Liars don't always lie – using layer logic?
Hello,
the liars sentence is a famous antinomy / paradox of classical logic:
L:= „This statement is not true“
If we assume, that L is true, then because of the definition of L it has to be not true (=false).
If we assume, that L is not true, then because of the definition of L it has to true
So in both cases L is true and false at the same time, and that is not allowed.
Most solutions of the problem do not allow sentences like L and especially no selfreferences.
But there is an other posiibility: A new logic called „layer logic“ or should I say „liar logic“?
The idea with layer logic is, that every determination of a truth value
belongs to a layer (0,1,2,3,...),
and different truth values are allowed in different layers (whatever layers are – see beyond).
As layers are „blind“ to themselfes and to higher layers,
we always have to use a higher layer if we „talk“ about a layer.
In the rules of layer logic we have this formulation of the liar L:
„For all k=0,1,2,3,...: This proposition LL is true in layer k+1, if LL is not true in layer k
and LL is false in layer k+1 else.“
In layer logic all propositions are „undefined“ in the lowest layer 0.
Therefore LL is undefined in layer 0.
Layer 1: This proposition LL is true in layer 0+1, if LL is not true in layer 0
and LL is false in layer 0+1 else..“
Therefore LL is true in layer 1.
Layer 2: This proposition LL is true in layer 1+1, if LL is not true in layer 1
and LL is false in layer 1+1 else..“
Therefore LL is false in layer 2.
Layer 3: This proposition LL is true in layer 2+1, if LL is not true in layer 2
and LL is false in layer 2+1 else..“
Therefore LL is true in layer 3.
Layer 4: This proposition LL is true in layer 3+1, if LL is not true in layer 3
and LL is false in layer 3+1 else..“
Therefore LL is false in layer 4.
We see that LL has an alternating truth value with increasing layers – there is no contradiction.
As propositions belong to all layers, LL is self-referential but not within a layer.
And what is with liars that speak about all layers?
„LA:= This proposition is not true in all layers“
In layer logic meta propositions about layers and truth values have to be nearly classic:
They can only be true or wrong and have to have the same truth value in all layers >=1.
So if LA is true in layer 1 it has to be true in all layers, so LA is false – in all layers, that would be a contradiction.
If LA is not true in layer 1 it has to be not true in all layers, so LA is true – in all layers, that would be a contradiction.
Therefore LA is not an allowed meta proposition in layer logic.
Maybe there are better examples for extended liars?
The layers may look somehow strange at first glance.
The layers were first just a formal parameter to differentiate truth values
(for example in the first part of a proof and the second).
Meanwhile I see them as a kind of a new dimension, of meta levels or cause and effect order
or a new part of time.
Even without knowing exactly what a layer is, we can use layer logic and layer set theorie,
as the rules for using them are mostly independent of this.
In a restricted way Professor Ulrich Blau in Munich invented a logic with layers
some 20 years before me, the reflection logic.
He counted how often we reflected about the truth value of a proposition
and those meta levels were his layers.
Here links with more detailed information about layer logic and layer set theory:
layer logic on researchgate
layer logic on The Philosophy Forum
In German:
www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/#post-492741
About Professor Ulrich Blau:
https://ivv5hpp.uni-muenster.de/u/rds/blau_review.pdf
In German:
https://link.springer.com/chapter/10.1007/978-94-017-1456-3_20
https://books.google.de/books?id=9xIxX206r5IC&pg=PA113&lpg=PA113&dq=reflexionslogik+blau&source=bl&ots=j5toa2ZhRK&sig=QWOej44nO1Upckoc9lEyfuZ1UEc&hl=de&sa=X&ved=2ahUKEwih-4Hm0erfAhVLaVAKHTwWDCsQ6AEwAXoECAkQAQ#v=onepage&q=reflexionslogik%20blau&f=false
As it is unusual and bulky I can understand that not many are going to study layer logic,
but in my eyes the possible results – a new look on logic and the world
and a way out of many antinomies - it is worth the effort.
On the other hand I am interested to learn,more about the liar, extensions and the solutions?
Yours
Trestone
the liars sentence is a famous antinomy / paradox of classical logic:
L:= „This statement is not true“
If we assume, that L is true, then because of the definition of L it has to be not true (=false).
If we assume, that L is not true, then because of the definition of L it has to true
So in both cases L is true and false at the same time, and that is not allowed.
Most solutions of the problem do not allow sentences like L and especially no selfreferences.
But there is an other posiibility: A new logic called „layer logic“ or should I say „liar logic“?
The idea with layer logic is, that every determination of a truth value
belongs to a layer (0,1,2,3,...),
and different truth values are allowed in different layers (whatever layers are – see beyond).
As layers are „blind“ to themselfes and to higher layers,
we always have to use a higher layer if we „talk“ about a layer.
In the rules of layer logic we have this formulation of the liar L:
„For all k=0,1,2,3,...: This proposition LL is true in layer k+1, if LL is not true in layer k
and LL is false in layer k+1 else.“
In layer logic all propositions are „undefined“ in the lowest layer 0.
Therefore LL is undefined in layer 0.
Layer 1: This proposition LL is true in layer 0+1, if LL is not true in layer 0
and LL is false in layer 0+1 else..“
Therefore LL is true in layer 1.
Layer 2: This proposition LL is true in layer 1+1, if LL is not true in layer 1
and LL is false in layer 1+1 else..“
Therefore LL is false in layer 2.
Layer 3: This proposition LL is true in layer 2+1, if LL is not true in layer 2
and LL is false in layer 2+1 else..“
Therefore LL is true in layer 3.
Layer 4: This proposition LL is true in layer 3+1, if LL is not true in layer 3
and LL is false in layer 3+1 else..“
Therefore LL is false in layer 4.
We see that LL has an alternating truth value with increasing layers – there is no contradiction.
As propositions belong to all layers, LL is self-referential but not within a layer.
And what is with liars that speak about all layers?
„LA:= This proposition is not true in all layers“
In layer logic meta propositions about layers and truth values have to be nearly classic:
They can only be true or wrong and have to have the same truth value in all layers >=1.
So if LA is true in layer 1 it has to be true in all layers, so LA is false – in all layers, that would be a contradiction.
If LA is not true in layer 1 it has to be not true in all layers, so LA is true – in all layers, that would be a contradiction.
Therefore LA is not an allowed meta proposition in layer logic.
Maybe there are better examples for extended liars?
The layers may look somehow strange at first glance.
The layers were first just a formal parameter to differentiate truth values
(for example in the first part of a proof and the second).
