The Mysterious Connections Between Irrational Numbers - e, pi, and phi

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  • Start date
In summary, the conversation discusses the significance of irrational numbers such as e, pi, and phi, and their relation to each other and different systems. The concept of irrational numbers and their representation is also explored, with the idea that they cannot be accurately represented using natural number notations. The accuracy of irrational numbers is debated, with some arguing that they can be accurately represented through infinite series, while others question this definition of accuracy. The conversation also touches on the idea of the real line and the representation of numbers on it.
  • #71
Matt,

You ommited parts of it, so here is all of it:
Matt said:
mandelbrot's set doesn't have integer dimension...
WWW said:
I know it, but in 1-dim all you can get is the shadow of what you can find between 1-dim and 2-dim, isn't it?
Matt said:
still not defined uncertainty and redundancy, non-standard terms.
WWW said:
Please explain Why do you think they are not defined?

Your answer to the first part is:
Matt said:
that's at best wrong, at worst completely meaningless.
Please give more details why do you think so?

Your answer to the second part is:
Matt said:
where in this thread have you offered a definition of uncertainty or redundancy? and i don't mean their plain english meanings.
Please show us an example of how a definition of r and u when:
f=dead cat
t=alive cat
r=redundancy
u=uncertainty

When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
will look like?
 
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  • #72
you haven't said what you mean by "shadow between 1-d and 2-d". sorry but it makes no sense as a sentence, elaborate, explain and clarify. nor have you explained what you mean be representation of a julia set, and what that has to do with the ambient space. seeing as RxR and R are in bijective correspondence I can encode the points of a 2-dim space in a 1-dim one, and by induction, any n-dim space. And the same goes for fractals as it's just some subset of some space.

how can i tell you what a definition of u and r will look like. they're your objects to define.
 
  • #73
Matt,

seeing as RxR and R are in bijective correspondence I can encode the points of a 2-dim space in a 1-dim one,
You can ecode any {x} with any {x,y} so what.

By {x}_only 1-dim data you cannot represent {x,y} 2-dim data.
how can i tell you what a definition of u and r will look like. they're your objects to define.
Now it is clearly understood that you have nothing but a 'NO'_reflex in this r u case.

For example, your response to:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
was "what is v"? and from this question we can learn that your abstraction's ability totally depends on the standard way.

My abstraction's ability totally depends on my non-standard way.

So, the problem of translation is twice difficult in our case.

But I think that there is a deeper problem here, with is:

You simply do not understand my ideas, and therefore they are "non sense" for you.

So I think it is the time to say good bye to each other because I cannot help you and you cannot help me.
 
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  • #74
given you inability to explain clearly any of your objects i can't see how asking what the v is in the diagram is a bad thing in any sense.
 
  • #75
My logic is an included-middle yours is excluded-middle.

You say that included-middle can be defined by excluded-middle, I say it cannot, simply because probabilty is a first-order property in included-middle system, and in excluded-middle system it is not a first-order property.

If you can show how probabilty is a first-order property in excluded-middle logical system, then it will be the gate between our different worlds.

In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.


For example:

f=dead cat

t=alive cat

r=redundancy (more then one copy of the same value can be found)

u=uncertainty (more than one unique value can be found)


When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
Code: 
 f   t   
 |   |   
 |#__| 
 |

Simple as that.

For example:

Let XOR be #

Let AND be &

Let a,b,c,d stands for uniqueness, therefore logical forms of 4-valued logic is:

Code: 
              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  | <--(First 4-valued logical form)
    |  |  |  |
    |  |  |  |
    |&_|&_|&_|_
    |
    ={x,x,x,x}


   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  | <--(Last 4-valued logical form)
    |#____|  |      
    |        |
    |#_______|
    |
    ={{{{x},x},x},x}

