layer logic - alternative for humans and aliens?

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Hello,

I still do not really know what the „layers“ are,

but cause and effect seem to give a hint:

For cause and effect are (classically) in a hierarchic order,

i.e. the cause has influence on the effect,

but not the other way round.

Same with the layers in layer logic:

A statement in layer t has a truth value and can contribute

to the definition of a truth value of a statement in layer t+1,

but not vice versa.

So we can assign causes to lower levels (like t) and effects to higher levels (like t+1).

With cause-and-effect chains we can construct (almost) arbitrarily high levels.

If we want to start a cause-and-effect chain,

We can use two specialities of layer logic:

On the one hand there is layer 0, the ultimate zero point,

i.e. every chain in layer logic has a natural starting point there

(and no infinite regress necessary).

On the other hand:

How ever high we are in a layer logic chain (with a statement to layer t),

we can come down easily:

We just use the meta statement „W(A,t)=w“, and this statement belongs to layer 1.

(Regarding my last holiday I call this “the Irish slide”).

I think that this resembles in some parts my intuitive understanding of the mind-body relation,

but this here only as a side note.

So even if there remain doubts about concret cause and effect relations (think of Hume!),

those relations are the best examples for “real” layers that I can give today.

Yours

Trestone

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Hello,

Up to now I had defined natural numbers in layer logic and set theory

by the following successor function m+:

To every set m we define a successor m+:

W(x e m+, t+1) := W(x e m, t) v W(x=m,1) ( for t=0 without W(x e m, t) )

The (adjusted) Peano axioms hold for m+.

We can define 0, 0+, 0++ etc. this way.

The so defined “natural numbers” m are not constant over layers:

In small layers t<m m has less elements than in large (where it becomes constant)

and similar to the classical natural numbers.

But we can use an alternative definition, that is not so hierarchical:

If m is defined in layer t+1, we can use values regarding m and layer t+1

to define m´ in layer t+1:

W(x e m+, t+1) := W(x e m, t+1) v W(x=m,1)

This definition is nearer to the classical natural numbers and I think we get sets, that are not not layer dependent.

I have not checked all Peano axioms yet.

We might do this overall and reduce the use of layer hierarchie to critical cases

(like self reference and undefinedparts).

Whether we get back some of the classical problems

(like Gödel´s uncompleteness theorem)

by this way

I do not see so far …

Yours

Trestone

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Hello,

perhaps some examples will help to show how layer logic works

1) a perception statement

A := „I see a red car“.

This classical statement can be true or false: W(A)=w or W(A)=f.

In layer logic this statement has a truth value in every layer t:

W(A,t) = w or f or u.

In practice we will not need different values in different layers for concrete perceptions

(and we will not need the third value u).

Therefore we have the spezial case in layer 0 as always: W(A,0)=u

and W(A,t)=w or W(A,t)=f for all t>0.

This is an explanation why we do not have to agree to a layer at concrete statements when determing a truth value.

2) Implicit undefined or self-referencing statement

B:= „This statement is not true“

In classic logic this statement is neither true nor false.

In layer logic we have to modify it slighty, as true is only valid with a layer:

SB:= „This Statement ist true in layer t+1, if it is not true in layer t (and false else)“

It is W(SB,0)=u. Therefore W(SB,1) = W(W(SB,0) -= w,1) = w

Therefore W(SB,2)= W(W(SB,1) -= w,1) = f , W(SB,3)= w, W(SB,4)= w etc.

Here we have a dependency on layers.

In practice the second kind of statements seems to be rather rare,

therefore layers seemed to be unnecessary,

and problems or paradoxes appeared only at the borders of the system.

Wether a logic with layers and less problems or paradoxes is a worthwile proposition

(at least for specialists) may be a matter of taste …

Yours

Trestone

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Hello,

Imagine,

the SETI-Project has reached contact to something about 5 light years
in distance
and we switch some of the first years, so the communication with the aliens
is in English.

One of the researchers (“SEARCH”) is logician and mathematician,
as those fields are supposed to be of universal validity.

Here the protocol of the (nearly) first contact.

Yours
Trestone:

SEARCH: “Hello ALIEN, we are especially interested in your logic
and mathematics
and wether they are different to ours?”