Meanwhile I see them as a kind of a new dimension, of meta levels or cause and effect order
or a new part of time.
Even without knowing exactly what a layer is, we can use layer logic and layer set theorie,
as the rules for using them are mostly independent of this.
In a restricted way Professor Ulrich Blau in Munich invented a logic with layers
some 20 years before me, the reflection logic.
He counted how often we reflected about the truth value of a proposition
and those meta levels were his layers.
Here links with more detailed information about layer logic and layer set theory:
layer logic on researchgate
layer logic on The Philosophy Forum
In German:
www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/#post-492741
About Professor Ulrich Blau:
https://ivv5hpp.uni-muenster.de/u/rds/blau_review.pdf
In German:
https://link.springer.com/chapter/10.1007/978-94-017-1456-3_20
https://books.google.de/books?id=9xIxX206r5IC&pg=PA113&lpg=PA113&dq=reflexionslogik+blau&source=bl&ots=j5toa2ZhRK&sig=QWOej44nO1Upckoc9lEyfuZ1UEc&hl=de&sa=X&ved=2ahUKEwih-4Hm0erfAhVLaVAKHTwWDCsQ6AEwAXoECAkQAQ#v=onepage&q=reflexionslogik%20blau&f=false
As it is unusual and bulky I can understand that not many are going to study layer logic,
but in my eyes the possible results – a new look on logic and the world
and a way out of many antinomies - it is worth the effort.
On the other hand I am interested to learn,more about the liar, extensions and the solutions?
Yours
Trestone
Comments (65)
Statement A: Statement B is true.
Statement B: Statement A is false.
here the definitions (AL and BL) for statements A and B in layer logic:
For all k=0, 1,2,3, …: Statement AL is true in layer k+1 if statement BL is true in layer k
and statement AL is false else.
Statement BL is true in layer k+1 if statement AL is not true in layer k
and statement BL is false else.
In layer 0 both AL and BL are „undefined“ (as always in layer 0).
Layer 1: Statement AL is true in layer 0+1 if statement BL is true in layer 0
and statement AL is false else.
Statement BL is true in layer 0+1 if statement AL is not true in layer 0
and statement BL is false else.
Therefore statement AL is false in layer 1 and BL is true in layer 1.
Layer 2: Statement AL is true in layer 1+1 if statement BL is true in layer 1
and statement AL is false else.
Statement BL is true in layer 1+1 if statement AL is not true in layer 1
and statement BL is false else.
Therefore statement AL is true in layer 2 and BL is false in layer 2.
So AL and BL have alternating truth values in the layers and are no problem.
As shown layer logic can also handle antinomies without direct selfreference.
Yours
Trestone
yes, I use my layers for all statements and their truth values.
For the layers I do not really need „numbers“, I only need a set that is inductive,
multiplication is not needed for the layers.
As the proof for Cantor´s diagonalization is valid no more with layer logic,
one kind of infinity (that of the natural numbers) is enough.
But the situation for mathematics is not all good:
As the proof for the uniqueness of the prime factorization is no more valid,
there might be different factorizations in different layers.
This could show a way to proof or falsicate layer logic experimentally,
but the needed numbers could be astronomically large.
Layer logic is selfrefertial as the truth value of statements can be defined with the help
of truth values of this statements.
But layer logic is not fully self referential, as statements are not allowed to use truth values
of the same (or higher) layers.
The exciting question is, if this layer selfreference is needed.
With layer logic I could define a set theory and natural numbers with arithmetics
that followed the rules of layer logic.
So the the approach carries further than I thought in the beginning ...
Yours
Trestone
At first glance I might think, How absurd. But these days there are lots of things in highly abstract mathematics that are beyond my pale, so maybe there is something to your ideas. I'm curious what fishfry and fdrake might think of it. In a faint way it resembles Schrödinger's cat. Nice article on Blau.
Quoting Trestone
If you are fine with disallowing statements in Layer Logic for which a truth value cannot be assigned, why not skip this added layer of complexity and disallow the original liar's statement in Classical Logic?
in Layer Logic I have to “disallow” only some meta statements.
If I would disallow the original liar's statement (and renounce layer logic)
I would loose a lot of new solutions and possibilities that come with Layer Logic:
Besides the liar for example Cantors Diagonalization, Russell`s Paradox,
probably also the Uncompleteness Sentences of Gödel
and the Halting Problem of Informatics.
And there are new possibilities for the philosophy of mind and probably for physics.
So the „little effort with the layers“ should be worth the effort ...
Yours Trestone
Thank you for your comment.
To „Schrödinger`s cat“:
I have tried to apply layer logic on Quantum theory (for double slit and entanglement).
I constructed a nice model with inverse time for virtual particels
(because they move in an „invisible“ layer),
but as I am not a physicist it is mostly speculation.
Yours
Trestone
So my understanding is that your resolution to the Liar's Paradox is not Layer Logic, but disallowing problematic statements.
As for Russell's Paradox, isn't that what is basically done with ZF, where we 'disallow' the set of all sets that are not members of themselves? With this approach Russell's Paradox has been resolved and there's no need for Layer Logic.
Personally I think this sort of disallowing problematic objects sweeps the problems under the rug.
my motivation for developing Layer Logic was to look after the things „under the rug“.
Layer set theory is rather nice: It has only one kind of infinity, the Russell set
and the set of all sets are ordinary sets.
Layer Logic and layer set theory are not nescessary, but they offer a new perspective
for the things „under the rug“.
Yours
Trestone
For any two-place predicate F (whether F is the membership relation of any other relation) we have a theorem of first order logic:
~ExAy(Fyx <-> ~Fyy)
So set theory does not need to add axioms that prove [read 'e' as epsilon]:
~ExAy(yex <-> ~yey)
What set theory does need is to be careful not to add axioms that prove:
ExAy(yex <-> ~yey)
here some points about layer set theory and the Russell set,
so you can check for yourself what is being swept „under the rug“:
S1: Definition of layer sets
The (layer set) x is in layer t+1 an element of the (layer) set M if and only if
x has the property P(x) in layer t.
In formulas:
The truth value of „x e M“ in layer t+1
is the truth value of P(x) in layer t.
Vt >=0: Vx VM: W(x e M,t+1):= W(P(x),t)
S2: Layer sets to propositions:
For every layer logic proposition P(x) about any layer set x
there exists a layer set M which fulfills for all layers t=0,1,2,3,...
the element equation:
W(x e M, t+1) := W(P(x), t)
That means:
The truth value of „x e M“ in layer t+1 is the same as
the truth value of P(x) in layer t.