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |&_|_ |  |       |#_|  |  |       |&_|_ |&_|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |&_|&_|&_|_      |&____|&_|_      |&____|&_|_      |&____|____
    |                |                |                |
    {x,x,x,x}        {x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |#_|  |&_|_      |#_|  |#_|       |  |  |  |       |&_|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |&_|&_|_ |       |#____|  |
    |     |          |     |          |        |       |        |
    |&____|____      |&____|____      |#_______|       |#_______|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  |  
    |#____|  |      
    |        |
    |#_______|
    |    
    {{{{x},x},x},x}
[/b]

A 2-valued logic is:

Code: 
    b   b 
    #   #    
    a   a     
    .   .   
    |   |   
    |&__|_   
    | 
 [b]   
    a   b     
    .   .   
    |   |  <--- (Standard Math logical system fundamental building-block) 
    |#__|   
    |
[/b]

If this time your response is "non-sense" then good-bye.
 
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  • #76
WWW said:
What is 'distance' from your point of view?

For me 'distance' is the preventing side of some perevent/complement system for example: http://www.geocities.com/complementarytheory/4BPM.pdf

By Complementary Logic any existing element that can be changed, is the result of at least two opposites that simultaneously pereventing/defining each other.

Distance is a property between objects in space. Space is a structure, which is constructed of discrete units. The structure of space is a distributive lattice. A set of properties, being a "complementary logic?", expressing difference in wholeness.
 
  • #77
Distance is a property between objects in space...
So to define distance we need at least two states local(= a unique object) and global(=a space).

Therefore any existing thing is at least a product of the interactions between the local and the global.

When we research a QM product then this is exactly what we find: a product which is both particle(=strong locality) and wave(=strong non-locality).

Shortly speaking, Complementary logic is the logic of interaction between opposite properties, which means: any distance (logical or physical) is the preventing property, where any non-distance is the complementing property.

Form this point of view, the evolution of concessions is the story of the increasing ability of communication between the global and the local in a cybernetic way, for example:

http://www.geocities.com/complementarytheory/CK.pdf
 
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  • #78
WWW said:
My logic is an included-middle yours is excluded-middle.

You say that included-middle can be defined by excluded-middle, I say it cannot, simply because probabilty is a first-order property in included-middle system, and in excluded-middle system it is not a first-order property.

If you can show how probabilty is a first-order property in excluded-middle logical system, then it will be the gate between our different worlds..[/B]


i'd like to see you explain where i said any of that.

the rest is starting to be readable. see what happens when you actually explain the meanings of the terms you use?

now, the main thing you need to demonstrate is that there is any point to all this.

For instance, are the axioms of ZF(C) consistent in this "logical world"
 
  • #79
f=dead cat

t=alive cat

r=redundancy (more then one copy of the same value can be found)

u=uncertainty (more than one unique value can be found)


When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
Code: 
 f   t   
 |   |   
 |#__| 
 |

Simple as that.

For example:

Let XOR be #

Let AND be &

Let a,b,c,d stands for uniqueness, therefore logical forms of 4-valued logic is:

Code: 
              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  | <--(First 4-valued logical form)
    |  |  |  |
    |  |  |  |
    |&_|&_|&_|_
    |
    ={x,x,x,x}


   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  | <--(Last 4-valued logical form)
    |#____|  |      
    |        |
    |#_______|
    |
    ={{{{x},x},x},x}

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |&_|_ |  |       |#_|  |  |       |&_|_ |&_|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |&_|&_|&_|_      |&____|&_|_      |&____|&_|_      |&____|____
    |                |                |                |
    {x,x,x,x}        {x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |#_|  |&_|_      |#_|  |#_|       |  |  |  |       |&_|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |&_|&_|_ |       |#____|  |
    |     |          |     |          |        |       |        |
    |&____|____      |&____|____      |#_______|       |#_______|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  |  
    |#____|  |      
    |        |
    |#_______|
    |    
    {{{{x},x},x},x}
[/b]

A 2-valued logic is:

Code: 
    b   b 
    #   #    
    a   a     
    .   .   
    |   |   
    |&__|_   
    | 
 [b]   
    a   b     
    .   .   
    |   |  <--- (Standard Math logical system fundamental building-block) 
    |#__|   
    |
[/b]
Can you show us a ZF(C) axiom whare probability included?

