Hello!
Yes, I am one of so many amateurs
who play around with logic
and try to find solutions for the “liar sentence”.
And as my university time (with mathematics, philosophy and informatics)
was some 20 years ago and as (in spite of once excellent marks)
I never was a profound mathematician,
there was little probability that of all people
I should discover something relevant there.
Yet “even a blind hen sometimes finds a grain of corn”
and today I am not sure whether I have found “something” or not.
What makes it more suspicius is that I had no genial idea
but simply combined and modified some existing ideas
and added one or two small ideas of my own.
The so modified logic I called “layer logic” (in German “Stufenlogik”),
and discussed it in internet forums (like this one).
Here I met a problem:
For interested laymen it was to mathematical and formal,
for scientists it was not formal and mathematical enough.
This may be the main reason,
why up to now no fundamental error or inconsistency was found.
My belief in layer logic grew,
when I learned that a German professor
(Prof. Dr. Ulrich Blau, reflection logic, “Reflexionslogik”),
had used a similar approach (some ten years earlier than me),
but only for logic and not using and expanding it for set theory and mathematics
as I have done.
Link to a book of Prof. Ulrich Blau (in German):
http://www.synchron-publishers.com/texte/05-philos/0508blau-t.html
Especially the “layer set theory” is very nice:
We have less axioms then in the classical Zermelo-Fraenkel set theory (ZF)
and the “set of all sets” and the “Russell set” are ordinary sets
and only one kind of infinity (that of the natural numbers) is needed.
The set of all natural numbers can be defined and an arithmetic is possible.
As I nowhere read about such a set theory,
some of my layer logic ideas could be new,
in spite of the first impression of many,
that it is “the same old story” of Russell,Tarsky and others
in perhaps “new bottles”.
Before going into details
let´s have a look into the advantages of layer logic:
What more could we do,
if the ideas of layer logic are valid?
There is a good chance that in layer logic and arithmetic
the incompleteness proofs of Kurt Gödel are no longer valid,
so that the layer theory can be consistent
and yet for every true statement in layer arithmetic a proof could exist.
There could be a new definition for (layer-)algorithms and (layer-)computers,
and the halting problem could be no more irresolvable.
(For practically solving it we probably would need a halting programm,
which to find even with layer logic will be a hard piece of work).
As there are much less contradictions in layer logic than in classical logic
(as different truth values in different layers are no problem)
most indirect proofs in any science would hold no more.
Such proofs are used for example in philosophy and physics and of cause in mathematics.
So I understand that science doesn´t wait with open arms for such a new logic.
(David Hilbert probably wouldn´t have liked the layer set theory
with only one kind of infinity and no more “Cantor´s paradise”,
he might have liked the new possibilities to handle the incompleteness proofs of Kurt Gödel).
As I still think that “layer logic” it is not ripe enough for universities or professional journals,
I once more try to discuss and improve it in internet forums.
But why do I show all this “gold and spices” to you?
Like Columbus I am not able to organize the journey
to the new continent of layer logic all by my own,
I need (mathematical and philosophical) craftsmen and ships,
and there you are “King Ferdinand and Queen Isabella”
or a daring crew and captain to me.
And like Columbus (and the early Heisenberg) I have some guessed ideas
and can formaly calculate whith layer logic,
but I do not fully understand what I have found,
I lack the meaning and interpretation of layer logic
and professional methods.
So much for openers,
now to layer logic itself:
Layer logic is a three valued logic (but that is not very important),
and meta-levels (or something similar) called “layers” are part of it (very important).
Here my axioms/rules for layer logic:
Axiom 0:
There is a inductive set T of layers: t=0,1,2,3,…
(We can think of the classical natural numbers, but we need no multiplication)
Axiom 1:
Statements A are entities independent of layers,
but get a truth value only in connection with a layer t, referred to as W(A,t).
(The layers are like an additional logical dimension)
Axiom 2:
All statements are undefined (=u) in layer 0.
VA: W(A,0)=u
(We need u to have a symmetric start.)
Axiom 3:
All statements in layers have either the truth value ´w´ (true)
or ´f´ (false) or ‘u’ (undefined).
Vt>0:VA: W(A,t)= either w or f or u.
Axiom 4:
Two statements A and B are equal in layer logic,
if they have the same truth values in all layers t=0,1,2,3,...
VA:VB: ( A=B := Vt: W(A,t)=W(B,t) )
Axiom 5:
(Meta-)statements M about a layer t are constant = w or = f for all layers d >= 1.
For example M := ´W(f,3)= f´.
Then w=W(M,1)=W(M,2)=W(M,3)=...
(Meta statements are similar to classic statements)
Axiom 6:
(Meta-)statements about ´W(A,t)=...´ are constant = w or = f for all layers d >= 1.
Axiom 7:
A statement A can be defined by defining a truth value for every layer t.
This may also be done recursively in defining W(A,t+1) with W(A,t).
It is also possible to use already defined values W(B,d)
and values of meta statements (if t>=1).
