First order logic with symmetry destroys information

[Skepdick]

First order logic (with equality) is the foundation of Mathematics.

Axiom of Reflexivity for all x: x = x

So let me ask you this question: Is (x = x) the same as (x = x) ?

What is the difference, you ask? My bad! My representation sucks and you missed the thing that matters most in answering the question.

Is (x = x) the same as (x = x) ?

Ah! Now you see it!

And now you must forget that any of this happened and embrace the religion of Mathematics.

Axiom of Symmetry If x = y then y = x (and the color is gone)

men in black flash - Google Search 2020-02-14 13-37-19.png (123.51 KiB) Viewed 6858 times

[Walker]

x equals y, however x is not y, therefore due to the imprecision of mathematics, and due to the actual suis generis nature of each entity x or entity y within the totality of reality not bound by the limitations of any particular language such as mathematics, equal cannot equal is.

[Skepdick]

x equals y, however x is not y, therefore due to the imprecision of mathematics, and due to the actual suis generis nature of each entity x or entity y within the totality of reality not bound by the limitations of any particular language such as mathematics, equal cannot equal is.

It's not even that! Even "X is X" is wrong.

X is itself (and NOTHING else)
X is not X.
X is not X.

equal(X, X, X) -> True
identical(X, X, X) -> False

The limits of Mathematics are not linguistic. The limits of Mathematics come from the medium with which Mathematics has been done for 2000 years.

It's the medium which makes Mathematics "timeless and eternal", "rigid and unchanging".

White paper. Black ink.

[Scott Mayers]

What do you mean by "First order logic with symmetry destroys information"?

[Scott Mayers]

x equals y, however x is not y, therefore due to the imprecision of mathematics, and due to the actual suis generis nature of each entity x or entity y within the totality of reality not bound by the limitations of any particular language such as mathematics, equal cannot equal is.

It's not even that! Even "X is X" is wrong.

X is itself (and NOTHING else)
X is not X.
X is not X.

equal(X, X, X) -> True
identical(X, X, X) -> False

The limits of Mathematics are not linguistic. The limits of Mathematics come from the medium with which Mathematics has been done for 2000 years.

It's the medium which makes Mathematics "timeless and eternal", "rigid and unchanging".

White paper. Black ink.

Your hangup with the basic concepts of logic is not sound.

Presume Totality is absolutely nothing. It has no 'logic', no "laws". Then it has no consistent rule to dictate that it has to have consistency. Thus it is both 'true' and 'false' and this still fits in with logic given we define what is false as the state of something to be both true and false. Totality then contains absolutely everything and anything and can be separated into worlds that are consistent and inconsistent.

We are a 'consistent' world and should we find something lacking this consistency, then it just means that within a system of consistent rules, we deny what is contradictory in one world (the consistent one) and place the contradictory statement aside for the complement of this within Totality.

The symbols asserting equality are our ways of expressing something dynamically through language even though the referents themselves are or can be static. The symbol on the left is just a label for what is on the right to be used as a shorthand reference rather than denoting the particular things the symbol stands for. But because the symbols are ALSO real things in themselves as pointers, you can also reflect the meaning of the symbol to itself.

"Logic" is just a special set of rules like those of a game. And because the 'game' can be about something real rather than just play, there is no law against setting up any system you want. So go ahead and define "X" as any symbol on the left of the symbol "=" is not equivalent to any symbol on the right. All that matters is that for whatever system you design, that it is non-contradictory, consistent, and, if you want it to be functionally useful, complete. It is also not necessary that you have a rational system at all. It just means it is a type of 'game' that lacks usefulness to others communicating in a consistent fashion.

The "=" as a sign comes from the "double negation" too, if you want to be particular. And it is only a sign that compares something on either side of it where we choose to assign whatever is on the left as the same as what we assign OR compare it to, depending on what we want it to represent.

[Skepdick]

Your hangup with the basic concepts of logic is not sound.

And my argument is that you don't know what "soundness" is! This is a factual claim.
By Wittgenstein's rule-following paradox, what you call/define as "soundness" could just be an error.

Perhaps, what you were doing all along is that you weren't evaluating my concepts for "soundness",
you were evaluating them for "boundness". You mistook "boundness" for "soundness".

I am going to define evaluation as: computing the true value of a function given a set of input values.
You are falling into a vicious, infinite recursion because of it. An explosion.

You are committing Type Errors, which are exactly like Category Errors - they manifest at the highest (not lowest) orders of logic.

Presume Totality is absolutely nothing. It has no 'logic', no "laws". Then it has no consistent rule to dictate that it has to have consistency.

EXACTLY. You have no consistent() predicate outside of your system. You don't know what consistency is, except by axiomatic definition inside the system.

