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Trestone

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  1. Hello studiot, I do not know all about layer logic and layer arithmetics, as I just made some definitions to have a layer atrithmetic - but I have no practice in using it. I looked if with layer logic a solution for your Chinese problem would be possible with natural numbers (and different layers) - but I do not know enough of layer arithmetics. Within one layer the solution is the same as in classical arithmetic and we get three integer fractions as solutions, but we learn nothing about layers this way, as there are the same rules for arithmetics within one layer as in classical arithmetic. Yours Trestone
  2. Hello, I found a link on English to Prof. Ulrich Blau´s work (numbers, paradoxes and relexion logic) (there are not many): (It is formally more correct then my work, but really "hard stuff", most about regions of infinity that my layer logic no longer needs) https://books.google.de/books?id=Xg6QpedPpcsC&pg=PA311&lpg=PA311&dq=Reflexion+logic+Ulrich+Blau&source=bl&ots=HZNNGnEARu&sig=ACfU3U2Hunv0FZTcdq9T5TDfVa7XArq9VA&hl=de&sa=X&ved=2ahUKEwjf9ZSVnczpAhVERxUIHXhHAp44ChDoATALegQICBAB#v=onepage&q=Reflexion%20logic%20Ulrich%20Blau&f=false Yours Trestone
  3. Hello studiot, it is ok to be puzzled by layer logic. I myself was puzzled for years and still am in some questions. When constructing propositions like the layer liar, the connection between layers is easy and clear. And the use of recursions helps to connect truth values of different layers. (And layer logic helps with induction/recursion, as there is always a "free" start with "all propositions have truth value undefined in layer 0") But I am not sure if we have such connections between layers for all propositions. If there are layers in the (logical) world, they could be also independent from each other. So a layer could be like a world of its own. There is a hierarchy with the layers, but this does not mean, that a truth value in a higher layer is “more true” or “more important” than a truth value of a lower level. All “exist” simultanously and equally. As we all perceive a similar world and do seldom discuss if there is an object or not (for example because of different properties in different layers) we all seem to live in the same layer, at least with our perception. As layers increase with cause and effect, this layer is dynamic. That was one of the reasons why I believed, that there is one layer for the whole universe, and that it increases with every interaction (except gravitation) of objects in it. I have learned to give a layer to every observer frame system. But here we have left the reign of pure logic and changed to human perceiption and physics. In your chinese problem I do not see the “12 layers”? If a solution in natural numbers is looked for, the last line could be connected to layer logic: 4y + 8 z = 39 . Classical the left side is even, the right side not. In the arithmetic of layer logic, a number can be even in one layer and not even in another, so there could be a layer were there exists a solution in natural numbers. (But I do not know numbers, that are even and not even in layers – and I have not solved the puzzle.) All in all layer logic is in some respects similar to the “ Many World Theory”, but I hope not too much, as I like this theory not at all. Yours Trestone Hello Dord, I think parts of classical logic were developed in ancient Greek for speaking at court and searching for truth. As there is not “one truth” in layer logic (but one for each layer), it could help to be more open minded and to tolerate even contradictory statements. But I do not think that layer logic will help us in real live with witnesses at court: I believe, that all humans are “in” the same layer when perceiving, so the differences do not come by different layers but different people. And humans are a greater mystery then logic... Yours Trestone
  4. Hello, in my definitions for layer logic a statement does not belong to a layer, but is independent of all layers. The statements have a truth value in each layer (sometimmes different in layers) and are mostly defined by recursion. An example is the (layer) liar L: For all t= 0,1,2, ...: The liar L is true in layer t+1 if it is not true in layer t – and false in layer t+1 else. The layer truth vector of a layer statement is an infinite vector for t=0,1,2,3,... . For the liar L it is (undefined, true, false, true, false, ...) The same is with layer algorithems or programms P, they are independent or comprehensive of layers - and can stop in one layer t and not stop in an other t+1. It is possible, that a Halting programm H exists, that gives a true in layer t+1 for every layer programm P, if P stops in layer t with input X. (Important: The same H will give a true ore false for P and X for layer t+2). So I think that there are not infinite Halting programms in layer logic. (But of course I do not have H explicitely) Yours Trestone Hello studiot, in the TAO-TE-CHING (Lao Tzu) is the saying: “A journey of a thousand miles begins with one step”. May be layer logic and “layer model' in computing have this one step together. But then the journey in my eyes goes different ways: In layer logic I have infinite layers (0,1,2,3,...) and a strict hierarchy of layers: In a lower (or equal) layer no information of a higher layer is accesible, they are “blind” for all above. And the truth values in the layers are recursivly used to define the truth values of a layer statement for all layers, so layer statements are independent of layers (= defined for all layers). And mainly of course it is a logic. Layer logic has four fathers: - Classic proposional logic (at least 65 % are the same) - Three-valued logic (Łukasiewicz logic) (using three truth values, 5%) - The Logic of reflection by Prof. Ulrich Blau (using layers, only for reflecting proposals, 10%) - Layer Logic of Trestone (expanding layers to all proposals, layer set theory, 20%) It can be used very similar to classical logic: For logic itself, for doing set theory and math, for computer science, for philosophy, for physics and other science (or here for fiction). Yours, Trestone