ALIEN: “Hello SEARCH, we do not have one logic or mathematics.
We use different ones for different purposes.”

SEARCH: “Can you give me an example for such a logic?”

ALIEN: “Just give me some problems you want to handle, and we will find
a suitable logic for you.”

SEARCH: “First all statements should be either true or false and implications
can be evaluated by analyzing the components.
It should help for consistant argumentation and reasoning.”

ALIEN: “Human classical logic would be a good choice,
but not all statements
would be either true or false.

By the way we use this logic in communicating with you.”

SEARCH: “With the execeptions, do you think of statements like
the liar statement:“This statement is not true”?”

ALIEN: “Yes, and with this logic you will have mathematical restrictions like
the incompleteness theorems of Kurt Gödel or the set of all sets
being no set.”

SEARCH: “You know Kurt Gödel?”

ALIEN: “We studied all that you have sended to us.”

SEARCH: “As we tried to do. Could you show me a logic without
the restrictions you mentioned?”

ALIEN: “You could do it easily yourself: The logic “everything is true””.

SEARCH: “Ok, that is true, but I meant a more useful example for practical
purposes?”

ALIEN: “We tried a “joke”!
A logic of the kind you asked for is not to complicated
but a little bit technically boring.
You have to use additional dimensions.
It is similar to solving the square root of -1
with complex numbers.”

SEARCH2: “Just try do explain it to me. By the way I am a new human being,
as my collegue died of old age.”

ALIEN: “Hello SEARCH2!
Perhaps we should give longer answers to you …
For analyzing all three problems indirect proof
is classically used.
So there are statements which would be simultaneously true
to their negations.
In the new logic these statements
(or more precisely their truth values)
are in another dimension than the negations.

We call this dimensions layers and the logic “layer logic”.
There are indefinitly many layers k=0,1,2,3,…
and every statement has a truth value in every layer.
The truth values can be different in different layers.

Classic statements are similar to layer statements
that are constantly true
(=T) or constantly false (=F) in all layers greater than 0.

In layer 0 all layer statements are undefined
(=U, a symmetrical starting)
and we have “undefined” as a third truth value in all layers.

All layer statements need a truth value in every layer
and truth values do only exist for the combination
of statements and layers.
Truth values can be defined recursivly using
and smaller layers.”

SEARCH2: “Let us try an example,
the statement “This statement is not true”.”

ALIEN: “First we have to add layers,
as a statement alone has no truth value:
“This statement L is not true in layer k”.
Now we have to define a truth value for L in every layer.
We do this by defining when L is true for every layer k+1
depending on the truth value of L in layer k:
For every k=0,1,2,…:
L is true in layer k+1 if L is not true in layer k
and L is false else.
With v(L,k)=T for “L has truth value true in layer k”:
v(L,k+1):=T IF ( v(L,k)=F or v(L,K)=U ) ELSE v(L,k+1):=F

We have v(L,0)=U as all statements are undefined in layer 0.
v(L,0+1):=T IF ( v(L,0)=F or v(L,0)=U ) ELSE v(L,0+1):=F
v(L,0+1):=T IF ( U=F or U=U ), therefore v(L,1)=T

v(L,1+1):=T IF ( v(L,1)=F or v(L,1)=U ) ELSE v(L,1+1):=F
v(L,1+1):=T IF ( T=F or T=U ) ELSE v(L,1+1):=F,
therefore v(L,2)=F

So we have v(L,0)=U, v(L,1)=T, v(L,2)=F, v(L,3)=T, v(L,4)=F, …

SEARCH2: “What does this mean for the original liar statement,
is it true or false?”

ALIEN: “Not all layer statements are classical statements,
the liar statement is one of those nonclassical statements.
It has no classical truth value, but is a normal layer statement
with alternating truth values.
It is like a complex number that is not real.
To get the benefits of layer logic you have to use it.

SEARCH2: “But it is not easy for me to change to a new logic,
for example if we talk about it we should use a known logic.”

ALIEN: “Fortunately we can use human classic logic
as this logic is the meta logic of layer logic.”

SEARCH2: “Is layer logic similar to the theory of types
by Bertrand Russell?”