For example: W(H,t+1) := W( W(H,t)=-w v W(H,t)=w,1)
A0-A7 are meta statements, i.e. W(An,1)=w.
Although inspired by Russell´s theory of types, layer theory is different.
For example there are more valid statements (and sets) than in classical logic
and set theory (or ZFC), not less.
And (as we will see in layer set theory) we will have the set of all sets as a valid set.
And self referring statements and sets are allowed in layer logic and layer set theory.
Now a look onto the liar in layer theory:
Classic: LC:= This statement LC is not true (LC is paradox)
Layer logic: We look at: ´The truth value of statement L in layer t is not true´
And define L by (1): Vt: W(L, t+1) :=w iff W(L,t) -= w and W(L, t+1):=f else.
Axiom 2 gives us: W(L,0)=u
(1) with t=0 gives us: W(L,0) = u-=w -> W(L,1) = w
(2) with t=1 : W(L,1) = w -> W (L,2) = f
(3) with t=2 : W(L,2) = f-=w -> W (L,3)= w ; W(L,4)= f; etc.
L is a statement with different truth values in different layers,
but L is not paradox.
Another layer liar statement is
(1a): Vt: W(LA, t+1) :=w iff Vd>0: W(LA,d) -= w and W(LA, t+1):=f else.
LA is a meta statement of Axiom 6 and would have to be constant.
Therefore it is not a well defined layer statement and therefore no problem.
Last not least a look on layer set theory:
The differences may become more clear in layer set theory,
my favourite part of layer theory:
The central idea is to treat “x is element of set M” (x e M) as a layer statement:
It is true in layer t+1 that set x is element of the set M,
if the statement A(x) is true in layer t.
Equality of layer sets:
W (M1=M2, d+1) = W ( For all t: W(xeM1,t) = W(xeM2,t) , 1 )
Especially: W (M=M, d+1)=w for d>=0.
The empty set 0:
W(x e 0, t+1) := W( W( x e 0, t ) = w , 1 ) = f for t>=0.
The full set All:
W(x e All, t+1) := W( W( x e All, t ) = w v W( x e All, t ) = u v W( x e All, t ) = f , 1 ) = w for t>0 and =u for t=0.
So other than in most set theories in layer theory the full set is a normal set.
Axiom M1 (assignment of statements to sets):
W(x e M, t+1) := W ( W ( A(x), t ) =w1 v W ( A(x), t ) =w2 v W ( A(x), t ) =w3 , 1 )
with w1,w2,w3 = w,u,f
For every layer set M there exists a layer logic statement A(x) witch fullfils for all t=0,1,2, …:
W(x e M, t+1) = W ( W ( A(x), t ) = w v W ( A(x), t ) = f , 1 )
W(x e M, 0+1) = W ( W ( A(x), 0) = w v W ( A(x), 0 ) = f , 1 )
= W (u=w v u=f, 1 ) = f
Axiom 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 t=0,1,2,3,… holds:
W(x e M, t+1) := W ( A(x), t ) (or the expressions of axiom M1).
Definition M3 (definition of meta sets):
If F is a logical function (like identity, negation or f.e. FoW(xeM1,t) = W(xeM1,t)=w )
then the following equation defines a meta set M: (M1=M is allowed):
W(x e M, t+1) := W ( F o W(x e M1, t), 1 )
Consequences of the axioms and definitions:
In layer 0 all sets are u:
W( x e M, 0 ) = u (as all statements in layer 0).
In layers > 0:
W(x e M, t+1) := w if W ( A(x), t ) = w else W(x e M, t+1) := f
For all x and (normal layer) sets M holds: W(x e M, 1) = u (as W(A(x),0)=u).
For all x and meta sets M holds: W(x e M, 1) = w or –w
Last not least let´s look upon the Russell set:
Classic definition: RC is the set of all sets, that do not have themselves as elements
RC:= set of all sets x, with x –e x
In layer theory: W(x e R, t+1) := W ( W ( x e x, t ) = f v W ( x e x, t ) = u , 1 )
W(x e R, 0+1) = W ( W ( x e x, 0 ) = f v W ( x e x, 0 ) = u , 1 ) = W (u=f v u=u , 1 ) = w
Therefore W(R e R,1) = w
W(R e R,2) = W ( W ( R e R, 1 ) = f v W ( R e R, 1 ) = u , 1 ) = W (w=f v f=u , 1 ) = f
And so W(R e R,3) = w, W(R e R,4) = f , …
R is a set with different elements in different layers, but that is no problem in layer set theory.
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):
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)=f )
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) = f (no contradiction, as in another layer)
Second case: W(x0 e F(x0),t)= f 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)=f ) = if ( W(x e x),t)=f )
This is the layer Russell set R (I omitted the ´u´-value for simplification) -
and no problem.