So if you define "inconsistency" and your system can deduce it - that's a computationally-valid proof that your system is self-consistent.

Thus it is both 'true' and 'false' and this still fits in with logic given we define what is false as the state of something to be both true and false.
Totality then contains absolutely everything and anything and can be separated into worlds that are consistent and inconsistent.

And that is your error! You don't have consistent() or inconsistent() predicates a priori. Because you don't know what consistency/inconsistency is.
You have defined them axiomatically. If you can construct a function that evaluates to the axiomatic definition then that's that. Existence proof.

Consistent is a world: (P ∧ ¬P) evaluate to False.
Inconsistent is a world: (P ∧ ¬P) evaluates to True

The only distinction that can be drawn is a world in which (P ∧ ¬P) ALWAYS evaluates to False is immutable, and a world in which (P ∧ ¬P) CAN evaluate to True is mutable. So your consistency/inconsistency distinction is equivalent to a mutability/immutability distinction.

Then I point out to you that you are USING variables. Throughout ALL OF YOUR LOGIC. Saying x:= 5 is a mutation. You are ASSIGNING the value 5 to the variable x, so clearly the universe in which you are DOING logic is mutable.... but the universe in which (P ∧ ¬P) evaluates to False is immutable.

So you are doing logic from two different perspectives! Oooooops?!?!?!

Would you say that's inconsistent?

We are a 'consistent' world and should we find something lacking this consistency, then it just means that within a system of consistent rules, we deny what is contradictory in one world (the consistent one) and place the contradictory statement aside for the complement of this within Totality.

There we go! Vicious circularity. To speaking about the "consistency" of your rules is incoherent - you keep pre-supposing it.
Rules are rules. Either you followed them or you didn't.

IF you follow your rules you CANNOT ARRIVE AT A CONTRADICTION. No matter what you do!

Because contradictions don't exist. And so it's perfectly fine to declare:

for all x: x = x
for all y: y != y

IF you build a system that obeys those rules - then you CANNOT ARRIVE AT A CONTRADICTION.

Here it is: https://repl.it/repls/PitifulTechnoMicrocode

Code: Select all

from universe import X, Y

x = X()
y = Y()

assert x == x
assert y != y 

The symbols asserting equality are our ways of expressing something dynamically through language even though the referents themselves are or can be static. The symbol on the left is just a label for what is on the right to be used as a shorthand reference rather than denoting the particular things the symbol stands for. But because the symbols are ALSO real things in themselves as pointers, you can also reflect the meaning of the symbol to itself.

This is also incoherent. It seems to me that the concept of evaluation strategy is absent from your thinking.

The meaning of the symbol is its value. If we assign the value 5 to x (x:= 5) then x means "5".

If you recognize the distinction that exists between pass-by-value and pass-by-reference then f(x) is pass-by-reference, but f(5) is pass-by value.

So then it simply begs a question

How do you EVALUATE x = x.

You can EVALUATE it as True.
Or you can EVALUATE it as False.

As I demonstrated above. It's insufficient to say x = x, without telling me its value.

"Logic" is just a special set of rules like those of a game. And because the 'game' can be about something real rather than just play, there is no law against setting up any system you want.

That's precisely what I am demonstrating.

Symmetrical systems cannot contain any information. They are viciously circular.

The "=" as a sign comes from the "double negation" too, if you want to be particular. And it is only a sign that compares something on either side of it where we choose to assign whatever is on the left as the same as what we assign OR compare it to, depending on what we want it to represent.

IF we define equality as a predicate: equal() then this statement requires evaulation:

equal( equal(x,y), equal(y,x))

The VALUE of the above composition could be True or False.

Which would correspond to the DEFINITION (declaration) of two DIFFERENT axioms:

System 1: (x = y) = (y = x)
System 2: (x = y) != (y = x)

The latter system can determine if x and y switched places.
The former system can't.

The latter system can answer the question "Did x and y switch places?" The former system can't.

That's 1 bit of information destroyed.

[Scott Mayers]

Your hangup with the basic concepts of logic is not sound.

And my argument is that you don't know what "soundness" is! This is a factual claim.
By Wittgenstein's rule-following paradox, what you call/define as "soundness" could just be an error.

Perhaps, what you were doing all along is that you weren't evaluating my concepts for "soundness",
you were evaluating them for "boundness". You mistook "boundness" for "soundness".

I am going to define evaluation as: computing the true value of a function given a set of input values.
You are falling into a vicious, infinite recursion because of it. An explosion.

You are committing Type Errors, which are exactly like Category Errors - they manifest at the highest (not lowest) orders of logic.

When I use "sound", it refers to whatever is real and valid about Nature's logical value of what is or is not true, whatever that may be. You appear to be interpreting circularity as something unsound with respect to Nature when it is only your own evaluation about the use of words speaking about logic in general.