ALIEN: “In the theory of types objects are splitted into differend types
and the types are used to avoid self reference within objects.
In layer logic the truth values are splitted into different layers
and the layers enable us to have self reference
within objects and statements.
So the answer is mostly no.”

SEARCH2: “Can you give an example for sets and self reference?"

ALIEN: “So let us have a look on layer set theory,
a rather nice piece of work.

The central idea is to treat “x is element of set S” (x e S)
as a layer statement:
It is true in layer k+1 that set x is element of the set S,
iff the statement A(x) is true in layer k.
v(x e S,k+1) :=T if v(A(x),k) = T (and F or U else).
And as in the original theory of Cantor
for every set statement A(x)
there exists a set.

We have the following two rules for sets:

Rule M1 (assignment of statements to sets):

For all k,sets x,set M exists a set statement A(x) which fulfills:
v(x e M, k+1) := v(v( A(x), k)=w1 v v(A(x), k)=w2 v v(A(x), k)=w3,1)
with w1,w2,w3 = T,U,F or one or two of them.

Rule M2 (sets defined by statements):

For every layer logic statement A(x) about a layer set x
there exits a layer set M so that for all k=0,1,2,3,… holds:
v(x e M, k+1) := v( A(x), k ) (or the expressions of rule M1).

The empty set 0:

We use “x e 0” as A(x)
For all k>=0: v(x e 0, k+1) := v(v( x e 0, k )=T,1) (=F for k>=0)

v(x e 0, 0+1) := v( v( x e 0, 0 ) = T, 1) = v( U = T , 1 ) = F
v(x e 0, 1+1) := v( v( x e 0, 1 ) = T, 1) = v( F = T,1) = F, etc.

The full set All:

v(x e All, k+1) := v( v( x e All, k ) = T v v( x e All, k ) = U v
v( x e All, k ) = F , 1 ) = T
for k>0 and =U for k=0.

v(x e All, 0+1) := v( v(x e All, 0) = T v v(x e All, 0) = U v
v v(x e All, 0) = F, 1 ) =
= v( U = T v U = U v U = F , 1 ) = T
v(x e All,1+1) := v(v( x e All, 1) = T v v(x e All, 1) = U v
v v( x e All, 1) = F , 1 ) =
= v( v( T = T v T = U v T = F , 1 ) = T, etc.

So other than in most set theories in layer theory
the full set is a normal set.”

SEARCH2: “What is with the Russell set, the set of all sets
that are not elements of themselfes?"

ALIEN: “We translate the definition of the Russell set R
to layer set theory:

v(x e R, k+1) := v( v( x e x, k ) = F v v( x e x, k ) = U , 1 )

v(x e R, 0+1) = v( v( x e x, 0 ) = F v v( x e x, 0 ) = U , 1 ) = T
(U=F v U=U , 1 ) = T ; therefore v(R e R,1) = T

v(R e R,2) = v( v( R e R, 1 ) = F v v( R e R, 1 ) = U , 1 ) = F
(T=F v F=U , 1 ) = F; therefore v(ReR,3) = T, v(ReR,4) = F, ...

R is a set with different elements in different layers,
but that is no problem in layer set theory, so R is a layer set."

SEARCH2: “I suppose that Cantor´s diagonalization in layer theory
is not valid any more?”

ALIEN: “You are right.
The set of all sets All is in bijection (via identity)
with its power set.
So we do not need different kinds of infinity
in layer set theory.

But let us have a look into the proof of Cantor,
transferred to layer theory:

Be S a set and P(S) its power set and F: S -> P(S)
a bijection between them (in layer d).
Then the set A with v(x e A, k+1) = T :=
if ( v(xeS,k)=T and v(xeF(x),k)=F )
is a subset of S and therefore in P(S).
So it exists x0 e S with A=F(x0).
First case: v(x0 e F(x0),k)=T , then v(x0 e A=F(x0), k+1) = F
(no contradiction, as in another layer)
Second case: v(x0 e F(x0),k)= F then v(x0 e A=F(x0),k+1) = T
(no contradiction, as in another layer)

If we have All as S and identity as Bijektion F
we get for the set A:
v(x e A, k+1) = T := if ( v(x e All,k)=T and v(x e x),k)=F ) =
= if ( v(x e x),k)=F )

This is the layer Russell set R
(We omitted the ´u´-value for simplification)
- and no problem.”