That should be enough for a start
For those who can read German here links with further details (all three with similar content):
http://philo-welt.de/forum/thread.php?threadid=6331
http://www.ask1.org/fortopic20575.html
http://www.philosophie-raum.de/index.php/Thread/25787-Die-Logik-%C3%A4ndern-Wie-und-warum/
(You also may search in the net with “logic trestone” or “Logik Trestone”)
Here some open questions:
Where is the definition of layer logic technically not correct and how can it be improved?
Or if you think that this layer logic ship will sink or never swim, can you give reasons why?
Advanced questions:
What does a layer mean?
And what is the meaning or relation to reality
of a statement or a property,
that is true in layer 1 and false in layer 2?
How can we use layer logic in physics?
Does layer logic help to understand mind, body and consciousness?
Yours
Trestone
Like Peter I couldn't quite get through the "question." It's not really a question, but a poorly formatted paper without an abstract and extra line spaces. I suggest you write the paper and post it for review. If you have a real question (not a request to review your paper), formulate it succinctly.
You should reference G. Spencer Brown's laws of form:
http://www.amazon.com/Laws-Form-G-Spencer-Brown/dp/0963989901
https://en.wikipedia.org/wiki/Laws_of_Form
and point out what you are adding. Also of course you need to reference Godel, Church and Turing and show new implications for these theorems. If there is no implication for these theorems, then you probably just have a rehash.
Personally I believe many modern theorems are flawed because they do not first prove the theorem is not a liar's sentence, as you call it ... a proposition which leads to a contradiction and so does its opposite.
Hello,
I fear I did undestand only little of The "Laws of Form" by G. Spencer Brown,
but for me it did not look similar to my layer theory,
as I found no layers an especially no new logic dimesion/parameter with layers.
To me layer logic and layer set theory are no typed theory, as the objects und terms
have no type or hierarchy.
The types or layers that are used are only used for a new dimension of logic
that is added to the truth values.
If we look at "x e M" in layer set theory, neither x nor M has a layer or type
(and layer set theory only has sets an needs no classes).
But we need a layer or type to give (the whole relation) "x e M" a truth value:
It may be true in layer 1 and false in layer 2.
My formal writing to this is W(x e M,1) = w and W(x e M,2) = f.
(The funktion W is from the German "Wahrheitswert").
It are the properties (like truth values, beeing element of, or having aphysical property) that are related with layers, not the objects or terms.
We also culd say: A relation between a object and a property is depending on the layer:
Statement A has truthvalue true in layer 1 and false in layer 2
or the relation between statement A and the truthvalue "true" is true in layer 1 ant false in layer 2 (like the relation "x e M"):
W(A is true,1) = w , W(A is true,2) = f .
That is the main reason, why layer theory deals rather good with selfreference and paradoxes:
They are not excluded but handled different in different layers.
The Russel set and the set of all sets are different in layer logic, but both are rather ordinary sets in layer set theory.
The meta-rules/axiom A5 and A6 help us speaking about all statements and layers ("for all", "there exists" ) without having tu use infinity hierarchies.
All in all to me this looks quite different to existing theories,
but as I told I am not an experienced logician.
Yours
Trestone
Hello Peter,
my problem is that i do not know what you mean by "syntax" or "sematics"
as I am no logician (and my studies are 20 years ago).
My axioms / rules are to complement the classic rules,
so the lacking rules and definitions are mostly the classical ones.
I am looking for someone who is able to do both:
Understand my ideas and my bad formalism -
and who understands modern logic and its formalism
and who finds my ideas interesting enough
to help me in spite of all.
Yours
Trestone
I could say the same about layer theory as you said about the laws of form ... From reading your comments I don't see what it is really offering.
My own opinion as an electrical engineer on the laws of form may differ from others. It roughly describes physical systems. We can and regularly do hook up a logic circuit in a "liars paradox" or similar type arrangement. It makes an oscillator. Mathematically if a definite solution is assumed, this assumption is invalidated by the occurrence of an imaginary number in its calculation. Mathematically, such imaginary numbers have come to mean that an assumed real exponential is actually an oscillating function, sine and cosine, which can be described with real numbers.
So what is the result of "layers?" If it is to identify and handle logic propositions with non-trivial truth values, and if it is to correspond to reality in some way, would it not have to produce the same result in reality as complex exponentials, and the laws of form? So it would be a different calculus for the same problem and result. The use of different calculi is common, such as wave and matrix mechanics. Sometimes one will be easier to visualize and the other more general.
It will be hard for you to pursue your idea and convince anyone it is different or new unless you comprehend the existing related literature and carefully demonstrate in a paper how it is similar or different. No one will do this for you. I have tried to give a general roadmap as to what parts of the literature you might look at, but you would need to do the work.
Hello,
I think I start with explaining my axioms or rules of layer logic:
Axiom 0:
There is a inductive set T of layers: t=0,1,2,3,…
The layers have to be inductive, so that we can describe with finite describtions
the values for infinite (all) layers.
Axiom 1:
Statements A are entities independent of layers,
but get a truth value only in connection with a layer t, referred to as W(A,t).
(The layers are like an additional logical dimension)
This is the very heart of layer logic.