In layer set theory „x e M“ can have three values: true, false and undefined.
S3: Empty layer set:
For the empty layer set 0 every set x is a Not-element,
that means „x e 0“ has the truh value false in all layers >0.
Vt>0: W(x e 0, t) := false and W(x e 0, 0) := undefined.
S4: The full layer set All:
For the full set All every set is an element in all layers >0.
Vt>0: W(x e All, t) := true and W(x e All, 0) := undefined.
Russel set with layers:
We define the Russell layer set R as follows:
x is element of R in layer t+1 if x is not element of x in layer t
and not an element in layer t+1 else.
Now we look on „R e R“ in different layers:
In layer 0 „R e R“ is undefined (as always in layer 0 in layer logic).
As undefined, R is not element of R in layer 0.
Therefore „R e R“ is true in layer 1 (=0+1).
Therefore „R e R“ is false in layer 2 (=1+1, not an element becaus of the „else“)
Therefore „R e R“ is true in layer 3.
Therefore „R e R“ is false in layer 4. And so on.
As alternating truth values in the layers are allowed in layer set theory,
the layer Russell set is an allowed set and not paradox.
Yours
Trestone
as I left university some 30 years ago and have developed layer logic only as a hobby,
I have no systematic theory, I mostly developed it in (German) discussions in threads like this.
Most details in English can be found here:
Layer logic at researchgate
Some more details in German here:
Layer Logic in German at ask1.org
I would be glad if someone would start a systematic work on Layer Logic.
Yours
Trestone
I had created a thread with a similar topic
https://thephilosophyforum.com/discussion/10641/how-the-greatest-lies-contain-the-greatest-truths
I am going to copy the relevant part. It would be interesting to hear your thoughts on it.
Quoting maytham naei
I can't make sense of your essay - from the very start where you begin by flinging around terminology used in a personal way but that you don't define nor declare as primitive.
If you are truly interested in people spending their valuable time to grasp your ideas, then you would be well served by formulating them systematically rather than merely hoping that Internet passerbys would try to figure out the meaning of your thrown-together verbiage for you.
One can say 'Everything I say is false'.
But one can't say it without self-contradiction.
(By the way, for purposes of paradox, it's clearer to refer to 'truth' and 'falsehood' rather than 'lying' and 'not lying'. Because someone can state a falsehood with lying (the person may not know that their statement is false).
In particular, you conflated lying with not telling the truth when you made the alternatives:
Lied vs told the truth.
The proper alternatives are either
Lied vs did not lie
or
Told truth vs didn't tell truth.
Quoting maytham naei
Yes, 'I mostly lie' is not self-contradictory. So what?
Quoting maytham naei
For four years the people of the United States had the 45th president in their lives who lied as many times a day as most people breathe in and out. Probably he even lied in his sleep.
I think the logic problems with „I always lie“ are similar to „This statement is not true“.
With classic logic they are true and false – what is not allowed.
In layer logic we would formulate for example:
LA is true in layer k+1 if LA:= „All I say“ is not true in layer k and LA is false else.
In layer 0 LA is undefined (as all layer logic propositions).
In layer 1:
LA is true in layer 0+1 if LA:= „All I say“ is not true in layer 0 and LA is false else.
Therefore LA is true in layer 1.
In layer 2:
LA is true in layer 1+1 if LA:= „All I say“ is not true in layer 1 and LA is false else.
Therefore LA is false in layer 2.
So “ I always lie“ or „All I say is not true“ in layer logic are allowed statements
that have alternating truth values „true“ and „false in the layers >0.
Yours
Trestone
Lenin is quoted with the following syaing:
“If these Germans want to storm a train station, they first buy a platform ticket!”
I am myself feeling like a German revolutionary / explorer.
But not so much as a scientist but more like Christopher Columbus:
I detected Layer Logic by chance as I am more on a philosophical than a logical journey.
My goal is to pass on the revolutionary intuition associated with the Layer Logic.
And the most important points thereby are not “platform tickets” or even Layer Logic,
but to open new ways of thinking besides classic logic,
Layer Logic itself to me looks not so complicated to understand,
but here some more explanations that might help:
The only new components are the layers.
Eight things are important about layers:
A) The layers are elements of an inductive set with elements 0,1,2,3, …
(multiplicative properties not needed)
B) All propositions P have truth values W only in combination with a layer k: W(P,k).
C) There are 3 possible truth values W(P,k): true (=t) , false (=f) and undefined (=u)
D) In layer 0 all Propositions P are undefined: For all P: W(P,0)=u
E) The truth value of a proposition P is the vector of all the truth values in all layers.
W(P) = (u, W(P,1), W(P,2), W(P,3), …)
F) A proposition P is well defined, if the truth values W(P,k) for all layers k
are well defined (one value for every layer).
G) When defining values for W(P,k+1) for proposition P all defined propostions and values
of smaller values (k or smaller) can be used - even W(P,k).
H) Layers and Propositions in layer k are „blind“ for this layer k and higher layers.
So when speaking about a property or using a value we have to change from layer k to k+1.
Analysis of most classical indirect proofs show, that with layer logic we have because of G) and H)
to use two different layers.
As true and false in different layers is allowed in layer logic there is no more
a contradiction and the indirect proofs are valid no more.
(Within a layer different truth values are still not allowed).
That is the revolutionary part of Layer Logic!
With all this formalization we still do not know what layers are
and why we did not notice them (or the new dimension) in the last 2000 years?
Well, I have already used and showed layers with the liar and with Russell`s set.
In everyday use most propably layers make no difference,
as properties may change with layers but they do not have to.
So layers mostly make a difference with infinity, selfreference and the start of cause - effect chains.
And in everyday life we all can be in the same layer that may change (simultanously)
with every physical interaction (besides gravitation) – but that is very speculative.
So that if two people look at an objekt at the same time,
they see the same propositions in the same layer.
But may be the main reason why we do not perceive layers could be,
that they don't fit into our view of the world ...
Prof. Ulrich Blau gave a more formerly definition of his reflexion logic
and his layers as „levels of reflection“ -
and he wrote a (German) book with about 1000 pages around it.
About Professor Ulrich Blau:
review about Prof. U. Blau
In German:
German Link 1 about Prof. Blau Reflexionslogik
German link 2 on Prof. Blau Reflexionslogik
His reflexion logic is only for a small part of all propositions, the reflective propositions,
where as Layer Logic treats all propositions in the new way –
as a full new logic with a new dimension, the layers.
By the way:
When I went to my first demonstration in 1989 UniMut in Berlin,
I actually bought a subway ticket before to get to the KuDamm,
where our students demonststration took place.
