What is the basis for reason? And mathematics?

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surreptitious57
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Logic wrote:
Do you know what an error is ?
An inconsistency in the system such as false / incomplete / contradictory information
So for a system to be error free it must not contain any of these faults or any others
Logik
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surreptitious57 wrote: Sun Mar 17, 2019 12:06 pm An inconsistency in the system such as false / incomplete / contradictory information
Why are inconsistencies errors?
surreptitious57 wrote: Sun Mar 17, 2019 12:06 pm So for a system to be error free it must not contain any of these faults or any others
Translation of the above without using the word "error".

A logical system must be free from contradictions, incompleteness and falsities.

1. Why a logical system OUGHT to be that way?
2. How do you state the law of non-contradiction without using falsities?

Go ahead and rewrite this in a way that doesn't contain falsities: P ∧ ¬P ⇔ False
surreptitious57
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Logic wrote:
Why are inconsistencies errors ?
Logic has to be sufficiently rigorous for it not to be inconsistent or contradictory
Any system that has these errors in it to any significant degree cannot be logical
Logik
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surreptitious57 wrote: Sun Mar 17, 2019 12:21 pm Logic has to be sufficiently rigorous for it not to be inconsistent or contradictory

In what framework (if not logic) are the rules one needs to adhere to decided in ?
surreptitious57 wrote: Sun Mar 17, 2019 12:21 pm Any system that has these errors in it to any significant degree cannot be logical
What's "logical" is true by definition. It's tautological.
In what framework have you decided on the "correct definition" for logic?
surreptitious57
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Joined: Fri Oct 25, 2013 6:09 am

Logic wrote:
In what framework have you decided on the correct definition for logic ?
In a syllogism each premise must be logically consistent with the previous one
In a proof each calculation must be logically consistent with the previous one

Logic therefore is about the relationship between different parts of the framework / system
There must be sufficient rigour / consistency within it or else it is rendered invalid / unsound
Logik
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surreptitious57 wrote: Sun Mar 17, 2019 1:14 pm
Logic wrote:
In what framework have you decided on the correct definition for logic ?
In a syllogism each premise must be logically consistent with the previous one
In a proof each calculation must be logically consistent with the previous one

Logic therefore is about the relationship between different parts of the framework / system
There must be sufficient rigour / consistency within it or else it is rendered invalid / unsound
You are not answering the question. You are laying down rules.

I am asking you how you came to produce these rules, and why you CHOSE those particular rules.

You could've just as well chosen "in a syllogism each premise must notbe logically consistent with the previous one", "In a proof each calculation must not be logically consistent with the previous one".
surreptitious57
Posts: 4217
Joined: Fri Oct 25, 2013 6:09 am

Logic wrote:
You could have just as well chosen
in a syllogism each premise must not be logically consistent with the previous one
In a proof each calculation must not be logically consistent with the previous one
Those rules would be completely useless in practice because they are too arbitrary
A logical system has to be sufficiently rigorous in order to produce effective results
Logik
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surreptitious57 wrote: Sun Mar 17, 2019 2:05 pm Those rules would be completely useless in practice because they are too arbitrary
A logical system has to be sufficiently rigorous in order to produce effective results
I think you have erroneously conflated consistency and rigour with utility.

I have SHOWN you a computational logic where neither the laws of identty or non-contradiction apply.

Are you telling me computers aen't useful because they are inconsistent?

I think you are over-stating the utility of the Aristotelian axioms
surreptitious57
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Logic wrote:
I have SHOWN you a computational logic where neither the laws of identity or non contradiction apply
This is only true for certain examples as these laws still apply so are useful for that reason
They will only become redundant when they have absolutely no practical application at all
Logik
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surreptitious57 wrote: Sun Mar 17, 2019 2:31 pm This is only true for certain examples as these laws still apply so are useful for that reason
They will only become redundant when they have absolutely no practical application at all
OK. They have no practical application for me. If you find them useful - use them.