The layers are a kind of new dimension,
that I add to logic,
like the imaginary/complex numbers to the real numbers.
We now have not only true or false statements,
but a lot of more possible (layer) logic statements:
For example a statement can be “undefined” in layer 0, be “true” in layer 1 and “false” in layer 2 and so on.
Axiom 2:
All statements are undefined (=u) in layer 0.
VA: W(A,0)=u
(We need u to have a symmetric start.)
As we usually use induction to define the truth value of a statement
(with the help of already defined other statements or lower levels),
This axiom or rule helps us to get always a starting point.
As the value “true” or “w” and also the value “false” or “f” would be an unsymmetric starting point,
I choose to start wit “undefined” or “u”.
Therefore I have to use a logic with three truth values (but this is not so essential like the layers).
Axiom 3:
All statements in layers have either the truth value ´w´ (true)
or ´f´ (false) or ‘u’ (undefined).
Vt>0:VA: W(A,t)= either w or f or u.
Within a layer the truth value should be unique and exactly one of the three posibilties.
Axiom 4:
Two statements A and B are equal in layer logic,
if they have the same truth values in all layers t=0,1,2,3,...
VA:VB: ( A=B := Vt: W(A,t)=W(B,t) )
I am not so sure about this definition of equality.
If all truth values in all layers are equal, two statements should be equal.
Axiom 5:
(Meta-)statements M about a layer t are constant = w or = f for all layers d >= 1.
For example M := ´W(f,3)= f´.
Then w=W(M,1)=W(M,2)=W(M,3)=...
(Meta statements are similar to classic statements)
Here I have a look at statements about layer logic statements an about layers.
The idea is, that not all is relative to layers, but that for example the truth value in a layer is kind of “fixed”
and not dependent of a layer himself but true for all layers >= 1.
Therby I am trying to use a kind of classical logic for speaking about layers
and the meta statements.
Axiom 6:
(Meta-)statements about ´W(A,t)=...´ are constant = w or = f for all layers d >= 1.
Same as at axiom 5.
Axiom 7:
A statement A can be defined by defining a truth value for every layer t.
This may also be done recursively in defining W(A,t+1) with W(A,t).
It is also possible to use already defined values W(B,d)
and values of meta statements (if t>=1).
For example: W(H,t+1) := W( W(H,t)=f v W(H,t)=w,1)
This is the inductive or recursive definition of new layer logic statements I was talking about.
For we have to give a truth value for every layer t= 0,1,2,3,…
And that are infinitely many but I want to use only finite definitions.
And the statements and truth values used in this definitions should be well defined earlier.
My description may not be formally correct, but I hope it gives the idea.
In the example we have W(H,0)=u because of axiom 2.
W(H,0+1) := W( W(H,0)=f v W(H,0)=w,1) = W( u=f v u=w ,1 ) = W( f v f, 1 ) = f
W(H,1+1) := W( W(H,1)=f v W(H,1)=w,1) = W( w v f, 1 ) = w
W(H,2+1) := W( W(H,2)=f v W(H,2)=w,1) = W( f v w, 1 ) = w
and W(H,t)=w for t >=2.
A0-A7 are meta statements, i.e. W(An,1)=w.
As in axiom 5 this means, we can speak with a kind of classical (or layer constant) logic
about the axioms.
Yours
Trestone
Hello Peter,
thank you for explaining what is meant by "syntax" and "semantics".
But I doubt that I will be willing and able to do all this,
as I am not intending to produce a scientific correct paper,
as I just am a layman with an idea.
So I will have to wait for the (unlikely?) case of somebody,
who is interested enough to do the "real work".
At least I gave it a try ...
Yours
Trestone
Hello,
Here a new try to explain the basic idea of my layer logic
(sorry, still rather informal)
by the example of the “liar sentence/statement”:
In classic logic a statement is logically equivalent to the “true” – statement or the ”false”-statement
(or to the “undefined”-statement if we use three truth values).
In layer logic we have a more complex kind of truth values (as we additionally use layers):
The thruth value of a statement is a kind of infinite tupel or vector of (classical) truth values:
There is a (classical) truth value for every layer 0,1,2,3, …
For example the tupel (undefined, true, false, true, false, …) for the liar statement,
or with my abbrevations u:= undefined, w:=true and f:= false
the vector (u,w,f,w,f,w,f,…).
We have a lot more of different logical statements this way,
But it is not so easy to define those statements, as they have potentially infinite parameters.
My way out is to allow only finite definitions of statements,
usually by induction in layers or by recursive definitions (see axiom/rule 7).
(As I need no arithmetics within the layers, the “set” of all layers 0,1,2,3, … is (hopefully) not to complex).
Now to the liar statement:
The classical formulation “L:= This statement L is not true”
is not a layer logic statement,
as we need a layer for speaking about a truth value like true (or not true).