A revolutionary student theatre had agitated me.
Later I voted in Marburg against student strikes, but took a prominent part later.
With philosophy students and professors we performed a play of me
(“The death of Sokrates”) with also contained a (Sophistic) saying about logic:
“If logic does not apply, it can confidently continue to apply -
and that is also still thought of logically!”
So you see, I've been dealing with platform cards and logic for 30 years.
Unfortunately, my creativity and intuitions are dwindling
so I have to talk about my ideas from 30 years ago ...
Yours
Trestone
I haven't had the time to read through your messages thoroughly so I don't expect a response if you've already addressed my comment in your earlier posts.
You disallowed "This proposition is not true in all layers" in Layer Logic. Is there something analogous in Layer Set Theory, perhaps something like "x is an element of R if it is not an element of R in all layers"?
If you have in mind the famous proofs regarding a universal set, uncountablity, incompleteness, Tarski's theorem, and the halting problem, then these have direct proofs. Any proof of the form.
Show ~P.
Assume P.
Derive contradiction.
Conclude ~P.
has a structure of direct proof.
Indirect proof is of the structure:
Show P.
Assume ~P
Derive contradiction.
Conclude P.
The non-existence of a universal set, uncountability, incompleteness, Tarski's theorem, and the halting problem do not rely on that structure.
the statement "x is an element of R if it is not an element of R in all layers"?
is not allowed as a layer theory statement, as there is no layer given
where "x is an element of R" should be true.
if we add an layer k, the "all layers" will break the rule, that in definitions only
smaller layers are allowed.
So the forbidding of statements is according to rules.
Yours
Trestone
perhaps the formulation „indirect proof“ was misleading.
I better could say „proof by contradiction“.
The point is, however we call those proofs, the constructed contradiction in them
is not valid any more when we transfer the proofs to layer logic.
The reason is, that by constructing the contradictions we have to use different layers,
and different truth values in different layer are not a contradiction in layer logic.
And yes, all the proofs you named are valid no more
and probably also the uncompleteness sentences of Gödel
(I did not proof this completely with layer logic so far).
Yours
Trestone
Show P.
Assume ~P
Derive contradiction.
Conclude P.
Quoting Trestone
In ordinary mathematical logic, contradictions are syntactical, not requiring assignment of truth values. Meanwhile, as far as I can tell, your layer logic is described primarily semantically in terms of truth values; I don't know the syntax of whatever proof system you have in mind, so I can't evaluate the means by which you would prevent (syntactical) contradictions. You could assert that provability entails soundness, but we need to prove that, not just assert it, and you can't prove it without first stating what the proof system is.
I would guess that layer logic does disprove contradictions. That is, layer logic disproves all formulas of the form 'P & ~P' (where they "reside" (or\e whatever way you say it) in the same level).
/
How does layer logic disallow this proof (which is not indirect)?:
Show: There is no function from a set onto its power set.
Proof ['P' for power set, 'e' for element]:
Let f be function from S to PS. Let d = {x | xeS & ~xef(x)}.
dePS.
If d is in range(f), then for some x in S we have d=f(x).
If xef(x), then ~xed, so ~xef(x).
If ~xef(x), then xed, so xef(x).
Contradiction. So d is not in the range of f. So f is not a function from S onto PS.
/
How does layer logic disallow this proof (which is not indirect)?:
Show: ~ExAy yex.
Let Ay yex.
Let d = {x | xey & ~xex}.
If ded, then ~ded.
If ~ded, then ded.
Contradiction. So ~ExAy yex.
/
Remember, I never mentioned 'truth' or 'falsehood'. I merely gave syntactical proofs. So you haven't shown how those proofs are disallowed by merely saying that truth and falsehood may alternate on levels.
As for semantics, basically, what you say is that there are three truth values and that statements are evaluated at different levels. You haven't given even the starting point: description of evaluation of truth and falsehood for atomic sentences, compound sentences, and quantificational sentences.
here I will show how I handle the proof of the halting problem:
Here a classical proof of it (from https://wiki.c2.com/?HaltingProblemDiscussions):
[i]"Assume I have a program P that can tell you whether any program halts or not for given input data.
I construct a program Q based on P which, if P says its input program doesn't halt, immediately halts, and if P says the program halts, goes into an infinite loop.
Feeding Q(Q) to P I can see that if P says it halts, it won't, and if P says it doesn't halt, it will.
Therefore I don't have any such program P and anyone else who says they do is full of it."[/i]
Now with layer logic we have to add layers if a program has to give a value/result:
A given program halts or not in layer k for given input data.
And the program P has to tell about the halting in layer k+1,
as only values oflower layers can be worked on.
Now when constructing Q, which if P says in layer k+1 its input programm
does not halt in layer k it immediately halts, that halt of Q will be in layer k+2,
as values of layer k+1 are needed.
And the infinite loop of Q in the other case will also be in layer k+2.
We feed now Q(Q) to P :
We look at Q in layer k. If P says in layer k+1 that Q halts on Q in layer k,
the construction of Q says, that Q goes to an infinite loop in layer k+2.
So Q halts in layer k and Q goes to an infinite loop in layer k+2.
That is no contradiction, as programms can have different outcomes in different layers.
(That is not the complete layer logic proof, but the main idea is given.)
Therefore it is possible, that a Halting programm H exists,
that gives a true in layer k+1 for every layer programm P,
if P stops in layer k with input X.
The layers make the difference.
Yours
Trestone
the proof about the power set can be similary be "unproofed" like the halting problem
by adding layers and layer logic.
There is even a more simple "proof":
In layer set theory the set of all sets (called All) is a set.
The power set of All is All.
So there is a bjection from a set (All) to its powerset (All) : the identity x->x.
That is one of the things why I like the layer set theory.
Yours
Trestone
I have found my earlier handling of Cantor´s diagonalization and proof in layer logic:
As All, the set of all sets, is a set in layer theory, it is no surprise,
that the diagonalization of Cantor is a problem no more (I just give the main idea):
(t marks the layers, W(x,t) ist the truth value of x in layer t, -w stands for „not true“ or „false“
ther value „undefined“ I left out to make things easier).
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 is defined 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 …
A important remark: I do not say that the classic proofs are false, they are perfectly right.
But with layer logic and layer set theory we are in a new world.
All terms have to be transferred into the layer world
and only there most of the proofs are valid no more.
It is a little like in Plato's allegory of the cave:
If someone has been out of the cave and seen the real world,
he has to learn new rules and the rules of the old shadow world
will no longer fit.