SEARCH2: “And can we still do arithmetics?”

ALIEN: “Yes, mostly as usual, sometimes in a special way.

We can define the successor m+ of a set m in the following way:
v(x e m+, k+1) := v(x e m, k) v v(x=m,1)
For k=0 without v(x e m, 0): v(x e m+, 1) := v(x=m,1)

v(0+,1) = v(x=0,1): In layer 1 the only element of 0+ is 0.
v(x e 0+, 1+1) := v(x e 0, 1) v v(x=0,1) = F v v(x=0,1).
v(x e 0+,2+1) :=v(x e 0,2) v v(x=0,1)= F v v(x=0,1) = v(x=0,1)
So 0+ is a set with only element 0 in all layers >=1.

Now we look at m=0+
v(x e 0++, 1) := v(x=0+,1):
In layer 1 the only element of 0++ is 0+.
v(x e 0++, k+1) := v(x e 0+, k) v v(x=0+,1)
In all layers >1 the only elements of 0++ are 0 and 0+.

So we find:
n+ contains in layer 1 exactly the element n
n+ contains in layer 2 exactly the elements n, n-1
n+ contains in layer n exactly the elements n, n-1, …,1
n+ contains in layer k>n exactly the elements n, n-1, …,0

For large k the natural numbers of layer set theory
are therefore similar to the classical natural numbers.

The (adjusted) Peano axioms hold for m+.

We can define 0, 0+, 0++ etc., (the natural numbers) this way.

The addition of numbers we define using the successors:

v(x e n + m+, k+1) := v(x e (n+m)+, k+1) =
= v(x e (n+m),k) v v(x=(n+m),1)

Multiplication:

v( x e n*m+, k+1 ) := v( x e n*m + n, k+1) =
= v(x e (n*m + n-1)+, k+1 ) =
= v( x e (n*m + n-1), k) v v(x = (n*m + n-1),1)

v(x e 2*2+, k+1 ) =v(x e 2*2+2, k+1 ) =v(x e (2*2+1)+, k+1)=
= v( x e 5, k) v v(x=5,1)"

SEARCH2: “Can you give me more details in a special paper?”

For first fundaments look at a Review of the logic
of Prof. Ulrich Blau
( as it is a pdf-file, you may have to put this URL directly in your browser:

https://wwwmath.uni-muenster.de/u/rds/blau_review.pdf )

and for layer logic at a thread by Trestone at ResearchGate:

Or you may search “the net” with “layer logic “Trestone”“
or with “Stufenlogik Trestone” (in German).

The symbolization there is slightly different:
W(A,t) is used instaed of v(A,k).

There still is no academic paper for layer theory –
perhaps someone is interested to do this?”

SEARCH2: “It will probably not be me, as my time is fading out …"

AlIEN: “Hello SEARCH2, you did not ask a question?”

ALIEN: “?”

ALIEN: “Here an aspect that might be interesting for philosophers:
The Münchhausen foundation trilemma
(Agrippa`s trilemma),
that there are only three poor choices to fundament
and start our argumentations
gets a new option with layer logic:
If we assume that a reason has to be true
in a higher level than the founded,
the reasoning can go back not further than to layer 1.

As every reasoning reduces the layers at least for 1,
starting at an arbitrary layer we reach layer 1
after finite steps.”

ALIEN: “?”

ALIEN: “Hello, is there anybody out there
interested to continue this communication?”

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Hello,

I am back from a holidy in Finland at the polar circle.

P.S. You do not have to wait ten years with your answers ...

Yours
Trestone

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Hello studiot,

My question is, if layer logic is a consistent alternative to classic logic - or if there are some deeper faults or incomprehensible parts.

The alien story is a kind of "advertising", as there seems to be little interest in discussing a new logic ...

Yours
Trestone

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Non-boolean logic exists. Can you concisely explain your idea?

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Hello endy0816,

the main new idea of my logic is that truth has (or can have) different layers.
Therefore a statement can be true in one layer and false in another.

Or with another view: Properties can be layer dependent -
in every layer we can have a different world.