  5. Hallo strange, my proof is finished at the line: “So there could be a holding program H with layer logic.” Your quote " The next processing layer ... It would be labeled t + r (t)." is part of later speculations. I did not solve the Halting problem but proofed, that with layer logic the proof of the Halting problem is no longer valid. In logical and mathematical layer logic I only use finite layers. The infinite layer (with layer logic there is only one infinity) I use for philosophy and there as layer of the mind. How nature and math are connected (for example pickets and natural numbers) I do not know, but some kind of connection there seems to be. And a logic that would have no connection to our reality/nature would be a strange and not very useful thing. (Sorry for my unclear and clumsy notation, but my university time is past more than 30 years). Yours Trestone
  6. Hello Strange, my main point is, to imagine, that classical logic may be not the real logic for our world and to construct an alternative. I found layer logic as a good possiblity. Of course it is very similar to classic logic (that sufficed for 2000 years), and as the meta logic of layer logic I still use classical logic. But with one parameter more (the layers), I can avoid almost all classical logical paradoxes and most indirect proofs are valid no more, even as layer logic does allow indirect proofs (within one layer). Math and computing science are still possible, but they are different in some points. Here a short analysis of the holding problem from computer science and on "layer algorithms" from the view of my “layer logic”: In the layer logic, a new parameter is added to the programs, layer t. A hierarchy applies: If a program wants to evaluate / use a value from another program from layer t, it can only do so at level t + 1 or higher (= t + r). We are looking for a (layer) program H that decides on each program P with (string) input X in layer t + r, whether this ever stops in step t or runs endlessly (e.g. due to a continuous loop). Definition / basic property H (P, X, t + r): The following applies to all programs P and inputs X: IF P (X, t) stops THEN H (P, X, t + r): = true ELSE H (P, X, t + r): = false In this case, r> = 1 must be selected, since layer t (at P) is used when calculating H. More precisely, the next universal layer t + r (t) is to be set (see below). We can now try to understand the classic counter-proof of the existence of H (P, X) and have to add the levels: Suppose H (P, X, t + r) exists with the property required above. (This is a hypothesis!) Then we use H to construct a "strange" program S: Definition S (P): The following applies to all P: S (P, t + r + k): = IF H (P, P, t + r) = true THEN loop ELSE S (P, t + r + k) = true; STOP (In contrast to the meta / colloquial formulation at H, S (P) can be written as a real program if the code of H is available. "loop" stands for a continuous loop) Here k> = 1 should be selected, since the layer t + r (at H) is used when calculating S. More precisely, the next universal layer t + r (t) + k (t + r (t)) has to be applied (see below). S therefore uses the result of the holding program when applying a program P to its own source code as input. Now we consider the self-application of S, i.e. we take the code for S as input for S S (S, t + r + k) = IF H (S, S, t + r) = true THEN loop ELSE S (S, t + r + k) = true; STOP Since H (S, S, t + r) = true exactly when S (S, t) stops, it is no longer paradoxical or contradictory: S (S, t + k + r) loops when S (S, t) stops and stops when S (S, t) does not stop! The following applies: t + k + r is not equal to t, i.e. two different layer calls from S. So S is a program with different values at different layers, but not necessarily paradoxical. S and H can therefore exist. So there could be a holding program H with layer logic. Why have I left r and k indefinite and not chosen 1 each? Now I suspect that the layer of step programs does not only depend on the subroutines called (they must be larger in each case), but also of the interactions in the universe (= the “layer of the universe " or at least of the “layer of the reference system”😞 My speculation: Every interaction (except through gravitation) increases the layer counter in the universe (also in computers) simultaneously, therefore the layers grow constantly and very quickly (and unfortunately hardly controllable). The next processing layer for layer t can only be narrowed down (at least 1 higher), but do not determine exactly. It would be labeled t + r (t). And we couldn't call computer programs twice with the same parameters ("don't go into the same river twice"), because the universal step counter “flows” constantly, and we cannot enter the step t, but we find it again and again (and higher). (Maybe it could be possible in the event horizon of a Black hole.) If the layer logic applies, then today's computer programs probably only work because they are limited to layer-independent programs, which is only a small part of the conceivable programs. Despite the problems outlined, this does not have to stay that way and maybe one or the other "layer" surprise is also possible in computer science ... E.g. one could refute the "Curch-Turing thesis" with layer computers: (i.e. calculate something new): If you implement an algorithm P (X) on a (normal) computer, the current layer t of the universe is implicitly used for each calculation, i.e. the computer calculates P (X, t). When P (X) is recalculated later, it calculates P (X, t + r). If P (X, t) is a step-dependent function, P (x, t) and P (X, t + r) could be different, although classically it should only be one value. Such a function P (X) could be the prime number decomposition of X, which could be layer-dependent for "large" X and t. However, the numbers would probably have to be so large that a practical review in this way is not yet possible. Maybe someone will try the experiment anyway or has an idea ... Yours Trestone