If someone returns from the sun to the cave,
nobody will listen to him or understand him.
“Speak within our shadow rules or be quiet.”
His new world is pure nonsense and fantasy for the Cave people.
Yours
Trestone
My impression remains that you're adding on layers as a way to justify disallowing problematic statements/sets from existing within Layer Logic. My take is that if we wanted to avoid the problem by disallowing problematic statements we might as well do that within the simple framework of Classical Logic...but that's just my view. I hope you can find others to bounce ideas off of. If you can convince TonesInDeepFreeze that will be a good sign!
Quoting TonesInDeepFreeze
That stands without your response.
Quoting TonesInDeepFreeze
Is my guess correct?
And does layer math prove the following?:
~0=1
and
~Ex (x is a natural number & x>x)
And you admit that layer math does not prove the fundamental theorem of arithmetic. So layer math would not seem to offer much as a mathematical foundation anyway.
Quoting TonesInDeepFreeze
That stands without your response.
Quoting Trestone
You begin with:
Quoting Trestone
But no axioms or rules of inference by which to claim that.
So to follow along with you in your layer math, one just has to accept the arbitrary lines in your arguments as given by you personally (there is no objective codification). You do not provide one with a way to check whether the lines you put forth are axioms or theorems of layer math but instead one must rely solely on your dicta as to what constitutes a valid line or inference in an argument.
Quoting Trestone
You begin with:
Quoting Trestone
In ordinary logic, truth values apply to sentences. It seems that had previously been the case in your discussion of layer logic too. Here you mention the truth value of x, So I take it that x ranges over sentences there. But then we find x ranging over prospective members of the set A. So which is it? x ranges over sentences or x ranges over prospective members of sets? So far, what you've given is pseudo-math or gibberish dressed up with undefined math/logic-sounding verbiage.
Also, you mention things (which I guess are sentence) as being true or false in layers, but now here we find that functions too are things in layers. But you've not stated what a layer sis or what kinds of things can be in layers or, as I mentioned earlier, how it is determined a given atomic, compound, or quantificational sentence is true or false in a layer.
Quoting Trestone
Are you there asserting that there exists such an F? If you are, but without first proving the existence of such an F, it would seem to be question begging, since by supposedly refuting Cantor's theorem, you're claiming to prove that there does exist such an F.
Quoting Trestone
Just to be clear, these are all distinct:
(1) A proof of ~P in a given system..
(2) A meta-proof that P is not a theorem of given system.
(3) Pointing out a line in a purported proof of a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).
(4) A meta-proof that P is false in a given model of a given theory.
So, letting P = Cantor's theorem, do you you claim either (1) or (2) regarding layer math? (I take it that you do claim (4) or something like it.)
Quoting Trestone
That's question begging. One can just as well say you've not left your own cave, as you are not familiar with the logic and mathematics that has been explored by generations of logicians and mathematicians who have themselves studied alternatives including types, orders, levels in set theory, quantification over theories themselves, modalities, possible world semantics, topological semantics, and even para-consistency.
Quoting TonesInDeepFreeze
I am sorry that I can not answer most your questions to formal details.
As I said, my studying math is over thirty years ago.
The idea with layer logic is, that all is as in classic (three valued) logic,
only that layers have to be added, if a truth value is used or looked upon.
Professor Ulrich Blau has done all the formal work for his reflexion logic,
so i assume the formal part for layer logic should be possible.
Now again to the proof of Cantor:
In layer math we have a different kind of sets, the layer sets.
And if something is "true" or "false" we have to give a layer.
So we look onto a proof were all terms are transferred to layer math - we are in the "new world".
Now I transfer this to layer math: F is the layer function for f.
M is the layer set for S.
And A is the layer set for d.
Be M a set and P(S) its power set and F: M -> P(M) a bijection between them (in layer k)
Then the set A is defined by 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).
A e P(M) (analog to d e PS).
Transferred: If A is in range(F), then for some x0 in M we have A=F(x0).
Now comes the part, where layers make the difference:
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 t and t+1 are different layers)
Second case: W(x0 e F(x0),t) = -w then W(x0 e A=F(x0), t+1) = w
(no contradiction, as t and t+1 are different layers)
So in layer math, the existence of F does not lead to a contradiction.
(And the set All with identity to P(All)=All even is an example in layer set theory).
I hope you have a little understanding for a "Columbus",
who accidently sailed to a new world, but is neither a governor nor a cartographer
and if asked where he was, tells something obscure about "India".
Nevertheless there could be a new world ...
Yours
Trestone
[b]Trestone:
"You will know them by their fruits" (Matthew 7:15-20)
Even if Matthew warns here of bad people,
I am astonished what new fruits (not only in mathematics)
are in range with layer logic.[/b]
Yours
Trestone
You're lacking not just all the formal details, but even a coherent outline.
Quoting Trestone
Again, you can't assume that there is bijection from M onto PM. You are purporting to prove there is such a bijection, so you can't do that by assuming there is one. Unless layer logic is so meaningless that it allows proof by question begging.
The problem is not just a lack of familiarity with the technicalities of mathematical logic, but that you don't understand even the very basics of even informal logical reasoning.
Quoting Trestone
You don't know that unless you've proven the consistency of layer math. But, of course, you can't prove the consistency of something that is not formed with sufficient determinateness to tell what is a theorem and what is not.
Quoting Trestone
That would be the Columbus who brought widespread fatal disease, subjugation, and genocide. Meanwhile, you bring confusion, ignorance, and misinformation. Not as bad, so I wouldn't insist on the comparison.
Those are utterly basic to the subject. If you don't even know what proof is, then you can't very well explain whatever alternative system you have in mind.
Quoting Trestone
That's kind of okay in an everyday sense. But a contradiction is actually a formula of the form
P & ~P
(Or, more generally, a set of formulas that implies a formula of the form P & ~P.)
Quoting Trestone
I've informed you twice already that you are misusing that terminology. The particular proofs we are talking about in ordinary mathematics are not indirect (and 'proof by contradiction' is another term for 'indirect').
Quoting TonesInDeepFreeze
Quoting Trestone
So, for example, within a single layer you don't prove "It is not the case that both 2 is even and 2 is not even".
Quoting TonesInDeepFreeze
Really, you think it is false that 0 does not equal 1?
Quoting TonesInDeepFreeze
Really, you think there is a natural number that is greater than itself?
Quoting Trestone
You haven't said how, in general, one evaluates the truth of atomic sentences, compound sentences, or quantificational sentences in any layer.
