Like all logic - humans invented and perfected it!
You just haven't learned how to speak Mathematics.
You DO know that another name for constructive mathematics is intuitionistic logic, right. RIGHT???
And it's not logic.It's mathematics. What's new here?
Logik wrote: Sun Feb 24, 2019 7:45 pm Python is a CONSTRUCTIVE logic and so it is far closer to the flexibility of human intuition than the rigid and archaic first-order logic.
And still no concrete example.
Logik wrote: Sun Feb 24, 2019 7:45 pm Secondly - you can't model complex reality in first-order models!
Logic was never thought of as a method to model reality. Boole called logic "the laws of thought".
You've just missed the boat.
Logik wrote: Sun Feb 24, 2019 7:45 pm Not only does it give the same results as human logical reasoning, in some cases it performs even BETTER than humans!
Well, again, no concrete example. Should be easy but no, you've never provided any concrete example of anything. I wonder why this is all in the abstract.
Logik wrote: Sun Feb 24, 2019 7:45 pm That's why Artificial intelligence is a thing you know...
You know who the best chess player in the world is, right? It's this program: https://github.com/official-stockfish/Stockfish
You know that there is an army of people in poor countries paid to "click" to teach those idiotic artificial intelligences!
You are an ignoramus and a fraud.
EB
Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Speakpigeon wrote: Mon Feb 25, 2019 6:47 pm
Logik wrote: Sun Feb 24, 2019 7:45 pm How do you define and interpret "=" in classical logic? However you want to, I guess.
Classical logic doesn't need to define equality because equality is not a logical connective.
That I need to explain this to you shows how ignorant of logic you are.
This is also why an argument using the symbol "=" has to contain premises giving the interpretation of it in logical terms. For example, you could have a premise saying something like (A = B) ⇔ (A ⇔ B). I'm sure you're going to misunderstand this but, still, that's what's needed.
The notion of equality is not used at all in logic because it's not a logical function and it's not a logical function because it's not a truth-functional function. So, all methods that use the notion of equality are mathematical theories, not methods of logic.
So, how is "=" defined in Python?
Remember, we need a logical definition, all based on logical connectives, the conjunction the disjunction etc.
Logik wrote: Sun Feb 24, 2019 7:45 pm That is a total fucking lie. Unlike Classical logic every single operator in Python is defined and documented!
Ok, good, so you'll have no difficulty producing the definition of the equal sign "=" in Python. Yes?
Logik wrote: Sun Feb 24, 2019 7:45 pm And it is all PUBLIC KNOWLEDGE! Much unlike the contents of your mind.
At least I'm here to explain and justify what I say. You, on the other hand, have proved yourself pathologically unable to explain anything or justify your position. All you can do is provide links that never have the information needed.
Logik wrote: Sun Feb 24, 2019 7:45 pm You can examine the definition of ANY function/operator you like over here: https://github.com/python/cpython
I couldn't find the definition of "=". It's not even mentioned.
Logik wrote: Sun Feb 24, 2019 7:45 pm And you can read the documentation over here: https://docs.python.org/3/
Yeah and if you read the Bible you'll find the proof God exists.
Logik wrote: Sun Feb 24, 2019 7:45 pm Oh it sure is human logic. Like all logic - humans invented and perfected it!
You just haven't learned how to speak Mathematics.
You DO know that another name for constructive mathematics is intuitionistic logic, right. RIGHT???
And it's not logic.It's mathematics. What's new here?
Logik wrote: Sun Feb 24, 2019 7:45 pm Python is a CONSTRUCTIVE logic and so it is far closer to the flexibility of human intuition than the rigid and archaic first-order logic.
And still no concrete example.
Logik wrote: Sun Feb 24, 2019 7:45 pm Secondly - you can't model complex reality in first-order models!
Logic was never thought of as a method to model reality. Boole called logic "the laws of thought".
You've just missed the boat.
Logik wrote: Sun Feb 24, 2019 7:45 pm Not only does it give the same results as human logical reasoning, in some cases it performs even BETTER than humans!
Well, again, no concrete example. Should be easy but no, you've never provided any concrete example of anything. I wonder why this is all in the abstract.
Logik wrote: Sun Feb 24, 2019 7:45 pm That's why Artificial intelligence is a thing you know...
You know who the best chess player in the world is, right? It's this program: https://github.com/official-stockfish/Stockfish
You know that there is an army of people in poor countries paid to "click" to teach those idiotic artificial intelligences!
You are an ignoramus and a fraud.
EB
Sophist you are missing the point. Like A LOT.

