Mathematical proof for UPB

[crafn]

Hello everyone, you might remember me from last year: https://board.freedomainradio.com/topic/46766-mathematical-aid-for-upb-based-ethics/

I've finally managed to define UPB in exact terms and verify the most important conclusions of it rigorously. Here's A formal proof of universal and objective ethics.
Disclaimer: This proof has not been reviewed or confirmed by anyone other than me. It most likely contains minor errors, but since the result is mathematically simple and intuitive, I would be surprised if a major flaw in the math was found. However, I suggest to take it with a grain of salt at this point.

While this paper may not be important or persuasive for the ordinary person, I believe it's the most heavyweight intellectual firepower that can exist in support of universal and objective ethics.

I'll be happy to hear comments, corrections and objections. 

Related note:

When one understands the 5-page proof, he must conclude that the verbal proof presented in the UPB book contains assertions and paths of reasoning that are not required by, or included in the formal proof. To take an example, claims such as
    "No moral theory can be valid if it argues that a certain action is right in Syria, but wrong in San Francisco."
or
    "The moral proposition “eating fish is evil” thus fails the test of universality because it is too specific --"
are not made in the formal proof. This kind of generality of moral rules is not required to prove the validity of universal and objective ethics. In fact, I have no clue how to define the generality of a proposition in non-arbitrary terms.

I do not understand the book well enough to show that this excessivity would be incorrect, or even undesirable, but the fact that it took me over a year of mental wrestling and dozens of retries in the conceptual weeds of UPB to understand what I believe is the essence of it, may suggest otherwise. With the lack of arguments, this is not a critique, but a heads-up, that the current verbal argument may not be presented in a clear and condensed form (which the formal proof is also not, for most people.)

[ofd]

Why did you restrict that to finite sets? Doesn't that collide with the defitnition of the set being universal?

[crafn]

1 minute ago, ofd said:

Why did you restrict that to finite sets? Doesn't that collide with the defitnition of the set being universal?

Finite sets makes some things easier to understand and write down. All of the proofs should be essentially identical with infinite sets.

The universality condition works with finite and infinite sets -- I don't know what you mean.

[ofd]

Quote

The universality condition works with finite and infinite sets

There are several problems. Finite sets can contain infinite sets if you don't delineate those two. Plus, no first order predicate logic and describe all the finite sets for that reason.

 

Quote

All of the proofs should be essentially identical with infinite sets.

Except that the two sets are not bijective. If you have a universal description it must necessarily include countable and uncountable sets.

You have begun a monumental task, which is laudable. But why bother with sets (which have their pitfalls) when there is already a logic to describe ethics in a formal way? It's easier to modify S5 so it can describe quantors that occur in UPB than to use set theory.

[crafn]

9 hours ago, ofd said:

There are several problems. Finite sets can contain infinite sets if you don't delineate those two. Plus, no first order predicate logic and describe all the finite sets for that reason.

Could you be more specific? Which line in which definition or proof is wrong in what situation? They're numbered for this exact reason. I don't see at all how your claim about the possibility of finite set containing infinite set (which is true), is relevant. I don't understand your third sentence. There is no need or attempt to describe "all the finite sets".

9 hours ago, ofd said:

Except that the two sets are not bijective. If you have a universal description it must necessarily include countable and uncountable sets.

You're imposing your own definition of universal on the proof, which has its own, very narrow definition of universality (the universality condition). This is irrelevant.

9 hours ago, ofd said:

But why bother with sets (which have their pitfalls) when there is already a logic to describe ethics in a formal way? It's easier to modify S5 so it can describe quantors that occur in UPB than to use set theory.

I've taken look at S5 modal logic superficially. I don't how that would help in proving universal and objective ethics at all. I'll dismiss your claims until you show how much easier it's to not bother with sets and use S5 instead to prove universal ethics in simpler terms and in less than 5 pages.

I have to wonder, do you understand my proof, or are you just talking?

[ofd]

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Could you be more specific?

You haven't shown that the universal set is universal and finite. You just assumed it, unless I missed that part.

 

Quote

I don't how that would help in proving universal and objective ethics at all.

Because of the many world aspect of it. You can distinguish between true and necessarily true right away. Furthermore, the theory is already established and well researched unlike set theory for ethical systems.

[crafn]

9 minutes ago, ofd said:

You haven't shown that the universal set is universal and finite. You just assumed it, unless I missed that part.

It's true I haven't, because there is no universal set. You probably mean the set of universe descriptions. It has no requirement on being "universal". It is filled with appropriate scenarios that one wishes to examine. If you wish to fill it with infinite amount of universe descriptions, you can do it.

13 minutes ago, ofd said:

You can distinguish between true and necessarily true right away.

And you can distinguish between true and necessarily true with the machinery presented on the first page of my proof. (If the extension of a proposition is the whole universe set, then it's necessarily true, otherwise not). There is even an example of this feature (example 2). Furthermore, the distinction between true and necessarily true is irrelevant for the proof.

17 minutes ago, ofd said:

Furthermore, the theory is already established and well researched unlike set theory for ethical systems.

No "set theory for ethical systems" is being used. The set theory I'm relying on is pretty well established; It's the most common foundation of mathematics, ZFC.

This will be my last answer for you, because the gap of disconnect does not seem to be closing after three posts.

[ofd]

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This will be my last answer for you, because the gap of disconnect does not seem to be closing after three posts.

Your unwillingness to see the problems prevents you also from thinking about the fundamental problems behind them. You assumed by assertion that ZF can easily be translated to PA. That proposition is anything but clear and requires some research and thinking. That question alone is more worthwhile pondering than your drivel. https://arxiv.org/pdf/1207.6357v1.pdf

[anjumahmed]

I'm just here to say that I love LaTeX formatted documents.

[TDB]

"Example 3. Propositions p and p′ can't be both true is equivalent with p ∩ p = ∅. "

 

Is is that a typo? Should be

Example 3. Propositions p and p′ can't be both true is equivalent with p ∩ p' = ∅. 

[crafn]

7 hours ago, TDB said:

"Example 3. Propositions p and p′ can't be both true is equivalent with p ∩ p = ∅. "

 

Is is that a typo? Should be

Example 3. Propositions p and p′ can't be both true is equivalent with p ∩ p' = ∅. 

It is, thank you! That, among some other minor stuff, will be fixed shortly. (Version 1.01, same url)