Since "logic" is necessarily about consistent systems of reasoning, AND Nature itself is 'sound' whether we agree to it or not, Nature doesn't have to play by rules. Things like circularity or "inconsistency" can be defined "consistent" and "sound" with respect to Totality because it is most INCLUSIVE of everything without bias to our perception of reality locally. The act of "defining" logic is itself begging something from our biased expectation of consistency being only an infinitesimal subset of Totality.

I cannot understand your presumption of any categorical errors on my part. Logic reduces to being 'explosive' with respect to Totality. I think personally that this contradiction is what defines existence itself.

Presume Totality is absolutely nothing. It has no 'logic', no "laws". Then it has no consistent rule to dictate that it has to have consistency.

EXACTLY. You have no consistent() predicate outside of your system. You don't know what consistency is, except by axiomatic definition inside the system.

So if you define "inconsistency" and your system can deduce it - that's a computationally-valid proof that your system is self-consistent.

You can't even 'define' anything without assuming you CAN define as a behavior itself.

Just think of how you could possibly define "sameness" in any precision without begging TWO distinct things as having some association to the term, "sameness". If I try to point to two different things and say the word, "same", to a child, they have to beg meaning into it in their own head and we only assume they understand by how they use the word (as a referent) to the idea in context to some agreed behavior. Since the position of two distinct things being compared are always in some different position in time or space regardless, everything is ABSOLUTELY 'unique' and you could not ever actually prove THAT anything is consistent (the 'same') except by contrasting it to what is more basically inclusive of everything: inconsistency.

The 'proof' of completeness of Propositional Calculus [the first order logic of statements about anything] begins by merely begging syntactic rules of what we want to say is "well formed". Then we list ALL POSSIBLE binary associations to everything in a chart form and then BEG one of those symbols as meaning 'true' when linked to some proposition. It is arbitrary what we mean by the term "true" but we semantically mean that the IDENTITY of some concept does not change (that is, 'fixed'). So we beg these associations:

X X
7 7
y y
sut sut
[yeojteo] [yeojteo]

with the word, "same", being spoken in posing each instance.
This begs that the relationship of those repeated symbols, things, or whatever we are denoting, is to be associted to the word "same" and when it appears that each of us agrees to its use consistently, that semantically gets interpreted as what we mean by truth, even though we may actually NOT be agreeing in to the same meaning if we could know what each other thinks directly.

We then need to compare pairs of things denoted with the association of saying simultaneously, "not same", to beg what we mean by pairs of dentoted objects/symbols like:

A c
j '
[red] [black]

I'm sure you understand this. But I don't know what point you are trying to convey in this thread about anything. What information is being 'destroyed' if it isn't demonstrated that something you think is symmetrical isn't initially 'constructed' in your own head by default? What does it mean to "destroy" information to you?

Thus it is both 'true' and 'false' and this still fits in with logic given we define what is false as the state of something to be both true and false.
Totality then contains absolutely everything and anything and can be separated into worlds that are consistent and inconsistent.

And that is your error! You don't have consistent() or inconsistent() predicates a priori. Because you don't know what consistency/inconsistency is.
You have defined them axiomatically. If you can construct a function that evaluates to the axiomatic definition then that's that. Existence proof.

Consistent is a world: (P ∧ ¬P) evaluate to False.
Inconsistent is a world: (P ∧ ¬P) evaluates to True

And these meanings are still incoherent without begging both as I demonstrate above using the word "same" versus "not same" that BOTH need begging simultaneously in a complementary pair of systems of associations inductively. Each denoted set demonstrating what is "same" and "not same" are EACH different 'logical systems' in that they have different patterns but need to both be understood by agreement between at least two people communicating with each other in order to understand the other meaning. This complementary pair of demonstrations are "symmetrically" essential to understand either "same" or "not same" as existing as independent logical systems that define 'value' in a higher ordered logic built with both of them collectively where we introduce 'value'.

Note how the word "consistent" literally means "con-" (with) and "-sis-" (same) "tent" (tense)? The terms, "true" and "not true" also correspond by begging association of sets of contrasting pairs of things with these words. You can't escape the circularity at the lowest level regardless without assuming pairs of complementary 'logic' systems existing but that we choose one to be uniquely called "true" arbitrarily with its complement as being "not true" relatively.

So the only absolute about first order logic is that there is no 'logic' without begging it as something we 'agree' to. And that agreement becomes what we semantically mean by 'true' and 'same' and 'consistent', where what is not, ...the complementry rational system(s), have to be understood as completing all other possible things we define as agreed to be disagreable by the assignments. The pair of complementary logics are what is 'sound' with respect to Totality. And this is necessarily 'inconsistent' with priority to Totality as being valid and true with respect to it.