  7. Hello studiot, layer logic is an outsider (mine) theory to logic. About 15 years before me Ptrofessor Ulrich Blau in Munic had similar ideas, he called it “reflexion logic”. For him layers were the times we reflected about a sentence (like the liar L “this sentence is not true”. Layer 0: no reflection. L has the truth value “undefined”. Layer 1: We reflect, that L was undefined in layer 0, therefore it is true. Layer 2: We reflect on our reflection: L is false. Layer 3: L is true. And so on. I defined layers for all kind of logic sentences (proposals). But I do not know so exactly, what my layers are: Are they meta layers of logical speech, layers of causality or a new dimension or something else? Anyway as an idea they open a new look on logik and the world. And most famous proofs as by Cantor and Gödel or Turing are not valid with layer logic anymore. I see no connection to the “layer model” in computing except the name. But perhaps it is interesting for you, that a computer that would use layer logic would not be limitid by the Halting Problem. In the indirect proof we get different layers – and so there is no more a contradiction. Unfortunatelly I do not know how to built a layer computer, so layer logic is more a philosophical theory (and that was what i intended when I started it 15 years ago ...) Yours Trestone
  8. Hello, here my proof that Cantors diagonalisation ore different infinities are no more valid with layer logic. As All, the set of all sets, is a set in layer theory, it is no surprise, that the diagonalisation of cantor is a problem no more (I just give the main idea, more details in the link below) (t marks the layers, W(x,t) ist the truth value of x in layer t). 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 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 … More details at this link: https://www.scienceforums.net/topic/59914-layer-logic-a-new-dimension/?tab=comments#comment-627045 Yours Ttrestone
  9. Hello joigus, the observer-dependency was not my invention, but is a consequence of relativity. From a physical view it is not relevant if observers are “amoebe” or “humans”, as long as they are in the same frame of reference. Of course it would be nice, if we could compute the differences between frames of refernce also for my interaction count – and what consequences this has. As I do not know most consequences of different counts, I am far away from this. (And I do not know, how an amoeba experiences the world). Yours Trestone
  10. Hello swansont, with layer logic math is different. The proofs of Cantor and Gödel are valid no more and also the proof, that the prime factoriztion is unique, as there could be different factorizations in different layers. More details at this link: https://www.scienceforums.net/topic/96134-layer-logic-alternative-for-humans-and-aliens/?tab=comments#replyForm (The parts about logic and math are more serious than this speculations about physics, but there nobody answered for a long time) Yours Trestone
  11. Hello Strange. You are right, relativity is a problem to my theory. Perhaps we can assume a universe for every observer. If the observer notices an interaction, the count is increased in his universe. If an other observer does not notice an interaction, the count stays unchanged. So we have a (universal) count for every observer. Wether this approach is working, I do not know. Yours Trestone Hello swansont, I don´t believe that my theory helps to tell electrons apart. Near to the end of my text I described a possible test: „We assume that our computers (with the exception of errors) work independently of time. So if we determine the prime factorization of a large number, the same result should be delivered yesterday, today and tomorrow. Now I was able to show that logic and mathematics may apply (layer logic), where the prime factorization could be layer-dependent, and where in reality the layer could correspond to the levels of the interaction counters. So it would be conceivable that if the computer tried again to determine the prime factors of a large number, would suddenly produce different results. Unfortunately, in this experiment we may have to wait until the end of the universe ...” Yours Trestone
  12. Hello, in my research to a new logic I also encountered the following more physical problem: According to my theory it could be that the interaction of two particles was different (or not possible) if both have previously had different interactions (e.g. since the Big Bang or on a pool table). In practice, particles or billiard balls can always interact with each other, no matter how many and different interactions preceded. Ockham's razor therefore says that the interaction stories are not relevant. I counter this with the "pocket knife of my imagination": We can also assume that all interactions in the universe are always counted, and this counter is available universally and simultaneously (= not locally) in the universe. It can only increase, so it has a direction. And only particles with the same interaction counter reading can interact and increase the counter by 1. We see that this counter is very similar to classic Newtonian time. Therefore, I consider it a new dimension of time / impact /interaction. Of course the whole thing would be superfluous if it only explained interactions, for which you don't need it at all. But you can (along with a local time reversal) explain many of the curiosities of quantum theory. First of all, I have to clarify that when interacting gravity (or space-time curvature) is excluded. One reason is that it is less easy to