Of course we do not experience this layer differences in our everyday life
and most things seem to be constant over layers and only one world.

But with borderline phenomena like infinity, mind - body interaction, consciousness, etc. this could be different.

The layers open a lot of possibilities to avoid contradictions
which restrict classical logic.

The main definitions I have given in my opening thread and the link at its end.

Like with complex numbers we can sovle problems with layer logic,
that can not be solved by classic logic (or other logics I know).
For example Gödels incompleteness theorems are not valid any more,
but natural numbers and a (in some parts different) arithmetics are definable.

As my studying at university is about thitrty years ago,
I can not give a representation of layer logic in modern state of the art,
perhaps somebody will try this?

Yours
Trestone

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Hello,

it looks as layer logic is "too human for aliens and too alien for humans."

I take this as an encouragement.

Yours,

Trestone.

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How does this differ from modal logic?

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Hello ydoaPs,

Layer logic is more similar to (a modified) classical propositional logic than to modal logic.
It does not handle modalities or possibilities.
It uses three truth values „true“, „false“ and „undefined“.
The most important feature are the layers.
A propostion in layer logic does not have one truth value but has a truth value in every layer 0,1,2,3,... ,
and different truth values in different layers are allowed.
So in a way every proposition has an infinite truth vector.
The layers are like additional new dimensions and allow new handlings of contradictions.

Yours
Trestone

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it looks as layer logic is "too human for aliens and too alien for humans."

I take this as an encouragement.

Really? That makes it sound useless to me: no use to aliens and no use to humans. Why is that encouraging?

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There's Ternary and above.

https://en.wikipedia.org/wiki/Three-valued_logic

There's Gödel's incompleteness theorems to consider, so I'm not sure this gets us anywhere new.

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Really? That makes it sound useless to me: no use to aliens and no use to humans. Why is that encouraging?

Hello Strange,

the fewer use layer logic has for humans and aliens the more free I am in my research.

And as humans, aliens and layer logic are probably changing,

it is not impossible that sometimes it will become usefiull ...

Yours

Trestone

There's Ternary and above.

https://en.wikipedia.org/wiki/Three-valued_logic

There's Gödel's incompleteness theorems to consider, so I'm not sure this gets us anywhere new.

Hello Endy0816,

layer logic uses three truth values, but that is not so important.

More important is the use of layers.

That layers give a new look on indirect proofs:

If different layers are involved there are no longer contradictions.

The diagonalization of Cantor does not work with layer logic.

I have strong indications that Gödel´s incompleteness theorems are not valid with layer logic,

but I did not proof this up to now (help welcome).

Changing of logic means to change a lot of things …

Yours

Trestone

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Hello ydoaPs,

Layer logic is more similar to (a modified) classical propositional logic than to modal logic.

It does not handle modalities or possibilities.

It uses three truth values „true“, „false“ and „undefined“.

The most important feature are the layers.

A propostion in layer logic does not have one truth value but has a truth value in every layer 0,1,2,3,... ,

and different truth values in different layers are allowed.

So in a way every proposition has an infinite truth vector.

The layers are like additional new dimensions and allow new handlings of contradictions.

Yours

Trestone

If there's no relating the layers to each other, what's the point? If there is, then it's just modality with another name.

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If there's no relating the layers to each other, what's the point? If there is, then it's just modality with another name.

Hello ydoaPS,

in most cases the truth values of propositions in a layer are defined

using the truth values of propositions in lower levels.

But it is possible to define the truth value of a proposition in some layers independently.

As I do not see the similarity of layer logic to modal logic,

can you give some details?

Yours

Trestone

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Hello,

merry Christmas
and a new year 2017 (on earth)
and 2022 (on your alien planet)
full of peace

Yours
Trestone

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Hello,

more than a year has gone,

but I am still exploring layer logic, mostly in German.

Here an older link for layer logic at a thread by Trestone at ResearchGate:

Or you may search “the net” with “layer logic “Trestone”“
or for more actual sides with “Stufenlogik Trestone” (in German).

Yours,
Trestone

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Hello,

in the meantime I developed layer logic further and tried to apply it to philosophical and physical questions.

