E!x x^ — GrandMinnow

Hello,

here the sight with layer logic instead of classical logic:

In layer logic you have to give a layer to the statement that should be possibly true:
In layer k the statement E! x^2 = 2 is true,
in another layer m the statement E!x x^2 = 2 may be also true, but with another x (say y).
So we can have x^2 = 2 is true in layer k and x^2 = 2 is false in layer m and y^2 = 2 in layer m.

Therefore the square root of 2 in layer logic is most probably not one irrational number
but many rational fractions in different layers (times).

Yours
Trestone

Hello Ryan O'Connor,

I do agree, that we have not proved the existence of irrational numbers.

My argument is not only the problem of indirect proofs, but a deeper one:
We can not be sure that our logic is totally right.

It is astonishly easy to construct an alternative logic that gives in finite cases mostly the same results as classic logic, but has totally other results with infinite cases or indirect proofs or antinomies.

I myself constructed such a logic, the layer logic.
There we have an additional parameter, the layer,
and statements are not right or wrong but have a truth value in every layer.
And this values can be different in different layers without making a contradiction.

Therefore in indirect proofs (as for the irrationalioty of root 2) we need to get the contradiction in the same layer. But analysis shows, that different layers are used,
and therefore with layer logic we do have no contradiction and no proof anymore.

The same way we can show, that the diagonalization of Cantor does not work anymore.
So we need only one kind of infinity (that of the natural numbers) in layer logic or layer set theory.

But not all is better with layer logic:
The uniqueness of the prime decomposition (over all layers) can not be proved.

So we have a nice layer set theory where the set of all sets is a set,
but arithmetics may partly be time dependent:

My newest guess is, that there is a layer for all objects (quants) that can interact
(except interacting with gravity),
and if some interact than the layer for all objects is increased.
This way we get in the layers a kind of time arrow since the big bang,
and properties (even in math like prime decomposition)
can depend from it and change with time.

Most mathematicans will not like such a world,
but it seems possible to me.

The square root of 2 could be a rational quotient with different numinator and denominator
in different layers (or times).

Some details to layer logic you can find in:
https://thephilosophyforum.com/discussion/1446/layer-logic-an-interesting-alternative

Even more (in German) here:
https://www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/

Yours
Trestone

Hello unenlightened,

the book of Doris Lessing looks interesting,
it reminds me partly on The_Three-Body_Problem of Liu Cixin.

Yours
Trestone

Hello Eremit,

I have only thought of one layer for the body and one for the mind.
The layer of the body is finite and increasing with every interaction (except gravity).
In layer theory there can be different properties in different layers -
and as mind and body are different I tried layer logic for an explanation.

Yours
Trestone

Hello,

two years are gone,

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

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

https://www.researchgate.net/post/Is_this_a_new_valid_logic_And_what_does_layer_logic_mean

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

For example: https://www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/


Yours,
Trestone

Hello szardosszemagad,

you write that you are "too old to learn new tricks".
My experiance is, that most younger ones also seem not to like to learn new tricks,
especially that of my layer logic.

I do not complain about this as I myself probably would not give layer logic a second glance
if it was presented to me by someone else.

So layer logic in the past ten years has become a rather personal thing to me.
And although logic is the main subject, it were my wishes and discontentedness that startet the project.
Of course I hoped and still am hoping to discover something of objective validity,
But to handle logic as a tool that has (at least partly) to obey my wishes
is one of the new aspects I have to offer.

When I had selected some basic rules for my layer logic it started a life of its own.
And as it is not easy to think consequently in a new logic,
I often add that something is "my personal view or interpretation of layer logic"
especially if it is a conclusion outside logic and mathematics.

For in the end I hope to use my tool to solve (or start solving, or inspire others to start solving) philosophical problems like mind-body, free will and consciousness.

Yours
Trestone

Sometimes this board seems like the philosophy equivalent of a physics board where people mostly present their "free energy" and "time cube" proposals.

Hello Terrapin Station,

I now can present a kind of "time cube proposal", that was inspired by layer logic:

Extension of time and evolution

15 billion years is not enough for the kind of history I am thinking of.

An easy way of expanting time is to propose a chronologically earlier parallel world.
I.e. we suppose a further world parallel to our world, where (all) the time is prior to the time in our world.
Both worlds could have their own big bang and evolution.

If Informations of the older world could come to our world,
our possibilities would increase immensly.

Those informations would come from a very distant past,
but this past could be more similar to our distant future
than to our local past back to our big bang,
where we mostly look for causes.

This informations could perhaps be placed directly at a particular time in our world
and would not have to go over our big bang,
as any time in our world is later than any time in the previous world.

This gives nice possibilities for directed panspermia:

Our predecessors in evolution in the previous world
could develop for billions of billions of years
(perhaps up to shortly before the end of their world)
until they send their „message in a bottle“ to our world
(perhaps shortly before the developement of galaxies),

We ( or our successors) could pay back by giving similar informations tio the next world.

Wether this would be good or bad is propably a question of perspective (or layer) ...


Yours
Trestone

Is the liar sentence in the layer logic still the "original" liar sentence or something different?

Hello Pippen,

to me not only the liar sentence is different in layer logic but the whole world:

Like with complex numbers there are more and new possibilities, new dimensions.

Which of those worlds is more "real"?

To me this is an open question.

Of course we are more used to the classic logic and world,
but how can we be sure that there are not additional layers and dimensions
that we did not notice up to now?

