Trestone

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  1. Three hairs in Loki`s soup In former times humans were on more familiar terms with gods and they could even bound Loki to serve them a soup of the gods. But Loki wouldn´t have been Loki if he thereby did not plot some tricks: He added three hairs to the soup, which became eatable only if all were found and removed. 1) The first hair was discovered by Pythagoras and Euclid: “ The square root of two is irrational”, i.e. it is not representable as a fraction of two integers - and there is a infinite amount of such numbers. As they could not remove this hair, the humans got used to swallow this irrational lumps in their soup for over two thousand years. 2) The second hair was detected by Georg Cantor: “The cardinal number of the power set is higher than that of the set” i.e. it has a greater kind of infinity and we can build a never ending sequence of infinities. He could not remove this hair, but a lot of mathematicians are pleased, that so the soup will never be spooned up. About one hundred years later this is a bit boring and some would like to know what is waiting at the bottom of the soup – instead of spooning forever. 3) The third hair of Loki was discovered by Kurt Gödel in 1931, it is named “Gödel's incompleteness theorem”: In every axiomatic system including arithmetic there are true propositions which are not provable in this system. (And there are much more true propositions than proofs). This hair was difficult to find, up to now it is also not removeable. If we imagine the provable propositions swimming on top of the soup, we are just skinning the soup with our proof-spoons and never come to deeper regions. There is a solution key as always in fairy tales: With the soup Loki gave a spoon called “logic” to the humans and they liked to use it. With this spoon they could eat a little of the soup, but the hairs could not be removed with it. Who wants to get to the bottom of the soup has to carve a new one. A first try which helps to handle the first and the second hair provides a search with “layer logic Trestone”. The third hair being a little more complicated, the help of a mathematician would be welcome, but chances are good. If the soup is empty (all hairs removed) I have to find more details for what is to be found at the bottom of this super bowl … Yours Trestone
  2. layer logic - alternative for humans and aliens?

    Hello, merry Christmas and a new year 2017 (on earth) and 2022 (on your alien planet) full of peace to all reading this. Yours Trestone
  3. layer logic - alternative for humans and aliens?

    Hello ydoaPS, in most cases the truth values of propositions in a layer are defined using the truth values of propositions in lower levels. But it is possible to define the truth value of a proposition in some layers independently. As I do not see the similarity of layer logic to modal logic, can you give some details? Yours Trestone
  4. layer logic - alternative for humans and aliens?

    Hello Strange, the fewer use layer logic has for humans and aliens the more free I am in my research. And as humans, aliens and layer logic are probably changing, it is not impossible that sometimes it will become usefiull ... Yours Trestone Hello Endy0816, layer logic uses three truth values, but that is not so important. More important is the use of layers. That layers give a new look on indirect proofs: If different layers are involved there are no longer contradictions. The diagonalization of Cantor does not work with layer logic. I have strong indications that Gödel´s incompleteness theorems are not valid with layer logic, but I did not proof this up to now (help welcome). Changing of logic means to change a lot of things … Yours Trestone
  5. layer logic - alternative for humans and aliens?

    Hello ydoaPs, Layer logic is more similar to (a modified) classical propositional logic than to modal logic. It does not handle modalities or possibilities. It uses three truth values „true“, „false“ and „undefined“. The most important feature are the layers. A propostion in layer logic does not have one truth value but has a truth value in every layer 0,1,2,3,... , and different truth values in different layers are allowed. So in a way every proposition has an infinite truth vector. The layers are like additional new dimensions and allow new handlings of contradictions. Yours Trestone
  6. layer logic - alternative for humans and aliens?

    Hello, it looks as layer logic is "too human for aliens and too alien for humans." I take this as an encouragement. Yours, Trestone.
  7. layer logic - alternative for humans and aliens?

    Hello endy0816, the main new idea of my logic is that truth has (or can have) different layers. Therefore a statement can be true in one layer and false in another. Or with another view: Properties can be layer dependent - in every layer we can have a different world. Of course we do not experience this layer differences in our everyday life and most things seem to be constant over layers and only one world. But with borderline phenomena like infinity, mind - body interaction, consciousness, etc. this could be different. The layers open a lot of possibilities to avoid contradictions which restrict classical logic. The main definitions I have given in my opening thread and the link at its end. Like with complex numbers we can sovle problems with layer logic, that can not be solved by classic logic (or other logics I know). For example Gödels incompleteness theorems are not valid any more, but natural numbers and a (in some parts different) arithmetics are definable. As my studying at university is about thitrty years ago, I can not give a representation of layer logic in modern state of the art, perhaps somebody will try this? Yours Trestone
  8. layer logic - alternative for humans and aliens?

    Hello studiot, thank you for asking! My question is, if layer logic is a consistent alternative to classic logic - or if there are some deeper faults or incomprehensible parts. The alien story is a kind of "advertising", as there seems to be little interest in discussing a new logic ... Yours Trestone
  9. layer logic - alternative for humans and aliens?

    Hello, I am back from a holidy in Finland at the polar circle. So I am refreshed and ready to answer your questions. P.S. You do not have to wait ten years with your answers ... Yours Trestone
  10. 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?”
  11. layer logic: a new dimension?

