Hello,

Imagine,

the SETI-Project has reached contact to something about 5 light years

in distance  

and we switch some of the first years, so the communication with the aliens

is in English.

One of the researchers (“SEARCH”) is logician and mathematician,

as those fields are supposed to be of universal validity.

Here the protocol of the (nearly) first contact.

Yours

Trestone:

SEARCH: “Hello ALIEN, we are especially interested in your logic

                  and mathematics

                  and wether they are different to ours?”

ALIEN:      “Hello SEARCH, we do not have one logic or mathematics.

                   We use different ones for different purposes.”

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

ALIEN:      “Just give me some problems you want to handle, and we will find 

                   a suitable logic for you.”

SEARCH: “First all statements should be either true or false and implications

                  can be evaluated by analyzing the components.

                  It should help for consistant argumentation and reasoning.”

ALIEN:      “Human classical logic would be a good choice,

                  but not all statements

                  would be either true or false.

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

SEARCH:  “With the execeptions, do you think of statements like

                  the liar statement:“This statement is not true”?”

ALIEN:       “Yes, and with this logic you will have mathematical restrictions like

                    the incompleteness theorems of Kurt Gödel or the set of all sets 

                    being no set.”

SEARCH:   “You know Kurt Gödel?”

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

SEARCH:   “As we tried to do. Could you show me a logic without

                   the restrictions you mentioned?”

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

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

                    purposes?”

ALIEN:         “We tried a “joke”!

                     A logic of the kind you asked for is not to complicated

                     but a little bit technically boring.

                     You have to use additional dimensions.

                     It is similar to solving the square root of -1

                     with complex numbers.”

SEARCH2:    “Just try do explain it to me. By the way I am a new human being,

                       as my collegue died of old age.”

ALIEN:           “Hello SEARCH2!

                        Perhaps we should give longer answers to you …

                        For analyzing all three problems indirect proof

                        is classically used.

                        So there are statements which would be simultaneously true

                        to their negations.

                        In the new logic these statements

                        (or more precisely their truth values)

                        are in another dimension than the negations.

                        We call this dimensions layers and the logic “layer logic”.

                        There are indefinitly many layers k=0,1,2,3,…

                        and every statement has a truth value in every layer.

                        The truth values can be different in different layers.

                        Classic statements are similar to layer statements

                        that are constantly true

                        (=T) or constantly false (=F) in all layers greater than 0.

                        In layer 0 all layer statements are undefined

                        (=U, a symmetrical starting)

                        and we have “undefined” as a third truth value in all layers.

                        All layer statements need a truth value in every layer

                        and truth values do only exist for the combination

                        of statements and layers.

                        Truth values can be defined recursivly using

                        already defined statements

                        and smaller layers.”

SEARCH2:     “Let us try an example,

                       the statement “This statement is not true”.”

ALIEN:           “First we have to add layers,

                       as a statement alone has no truth value:

                        “This statement L is not true in layer k”.

                       Now we have to define a truth value for L in every layer.

                       We do this by defining when L is true for every layer k+1

                       depending on the truth value of L in layer k:   

                       For every k=0,1,2,…: 

                        L is true in layer k+1 if L is not true in layer k

                         and L is false else.

                       With v(L,k)=T for “L has truth value true in layer k”:

                       v(L,k+1):=T IF ( v(L,k)=F or v(L,K)=U ) ELSE v(L,k+1):=F

                       We have v(L,0)=U as all statements are undefined in layer 0.

                       v(L,0+1):=T IF ( v(L,0)=F or v(L,0)=U ) ELSE v(L,0+1):=F

                       v(L,0+1):=T IF ( U=F or U=U ), therefore v(L,1)=T

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

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

                        therefore v(L,2)=F

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

SEARCH2:      “What does this mean for the original liar statement,

                        is it true or false?”

ALIEN:            “Not all layer statements are classical statements,

                         the liar statement is one of those nonclassical statements.

                         It has no classical truth value, but is a normal layer statement

                         with alternating truth values.

                         It is like a complex number that is not real.

                         To get the benefits of layer logic you have to use it.

SEARCH2:       “But it is not easy for me to change to a new logic,

                         for example if we talk about it we should use a known logic.”

ALIEN:             “Fortunately we can use human classic logic

                         when talking about layer logic,

                         as this logic is the meta logic of layer logic.”

SEARCH2:       “Is layer logic similar to the theory of types

                         by Bertrand Russell?”

ALIEN:             “In the theory of types objects are splitted into differend types

                         and the types are used to avoid self reference within objects.

                         In layer logic the truth values are splitted into different layers

                         and the layers enable us to have self reference

                         within objects and statements.

                         So the answer is mostly no.”

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

ALIEN:              “So let us have a look on layer set theory,

                         a rather nice piece of work.

                         The central idea is to treat “x is element of set S” (x e S)

                         as a layer statement:

                         It is true in layer k+1 that set x is element of the set S,

                         iff the statement A(x) is true in layer k.

                         v(x e S,k+1) :=T if v(A(x),k) = T (and F or U else).

                         And as in the original theory of Cantor

                         for every set statement A(x)

                         there exists a set.

                         We have the following two rules for sets:

                         Rule M1 (assignment of statements to sets):

                         For all k,sets x,set M exists a set statement A(x) which fulfills:

                 v(x e M, k+1) :=  v(v( A(x), k)=w1 v  v(A(x), k)=w2 v v(A(x), k)=w3,1)

                          with w1,w2,w3 = T,U,F or one or two of them.