Yours
Trestone

In layer logic it is not paradox, but an ordinary statement. — Trestone

Or, someone could simply say that a paradox is a statement that has alternating truth values in "layer logic."
And then you've got the same problem you had prior to layer logic.

Hello Terrapin Station,

to me layer logic is similar to complex numbers:

If you keep thinking classic, you can say: "x*x=-1" has no real solution.
And being unsolvable is almost equivalent
to having a complex solution with an non-zero imaginary part..
On the other side mathematicans would agree:
From the point of view of complex numbers a lot of the classic problems
(like the square root of -1) do not exist as problems anymore.

Yours
Trestone

Hello Pippen,

For instance what would be the status of your Liar sentence in layer logic? You couldn't tell, just for a particular layer which by definition just describes a tiny fraction of the Liar sentence. So the Lair sentence as a whole would be as "paradox" as in classical logic, jumping between true and false. Your logic merely "visualizes" this back-and-forth-jumping.

In layer logic the "status" or "truth value" of the Liar sentence is an infinite truth vector v=(u,t,f,t,f,t,f,...)
and every proposition has an indefinite truth vector in layer logic.
Only those with constant truth vectors can be related to a classic truth value,
the liar sentence is not such a proposition.
Of course it would be nice if there would always be a classical interpretation,
but if we take layer logic seriously as a new logic with more possibilities for truth (vectors),
it is not asthonishing that it allows new kinds of propositions.
The "old" truth may become less important if we get used to the new.

Yours
Trestone

Hello Pippen,

as layer logic is a new kind of logic with a new kind of truth all propositions about truth are different to classic propositions.
Therefore you are right, the liar sentence in layer logic is not the same as the classic liar sentence.

On the other side layer logic helps us to speak about self refering propositions without contradictions.
If we would switch to layer logic the classic propositions would not be important anymore.

Even I am not that radical, I see layer logic as an interesting alternative that shows us,
that things could be quite different ...

Yours
Trestone

Hello alan1000,

thanks for giving feed back.

To be precise, I did not prove Cantor´s Diagonal Argument to be wrong.
I just changed the rules and constructed a logic and a set theory
where it can not be proved any more.

Yours
Trestone

Hello,

imagine, there´s a new kind of logic,
but nobody has a look ...

Yours,
Trestone

Hello,

if you are interested in solutions to the liar statement,
you could have a look at "layer logic":
It is a three-valued logic that uses a new additional dimension of layers.
More details at https://thephilosophyforum.com/discussion/1446/layer-logic-an-interesting-alternative

Yours
Trestone

The idea here is simple. If giving the LP either the value of "true" or the value of "false" results in an inescapable contradiction, we can avoid the Paradox by saying that the LP has neither value, thus preventing the contradiction. There are many arguable problems here. Firstly, this would seem to require abandoning the Law of the Excluded Middle or else the Principle of Bivalence. Now, I've no qualms with dropping Classical Logic in favor of a Non-Classical Logic, but I get the feeling many people would not like that. — MindForged


Do we have to abandon classical logic when we claim that the sentence "go away" is neither true nor false? — Michael

Hello,

I do not think that we have to abandon classical logic because of the "liar",
but I like playing around with non-classical logics.
One exotic logic is my favorite:
The "logic of reflection" by Ulrich Blau
which i developed and extended to an alternative logic, the "layer logic".

Link to the logic of Ulrich Blau:
https://ivv5hpp.uni-muenster.de/u/rds/blau_review.pdf

Link to "layer logic" "Trestone":
https://www.researchgate.net/post/What_do_you_think_of_layer_logic-and_the_use_of_a_new_dimension_to_come_around_contradictions

In layer logic a hierarchy of truth layers is used (similar to the hierarchy of types of Bertrand Russell).
Proposals to not have truth values any more,
but only proposals in connection with a layer have a truth value.

The liar statement here has the following form:
L:= For all n= 0,1,2,3,... "This statement is true in layer n+1 if it is not true in layer n and else it is false"

As all statements are u=undefined in layer 0,
we get for layer n=0:
"This statement is true in layer 0+1 if it is not true in layer 0 and false else"
Therfore: "This statement is true in layer 1"
For n=1: "This statement is true in layer 1+1 if it is not true in layer 1 and false else"
Therfore: "This statement is false in layer 2"
We see: The liar statement L is undefined in layer 0, true in layer 1,3,5,7,...
and false in layer 2,4,6,8,...

So the truth-value is alternating with the layers.
A classical statement would have only one truth-value in all layers 1,2,3,4,5,...,
so we can see, that L is a non-classical statement.
In layer logic it is not paradox, but an ordinary statement.

Layer logic is a little bit cumbersome,
but has amazing advantages:
Nearly all paradoxes can be solved similiar to the liar.
With layer logic a "layer set theory" can be defined,
where the diagonalization proof of Cantor is no longer valid
and where the Russell set and the set of all sets are ordinary sets.
Even natural numbers and an arithmetic can be defined.
A small set back: The prime factorisation of natural numbers could differ in layers.
But on the other hand it is probable, that the proofs of Gödel`s incompleteness theorems are valid no more.

Layer logic is a kind of a third way between classic logic and constructivistic/intuitive logic.
It is a logic with three truth-values, but the layers are the most important part.
Indirect proofs are allowed in layer logic, but only within a layer.
As in most classic proofs there are different layers involved if transformed to layer logic,
those indirect proofs are mostly not valid any more.

Unfortunately nobody up to now has seen a layer in reality,
but as long as the idea does not lead to contradictions it remains an interesting idea.

Yours
Trestone