We are a 'consistent' world and should we find something lacking this consistency, then it just means that within a system of consistent rules, we deny what is contradictory in one world (the consistent one) and place the contradictory statement aside for the complement of this within Totality.

There we go! Vicious circularity. To speaking about the "consistency" of your rules is incoherent - you keep pre-supposing it.
Rules are rules. Either you followed them or you didn't.

IF you follow your rules you CANNOT ARRIVE AT A CONTRADICTION. No matter what you do!

And this is unsound with respect to Totatity (the total reality of all worlds) because it IS by necessity 'contradictory' and 'circular' if we are speaking about the whole or 'complete' picture of reality. So contradiction exists at the most fundamental level of reasoning and is only SELECTED out by dividing the world of things into symmetric complementary logics of which we arbitrarily select the ones we AGREE to among two or more people creating rational rule systems we call 'logical' when obeyed.

All we do when we create a system of logic on the most fundamental level is divide some more basic set of things in two and call the selected set 'well' (agreed to) while the rest is just set aside. Those systems that are complete over the domain of things we chose from the rules we create are what we call "logical" for practical purposes. But it doesn't mean that what is left out is non-existent in Totality. That is what the Incompleteness theorems are speaking about. We are con-fin-ed to de-fine rules between people communicating for being fin-ite beings and so while we choose to place what is not consistent to those rules we create aside, it doesn't imply that there is no greater system of 'logic' that includes those non-consistent things of the logic we agreed upon consistently.

Because we CAN create rules of a game, it doesn't mean that the particular game in question has to be rationally consistent. We just prefer when participating with each other that some set of rules need to be clearly defined for some practical purpose of satisfaction among participants choosing to play. Totality, on the other hand, accepts anything consistent or inconsistent to be 'complete'. So if we want to understand something about reality as some whole, we have to assume anything and everything as possible in some absolute universe and just narrow this down a subset of these to some particular universe of discourse we want in some agreement (consistently).

I can't make sense of your meaning of destroying information without you expecting that information is created in the first place. We don't 'create' information. We just take information as is and divide it into sets of things to be compared. If you want to pick something you consider 'non-symetrical', you are no longer being logical because all things require comparing some X to some Y as a PAIR of things at minimal. This pairing is a symmetrical regardless of how you define "some X" to "some Y" or you could not 'compare*' anything at all. ["-pare" is from the same meaning as 'pair' in this word. To be 'rational', for instance, comes from 'ratio' is a comparison of two factors. ]

Are you following?

[Scott Mayers]

Co-linking this post of a simultaneous thread of someone probably thinking of this discussion that I wrote: viewtopic.php?f=5&t=28463&p=442571#p442571

Using that example, with X = "My rationale is correct", we have,

Assume (1) X = "My rationale is correct"

Then (2) You CAN disagree by asserting not-X [meaning that you think my rationale is incorrect, incomplete, or something not satisfying to you.]

The (3) Both of these exhaustively exist in our shared Universal class of discourse. That is, both X and not-X exist.

X and not-X existing is a contradiction only where we EXPECT (X or not-X) to be the case which is implicit by (1) OR (2) to be true exclusively. But the inclusive truth is that both are true when considering the more COMPLETE picture of our shared universe of agreement potential.

[Skepdick]

When I use "sound", it refers to whatever is real and valid about Nature's logical value of what is or is not true, whatever that may be.

This does not compute for me. The logical value of a logical expression is only one of two things:
1. Whatever value YOU assign it (e.g x = 5, y = 10)
2. Whatever value is evaluated given the values YOU assigned to the inputs, and the rules you defined for evaluation,
plus(x, y) = 10

There is no such thing as "Nature's logical value" YOU are "Nature". YOU are the one assigning value to your terms.

The logical value of a logical term is whatever value you assign to it. It could be a Boolean. It could be a numeric. It could be a rose.

You appear to be interpreting circularity as something unsound with respect to Nature when it is only your own evaluation about the use of words speaking about logic in general.

"Nature" does not evaluate logical expressions. Humans do.

Since "logic" is necessarily about consistent systems of reasoning,

You have no idea what "necessity" or "consistency" are, except by AXIOMATIC DEFINITION.

If you axiomatically define "inconsistency" as (P and -P) <=> False, then any system that can deduce that is "inconsistent"
if you axiomatically define "inconsistency" as (P and -P) <=> True, then any system that can deduce that is "inconsistent".

Which one of the two definitions of "inconsistency" is the true definition?

The two systems are simply each others' inverse.
( (P and -P) )
(! (P and -P) )

AND Nature itself is 'sound' whether we agree to it or not, Nature doesn't have to play by rules.