Quoting Trestone
Pray tell, what class of algorithms do you have in mind that are not Turing machine computable? Actually, to reduce even more confusion and misinformation than you've already posted, pray don't tell.
Quoting TonesInDeepFreeze
Quoting Trestone
You are completely confused.
Cantor proves that such a bijection does not exist. You purport to prove that such a bijection does exist.
To prove that such a bijection does not exist, we may assume one does exits, then derive a contradiction, then conclude that such a bijection does not exist (and that is not indirect proof).
But to prove that such a bijection does exist, you can't start by assuming that such a bijection does exist. That is question begging.
Moreover, we don't even need to assume such a bijection exists and derive a contradiction. Rather, we argue from universal generalization: Let f be any function from S to PS. Then we show that S is not a surjection.
Quoting Trestone
You skipped the problem I mentioned. First you use 'x' to stand for a sentence, then 'x' to stand for a set. That, like virtually everything else you say, is nonsense.
Quoting TonesInDeepFreeze
"unproved" is nonsense terminology unless you specify which of these you mean:
1) A proof of ~P in a given system..
(2) A meta-proof that P is not a theorem of given system.
(3) Pointing out a line in a purported proof in a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).
What you seem to think is that you have accomplished a variation of (1) by showing that in your own system it is not the case that there is not a bijection between S and PS. But you fail from the very start by assuming that there is such a bijection.
Quoting Trestone
For crying out loud, you compare yourself to Lenin, the hero of the Cave, Christopher Columbus, and Matthew the Apostle. Maybe get a grip and cut back on the self-aggrandizement.
Anyway, the fruitfulness of set theory is in having a recursively axiomatized theory for the mathematics of the sciences, conceptualization and formalization of computability theory that enables, among other things, the computer technology we all use to type things on the Internet, a treasure of peer-reviewed articles that provide reading about fascinating intellectual accomplishments, and a starting point for approaches that differ from classical mathematics, including multi-valued logics, constructive mathematics, paraconsistent logic, and others that can be quite exotic. Meanwhile, the fruit of your postings is from the poisoned tree of ignorance.
thank you for still answering me!
I do not think that we will in near time agree on the main points of Layer Logic,
but to have a look from different sides helps me to understand,
where open questions and points might be.
With the two examples with the “~” I made a mistake, as you noticed:
I thought the “~” refered only to the next sign, and not to the whole term.
~0=1 Trestone: true in layer math and
~Ex (x is a natural number & x>x) Trestone: true in layer math
Perhaps this is exemplary: If we are not used to the language of the other,
even small misundestandings can create totally wrong interpretations.
(And probably most times I am wrong).
Looked upon only as a theory, it is not nice, that natural numbers might have
different prime decompositions in different layers.
But it could help my theory without being fully formalized:
If (perhaps in 100 years) we handled big enough numbers (with computers),
the following could happen:
On one day we get for a number n the prime decomposition P1.
One week later we get on the same computer with the same program for n
another prime decomposition P2 (and similar disturbing results with other computers).
(After a speculation of me, there are layers in the real (physical) world,
and they increase with every physical interaction (except gravity).
So to increase layers we only have to wait a little time.)
Most probably this will be handled as computer bugs
and nobody will remember Layer Logic.
But by chance somebody might check even old articles -
and Layer Logic could be one of the candidates to explain the surprising effect.
As I can not tell how large n and the layers have to be,
I can not say how long we will have to wait for an Arthur Stanley Eddington.
(Yes, now I have compared myself with Albert Einstein,
but he is already part of my nickname)
So “sit and wait” is maybe not the worst strategy to develop Layer Logic ...
Yours
Trestone
What is the proof in layer math?
Quoting Trestone
What is the proof in layer math?
In general, without stating axioms and a proof system, how do know what is true in layer math? Is there a layer math phone hotline you call and they tell you what's true or false?
in the German link to layer logic there are some more definitions.
Layer Logic in German at ask1.org
For example this to the natural numbers (here in English):
N1: Definition of successor function M+ for level set M
(for the construction of natural numbers):
Vt> 0: W (x e M+, t + 1): = W (x e M, t + 1) v W (x = M, 1)
Let us consider the 0: W (x e 0, t) = f for t> 0.
“Zero” is therefore empty from t = 1 (independent of the level).
1 = 0 +: W (x e 0+, t + 1) = W (x e 0, t + 1) v W (x = 0.1) = W (x = 0.1)
From t = 1, "one" contains exactly the one element "zero".
(Therefore 0 is not the same as 1).
General: n + contains exactly the elements n, n-1,…, 1 in level t> 0
The addition can now also be defined in the same way as the classic procedure:
W (xen + m+, t + 1): = W (xe (n + m)+, t + 1) = W (xe (n + m), t) v W (x = (n + m), 1 )
As natural numbers are layer sets in layer set theory,
I have to define what n>m means (and to be more exact, what it means in layer k).
I propose, that a natural number set n is greater than a number set m in layer k,
if n has more elements than m in layer k.
As x always has the same elements as x in layer k>0
we can say, that in all layers k>0 there is no x that is a layer natural number
and that fulfills x>x.
Yours
Trestone
Do you understand the notion of either using commonly known notation or explaining our own personal notation starting at its most simple?
Why would another prime decomposition, P2, arise if not for a computer implementing this layer logic? Thus, avoid disturbing results by ignoring layer logic.
Sorry. I see things from weary old eyes. :roll:
there are two problems:
First, even having studied mathematics 30 years ago,
I never liked the formalisms.
And when studying philosophy, my feeling was,
that the (formal) approach to logic was not good.
In oppsition to all this I developed my new logic
using and developing my own notations over almost twenty years –
in (mostly German) discussions and not in a very systematic way.
Over the years less and less reactions came to my writings,
so now I am used to being “a voice crying in the wilderness”.
The second problem is, that meanwhile I can hardly take
the position of someone to whom Layer Logic is new.
To me all looks easy and clear – as I have lived with it for so long time.
Here questions can help, but point one is a problem.
And maybe unconsciously I want to be the only one
who understands Layer Logic,
so that I am the only one who can play with it ...
Yours
Trestone
in my eyes you do not have to install Layer Logic on a computer:
It`s already there.
(Of course this is only a daring hypothesis of me.)
It is the same as with the General Theory of Relativity,
Eddington had not to install it to the sun andf the light rays,
it was already part of the world (if it was true).
In everyday life we do not notice that we have Layer Logic instead of classic logic,
as most propositions are not layer dependent.
But prime decompositions of very large numbers could be.