Therefore we have the corresponding layer logic liar statement:
“For all layers t= 0,1,2,3, …: LL:= This (layer logic) statement LL has in layer t+1 the truth value “true=w”,
if LL has in layer t not the truth value “true=w”.
and LL has in layer t+1 the truth value “false=f in all other cases (that is if it is true).
Short: For all t>=0: W(LL,t+1) := w (if W(LL,t) = u or f ) and
W(LL,t+1) := f else (that is if W(LL,t)=w).
As with axiom/rule 2 there is W(LL,0)=u, we have:
W(LL,1) = w if ( (W(LL,0) = u or f ) = w if ( u=u or f ) = w
W(LL,2) = w if ( ((W(LL,1) = u or f) and W(LL,2) = f if W(LL,1=w) = f
W(LL, 3) = w; W(LL,4)=f) etc.
The definition of LL is still a mixture of layer logic and classic logic,
but motivated by axioms/rules 5 and 7.
A more precise (but not more clear) definiton for LL (using only layer logic).
leading to the same results:
W(LL,0):=u
For all t>=0:
W(LL, t+1) := w if W (W(W(LL,t) = u, 1)=w or if W(W(LL,t)= f, 1)=w),1)=w ) else
W(LL, t+1) := f (that is if W(W(LL,t)=w,1)=w)
So the layer liar statement LL has the (alternating) layer truth vector (u,w,f,w,f,w,…)
and is an rather ordinary layer statement.
Classic statements have constant truth vectors (u,w,w,w,w,w,w,..) or (u,f,f,f,f,f, …) or (u,u,u,u,u,u, …).
Using the truth vectors and layers in logic we have a lot more possibilities,
similar to using complex numbers instead of natural or real numbers,
and most paradox or impossible statements (as often used in famous indirect proofs)
are a little more complex than ordinary statements but paradox or impossible no more …
Changing logic is of course like “biting the hand that feeds you”,
but trying new ways of thinking (especially for over-fifty-year-old like me)
is an interesting challenge.
Yours,
Trerstone
Hello,
yes, self-referencing statements like the liar statement are no problem to layer logic,
because it uses layers
(which on first look seem similar to the types of Russel, but are fundamentally different!).
There is one big "but":
It only works, if layer logic is taken serious and the whole argumentation is done with this new logic.
For example the diagonalization of Cantor is no more valid with layer logic.
And there are no more "true statements":
Layer logic statements are true or false (or undefined) only in combination with a layer.
The layer liar statement is alternating true and false in different layers,(not classically true or false or undefined)
and a welldefined layer logic statement.
It would of course be good to have a better formalization for layer logic,
but I am rather sure that my results will hold
if somebody is able to make up for it.
Yours
Trestone
Hello Peter,
thanks for the link to the "ten signs":
My posting seems to fail by #1, #5, #9 and #10 and partly at all other points,
so there is good reason to spare the time reading it - and I myself would probably not read it thoroughly too ...
Of course Cantor´s diagonalization is a valid (indirect) proof in classic logic and set theory. Same with the proof of Euklid, that the square root of 2 is no rational number.
What I ws trying to say is, if we switch our thinking to using (only) layer logic and layer set theory,
both proofs are valid no more.
And we would have to check all classic results,
whether the results and proofs are still valid - or if there are new and other results.
That`s "boone and bane" of a new logic.
And as long as it is not in contradiction with erveryday life,
I think it is worth a try.
Yours
Trestone
P.S. Is it better to read this way?
Hello!
(Here a "re-formated" try (no linefeeds) of my first postings:
(I originally imported them from a word file or answered directly here with the answer window -
except of some empty lines I could detect no problems when looking at the thread.
This time I tried using only shift-return)
Yes, I am one of so many amateurswho play around with logic
and try to find solutions for the “liar sentence”.
And as my university time (with mathematics, philosophy and informatics)
was some 20 years ago and as (in spite of once excellent marks)
I never was a profound mathematician,
there was little probability that of all people
I should discover something relevant there.
Yet “even a blind hen sometimes finds a grain of corn”
and today I am not sure whether I have found “something” or not.
What makes it more suspicius is that I had no genial idea
but simply combined and modified some existing ideas
and added one or two small ideas of my own.
The so modified logic I called “layer logic” (in German “Stufenlogik”)
and discussed it in internet forums (like this one).
Here I met a problem:
For interested laymen it was to mathematical and formal,
for scientists it was not formal and mathematical enough.
This may be the main reason why up to now no fundamental error or inconsistency was found.
My belief in layer logic grew, when I learned that a German professor
(Prof. Dr. Ulrich Blau, reflection logic, “Reflexionslogik”)
had used a similar approach (some ten years earlier than me),
but only for logic and not using and expanding it for set theory and mathematics
as I have done.
Link to a book of Prof. Ulrich Blau (in German):
http://www.synchron-publishers.com/texte/05-philos/0508blau-t.html
Especially the “layer set theory” is very nice:
We have less axioms then in the classical Zermelo-Fraenkel set theory (ZF)
and the “set of all sets” and the “Russell set” are ordinary sets
and only one kind of infinity (that of the natural numbers) is needed.