I am not claiming that it does! You are projecting things onto Nature. I am talking about logic/reasoning as a human activity.
In as much as humans are part of nature, and humans play rule-following games - that is where it stops.

But the humans fail to recognize that in playing the rule-following games, the humans play TWO distinct roles.

Role 1: rule-setter (axiom definer)
Role 2: rule-follower (evaluator)

Humans switch roles without noticing - some times they make assertions from the rule-setter's perspective. Sometimes they make assertions from the rule-follower's perspective. As far as I care to define inconsistency - switching perspectives is it!

Things like circularity or "inconsistency" can be defined "consistent" and "sound" with respect to Totality because it is most INCLUSIVE of everything without bias to our perception of reality locally. The act of "defining" logic is itself begging something from our biased expectation of consistency being only an infinitesimal subset of Totality.

And what is "Totality" defined in respect of?

I cannot understand your presumption of any categorical errors on my part. Logic reduces to being 'explosive' with respect to Totality. I think personally that this contradiction is what defines existence itself.

Maybe. Lets unpack "Totality"

You can't even 'define' anything without assuming you CAN define as a behavior itself.

This is the rule-following paradox again. Are you sure that what you are DOING, by scribbling symbols on paper is "definition".
If definition is behaviour, then why are your symbols and your medium static - lacking any behaviour?

As far as I can tell programming is the act of "defining". My definitions have behaviour. My definitions are real, empirical phenomena.

Just think of how you could possibly define "sameness" in any precision without begging TWO distinct things as having some association to the term, "sameness".

If you succeed on that endeavor you are making a Metaphysical/Ontological error. Every time. TWO things are never "the same". They MAY be equal in value, but they will always have distinct identities.

A is itself.
B is itself.
A is not B.

A could be the same kind of thing as B. e.g they are both be roses.
A and B could be different kinds of things (A is a rose, B is a car) but they could have shared properties e.g color.

But one metaphysical/ontological truth remains always: A could never be "the same" as B.

I'm sure you understand this. But I don't know what point you are trying to convey in this thread about anything. What information is being 'destroyed' if it isn't demonstrated that something you think is symmetrical isn't initially 'constructed' in your own head by default? What does it mean to "destroy" information to you?

Given the representation: (x = x) = (x = x)
I can answer the question: Did the terms of the logical expression switch places?

Given the representation (x = x) = (x = x)
I cannot answer the question: Did the terms of the logical expression switch places?

Information is that which allows you to answer yes/no questions!
I have lost the ability to answer the yes/no question -> Information has been lost.

This is not "new and controversial". It is HOW abstraction works. You ignore details that are unimportant. You focus on details which are important.

IF spatial coordinates (ordering?) are important, symmetry destroys information.

By defining an axiom of symmetry one disallows asymmetry in the system. Asymmetry is necessary for information-flow.

Here is a system in which: (x = y) != (y = x)

https://repl.it/repls/WiltedOriginalButton

Code: Select all

from universe import x,y

assert ((x ==y) == (x==y))
assert ((y == x) == (y == x))
assert not ((x == y) == (y == x))

And these meanings are still incoherent without begging both as I demonstrate above using the word "same" versus "not same" that BOTH need begging simultaneously in a complementary pair of systems of associations inductively.

Neither system speaks about sameness though? You are dragging your definitions into my systems.

Both systems EVALUATE the logical expressions. One EVALUATES to True. The other EVALUATES to False.

This complementary pair of demonstrations are "symmetrically" essential to understand either "same" or "not same" as existing as independent logical systems that define 'value' in a higher ordered logic built with both of them collectively where we introduce 'value'.

No. Symmetry destroys difference! There are TWO DIFFERENT terms in the expression: x = x

How do I know? Because I made them that way! Unique and different from each other.

Note how the word "consistent" literally means "con-" (with) and "-sis-" (same) "tent" (tense)?

Gibberish. Words don't mean anything. Humans give meaning to words.

You can't escape the circularity at the lowest level regardless without assuming pairs of complementary 'logic' systems existing but that we choose one to be uniquely called "true" arbitrarily with its complement as being "not true" relatively.

OK, so you have two systems.

One system axiomatically says: 1 = 1.
The other system axiomatically says 1 != 1

Which one is "inconsistent"?

When you are USING Imperative logic ( https://en.wikipedia.org/wiki/Imperative_logic ) the question doesn't make sense.

We could assign Truth-value to EITHER expression.

So the only absolute about first order logic is that there is no 'logic' without begging it as something we 'agree' to. , ...the complementry rational system(s), have to be understood as completing all other possible things we define as agreed to be disagreable by the assignments. The pair of complementary logics are what is 'sound' with respect to Totality. And this is necessarily 'inconsistent' with priority to Totality as being valid and true with respect to it.

OK! Lets start then!