I assume that the the change of layers is very easy in everyday life
(and in computers):
we just have to wait for some time.
(More exact: We have to wait for the next physical interaction
(not by gravity) in our environment.
And with every evaluation a computer has to use a new layer)
As usually propositions do not change with layers,
we do not notice this change of layers in our surroundings.
But the layers could be there all the time ...
Yours
Trestone
You have developed a personal philosophy based on LL. Congratulations.
Perhaps a naive notion here, but whether a statements/procedure can take a value or true or false depends on the context. I apologize in advance if I am going over stuff that you are familiar with.
1) Factual statements
Are we making a statement about the real physical world (AKA Kant's nuomenal world, existence, the universe, reality, everything that is the case, etc)? Not sure what country you're from, but in the USA if you give testimony in a court of law you swear to tell the truth - which basically means that if a statement accurately (or as accurately as possible) describes an event in the real world, then it is true. This is the correspondence theory of truth.
With this in mind, the sentence/proposition "This sentence is false" does not describe any event in the real physical world, and as such it cannot be assigned a truth value.
2) Logical/Mathematical Statements/propositions
Logic/math statements do not refer to any event (real or hypothetical) in the physical universe, but are only true or false depending on the rules within the particular mathematical/logical system framework being used. While there are many such frameworks, at the risk of over simplifying each of these logical frameworks works in a similar fashion. You start off with axioms which are defined to be true within the particular logical framework you are working in - and then you have rules for generating other statements. For example, here are the axioms for Peano Arithmetic. I am not an expert in this, but there are many people on this forum who are highly knowledgeable and might provide you with more details.
So now the question here is, within what logical framework are we asserting the proposition "This sentence is false"? You said it is a "paradox of classical logic". The term "classical logic" is a bit vague, but here is the SEP article on Classical Logic Perhaps there is a way to translate the sentence into classical logic syntax (it's beyond my capabilities) but I'm reasonably confident that even if the sentence could be formulated it would have a value of false
Put differently, the sentence "This sentence is false" does not express a coherent thought and thus cannot take a truth value. There are many examples of nonsense sentences in the philosophical literature. "Quadruplicity drinks procrastination". "Colorless green ideas sleep furiously." We recognize that these sentences are grammatically correct and we also recognize that they are nonsense sentences. So there is no need to construct an elaborate logic system to handle such statements.
But maybe I'm missing your point altogether.
To get out of the wilderness, you could you formulate your notions in a way that other people can follow them, step by step, from basic to more involved.
Quoting Trestone
You're doing a great job to make sure that you are.
originally the Layer Logic was only a theory and a new logic system
like others.
So I handled truth values with Layer Logic but I did not bother what truth really was.
Later I noticed, that I could descripe cause and effect with layers,
and now all measured properties are described by me with layers.
Here I am near to Factual Statements.
But why would we need the real physical world for "This sentence is false"?
It is a logical proposition to me and so part of the logical world
(but I can not decide wether it is true or false).
Transferred to Layer Logic the liar LL has different events/objects in different layers:
„For all k=0,1,2,3,...: This proposition LL is true in layer k+1, if LL is not true in layer k
and LL is false in layer k+1 else.“
The definition of the truth value of LL in layer k+1 only depends on values of layer k.
This values are like events/objects in layer k (here logical and not physical).
As I showed this definitions lead to alternating truth values u,t,f,t,f,t,...,
and there are no problemds with the truth v alues any more).
The idea with Layer Logic is to use a classic logic (like L3 of ?ukasiewicz)
and add layers 0,1,2,3,4,... to it – for every time a truth value is determined.
Using the layers hierarchical when defining propositions,
most paradoxes and antinomies can be avoided.
And proofs that use contradictions are mostly valid no more,
as true in one layer and false in annother layer are allowed truth values.
So the Layer Logic helps at much more problems than only the liar.
The use of self references is explicitely allowed in Layer Logic,
it is only not allowed within a layer.
LL is a sentence that makes perfect sense in Layer Logic,
it is an example of a sentence with changing truth values with different layers.
Classic logical sentences would have a constant truth value in all layers,
except in layer 0 where all layerr statements have the value “undefined”.
But there is a more factual use / interpretation of Layer Logic:
if an object in physics has a measured proposition,
whe can apply a layer to this measurement:Objekt O has proposition p in layer k
(= when measured in layer k).
Now I think that if there is a physical interaction (not gravity) around O,
the layer in the sourrounding will increase (say to k+1).
As there are physical interactions all the time,
we only have to wait a very short time to get a new and higer layer
(for example in a computer).
So if we have a property, that changes from layer k to layer k+1,
we just have to wait untill both layers are reached one after the other.
Such a property could be the prime decompensation of a large number n.
And so we could see in a physical experiment that there are layers
(or something other strange).
Yours
Trestone
That is not universally accepted by all philosophers of mathematics.
Quoting EricH
Classical logic is exactly formalized. It's not vague.
Quoting EricH
The subject is explicated by Tarski's Theorem. If a theory has a truth-predicate for itself, then the theory is inconsistent, thus it has no model, thus there is not a model in which to evaluate the truth of falsehood of 'this sentence is false'.
Quoting EricH
'Express a coherent thought' is an informal notion. Without formalization, there is no effective procedure by which in general people may objectively and definitively determine whether or not something "expresses a coherent thought". Mathematical logic though deals with 'this sentence is false' in a formal way that is clear and objective.
Quoting EricH
This bears repeating: If a theory has a truth-predicate for itself, then the theory is inconsistent, thus it has no model, thus there is not a model in which to evaluate the truth of falsehood of 'this sentence is false'.
"All speech is lying."
Where it is true, it is also false (because there is an equal amount of everything in everything).
So whereas you might believe that some things are patently true or false, they are all true and false. It is due to the limitations of language (language's relationship with Reality) and all things intellectual that makes this the case.
Suggest you try to follow what @TonesInDeepFreeze is saying - they have deep knowledge of this topic.
Thank you for the intelligent response. It appears to me that we're more or less on the same side of this particular discussion - namely that the OP does not make much sense. My analysis was more informal - yours is clearly grounded in a deeper knowledge of math.
Quoting TonesInDeepFreeze
One question for you - when I said "the term classical logic is a bit vague" - I was referring to the way it was used in the OP. In my response, I pointed to the SEP article on Classical Logic Is this your understanding of the term classical logic? If not, could you point me in the right direction?
The poster refers to 'classical logic', but doesn't know what classical logic is.
Quoting EricH
I only glanced through that article just now, but it looks good to me. I have found that Stanford Encyclopedia of Philosophy articles on logic are excellent and beautiful to read.
you remind me of the border guard
who demanded the TAO-TE-KING from Laotse.
