Hello,

since Aristoteles some alternatives to classic logic have been constructed,
but other than in physics classic logic still seems to be the leading theory and practice.

That did not prevent me from looking for a new logic,
especially for philosophy and mathematics.

What disturbed me in philosophy were the problem of free will, the problem of body and mind,
the problem of consciousness, the problem of ultimate foundation
and the problem of cause and effect or first cause and - last not least - the liar sentence.

In mathematics I did not like the diagonal proofs of Cantor, the irrational numbers, Goedel´s incompleteness theorems and the halting problem of informatics.

In all these themes we use logic to argue or to prove something (mostly impossibilities).
My hope: With a new logic we could perhaps come to new results.

Of course the new logic should not be trivial ( like „all is true“)
and should fit to our reality.
That looks difficult, but it was surprisingly easy to find a promising approach.

I combined ideas of Russell (type hierarchy), the complex numbers („a new parameter“)
and quantum theory („only by measuring a property gets a defined value, else it is undefined“).

After some tries I got layer logic from this ingrediences.
I learned later, that professor Ulrich Blau in Munich, Germany, had some ten years earlier
constructed a similiar logic called „the logic of reflection“.

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

His logic is not for all kind of proposals, mainly for reflective ones.
So he can handle the liar sentence, but not most of the other problems from above.
Interestingly he seems to be the only logician with a similar approach to mine.
So layer logic could in most parts be a really new logic (but I do not know all kind of logics).
But for me its more important if it helps to solve my problems.

Layer logic uses three truth values: true, false and undefined (t,f,u).
But the main new part are the layers 0,1,2,3,4,5,... (not the natural numbers but an inductive set)

Proposals to not have truth values any more,
but only proposals in connection with a layer have a truth value.
And every proposal has a truth value in every layer simultanously.

If we have a proposal p it has a truth value T(p,k) in every layer k=0,1,2,3,4,5,...
So truth of a proposal p now is a kind of infinite vector: ( T(p,0), T(p,1), T(p,2), … )

And a layer logic proposal p (or sentence) is well defined only if we give the truth value T(p,k)
for every layer k.
We can do this directly or mostly by recursion.

If we observe the hierarchy of the layers, we so get proposals, that are free of inconsistencies or paradoxes.
Hierarchy of layers means: layers are blind to themself an higher layers, in a lyer only values of lower layers are accessible.

In layer 0 all proposals are undefined (For all p: T(p,0)=u) (symmetrical start) .

The liar statement here has the following form:

L:= For all k= 0,1,2,3,... : T(L,k+1) = true if ( T(L,k) is not true ) else T(l,k+1) = false

As all statements are u=undefined in layer 0,
we get for layer k=0:

For k= 0: T(L,0+1) = true if ( T(L,0) is not true ) else T(l,0+1) = false
T(L,1) = true if ( u is not true ) else T(l,1) = false
Therefore T(L,1) = true

With this we get for k=1:
For k= 1: T(L,1+1) = true if ( T(L,1) is not true ) else T(l,1+1) = false
Therefore T(L,2) = false

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,...
The truth vector of L is (u,t,f,t,f,t,f,...)

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.
(Perhaps someone may prove or disprove this in a dissertation?)

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.

Most of my mathematical and logical problems are solved or nearly solved with layer logic.
(If layer logic is consistent, what I have not formally proved, perhaps a diploma thesis?)

The philosophical problems are a bit harder, there I just discovered first steps of a maybe long journey.

Up to now layer logic is only discussed in laymen forums (english and german),
but if a real logician wants to engage in or against it,
I would welcome him or her.

But the opinions or comments of laymen are welcome as well,
for most times my postings remain without reply,
perhaps my layer logic is too technical or too exotic?
And classical logic is the hand that feeds us ...

Link to more details of "layer logic" "Trestone"
(You may also search in the net,
in German further material with „Stufenlogik“ „Trestone“):
https://www.researchgate.net/post/What_do_you_think_of_layer_logic-and_the_use_of_a_new_dimension_to_come_around_contradictions


Yours
Trestone