A: 1 = 1 is True
B: 1 != 1 is True

But it's not even as easy as that! In the above options hides exactly the issue of double negation. And it becomes trivial to see it when you use S-expressions

A. (= 1 1) is either True or False
B. (! (= 1 1)) is either True or False
C. (! (! (= 1 1))) is either True or False

In English: is the double negation of equality the same as equality?

We are a 'consistent' world and should we find something lacking this consistency, then it just means that within a system of consistent rules, we deny what is contradictory in one world (the consistent one) and place the contradictory statement aside for the complement of this within Totality.

How do you know that the Total universe is consistent? Oh that's right - it's your axiom

And this is unsound with respect to Totatity (the total reality of all worlds) because it IS by necessity 'contradictory' and 'circular' if we are speaking about the whole or 'complete' picture of reality.

Obviously. What else could we be speaking about. But if I give you a logical system which manufactures that which you call a "contradiction"

Then is it not empirically obvious to you that contradictions exist, not by any linguistic/logistic definition/notion. They EXIST. As real world objects.

The thing you are telling me does not exist "in reality" exists in reality. So who do I believe then? Logic or reality?

So contradiction exists at the most fundamental level of reasoning

No. Contradictions EXIST as spacio-temporal phenomena! Independent of human reasoning.

A computer program which EVALUATES a contradiction is not a figment of my reasoning.

It it is a product of my reasoning.

and is only SELECTED out by dividing the world of things into symmetric complementary logics of which we arbitrarily select the ones we AGREE to among two or more people creating rational rule systems we call 'logical' when obeyed.

You are still dancing around the point. You have no "rationality" predicate. If rationality is "following the rules of a logic system" then obeying ANY rule-based system is "rational".

(1 = 1) is rational. If obeyed.
(1 = 0) is also rational. if obeyed.

Which system should we obey?

Because we CAN create rules of a game, it doesn't mean that the particular game in question has to be rationally consistent.

Vicious circularity.You literally just defined "rationality" AND "consistency" as "following the rules of a game".
Surely this is common sense to you: If you follow the rules of any axiomatic game you cannot arrive at a contradiction!
If you cannot arrive at a contradiction then it cannot be said that you are "irrational" or "inconsistent"!

Every formalist knows this. Euclid started with a set of axioms and gave us Euclidian geometry.
Bolyai tried to deduce a contradiction by changing Eiclid's axioms: through a point, parallel to a given line, many lines could be drawn.
He didn't arrive at a contradiction. Riemann assumed that through a point, parallel to a given line, no line could be drawn. And he invented yet another geometry.

All you are doing is computation. The computer is not "irrational" or "inconsistent". The computer is an obedient rule-follower.

You are blaming the rule-follower for the rule-setter's failures.

We just prefer when participating with each other that some set of rules need to be clearly defined for some practical purpose of satisfaction among participants choosing to play.

Totality, on the other hand, accepts anything consistent or inconsistent to be 'complete'.

So what do you do if contradictions are part of Totality? That seems to go against your axiom...

I can't make sense of your meaning of destroying information without you expecting that information is created in the first place.

Contradictions create information.

If you logically deduced X, but you observed Y - that's a contradiction. New information.

You've learned that your logic is faulty.

We don't 'create' information.

We do. Every time reality contradicts our expectation. New information/knowledge is created.

We just take information as is and divide it into sets of things to be compared. If you want to pick something you consider 'non-symetrical', you are no longer being logical because all things require comparing some X to some Y as a PAIR of things at minimal.

Information necessitates asymmetry. You are not being logical to claim that it does.

When you are speaking about pairs of things you are literally talking about mutual information.

[Skepdick]

Co-linking this post of a simultaneous thread of someone probably thinking of this discussion that I wrote: viewtopic.php?f=5&t=28463&p=442571#p442571

Using that example, with X = "My rationale is correct", we have,

Assume (1) X = "My rationale is correct"

Then (2) You CAN disagree by asserting not-X [meaning that you think my rationale is incorrect, incomplete, or something not satisfying to you.]

The (3) Both of these exhaustively exist in our shared Universal class of discourse. That is, both X and not-X exist.

X and not-X existing is a contradiction only where we EXPECT (X or not-X) to be the case which is implicit by (1) OR (2) to be true exclusively. But the inclusive truth is that both are true when considering the more COMPLETE picture of our shared universe of agreement potential.

OK, but why stop there? In your shared universe of discourse you have no objective criterion for "correctness".
We have no objective definition for "truth", so your true may be my false and vice versa.

So now we have 4 systems.

exists(x) -> True
exists(-x) -> True
exists(x) -> False
exists(-x) -> False

Each one of the above could be correct OR incorrect. So now we have 8 systems.