    Hello, perhaps some examples will help to show how layer logic works (and hopefully inspire somebody to answer or ask a question): 1) a perception statement A := „I see a red car“. This classical statement can be true or false: W(A)=w or W(A)=f. In layer logic this statement has a truth value in every layer t: W(A,t) = w or f or u. In practice we will not need different values in different layers for concrete perceptions (and we will not need the third value u). Therefore we have the spezial case in layer 0 as always: W(A,0)=u and W(A,t)=w or W(A,t)=f for all t>0. This is an explanation why we do not have to agree to a layer at concrete statements when determing a truth value. 2) Implicit undefined or self-referencing statement B:= „This statement is not true“ In classic logic this statement is neither true nor false. In layer logic we have to modify it slighty, as true is only valid with a layer: SB:= „This Statement ist true in layer t+1, if it is not true in layer t (and false else)“ It is W(SB,0)=u. Therefore W(SB,1) = W(W(SB,0) -= w,1) = w Therefore W(SB,2)= W(W(SB,1) -= w,1) = f , W(SB,3)= w, W(SB,4)= w etc. Here we have a dependency on layers. In practice the second kind of statements seems to be rather rare, therefore layers seemed to be unnecessary, and problems or paradoxes appeared only at the borders of the system. Wether a logic with layers and less problems or paradoxes is a worthwile proposition (at least for specialists) may be a matter of taste … Yours Trestone
  12. layer logic: a new dimension?

    Hello, Up to now I had defined natural numbers in layer logic and set theory by the following successor function m+: To every set m we define a successor m+: W(x e m+, t+1) := W(x e m, t) v W(x=m,1) ( for t=0 without W(x e m, t) ) The (adjusted) Peano axioms hold for m+. We can define 0, 0+, 0++ etc. this way. The so defined “natural numbers” m are not constant over layers: In small layers t<m m has less elements than in large (where it becomes constant) and similar to the classical natural numbers. But we can use an alternative definition, that is not so hierarchical: If m is defined in layer t+1, we can use values regarding m and layer t+1 to define m´ in layer t+1: W(x e m+, t+1) := W(x e m, t+1) v W(x=m,1) This definition is nearer to the classical natural numbers and I think we get sets, that are not not layer dependent. I have not checked all Peano axioms yet. We might do this overall and reduce the use of layer hierarchie to critical cases (like self reference and undefinedparts). Whether we get back some of the classical problems (like Gödel´s uncompleteness theorem) by this way I do not see so far … Yours Trestone
  13. layer logic: a new dimension?

    Hello, I still do not really know what the „layers“ are, but cause and effect seem to give a hint: For cause and effect are (classically) in a hierarchic order, i.e. the cause has influence on the effect, but not the other way round. Same with the layers in layer logic: A statement in layer t has a truth value and can contribute to the definition of a truth value of a statement in layer t+1, but not vice versa. So we can assign causes to lower levels (like t) and effects to higher levels (like t+1). With cause-and-effect chains we can construct (almost) arbitrarily high levels. If we want to start a cause-and-effect chain, We can use two specialities of layer logic: On the one hand there is layer 0, the ultimate zero point, i.e. every chain in layer logic has a natural starting point there (and no infinite regress necessary). On the other hand: How ever high we are in a layer logic chain (with a statement to layer t), we can come down easily: We just use the meta statement „W(A,t)=w“, and this statement belongs to layer 1. (Regarding my last holiday I call this “the Irish slide”). I think that this resembles in some parts my intuitive understanding of the mind-body relation, but this here only as a side note. So even if there remain doubts about concret cause and effect relations (think of Hume!), those relations are the best examples for “real” layers that I can give today. Yours Trestone
  14. Beyond infinity

    Hello, the results of Cantor and Gödel are astonishing. But we still can ask, what we can do to avoid the consequences (and perhaps expulse some mathematicans from Hilbert`s paradise). As physics has showed us with time, space and matter, a small change in basic concepts may alter a lot. In mathematics we have the complex numbers that show how to solve unsolvable equations like x*x= -1. As (classical) logic is used in the proofs of Cantor and Gödel, I looked for alterations in logic, that would influence these proofs and give us other results. As I always was doubtful to indirect proofs (where there are two branches), I experimented with different “views” in logic, where every “view” has its own truth value for a proposition. (For example “the liar” is “true” from view1 and false “from” view2). Later I called the views “layers” and used (inductive) natural numbers t = 0,1,2,3,… Now a proposition or statement is not longer “true” (ex-)or false”, but has a truth value only in combination with a layer t, and the truth values in different layers can be different. The layers have some hierarchical order, but the whole is quite different to the “Hierarchy o Types” by Russel, as self reference is fully allowed. Of course this is not classic logic any more, but it has nice effects to set theory: In Cantor`s diagonalization proof we can see, that different layers are implied and therefore there is no contradiction. We therefore need no different kinds of infinity, the “set of all sets” is without paradoxes (and is its own powerset), even “Russell´s set” is existing witout contradiction. We even need less axioms than ZFC to form a set theory. More Details are in the neighbouring thread: http://www.scienceforums.net/topic/59914-layer-logic-a-new-dimension/ The Peano axioms and an arithmetic can be defined with layer logic (but the uniqness of the prime factorization might not hold), I suppose that Gödels proof for incompletness does not hold for layer logic and layer set theory, but beeing no professional mathematican I did not look deeply up to now. Perhaps someone might do this? (And formalize my more intuitive settings). Yours Trestone
  15. layer logic: a new dimension?

    Hello ydoaPs, all this argumention and the proofs are based on classical logic. As I showed, Cantor´s diagonalization does not work in "layer logic". I assume, that Gödel´s proof does not work with "layer logic" and "layer logic arithmetic", but this I have not proofed up to now (beeing no professional mathematican). Changing the rules of logic is of course a wild thing and a kind of "nasty trick", ("not playing fair"), but why not give it a try? Yours Trestone