(Again a comparison to a famous person ...)
Unfortunately, I am unable to write down my ideas in 1000 lines
of beautiful Classical Chinese.
Therefore, here a short poem by me in German and English
and quotations from the beginning of the TAO-TE-KING.
Logeric
Es pendelte ein Philosoph mit der Bahn
nahm sich dabei der Logiklücken an
verhedderte sich in Stufen
denn die Geister die er gerufen
verlachen Logik als Wahn.
A philosopher commuted by train
took care of logic gaps there in vain
tangled up in layers
called spirits in his prayers
that laught at logic in wane.
TAO TE KING (Laotse, Geman by Richard Wilhelm)
Der Sinn, der sich aussprechen läßt,
ist nicht der ewige Sinn.
Der Name, der sich nennen läßt,
ist nicht der ewige Name.
"Nichtsein" nenne ich den
Anfang von Himmel und Erde.
"Sein" nenne ich die Mutter der Einzelwesen.
...
Des Geheimnisses noch tieferes Geheimnis
ist das Tor, durch das alle Wunder hervortreten.
TAO TE CHING (Laotse, English by Stephen Mitchell)
The tao that can be told
is not the eternal Tao.
The name that can be named
is not the eternal Name.
The unnamable is the eternally real
Naming is the origin
of all particular things.
...
Darkness within darkness.
The gateway to all understanding.
Yours,
Trestone
maybe it helps if we try imagination:
First we add a layer (0,1,2,3,...) to every statement with a truth value.
For example “The liar statement is false – in layer 2” and
“The liar statement is true – in layer 3”.
All statements in layer 0 will have the value “undefinded”,
and in all higher layers every statement has exactly one of the values
“true”, “false” or “undefined”.
Then we define truth values of statements by using higher layers
for the defined values and lower layers for the defining ones.
For example “Statement LL is true in layer k+1, if LL is not true in layer k
and LL is false in layer k+1 else.“
And fourth we use classic 2-valued logic and statements as meta logic
if we talk about layer logic, especially if we talk about layers or truth values
(like in the rules above).
And last we imagine a world where we all live in the same invisible layer
that grows with time.
What consequences would all this have for logic and math?
I believe there would be fewer contradictions
and there could be a rather simple set theory.
Yours
Trestone
A better comparison is with the boy who declared, "The Emperor has no clothes."
about a month ago I wrote a fairytale about layer logic,
based on this story with The Emperor`s Clothes.
The role of the Emperor and the boy/child is played by other people in my version ...
Link to German fairytale:
https://www.leselupe.de/beitrag/der-logik-neue-kleider-146395/
In English:
The Logic´s New Clothes
Once upon a time there was an emperor who loved science.
He called many bright minds around him who were eagerly researching.
One day two logicians came and made their discipline palatable to him
with the following words: Our logic is not only two thousand years old,
it is also the basis of all science and only those who are stupid
and not suitable for true science cannot understand it, because it is very easy.
The Emperor did not understand why he needed this logic now,
and so he asked deserved statesmen and ministers what they thought of it.
They did not want to show any nakedness and emphasized the universal validity
and unquestionable truth of the new logic.
So the emperor dared to go to the streets with the new old logic.
Even when the root of 2 turned out to be irrational by means of logic,
one could suddenly find uncountable infinite sets and the arithmetic showed
that most true sentences could not be proven, this was seen and believed
as an indication of the sophistication of this logic.
"But the logic has no clothes on - it doesn't work!" finally called a little child.
Then they quickly threw logic a few layers over and ended the procession.
There is also a story where I compare myself with Kassandra,
the prophetess that nobody listens to.
German link:
https://www.leselupe.de/beitrag/die-logik-von-troja-146428/
In English:
The logic of Troy
Troy had withstood the siege of the Greeks for ten years.
The voices calling for Agamemnon to withdraw grew louder and louder.
Odysseus called for the carpenters and started one last trick.
The Greeks withdrew with all their ships and left behind on the beach
just a huge wooden horse.
It was supposed to grant Athena's blessing on the journey home
and was dedicated to her Logos.
With the Trojans, Laocoon and Kassandra warned against bringing
so calamity into the city.
But Laocoon and his sons were killed by a snake sent by Athena -
whoever kills is right, that is the logic of the Greeks and Trojans!
And although Kassandra always prophesied the truth, no one believed her anymore,
because Apollo, who had been rejected by her, had cursed her gift of vision this way.
"Do you really want to break down the walls of common sense for this Danaer gift?" -
They did it with joy and had a feast.
The Greeks who crawled out of the horse's belly late at night,
then celebrated an orgy of a completely different kind,
opened the gates for the returning warriors and slaughtered Troy with fire and sword.
And since nobody listens to Kassandra's calls for alternatives even today,
we still use the captivatingly simple logic of the Greeks and Trojans ...
Yours
Trestone
That's not an argument logicians would give for the worthiness of logic.
Quoting Trestone
What logician ever said that?
There is no sentence that can't be proven in some system. However for any system (of a certain kind), there are sentences not proven in that system.
Why don't you at least learn to properly identify that which you falsely comment on?
Quoting Trestone
Fear of not being accepted is not a basis on which logicians endorse work in the field of study. Or, if you claim it is, then please point to a logician of whom it is true.
Quoting Trestone
It's question begging just to claim that it doesn't work. Actually, logicians prove it does work in the sense of (1) proving soundness, (3) proving completeness, (3) displaying the development of mathematics from axioms.
An analogy so attenuated in connection with logic that it's ridiculous.
Is it good or bad to be ridiculous?
Yours
Trestone
Depends on the situation.
from a German philosophy TV discussion today I learned,
that ingeneering (= “ingenius engineering”) needs positive visions.
So maybe I should concentrate more on what new things are or could be possible
with Layer Logic than showing what no longer works.
Perhaps someone will help except saying “none”...
Yours
Trestone
Sure, always nice work to get others to paint the white fence for you.
Perhaps my mistake is, that I didn`t pretend enough
this work on Layer Logic is fun
and I didn´t wait for passers-by to offer me an apple
to replace me for some time ...
I also like the view of Pippi Longstocking:
"2 times 3 make 4
Widdewiddewitt and three makes nine
I’ll make the world
widdewidde the way I like it."
Yours
Trestone
there could be a closer connection between children and my math
than can be seen at first glance:
When Layer Logic spreads in 100 years
maybe children will have to learn it in school too
(and hate me).
Yours
Trestone