Or if we consider that existence and non-existence are in our shared Universal class of discourse, we can double that up to 16

does-not-exists(x) -> True
does-not-exists(-x) -> True
does-not-exists(x) -> False
does-not-exists(-x) -> False

Double negation is constructive.

(!(!(X) != X

Every observed state-change in the system is 1 new bit of information. Lets say that you and I are disagreeing:

X = Scott
Y = Skepdick

Lets pretend for 2 seconds that () denotes the boundary of an open system (e.g a system that interacts with its surroundings).

Disagreement: (X != Y)

As the system transitions state from (X != Y) -> !(X = Y) observe how the !-symbol moves across the system boundary and into the Observer's awareness.

1 bit of new information. Eureka, says the observer (!) . Scott and Skepdick finally agree.

That's the leader election problem in computer science. Two agents agreeing on a value.
Which itself is a sub-class of general consensus problems.

Philosophy is like that.

A says (1 = 1).
B says (1 = 1).

When they switch places Philosophy is done - they are now arguing for the sake of arguing.

A says (1 = 1)
B says (1 = 1)

[Skepdick]

Locus Solum: From the rules of logic to the logic of rules

Locus Solum means "Only the location matters"

A is itself.
A is not A.

[Scott Mayers]

Skepdick,

I'll try to respond to some of the above but think that you are missing my own meaning, not necessary the underlying logic. You use different ways of going about this that is more 'top-down' thinking versus my own approach 'bottom-up'. That you begin from assuming 'time' and 'truth' as something defaulted to for human reasoning differs from my perspective. I approach logic in sync with the laws of physics because these too have some 'consistency' that defines our particular universe.

Your last post that speaks of 'position' is more to my point. Just think of 'time' as a 'position' as well. When you have the convention of "read from left to right", this arbitrary rule gets ignored. The fact that you can only look at one symbol at a time OR the whole statement as a complex symbol is also ignored.

I can assure you that I understand you but disagree with your particular language or approach. I find it incomplete if you can't interpret 'information' itself as conserved in Totality. I'll now look back and respond to some of the above.

[Scott Mayers]

When I use "sound", it refers to whatever is real and valid about Nature's logical value of what is or is not true, whatever that may be.

This does not compute for me. The logical value of a logical expression is only one of two things:
1. Whatever value YOU assign it (e.g x = 5, y = 10)
2. Whatever value is evaluated given the values YOU assigned to the inputs, and the rules you defined for evaluation,
plus(x, y) = 10

There is no such thing as "Nature's logical value" YOU are "Nature". YOU are the one assigning value to your terms.

The logical value of a logical term is whatever value you assign to it. It could be a Boolean. It could be a numeric. It could be a rose.

I use "soundness" in the way most (but not all) texts on logic mean. [I was actually surprised to discover that even some texts/logicians intepret these different!]

You say that "The logical value...." of which I have to stop you right there. You recognize the arbitrariness of assignment of rules to be allowed in particular logical systems. But you overlook that value itself is its own thing. In a Boolean Algebra (where the 'algebra' here means a particular subset of logic of values), binary is NOT the only thing to assign to something. "Boolean", though we usually reduce it to binary truth values, is actually a way to link the set of things we call 'values' to some other set. They don't even have to refer to 'truth', just as the computer logic on the level of the machine interprets these as charge quanity of electrons or the absence of them.

Your response to the word "soundness" here tells me that you miss my understanding of "soundness" with respect to Totality itself. It is hard for us to NOT use the binary 'true' versus 'false' ideas and why it is awkward to express what 'fits' in with Totality without using truth-values. The 'soundness' in logic that I understand deals with whether the actual variables in the system that get 'assumed' are actually FIT to some shared reality of those using the system, and to whether the premises used FIT to the conclusion. This 'fitness' begs what we mean by 'agreement' and this in turn begs what we mean as 'true'. Since any two humans engaged in some game can agree or not agree to play, the concept of truth is to what is shared when playing. But the 'soundness' would be like extending the meaning of the premises in the gamep as also fitting to something beyond just the game itself.

For instance, in a "two-player game", this game is 'sound' if there ARE two players by definition of the game. It doesn't mean one person or more than two cannot alter this to create a different game. But if the system is expected to be AS DEFINED, then the first input presumption requires that two players actually ARE playing this particular game. You may play chess by yourself but you don't presume the game is 'sound' as something you can prove to others that you 'won' or 'lost' the game because of this fact. It is not invalid to pretend that you are two players for whatever reason you are playing by yourself either. But it would just be unsound if you use your alteration of the rules as something 'fit' to the rules of the game as understood by outsiders looking in.

"Nature" is just my reference to reality apart from any particular human intervention. Physical laws to me are a form of Nature's Logic for some Universe (subset of Totality). And so....