You are, however making the same error as every theist.

Between the option of prayer and hospital - you choose prayer. Because it "works for you".

I am showing you a tool that works better. Use it, don't use it.
Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Averroes wrote: Sun Mar 17, 2019 10:08 am There is no logic without the law of non-contradiction.
False. Lambda calculus. Turing-completeness. Logics without contradictions.

The True Intuitionistic logic

Averroes wrote: Sun Mar 17, 2019 10:08 am Yes, indeed, you made a logical error.
No I haven't. Proofs compute (Curry-Howard).

That which you call a "logical error" is just a false positive on your part. Your classifier is broken.
Averroes wrote: Sun Mar 17, 2019 10:08 am First of all, you have redefined equality and your redefinition of equality differs from both the mathematical definition and from common understanding as well.
I haven't redefined anything. Equality does not have a clear definition beyond the real numbers.
What does it mean for a cat to be equal to a cat?

You are simply extending set theory beyond its domain of applicability.
Averroes wrote: Sun Mar 17, 2019 10:08 am These are clearly contradictory statements!
The objective standard for mathematical proofs (Curry-Howard) suggests that you are clearly mistaken.

Averroes wrote: Sun Mar 17, 2019 10:08 am But more importantly in this case, for example, some humans are murderers and some humans are not murderers. Someone who has killed thousands of defenseless women and children is clearly not equal to someone who has not killed any human being. A murderer is clearly not equal to an innocent human being. But according to the redefinition of equality given in the program, a murderer is equal to an innocent human being! This is absolutely unacceptable from both a legal and a moral point of view.
Bullshit.

You fail to draw a distinction between the abstract class "Human" (a mere category) and a Human-instance e.g any particular human

Let me teach you a thing or a million about decision/complexity theory.
Because "murderer" is a value-judgment (for lack of a better term) you need system to answer the question: Is Person-X a murderer?
Now, in a million years we may well have AI sophisticated enough to answer that question all on its own.

in 2019 we use our legal system to do this. Effectively we have an oracle machine: https://en.wikipedia.org/wiki/Oracle_machine
And so "murderer" is simply a property of any individual human assigned by the legal system. Like this: https://repl.it/repls/IdleCalmCosmos

The word "equal" means NOTHING. Because murderers still have equal rights.
Averroes wrote: Sun Mar 17, 2019 10:08 am In set theory:
Who cares about set theory? Curry-Howard deals with type theory.
Averroes wrote: Sun Mar 17, 2019 10:08 am While the redefinition expressed in the Python codes given in the OP can be expressed as follows:
Given any x and y, x=y if and only if, there is at least one predicate P and there is at least one predicate Q such that, P(x) if and only if Q(y).
Symbolically: ꓯxꓯy(x=y ↔ ꓱPꓱQ(P(x)↔Q(y)))
You are translating a Turing-complete system into an incomplete one (set theory).

Isn't that an error?

Of course - a set theorist can't tell
Averroes wrote: Sun Mar 17, 2019 10:08 am The later is clearly different from the accepted mathematical definition of equality.
So here we have two different definitions of equality!
Well, actually that's precisely the problem.

According to the law of identity: x = x means BOTH "x is equal to x" and x is identical to x".

So the "=" sign means different things at the same time?!? it is overloaded. It means many different things in many different contexts.

It's a grammatical error
Averroes wrote: Sun Mar 17, 2019 10:08 am correct results are obtained.
The "correct result"? So.... you are telling us that you KNOW what results you WANT logic to produce?
If you already know the correct answer then what do you need logic for?

of course, you have fallen for the same trap every mathematician does. x = x is ASSUMED true, not DECIDED true
You actually have no algorithm to decide whether any two things are the same

1 = 1
8888888888888888881 = 888888888888888881

You can't assume equality. You gotta do the work

On and on we go. I will deconstruct any bullshit you bring to the table.

Logic is the domain of computer scientists now. Time to adapt or step down.