Matt said:
now, the main thing you need to demonstrate is that there is any point to all this.
Please give more details.
 
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  • #80
es, you keep reposting that but odn't actually demonstrate that it is useful at any point.

As you don't define what you mean by probability we cannot answer your last request. Seriously, mathematics is a formal construction; you cannot just informally use words and expect it to be meaningful. if we think of a proper quantum system (ie not the stupid cat experiment) then all of the things in it are modeled using properly defined mathematical objects. so why don't you demonstrate a way of producing pure states, say, within your system. hint, you'll have to construct the real numbers, the complex numbers, in fact everything if you want to do mathematics. if youy merely want to argue about philosophy then do so, but don't get angry and change your user name so that we might think you were pretending to do maths seriously.
 
  • #81
I did not choose to change my name. I actually had no choice because PF mentors shut me down twice in the last 2 years, and as you now, if you want to register again you have no choice but to do it under a new name.

QM element is naturally included-middle element, because it is based on two opposite properties that preventing from us to know exactly both of them simultaneously, as we can do in macro systems.

If we want to develop some formal language that deal with QM world, we have to do it by changing our logical reasoning from excluded-middle to an included-middle.

And here Complementary Logic entering to the picture, and using probability as first-order property.

Through CL (Complementary Logic) any n>1 has several variations of internal structures based on interactions between its integral side (root-like side) and its differential side (leaf-like side).

These internal structures can be ordered by their vagueness degrees, which vagueness is a combination between redundancy_AND_uncertainty properties that give us the "cloud of probability" of each ordered information form.

Redundancy exists if more then one copy of the same value can be found.

Uncertainty exists if more than one unique value can be found.

For example: http://www.geocities.com/complementarytheory/ComplexTree.pdf

Pay attention that I used the words "information forms" because these information forms, which are ordered by their vagueness degrees, can be used as general building-blocks that can help us to develop much more fine models that have to deal with included-middle problems.

From the pdf example we can learn that the standard base value expansion method is actually based on 0_redundancy_AND_0_uncertainty building blocks, which are a very small part of infinitely many different building blocks that can be used by us to construct and explode a very complex information models with variety of combinations of vagueness.

Another thing is that some experimental result is actually some single section which is cut out of 0_redundancy_AND_0_uncertainty building blocks that are ordered in several scales.

My system suggesting a much more complex information form as a result, as can be found in the last page of the pdf example.

Shortly speaking, because any information form in my system is at least structural/quantitative, it can be used straightly as it is, and we don't have to translate it to quantity before we can use it in our system.
 
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  • #82
why are all your information forms only numbers?
 
  • #83
Please see post #79 and also please read again my last post, thank you.

If I am more understood to you then please read:

http://www.geocities.com/complementarytheory/Complex.pdf

Why are all your information forms only numbers?
I need help to develop it, it is only in its first stages, and it is definitely cannot be done by a one person.
 
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  • #84
Actually, you should be able to post here as Organic.


QM element is naturally included-middle element, because it is based on two opposite properties that preventing from us to know exactly both of them simultaneously, as we can do in macro systems.

This is a common misunderstanding about QM. Things aren't simultaneously (classical) particles and (classical) waves... they're neither; instead they're some new quantum mechanical thing that given the right circumstances, can approximate a (classical) particle or a (classical) wave.

Since quantum mechanical things aren't classical particles, it should be unsurprising that they cannot be represented exactly as classical particles.


If we want to develop some formal language that deal with QM world, we have to do it by changing our logical reasoning from excluded-middle to an included-middle.

As I've tried to point out with your Mandelbrot example, one can use weaker systems (e.g. a "1-dim system") to build new systems (e.g. a "2-dim system"). Even if you are right, it is not necessarily the case that the old logical reasoning is incapable of building the new logical reasoning.


And here Complementary Logic entering to the picture, and using probability as first-order property.

In particular, you seem to indicate now that probability is the key ingredient in your vision of the new way to do things. Well, the old way has known how to do probability for a long time, why do you think it is inadequate now?
 