Not a little like A LOT.

Conjunction is this. Yes ? P ∧ Q

Now, Mr Sophist, you will forgive me if I am not going to take you through the history of computer science in one post, but observe that while conjunction is just ONE operation, through the constructive properties of Boolean primitives we can begin doing basic arithmetic:

https://en.wikipedia.org/wiki/Bitwise_o ... quivalents
https://en.wikipedia.org/wiki/Bitwise_o ... Bit_shifts

And so let me not bore you 60 years of computer science but through continuous construction and layering abstraction upon abstraction a modern-day processor has hundreds of instructions (many of which you probably haven't even heard of). https://en.wikipedia.org/wiki/X86_instruction_listings

I want to bring your particular attention to "CMP", "CMPSB" and "CMPSW" which are hardware-based instructions for COMPARING operands.

Now like any sophist trying to move the cookie jar to a shelf where I am not supposed to reach it. There are several layers of abstraction ABOVE the processor. You have the machine code, the Assembler layer, the operating system and only THEN comes the Python interpreter.

So forgive me if I am not going to translate a 13-steps removed abstraction into binary operators.
If you wish to climb the pyramid yourself (something which has taken me 10 years or so) you are welcome to.

And if you DON'T wish to climb the pyramid, it is sufficient for you to learn JUST the python basics. Because the Python language has been implemented in Python itself! https://bitbucket.org/pypy

Like I said, if you want to learn anything - you have to do the work yourself. I am not going to spoon-feed you.

We don't use logik to model reality. We use logic to model how we THINK about reality. And I can express the contents of my mind far more precisely and to the point wit ha high-order logic than you can in Boolean logic!

I can build far more precise model (maps!) of my own mind than you can.

You keep mistaking the concrete for the abstract. Working computer software is as concrete evidence as it gets.
Last edited by Logik on Mon Feb 25, 2019 7:34 pm, edited 1 time in total.
Speakpigeon
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Location: Paris, France, EU

Logik wrote: Mon Feb 25, 2019 7:27 pm Sophist you are missing the point. Like A LOT. Not a little like A LOT. Conjunction is this. Yes ? P ∧ Q Now, Mr Sophist, you will forgive me if I am not going to take you through the history of computer science in one post, but observe that while conjunction is just ONE operation, through the constructive properties of Boolean primitives we can begin doing basic arithmetic: https://en.wikipedia.org/wiki/Bitwise_o ... quivalents
https://en.wikipedia.org/wiki/Bitwise_o ... Bit_shifts
And so let me not bore you 60 years of computer science but through continuous construction and layering abstraction upon abstraction a modern-day processor has hundreds of instructions (many of which you probably haven't even heard of). https://en.wikipedia.org/wiki/X86_instruction_listings
I want to bring your particular attention to "CMP", "CMPSB" and "CMPSW" which are hardware-based instructions for COMPARING operands. Now like any sophist trying to move the cookie jar to a shelf where I am not supposed to reach it. There are several layers of abstraction ABOVE the processor. You have the machine code, the Assembler layer, the operating system and only THEN comes the Python interpreter. So forgive me if I am not going to translate a 13-steps removed abstraction into binary operators. If you wish to climb the pyramid yourself (something which has taken me 10 years or so) you are welcome to. And if you DON'T wish to climb the pyramid, it is sufficient for you to learn JUST the python basics. Because the Python language has been implemented in Python itself! https://bitbucket.org/pypy Like I said, if you want to learn anything - you have to do the work yourself. I am not going to spoon-feed you. Logic is not the laws of thought. Computer science is the laws of thought. But I can't convince you of this because you think you are right.You keep mistaking the concrete for the abstract. Working computer software is as concrete evidence as it gets.
In other words, it's not logic.
EB
Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Speakpigeon wrote: Mon Feb 25, 2019 7:32 pm In other words, it's not logic.
EB
What would convince you that you are wrong?
Atla
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Speakpigeon wrote: Mon Feb 25, 2019 6:47 pm So, how is "=" defined in Python?
Remember, we need a logical definition, all based on logical connectives, the conjunction the disjunction etc.
Apparently in Python you can for example redefine the equal sign any way you want. I edited two words in one of Logic's programs and I got this result:

A = B: Duck
A != B: Chicken

So now equality returns with 'Duck'. Guess that's why this moron insists on using Python.
Logik
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Atla wrote: Mon Feb 25, 2019 8:43 pm
Speakpigeon wrote: Mon Feb 25, 2019 6:47 pm So, how is "=" defined in Python?
Remember, we need a logical definition, all based on logical connectives, the conjunction the disjunction etc.
Apparently in Python you can for example redefine the equal sign any way you want. I edited two words in one of Logic's programs and I got this result:

A = B: Duck
A != B: Chicken

So now equality returns with 'Duck'. Guess that's why this moron insists on using Python.
Because that is precisely the point this "moron" is trying to make.

When you say "Jon is Human" and you answer "True". You expressed it using Set theory.
as John ∈ Human. GREAT! Please define what you mean by ∈. How does ∈ work exactly?

It means 'member of a set' you say.

So in order to make the statement "John is human." you HAD to answer the question "Is John a member of the set Human"?

Please explain to us how you determine whether something is or isn't a member of any particular set?
And when you ask that question you soon realize you NEED Iterators https://en.wikipedia.org/wiki/Iterator
And when I ask you what an iterator is you are going to give me ∀

And I am just going to play the define(X) game till you run out of logic symbols! And you will.

Because the one thing you fail to grasp is ALL logic operators can be modeled as Mathematical transfer functions.
And the one key thing about transfer functions is realizability ( https://en.wikipedia.org/wiki/Realization_(systems) )

And you know what realizability does? It grounds symbols!

When you realize an abstract Mathematical transfer function into a physical object you are turning the theoretical into practical.
It becomes a real, tangible thing that has causal effects on reality. And that's where things stop working like you've read in the books.

In theory there is no difference between theory and practice, but in practice there is...
Last edited by Logik on Tue Feb 26, 2019 5:51 pm, edited 1 time in total.
Eodnhoj7
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Logik wrote: Sun Feb 24, 2019 2:11 pm More from the realm of Constructive mathematics.

Here is a consistent logical system in which both A = B AND A != B are true.

This is SUPPOSED to be a logical error?!?!?

https://repl.it/repls/FantasticTenseDividend

#ShatteringTheAristotelianDream

Taking english and converting it to computer language, while admitting you have read and re-read these principles a million times, is beyond suspicious.

However I even argue, and the principle observes this, that the principle can be observed through multiple languages as all languages are merely axioms in and of themselves.
Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Eodnhoj7 wrote: Mon Feb 25, 2019 11:02 pm
Logik wrote: Sun Feb 24, 2019 2:11 pm More from the realm of Constructive mathematics.

Here is a consistent logical system in which both A = B AND A != B are true.

This is SUPPOSED to be a logical error?!?!?

https://repl.it/repls/FantasticTenseDividend

#ShatteringTheAristotelianDream

Taking english and converting it to computer language, while admitting you have read and re-read these principles a million times, is beyond suspicious.

However I even argue, and the principle observes this, that the principle can be observed through multiple languages as all languages are merely axioms in and of themselves.
And I already told you that you are welcome to take the work, publish it and claim it as your own. It is not even original - every computer scientist already knows this because this is 100 years old stuff. It's new to YOU because you come from a philosophy background.