In German you cand find here most details:

For example I found a solution of the mind - body problem
by doing a special interpretation of quantum theory:

If particels or quants have several possible ways from start to target,
invisible "virtuel possible" particles will go all the ways to the target,
and then (still invisible and virtuel) come back to the start, reverse in time,
bringing back informations about the (future) target.
As they are in the same layer as when started, this virtual informations can not be read by the start.
Therefore in physical quantum movements, one of the returning particles has to be selected blindly, and this will become the real particle.
So we unterstand quantum contingency better.

We already learned that the physical world has a universal layer, that increases with every interaction (except gravitation).

Now I assume, that the mind belongs to the infinite layer.
If body and mind are onnected in the nervous system,
the mind can "read" all quantum informations, especially the informations of the target.
He therefore can choose "conciously" and not "blind".

In this way the mind can act, but he can only choose possibilities,
that the body also could have chosen by chance.

Another point is to connect gravity (distortion in space-time) with the mind.
As there is mostly a combination of body and mind,
the gravity effects could be between minds (in the infinite layer).
Dark matter could be "pure mind".

But this all is rather speculative of course ...

Yours
Trestone

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Hello,

here my proof that Cantors diagonalisation ore different infinities are no more valid with layer logic.

As All, the set of all sets, is a set in layer theory, it is no surprise,

that the diagonalisation of cantor is a problem no more (I just give the main idea,
more details in the link below)

(t marks the layers, W(x,t) ist the truth value of x in layer t).

Be M a set and P(M) its power set and F: M -> P(M) a bijection between them (in layer d)

Then the set A with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w )

A is a subset of M and therefore in P(M).

So it exists x0 e M with A=F(x0).

First case: W(x0 e F(x0),t)=w , then W(x0 e A=F(x0), t+1) = -w (no contradiction, as in another layer)

Second case: W(x0 e F(x0),t)= -w then W(x0 e A=F(x0), t+1) = w (no contradiction, as in another layer)

If we have All as M and identity as Bijektion F we get for the set A:

W(x e A, t+1) = w := if ( W(x e All,t)=w and W(x e x),t)=-w ) = if ( W(x e x),t)=-w )

This is the layer Russell set R (I omitted the ´u´-value for simplification)-

and no problem.

(R is a regular set in layer set theory).

So in layer theory we have just one kind of infinity – and no more Cantor´s paradise …

Yours
Ttrestone

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On 9/18/2011 at 1:37 PM, Trestone said:

but true or false related to a viewing angel

Missed this. This really says it all. A viewing angel is telling me from nth layer that you're mistaken.

Cheers

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Posted (edited)

Hello Trestone,

I see that you first posted a proposal for something called layer logic before I was a member.

Today Search engines don'e sem to have given the idea much if any traction.

So perhaps I can ask if the layer logic is anything like what is called the 'layer model' in computing ?

I note your notation/presentation seems to be an adaptation of normal mathematics and (computer) programming.

Edited by studiot

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!

Moderator Note

Similar nonsense merged. And moved to Speculations.

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Posted (edited)

Hello studiot,

layer logic is an outsider (mine) theory to logic.

About 15 years before me Ptrofessor Ulrich Blau in Munic had similar ideas, he called it “reflexion logic”.
For him layers were the times we reflected about a sentence (like the liar L “this sentence is not true”.
Layer 0: no reflection. L has the truth value “undefined”.
Layer 1: We reflect, that L was undefined in layer 0, therefore it is true.
Layer 2: We reflect on our reflection: L is false. Layer 3: L is true. And so on.

I defined layers for all kind of logic sentences (proposals).
But I do not know so exactly, what my layers are:
Are they meta layers of logical speech, layers of causality or a new dimension or something else?
Anyway as an idea they open a new look on logik and the world.
And most famous proofs as by Cantor and Gödel or Turing are not valid with layer logic anymore.

I see no connection to the “layer model” in computing except the name.

But perhaps it is interesting for you,
that a computer that would use layer logic would not be limitid by the Halting Problem.
In the indirect proof we get different layers – and so there is no more a contradiction.
Unfortunatelly I do not know how to built a layer computer,
so layer logic is more a philosophical theory
(and that was what i intended when I started it 15 years ago ...)

Yours
Trestone

Edited by Trestone
dopple lines

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