  • #85
Good luck with convincing him that quantum objects such as photons are neither waves nor particles. I seem to remember posting a long sequence emphasizing the difference between "displaying wave like properties" and "being a a wave". I don't think it got through.
 
  • #86
Hurkyl,

Actually, you should be able to post here as Organic.
First, I really hope that it is not you who shut me down as Organic, because it is a west of time to speck with mentors which closing members because they have different point of view than them.
Hurkyl said:
...instead they're some new quantum mechanical thing that given the right circumstances,...
And this is exactly my point of view which is: wave/particle properties are under a probability state, and they are not physical realm until we change this probability according to our measurements tools, to some accurate particle-like XOR wave-like results.

Shortly speaking , I have Max Born's probability point of view ( http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/tourquan/born.htm ).
In particular, you seem to indicate now that probability is the key ingredient in your vision of the new way to do things. Well, the old way has known how to do probability for a long time, why do you think it is inadequate now?
The probability of convetional Math is not a first-order property, therefore a natural first-order system is much better in this case, even if the old way works.
As I've tried to point out with your Mandelbrot example, one can use weaker systems (e.g. a "1-dim system") to build new systems (e.g. a "2-dim system"). Even if you are right, it is not necessarily the case that the old logical reasoning is incapable of building the new logical reasoning.
The old logical reasoning is capable of building the new logical reasoning if probabilty is a first-order property of it.

If you don't think so then please show us how we can represnt Complementary Logic by an excluded-middle logical system.

To help you, please read this first:
http://www.geocities.com/complementarytheory/BFC.pdf
 
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  • #87
present here and now a rigorous explanation/definition of probability uaing only "first order" objects, whatever theyu may be. Or are you confusing the warm fuzzy idea of probability with its rigorous axiomatic abstraction?

can i suggest that the reasons you aren't allowed to post in the maths forum are your refusals to deal in mathematics and hijacking of threads to espouse your unmathematical views?
 
  • #88
wave/particle properties are under a probability state

The QM point of view is that wave / particle properties (when they appear) are approximate, not under a "probability state".
 
  • #89
can i suggest that the reasons you aren't allowed to post in the maths forum are your refusals to deal in mathematics and hijacking of threads to espouse your unmathematical views?
I was shut down after I put https://www.physicsforums.com/showthread.php?t=18972 in General Math.

Math , in my opinion, is not an unchagable monolitic objective state, but a living form of language.
present here and now a rigorous explanation/definition of probability uaing only "first order" objects, whatever theyu may be. Or are you confusing the warm fuzzy idea of probability with its rigorous axiomatic abstraction?
please give more details, because I do not really understand what do you looking for.
 
  • #90
The QM point of view is that wave / particle properties (when they appear) are approximate, not under a "probability state".
By saying "under a probability state" (sorry about my poor English) I speak about wave / particle properties before they appear through our experiment tools.
 
  • #91
please give more details, because I do not really understand what do you looking for.

He's looking for you to make a list of statements and say "These are the statements we are assuming to be true", and then for the subject at hand, to only make statements which can be derived from those assumed statements using rules of deduction.

e.g.

if one of the statements was "For any z: If P(z) then Q(z)", and another of the statements was "P(a)", then we can conclude "Q(a)" via:

Forall z: if P(z) then Q(z)
therefore
if P(a) then Q(a)

and

if P(a) then Q(a)
P(a)
therefore
Q(a)


And you should be able to do this (or at least indicate a way this can be done) for any statement you wish to claim true.


This is how mathematics is done. If you don't want to do it this way, then you're doing something other than mathematics.



And if you wish to rewrite logic, then you should list the legal rules of deduction as well. (since logic is simply rules of deduction, then if you want to change logic you have to present new rules of deduction)
 
  • #92
As you feel confident in saying that maths can only deal with probability as a higher order object, you must be able to state what you mean by probability.

We may all have some notion about things "possibly" happening and some things being more likely to occur, but, once more, you're confusing vague, fuzzy notions of real life with the abstraction on mathematics and saying they are the same.