define. A second is that the physicists have not yet come to equal treatment of the four basic forces and my new extra time could make space-time difficulties. Since I am not a physicist, we can also say that I separated it “intuitively”. My model for a quantum interaction looks like this: At a starting point (e.g. in front of a double slit) a particle has according to the uncertainty relation, several possibilities to reach target points where interactions could take place. These possibilities explore swarms of "virtual possibility particles". As long as the interaction at the target has not yet taken place, these are not real and can return to the start (with information from the target) inversely in time. At the start, the virtual target information cannot be read. Therefore, one of the options (including the virtual information) is selected blindly (quantum coincidence), this becomes real and triggers an interaction at the goal. This also increases the universal interaction counter by 1. The particle from the start can no longer interact due to the lower counter. (You can also interpret it as "it moved from start to target", however, the possibility paths are not particle paths and can e.g. go through two slits (interference)). Entanglement can also be explained with the interaction counter and inverse possibility particles. Here the virtual information about the polarization filter angle is brought back (time inverse) to the start, after selecting these angles go through and then become real with the interaction at the target (or the two targets) and the counter increases. The time reversal for virtual possibility particles was in these considerations maybe more important than the interaction counter, that makes an interaction real and fixes the result (quantum measurement). But there is also an application in which the influence of the impact counter could be shown more directly: We assume that our computers (with the exception of errors) work independently of time. So if we determine the prime factorization of a large number, the same result should be delivered yesterday, today and tomorrow. Now I was able to show that logic and mathematics may apply (layer logic), where the prime factorization could be layer-dependent, and where in reality the layer could correspond to the levels of the interaction counters. So it would be conceivable that if the computer tried again to determine the prime factors of a large number, would suddenly produce different results. Unfortunately, in this experiment we may have to wait until the end of the universe ... I do not want to hide one disadvantage of the interaction counter: Although I allow time travel "on a small scale", "on a large scale" I make it impossible: If my grandfather lived in front of me at interaction counter k, that's how i was born at k + r. So I can only interact with objects with interaction counter k + r or larger, but not with my grandfather in the past at k. So the grandfather murder was murdered. The lottery numbers are similar: If the drawing of the lottery numbers at counter k + r, this cannot provide information about counter k be brought before the drawing because there is nothing to interact with. So maybe you can go back in time, but that's only useful for quantums, because in the second time dimension interaction counter everything is ordered (macroscopically) monotonously. Anyone interested in details of the layer logic from which all of this started here the link: https://www.scienceforums.net/topic/96134-layer-logic-alternative-for-humans-and-aliens/?tab=comments#replyForm Yours Trestone
  13. Hello, in the meantime I developed layer logic further and tried to apply it to philosophical and physical questions. In German you cand find here most details: http://philo-welt.de/forum/thread.php?postid=458920 For example I found a solution of the mind - body problem by doing a special interpretation of quantum theory: If particels or quants have several possible ways from start to target, invisible "virtuel possible" particles will go all the ways to the target, and then (still invisible and virtuel) come back to the start, reverse in time, bringing back informations about the (future) target. As they are in the same layer as when started, this virtual informations can not be read by the start. Therefore in physical quantum movements, one of the returning particles has to be selected blindly, and this will become the real particle. So we unterstand quantum contingency better. We already learned that the physical world has a universal layer, that increases with every interaction (except gravitation). Now I assume, that the mind belongs to the infinite layer. If body and mind are onnected in the nervous system, the mind can "read" all quantum informations, especially the informations of the target. He therefore can choose "conciously" and not "blind". In this way the mind can act, but he can only choose possibilities, that the body also could have chosen by chance. Another point is to connect gravity (distortion in space-time) with the mind. As there is mostly a combination of body and mind, the gravity effects could be between minds (in the infinite layer). Dark matter could be "pure mind". But this all is rather speculative of course ... Yours Trestone
  14. Hello, more than a year has gone, but I am still exploring layer logic, mostly in German. Here an older link 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. Hello, it looks as layer logic is "too human for aliens and too alien for humans." I take this as an encouragement. Yours, Trestone.
  21. 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
  22. 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
  23. 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
  24. 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?”
  25. 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
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