                        Rule M2 (sets defined by statements):

                        For every layer logic statement A(x) about a layer set x

                        there exits a layer set M so that for all k=0,1,2,3,… holds:

                         v(x e M, k+1) :=  v( A(x), k )  (or the expressions of rule M1).

                        You asked for examples:

                        The empty set  0:

                        We use “x e 0” as A(x)

                        For all k>=0: v(x e 0, k+1) := v(v( x e 0, k )=T,1)  (=F for k>=0)

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

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

                       The full set All:

                        v(x e All, k+1) := v( v( x e All, k ) = T v v( x e All, k ) = U v

                                                         v( x e All, k ) = F , 1 )  = T

                          for k>0 and =U for k=0.

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

                                                      v  v(x e All, 0) = F, 1 )  =

                                                    = v( U = T v U = U v U = F , 1 ) = T

                           v(x e All,1+1) := v(v( x e All, 1) = T v v(x e All, 1) = U v

                                                     v v( x e All, 1) = F , 1 ) =

                                                    = v( v( T = T v T = U v T = F , 1 ) = T, etc.

                       So other than in most set theories in layer theory

                       the full set is a normal set.”

SEARCH2:     “What is with the Russell set, the set of all sets

                       that are not elements of  themselfes?

ALIEN:           “We translate the definition of the Russell set R

                       to layer set theory:

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

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

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

                        v(R e R,2) = v( v( R e R, 1 ) = F  v  v( R e R, 1 ) = U , 1 ) = F

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

                       R is a set with different elements in different layers,

                       but that is no problem in layer set theory, so R is a layer set.

SEARCH2:     “I suppose that Cantor´s diagonalization in layer theory

                        is not valid any more?”

ALIEN:            “You are right.

                         The set of all sets All is in bijection (via identity)

                         with its power set.

                         So we do not need different kinds of infinity

                         in layer set theory.

                         But let us have a look into the proof of Cantor,

                         transferred to layer theory:

                         Be S a set and P(S) its power set and F: S -> P(S)

                         a bijection between them (in layer d).

                         Then the set A with v(x e A, k+1) = T :=

                               if ( v(xeS,k)=T and v(xeF(x),k)=F )

                               is a subset of S and therefore in P(S).

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

                         First case: v(x0 e F(x0),k)=T , then v(x0 e A=F(x0), k+1) = F

                         (no contradiction, as in another layer)

                         Second case: v(x0 e F(x0),k)= F then v(x0 e A=F(x0),k+1) = T

                         (no contradiction, as in another layer)

                         If we have All as S and identity as Bijektion F

                         we get for the set A:

                         v(x e A, k+1) = T := if ( v(x e All,k)=T and v(x e x),k)=F ) =

                                                       = if ( v(x e x),k)=F )

                         This is the layer Russell set R 

                          (We omitted the ´u´-value for simplification)

                          - and no problem.”

SEARCH2:       “And can we still do arithmetics?”

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

                           Let us start with the Peano axioms:

                           We can define the successor m+ of a set m in the following way:

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

                           For k=0 without v(x e m, 0):  v(x e m+, 1) := v(x=m,1) 

                           We start with m=0,

                           v(0+,1) = v(x=0,1):   In layer 1 the only element of 0+ is 0.

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

                           v(x e 0+,2+1) :=v(x e 0,2) v v(x=0,1)= F v v(x=0,1) = v(x=0,1)

                           So 0+ is a set with only element 0 in all layers >=1.

                           Now we look at m=0+ 

                           v(x e 0++, 1) := v(x=0+,1): 

                           In layer 1 the only element of 0++ is 0+.

                           v(x e 0++, k+1) := v(x e 0+, k) v v(x=0+,1) 

                           In all layers >1 the only elements of 0++ are 0 and 0+.

                            So we find:

                            n+ contains in layer 1 exactly the element n

                            n+ contains in layer 2 exactly the elements n, n-1

                            n+ contains in layer n exactly the elements n, n-1, …,1

                            n+ contains in layer k>n exactly the elements n, n-1, …,0

                            For large k the natural numbers of layer set theory

                            are therefore similar to the classical natural numbers.

                           The (adjusted) Peano axioms hold for m+.

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

                           The addition of numbers we define using the successors:

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

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

                            Multiplication:

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

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

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

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

                             = v( x e 5, k) v v(x=5,1)

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

ALIEN:                 “You already have it:

                              For first fundaments look at a Review of the logic

                              of Prof. Ulrich Blau

                              ( as it is a pdf-file, you may have to put this URL directly                                 in your browser:

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

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

                          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 with “Stufenlogik Trestone” (in German).

                             The symbolization there is slightly different:

                              W(A,t) is used instaed of v(A,k).

                             There still is no academic paper for layer theory –

                              perhaps someone is interested to do this?”

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

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

ALIEN:                  “?”

ALIEN:                    “Here an aspect that might be interesting for philosophers:

                               The Münchhausen foundation trilemma

                                (Agrippa`s trilemma),

                                that there are only three poor choices to fundament

                                and start our argumentations

                                gets a new option with layer logic:

                                If we assume that a reason has to be true

                                in a higher level than the founded,

                                the reasoning can go back not further than to layer 1.

                                As every reasoning reduces the layers at least for 1,

                                starting at an arbitrary layer we reach layer 1

                                after finite steps.”

ALIEN:                   “?”

ALIEN:                   “Hello, is there anybody out there

                               interested to continue this communication?” 

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

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