You appear to claim that the axiomaitized probability theory of mathematics is inherent as a basic concept in your theory. That indicates that you do not understand the Kolmogorov version of probability theory. Please domonstrate that you somehow have an equivalent theory that is "fundamental". This must contain sets, measures and functions (ie cartesian products), as well as at least some mention of the real numbers.

Do not think that this axiomatic thing *is* the fuzzy concept of likelihood. It is, as tends to be the case, a mathematical construction.

Show in your elemental theory of probability that is as ontological simple as it gets, that the probability of obtaining 3 heads in three throws of an unbiased coin is 1/8.
 
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  • #93
I think irrational numbers have a place on the real number line because they do have a real and accurate value, it just can't be represented well as a ratio of two integers.
 
  • #94
I think I@ve found a way of expressing what I've been trying to sum up about this for a while now.

You let the properties of the objects and operations in you theory define the theory, whereas you should let the theory define the allowed objects and operations.

Hence your claim to specialize to boolean logic means that if you let the objects be the usual kind of statement and AND and XOR be the usual connectives, then you have Boolean logic. Yet you''ve not offered a generalization properly, becuase you have to redefine all the operations for each specialization; it isn't a genuine generalization. Imagine if you will, and you probably won't, that I am claiming I've got a general theory of Algebra. I don't have any axioms, rules or definitions, just things I call algbraic objects and operations i call algebraic operations. Now I claim that if I let these be groups and group maps I'm doing group theory, ie that is specializes to group theory. But I@ve offered nothing to back that up and it is a completely vacuous theory really. (Incidentally I can offer a generalized theory of algebra which does contain the groups as a special subset: what do you know about cocommutative Hopf algebras?)
 
  • #95
First, thank you for your positive attitude.

As a first step, I’ll try to explain what is probability through my point of view .

Let us take a piano with 4 possible different notes.

By using the word 'possible' I mean that in the first stage, any key can be anyone of the 4 notes, and we have no way to know what note each key has, before we are using it.

Each time when I press simultaneously on its all 4 keys, I get an accord.

Let us notate each unique note by a different letter, for example: a,b,c,d

Redundancy is (more then one copy of the same value can be found)

Uncertainty is (more than one unique value can be found)

Let XOR be #

Let AND be &

A 4-valued logic is:
Code: 
              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  | <--(First 4-valued logical form)
    |  |  |  |
    |  |  |  |
    |&_|&_|&_|_
    |
    ={x,x,x,x}
[COLOR=Blue][B]In the first case each accord can be one of 4^4 different possibilities.[/B][/COLOR]



   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  | <--(Last 4-valued logical form)
    |#____|  |      
    |        |
    |#_______|
    |
    ={{{{x},x},x},x}
[COLOR=Blue][B]In the last case each accord is a one and only one possibility.[/B][/COLOR]
The ordered possibilities between 4^4 and 1 is:

Code: 
[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |&_|_ |  |       |#_|  |  |       |&_|_ |&_|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |&_|&_|&_|_      |&____|&_|_      |&____|&_|_      |&____|____
    |                |                |                |
    {x,x,x,x}       {{x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |#_|  |&_|_      |#_|  |#_|       |  |  |  |       |&_|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |&_|&_|_ |       |#____|  |
    |     |          |     |          |        |       |        |
    |&____|____      |&____|____      |#_______|       |#_______|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  |  
    |#____|  |      
    |        |
    |#_______|
    |    
    {{{{x},x},x},x}
[/b]

r is (more then one copy of the same value can be found)

u is (more than one unique value can be found)

Let XOR be #

Let AND be &


When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
Code: 
 f   t   
 |   |   
 |#__| 
 |

If you understand what is probability by me, then try to translate it to the standard excluded-middle reasoning.
 
Last edited:
  • #96
Nowhere in there do you state what you mean by probability.

And you're using connectives AND and XOR as if they are the usual objects of boolean logic, when you say they aren't. Moreover it appears that you're saying that a,b,c,d are not events/statements but the possible "truth" values in the system of logic. That is waht you're getting at if you say you have a 4 valued logic system (otherwise you've not define what the logical values may be).