The set of all natural numbers can be defined and an arithmetic is possible.
As I nowhere read about such a set theory, some of my layer logic ideas could be new,
in spite of the first impression of many,
that it is “the same old story” of Russell,Tarsky and others in perhaps “new bottles”.
(Layer logic and set theorie also deals rather nice (and new?) with selfreference I think)
Before going into details
let´s have a look into the advantages of layer logic:
What more ore new could we do, the ideas of layer logic were valid?
There is a good chance that in layer logic and arithmetic
the incompleteness proofs of Kurt Gödel are no longer valid,
so that the layer theory can be consistent
and yet for every true statement in layer arithmetic a proof could exist.
There could be a new definition for (layer-)algorithms and (layer-)computers,
and the halting problem could be no more irresolvable.
(For practically solving it we probably would need a halting programm,
which to find even with layer logic will be a hard piece of work).
As there are much less contradictions in layer logic than in classical logic
(as different truth values in different layers are no problem)
most indirect proofs in any science would hold no more.
Such proofs are used for example in philosophy and physics and of cause in mathematics.
So I understand that science doesn´t wait with open arms for such a new logic.
(David Hilbert probably wouldn´t have liked the layer set theory
with only one kind of infinity and no more “Cantor´s paradise”,
he might have liked the new possibilities to handle the incompleteness proofs of Kurt Gödel).
As I still think that “layer logic” is (formally) not ripe enough for universities or professional journals,
I once more try to discuss and improve it in internet forums (as here).
But why do I show all this “gold and spices” to you?
Like Columbus I am not able to organize the journey
to the new continent of layer logic all by my own,
I need (mathematical and philosophical) craftsmen and ships,
and there you are “King Ferdinand and Queen Isabella”
or a daring crew and captain to me.
And like Columbus (and the early Heisenberg) I have some guessed ideas
and can intuitively and a little formaly calculate whith layer logic,
but I do not fully understand what I have found,
I lack the meaning and interpretation of layer logic
and professional methods.
So much for openers, now to layer logic itself:
Layer logic is a three valued logic (but that is not very important),
and meta-levels (or something similar) called “layers” are part of it (very important).
Here my axioms (or better called rules) for layer logic
(they are to substitute similar rules of classic logic,
all other classic rules or axioms are kept unchanged):
Axiom 0:
There is a inductive set T of layers: t=0,1,2,3,…
(We can think of the classical natural numbers, but we need no multiplication)
Remark: The layers have to be inductive, so that we can describe with finite describtions
the values for infinite (all) layers.
Axiom 1:
Statements A are entities independent of layers,
but get a truth value only in connection with a layer t, referred to as W(A,t).
(The layers are like an additional logical dimension)
Remarks: This is the very heart of layer logic.
The layers are a kind of new dimension,
that I add to logic, they also are a kind of perspective.
The truth value of a statement W(A,t) (W from German “Wahrheitswert”)
is depending from the statement and from the perspective / layer.
(Geetings to Immanuel Kant).
This new parameter/dimension “layer” is like the imaginary/complex numbers to the real numbers.
We now have not only true or false statements,
but a lot of more possible (layer) logic statements:
For example a statement can be “undefined” in layer 0, be “true” in layer 1 and “false” in layer 2 and so on.
Axiom 2:
All statements are undefined (=u) in layer 0.
VA: W(A,0)=u
(We need u to have a symmetric start.)
Remarks: As we usually use induction to define the truth value of a statement
(with the help of already defined other statements or lower levels),
This axiom or rule helps us to get always a starting point.
As the value “true” or “w” and also the value “false” or “f” would be an unsymmetric starting point,
I choose to start with “undefined” or “u”.
(Prof. Ulrich Blau did the same.)
Therefore I have to use a logic with three truth values (but this is not so essential like the layers).
Axiom 3:
All statements in layers have either the truth value ´w´ (true) or ´f´ (false) or ‘u’ (undefined).
Vt>0:VA: W(A,t)= either w or f or u.
Remark: Within a layer the truth value should be unique and exactly one of the three posibilties.
Axiom 4:
Two statements A and B are equal in layer logic,
if they have the same truth values in all layers t=0,1,2,3,...
VA:VB: ( A=B := Vt: W(A,t)=W(B,t) )
Remarks: I am not so sure about this definition of equality.
If all truth values in all layers are equal, two statements should be equal.
Axiom 5:
(Meta-)statements M about a layer t are constant = w or = f for all layers d >= 1.
For example M := ´W(f,3)= f´.
Then w=W(M,1)=W(M,2)=W(M,3)=...
(Meta statements are similar to classic statements)
Remarks: Here I have a look at statements about layer logic statements an about layers.
The idea is, that not all is relative to layers,
but that for example the truth value in a layer is kind of “fixed”
and not dependent of a layer himself but true for all layers >= 1.
Therby I am trying to use a kind of classical logic for speaking about layers
and the meta statements.