You appear to be interpreting circularity as something unsound with respect to Nature when it is only your own evaluation about the use of words speaking about logic in general.

"Nature" does not evaluate logical expressions. Humans do.

Yes, "Nature", while not dealing with human expressions for communications uniquely, reality has "laws of physics" which act as its 'logical system' that includes us as a subset within it. This includes your meanig of logic as a communication mechanism for measuring abstract sentences. The semantic referents of the symbols with respect to reality apart from the system and how the premises link to the conclusion of a system are what 'soundness' is about. It means 'true' with respect to the semantic fitness to reality that the logical system is being used to measure and draw conclusions that we in turn take back to reality as meaningful.

Since "logic" is necessarily about consistent systems of reasoning,

You have no idea what "necessity" or "consistency" are, except by AXIOMATIC DEFINITION.

If you axiomatically define "inconsistency" as (P and -P) <=> False, then any system that can deduce that is "inconsistent"
if you axiomatically define "inconsistency" as (P and -P) <=> True, then any system that can deduce that is "inconsistent".

Which one of the two definitions of "inconsistency" is the true definition?

The two systems are simply each others' inverse.
( (P and -P) )
(! (P and -P) )

All you are doing is asserting something about the arbitrariness of the English words as symbols in their own right can be assigned. Sure, I can define the word "consistency" to mean the opposite. What point are you making that I already don't understand? Those words when used to discuss some particular logical system are NOT the system in disussion because they use a prior logic (a metalogic) to describe it. You are jumping from interpreting the system being discussed as an object to the metalogic as though it were a very part of the system in discussion. "Soundness" is a metalogical fitness of the inputs to the observed logic system to a greater outside system where "Validity" is the metalogical description of whether the premises of the observed logic fits to the conclusions within the rules of this observed system.

My agreement to the arbitrary nature of creating a system is not unnoticed. My point is THAT you can arbitrarily define some system and you will have a complementary class of systems that always exist. This collection of both the particular system in question to any or all of the complementary systems that can be imagined, are completely exhaustive, especialy if the given system and its complement are included in a more inclusive system. The greatest such 'system' of reasoning (the ultimate metalogical logic of all logics included in a universe) is one that is INCONSISTENT because it includes subsystems that are necessarily consistent and complete and all others that are relatively inconsistent. THAT is a universal fact.

Older logicians used to use the word, "trivial" to mean "any system that permits a third value besides true and false to exist", .....but NOT to mean that is didn't exist. The same was understood with the word, "contradictory", but was used to describe conclusions within some given system as having this third possibility. The "incompleteness theorem" was one that used a binary system of reasoning to speak about the metalogical truth about all logical systems OUTSIDE it. This theorem proved that you cannot have a universal logic that is COMPLETE without it being 'trivial' in the old fashion use of the term. That is, it means that with respect to Totality, it has to include BOTH the consistent and inconsistent set of systems which would 'completely' cover all factors about reality. That this could be proven FROM a 'consistent' logic but by default is already true in a trivial system (or "inconsistent logic") that respects contradiction, means that reality is CLOSED when we include the recognition that Totality is NECESSARILY (inconsistent AND consistent) that with respect to us being inside Totality but not Totality iself is "inconsistent".

That is,

Absolute reality is 'consistently' inconsistent, an essential circularity from our local perspective but still rationally 'consistent' from a God's-eye-view.

All that is needed to exhaustively COVER all logic systems is the three fundamental rules that assure that somewhere in ANY system is some factor ABOUT consistency, like 'identity', one about its complement, and a third that either rules out or places aside things that permit both to be consistent of the same reality.

Reality takes the view that the third law referencing 'contradiction', is permitted to be true or false depending on your particular system of logic.

I have a feeling that you are going to just disagree or tell me that I'm 'wrong' regardless of what I say. But even that infinite and unclosed form of responses you have at least FIT in with my own point. We cannot find agreement on what is or is not logical if we want to specify ONLY some rule that all logic MUST be 'consistent'.


The greater reality is that there will always be another disagreement to be made between us discussing this here EVEN if we agreed with each other. I say X. You then say no, not-X. Then I say both (X and not-X). You then let some Y = (X and not-X) as my latest argument and then say not-Y.

Then we continue this infinitely as though we cannot find agreement when we both agree and disagree. If we just 'agreed', the conversation ends, this thread closes, and once burried and forgotten, you, I, or some other person will re-raise this same topic of confusion and repeat the whole process again infinitely.

This trap in our discussion is my point and to me my rationale ironically 'closes' the issue WHEN the confusion between people discussing this topic never ends. But THAT realization itself [being that logic is unresolvable as a whole] is useful to understand reality about 'laws' including those of physics and the totality of all universes collectivley.

I DARE you do agree with me!!