There is also the observation that you're using these & and # connectives (without offering their truth tables) inside these diagrams which we are told are a full set of values between xor and and, so the definition uses the object in its definition. I dont' see any recusive way to make the valid.
 
  • #97
Please forget for a moment the stantard excluded-middle point of view of AND(=&) and XOR(=#) and try to understand it as I wrote it in the previous post.

Can you do that?
Moreover it appears that you're saying that a,b,c,d are not events/statements but the possible "truth" values in the system of logic
Take each note as a "true" statement.

In this 4-notes piano, any given note is "true", because it is not an excluded-middle logical system.

Please look at the Complementary Logic diagram:http://www.geocities.com/complementarytheory/BFC.pdf

In an included-middle reasoning two opposites are simultaneously preventing/defining each other and the result is a middle(=included-middle).

In an excluded-middle reasoning two opposites are simultaneously contradicting each other and the result is no-middle(=excluded-middle).

An excluded-middle system is a private case in Complementary Logic, as you can see in the example of the 2-valued logic in the previous post.
 
Last edited:
  • #98
If all the statements must be true you're even omre off beam than you first appear.
 
  • #99
What if I knew that key #1 plays either A or B, and key #2 playes either B or C, and that key #1 and key #2 play different notes?
 
  • #100
What if I knew that key #1 plays either A or B, and key #2 playes either B or C, and that key #1 and key #2 play different notes?
1) there is no c but only a XOR b in a 2-valued system.

2) In the first case of a 2-valued system, each accord can be one of 2^2 different possibilities, and we cannot know what an accord we get until we actually pressing simultaneously on both keys (and this is exactly the meaning of probability here).

3) In the last case of a 2-valued system, each accord can be one of 1 different possibilities, and we get only an a,b accord when we are pressing simultaneously on both keys (there is no probability here).

4) In Complementary Logic there is no contradiction but only a simultaneos existencs of at laest two opposite that simultaneously preventing/defining each other, and the result is a middle(=included-middle).

5) In an excluded-middle reasoning two opposites are simultaneously contradicting each other and the result is no-middle(=excluded-middle).
If all the statements must be true you're even omre off beam than you first appear.
Please look again at: http://www.geocities.com/complementarytheory/BFC.pdf

You simply refuse to understand that there is no true XOR false and therefore no truth tables in NATURAL included-middle logical system like Complementary Logic.

Hyrkyl and Matt:

Please read and try to understend post #95 and #97, and if you try again to force an excluded-middle on them, then don't west your time.

---------------------------------------------------------------------------------------------------

If we go back to irrational numbers, then by CL (Complementary Logic) each irrational number is a unique_but_not_accurate element because of a very simple reason:

Each irrational number is a unique path (or cut) along infinitely many different scales, but this path is not accurate because it has no "right side".
 
Last edited:
  • #101
"Take each note as a "true" statement"

and now you contradict that...
 
  • #102
Take each note as a "true" statement
I wrote "true" and not true, which means that there is no true XOR false in my system.
 
  • #103
which highlights the fact that you've not explained what the possible truth values are in your systems. it appears that your diagrams just correspond to some constructions involving and and xor in some logic system that you're refusing to explain/ nor have you explained why these (ill-defined - xor is not assiciative so you can't use it without bracketing) diagrams are remotely important or useful.
 
  • #104
No exluded-middle point of view can understand Complementary Logic, and the reason is very simple:

In CL we have at least two simultaneous levels to a logical expression:

( Its differential side(= a XOR b) / its integral side(= a AND b) ), where a,b are opposites.

If you can't understand that truth tables are not used in Complementary Logic, then don't west your time.
 
Last edited:
  • #105
but now you're saying that you can't get boolean logic out of it since it has truth tables that govern it, and there must be some analogous result there. if you're not going to even offer some way of describing the truth value, be it in 0,1, or some fuzzy, or even trivalued F,T,U system then you can't do anything.

what on Earth do you mean by opposites? what is the opposiite of the function sin(x)? remember you've said in the past that anything is allowed to be some 'information form' to be explored.
 

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