It is important to me, that layer logic (with some special rules)
is also used for the meta languague, so that there is one logic for all.
Axiom 6:
(Meta-)statements about ´W(A,t)=...´ are constant = w or = f for all layers d >= 1.
Remarks: Same as at axiom 5.
Axiom 7:
A statement A can be defined by defining a truth value for every layer t.
This may also be done recursively in defining W(A,t+1) with W(A,t).
It is also possible to use already defined values W(B,d)
and values of meta statements (if t>=1).
For example: W(H,t+1) := W( W(H,t)=f v W(H,t)=w,1)
Remarks: This is the inductive or recursive definition of new layer logic statements I was talking about.
For we have to give a truth value for every layer t= 0,1,2,3,…
And that are infinitely many but I want to use only finite definitions.
And the statements and truth values used in this definitions should be well defined earlier.
My description may not be formally correct, but I hope it gives the idea.
In the example we have W(H,0)=u because of axiom 2.
W(H,0+1) := W( W(H,0)=f v W(H,0)=w,1) = W( u=f v u=w ,1 ) = W( f v f, 1 ) = f
W(H,1+1) := W( W(H,1)=f v W(H,1)=w,1) = W( w v f, 1 ) = w
W(H,2+1) := W( W(H,2)=f v W(H,2)=w,1) = W( f v w, 1 ) = w
and W(H,t)=w for t >=2.
A0-A7 are meta statements, i.e. W(An,1)=w.
As in axiom 5 this means, we can speak with a kind of classical (or layer constant) logic
about the axioms.
Although inspired by Russell´s theory of types, layer theory is different.
For example there are more valid statements (and sets) than in classical logic
and set theory (or ZFC), not less.
And (as we will see in layer set theory) we will have the set of all sets as a valid set.
And self referring statements and sets are allowed and easily handled in layer logic and layer set theory.
Now a look onto the liar in layer theory:
Classic: LC:= This statement LC is not true (LC is paradox)
Layer logic: We look at: ´The truth value of statement L in layer t is not true´
and define L by (1): Vt: W(L, t+1) :=w iff W(L,t) -= w and W(L, t+1):=f else.
(“-=w” means: is not = w, that means is=f or is=u)
Axiom 2 gives us: W(L,0)=u
(1) with t=0 gives us: W(L,0) = u-=w -> W(L,1) = w
(1) with t=1 : W(L,1) = w -> W (L,2) = f
(1) with t=2 : W(L,2) = f-=w -> W (L,3)= w ; W(L,4)= f; etc.
L is a statement with different truth values in different layers, but L is not paradox.
(just alternating)
(Prof. Ulrich Blau uses a similar approach.)
Another layer liar statement is or could be:
(1a): Vt: W(LA, t+1) :=w iff Vd>0: W(LA,d) -= w and W(LA, t+1):=f else.
LA is a meta statement of Axiom 6 and would have to be constant.
Therefore it is not a well defined layer statement and therefore no problem.
Here we see, how the meta rules help to make the whole theory consistent,
they seem to be new and my own invention.
Last not least a look on layer set theory:
The differences may become more clear in layer set theory,
my favourite part of layer theory:
The central idea is to treat “x is element of set M” (x e M) as a layer statement:
It is true in layer t+1 that set x is element of the set M,
iff the statement A(x) is true in layer t.
Equality of layer sets:
W (M1=M2, d+1) = W ( For all t: W(xeM1,t) = W(xeM2,t) , 1 )
Especially: W (M=M, d+1)=w for d>=0.
Remark: Layer 1 can be used because of the meta axioms.
The empty set 0:
W(x e 0, t+1) := W( W( x e 0, t ) = w , 1 ) = f for t>=0.
The full set All:
W(x e All, t+1) := W( W( x e All, t ) = w v W( x e All, t ) = u v W( x e All, t ) = f , 1 ) = w
for t>0 and =u for t=0.
So other than in most set theories in layer theory the full set is a normal set.
Axiom M1 (assignment of statements to sets):
W(x e M, t+1) := W ( W ( A(x), t ) =w1 v W ( A(x), t ) =w2 v W ( A(x), t ) =w3 , 1 ,
with w1,w2,w3 = w,u,f
For every layer set M there exists a layer logic statement A(x) witch fullfils for all t=0,1,2, …:
W(x e M, t+1) = W ( W ( A(x), t ) = w v W ( A(x), t ) = f , 1 )
W(x e M, 0+1) = W ( W ( A(x), 0) = w v W ( A(x), 0 ) = f , 1 ) = W (u=w v u=f, 1 ) = f
Axiom 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 t=0,1,2,3,… holds:
W(x e M, t+1) := W ( A(x), t ) (or the expressions of axiom M1).
Remarks: M1 ann M2 show, that layer set theory is near to the old (naïve) Cantor set theory, only the layers are new.
Definition M3 (definition of meta sets):
If F is a logical function (like identity, negation or f.e. FoW(xeM1,t) = W(xeM1,t)=w )
then the following equation defines a meta set M: (M1=M is allowed):
W(x e M, t+1) := W ( F o W(x e M1, t), 1 )
Consequences of the axioms and definitions:
In layer 0 all sets have all other sets als u-elemnts:
W( x e M, 0 ) = u (as all statements in layer 0).
In layers > 0:
W(x e M, t+1) := w if W ( A(x), t ) = w else W(x e M, t+1) := f
For all x and (normal layer) sets M holds: W(x e M, 1) = u (as W(A(x),0)=u).
For all x and meta sets M holds: W(x e M, 1) = w or –w
Last not least let´s look upon the Russell set:
Classic definition: RC is the set of all sets, that do not have themselves as elements
RC:= set of all sets x, with x –e x
In layer theory: W(x e R, t+1) := W ( W ( x e x, t ) = f v W ( x e x, t ) = u , 1 )
W(x e R, 0+1) = W ( W ( x e x, 0 ) = f v W ( x e x, 0 ) = u , 1 ) = W (u=f v u=u , 1 ) = w
Therefore W(R e R,1) = w
W(R e R,2) = W ( W ( R e R, 1 ) = f v W ( R e R, 1 ) = u , 1 ) = W (w=f v f=u , 1 ) = f
And so W(R e R,3) = w, W(R e R,4) = f , …
R is a set with different elements in different layers, but that is no problem in layer set theory.
Canor´s diagonalisation:
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):
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)=f )
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) = f (no contradiction, as in another layer)
Second case: W(x0 e F(x0),t)= f 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)=f ) = if ( W(x e x),t)=f )
This is the layer Russell set R (I omitted the ´u´-value for simplification) - and no problem.
That should be enough for a start
For those who can read German here links with further details (all three with similar content):
http://philo-welt.de/forum/thread.php?threadid=6331
http://www.ask1.org/fortopic20575.html
http://www.philosophie-raum.de/index.php/Thread/25787-Die-Logik-%C3%A4ndern-Wie-und-warum/
(For more details you also may search in the net with “logic trestone” (in eglish) or “Logik Trestone” in German))
Here some open questions:
Where is the definition of layer logic technically not correct and how can it be improved?
Or if you think that this layer logic ship will sink or never swim, can you give reasons why?
Advanced questions:
What does a layer mean?
And what is the meaning or relation to reality
of a statement or a property,
that is true in layer 1 and false in layer 2?
How can we use layer logic in physics?
Does layer logic help to understand mind, body and consciousness?
Where else can it be used in philosophy?
Where else can it be used?
Yours
Trestone
Hello,
now I have a try with my input in an an attached file.
Yours
Trestone
Hallo,
here again my statements to Axiom 5 an Axiom 6:
"
Axiom 5:
(Meta-)statements M about a layer t are constant = w or = f for all layers d >= 1.
For example M := ´W(f,3)= f´.
Then w=W(M,1)=W(M,2)=W(M,3)=W(M,4)=...=w
(Meta statements are similar to classic statements)
Remarks: Here I have a look at statements about layer logic statements and about layers.
The idea is, that not all is relative to layers,
but that for example the truth value in a layer is kind of “fixed”
and not dependent of a layer himself but true for all layers >= 1.
Therby I am trying to use a kind of classical logic for speaking about layers
and the meta statements.
It is important to me, that layer logic (with some special rules)
is also used for the meta languague, so that there is one logic for all.
Axiom 6:
(Meta-)statements about ´W(A,t)=...´ are constant = w or = f for all layers d >= 1.
That means W( W(A,t)=w, d) = constant w or constant f for all layers d >= 1.
Ecspecially W( W(A,3)=w,3) = w (or=f).
The evaluatuion ist true for all levels d >= 1, not only for higher ones.
In this point layer logic is more classical than hierarchical.
Yours
Trestone
Hello,
meanwhile I had a look at the proof of Gödel´s first incompleteness theorem.
A central point seems to be a diagonalization argument. This argument is not valid any more when layer logic is used.
Therefore the following is possible:
Classic mathematics and constructive mathematics are incomplete,
but a new mathematics based on layer logic could be complete.
Perhaps this may inspire somone to have a second look on layer logic (even if it is still rather raw and clumpsy).
Yours
Trestone
Hello,
I am also interested in not technical answers,
like what is difficult to understand about layer logik
or what assoziations are started by layer logic?
Yours
Trestone
Hello,
Layer logic is in some ideas similar to the type theory of Russell.
Type theory was for many too artifical and “ad hoc”-motivated.
My layer theory seems to share a similar fate.
As far as I know type theory however did not open a way around the incompleteness proofs of Kurt Gödel,
so layer logic might be worth a second look.
Yours,
Trestone
Hello,
after two years waiting I am still believing in layer logic as an interesting approach.
And I still do not know what the layers are:
Just mathematical parameters/dimensions like in complex numbers, planes of reflection, meta levels or something totally different?
What ever they are, they could be the key to new knowledge.
And it looks like that I have to search alone for it ...
Yours
Trestone
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