• TonesInDeepFreeze
    265
    Note: In my previous posts, anywhere I mistakenly wrote 'level' I meant 'layer', as I guess would be obvious anyway.



    In ordinary mathematical logic, contradictions are syntactical, not requiring assignment of truth values. Meanwhile, as far as I can tell, your layer logic is described primarily semantically in terms of truth values; I don't know the syntax of whatever proof system you have in mind, so I can't evaluate the means by which you would prevent (syntactical) contradictions. You could assert that provability entails soundness, but we need to prove that, not just assert it, and you can't prove it without first stating what the proof system is.TonesInDeepFreeze

    That stands without your response.

    I would guess that layer logic does disprove contradictions. That is, layer logic disproves all formulas of the form 'P & ~P' (where they "reside" (or\e whatever way you say it) in the same level [should be layer]).TonesInDeepFreeze

    Is my guess correct?

    And does layer math prove the following?:

    ~0=1

    and

    ~Ex (x is a natural number & x>x)

    And you admit that layer math does not prove the fundamental theorem of arithmetic. So layer math would not seem to offer much as a mathematical foundation anyway.

    what you say is that there are three truth values and that statements are evaluated at different levels [should be layers]. You haven't given even the starting point: description of evaluation of truth and falsehood for atomic sentences, compound sentences, and quantificational sentences.TonesInDeepFreeze

    That stands without your response.

    how I handle the proof of the halting problemTrestone

    You begin with:

    with layer logic we have to add layers if a program has to give a value/result:
    A given program halts or not in layer k for given input data.
    Trestone

    But no axioms or rules of inference by which to claim that.

    So to follow along with you in your layer math, one just has to accept the arbitrary lines in your arguments as given by you personally (there is no objective codification). You do not provide one with a way to check whether the lines you put forth are axioms or theorems of layer math but instead one must rely solely on your dicta as to what constitutes a valid line or inference in an argument.

    my earlier handling of Cantor´s diagonalization and proof in layer logicTrestone

    You begin with:

    (t marks the layers, W(x,t) ist the truth value of x in layer t, -w stands for „not true“ or „false“
    ther value „undefined“ I left out to make things easier).
    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 is defined with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w )
    Trestone

    In ordinary logic, truth values apply to sentences. It seems that had previously been the case in your discussion of layer logic too. Here you mention the truth value of x, So I take it that x ranges over sentences there. But then we find x ranging over prospective members of the set A. So which is it? x ranges over sentences or x ranges over prospective members of sets? So far, what you've given is pseudo-math or gibberish dressed up with undefined math/logic-sounding verbiage.

    Also, you mention things (which I guess are sentence) as being true or false in layers, but now here we find that functions too are things in layers. But you've not stated what a layer sis or what kinds of things can be in layers or, as I mentioned earlier, how it is determined a given atomic, compound, or quantificational sentence is true or false in a layer.

    F: M -> P(M) a bijectionTrestone

    Are you there asserting that there exists such an F? If you are, but without first proving the existence of such an F, it would seem to be question begging, since by supposedly refuting Cantor's theorem, you're claiming to prove that there does exist such an F.

    the proof about the power set can be similary be "unproofed" like the halting problemTrestone

    Just to be clear, these are all distinct:

    (1) A proof of ~P in a given system..

    (2) A meta-proof that P is not a theorem of given system.

    (3) Pointing out a line in a purported proof of a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).

    (4) A meta-proof that P is false in a given model of a given theory.

    So, letting P = Cantor's theorem, do you you claim either (1) or (2) regarding layer math? (I take it that you do claim (4) or something like it.)

    His new world is pure nonsense and fantasy for the Cave people.Trestone

    That's question begging. One can just as well say you've not left your own cave, as you are not familiar with the logic and mathematics that has been explored by generations of logicians and mathematicians who have themselves studied alternatives including types, orders, levels in set theory, quantification over theories themselves, modalities, possible world semantics, topological semantics, and even para-consistency.
  • TonesInDeepFreeze
    265
    What is the first line in each of the below proofs that is not allowed in layer math?


    Show: There is no function from a set onto its power set.

    Proof :

    Let f be function from S to PS. Let d = {x | xeS & ~xef(x)}.

    dePS.

    If d is in range(f), then for some x in S we have d=f(x).

    If xef(x), then ~xed, so ~xef(x).

    If ~xef(x), then xed, so xef(x).

    Contradiction. So d is not in the range of f. So f is not a function from S onto PS.

    /

    Show: ~ExAy yex.

    Let Ay yex.

    Let d = {x | xey & ~xex}.

    If ded, then ~ded.

    If ~ded, then ded.

    Contradiction. So ~ExAy yex.
    TonesInDeepFreeze
  • Trestone
    51
    Hello TonesInDeepFreeze,

    I am sorry that I can not answer most your questions to formal details.
    As I said, my studying math is over thirty years ago.

    The idea with layer logic is, that all is as in classic (three valued) logic,
    only that layers have to be added, if a truth value is used or looked upon.

    Professor Ulrich Blau has done all the formal work for his reflexion logic,
    so i assume the formal part for layer logic should be possible.

    Now again to the proof of Cantor:

    What is the first line in each of the below proofs that is not allowed in layer math?

    Show: There is no function from a set onto its power set.

    In layer math we have a different kind of sets, the layer sets.
    And if something is "true" or "false" we have to give a layer.
    So we look onto a proof were all terms are transferred to layer math - we are in the "new world".

    Proof :

    Let f be function from S to PS. Let d = {x | xeS & ~xef(x)}.

    Now I transfer this to layer math: F is the layer function for f.
    M is the layer set for S.
    And A is the layer set for d.

    Be M a set and P(S) its power set and F: M -> P(M) a bijection between them (in layer k)
    Then the set A is defined by 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).
    A e P(M) (analog to d e PS).

    If d is in range(f), then for some x in S we have d=f(x).

    Transferred: If A is in range(F), then for some x0 in M we have A=F(x0).

    If xef(x), then ~xed, so ~xef(x).

    If ~xef(x), then xed, so xef(x).

    Contradiction. So d is not in the range of f. So f is not a function from S onto PS.

    Now comes the part, where layers make the difference:

    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 t and t+1 are different layers)

    Second case: W(x0 e F(x0),t) = -w then W(x0 e A=F(x0), t+1) = w
    (no contradiction, as t and t+1 are different layers)

    So in layer math, the existence of F does not lead to a contradiction.
    (And the set All with identity to P(All)=All even is an example in layer set theory).

    I hope you have a little understanding for a "Columbus",
    who accidently sailed to a new world, but is neither a governor nor a cartographer
    and if asked where he was, tells something obscure about "India".
    Nevertheless there could be a new world ...

    Yours
    Trestone
  • Trestone
    51
    Note: In my previous posts, anywhere I mistakenly wrote 'level' I meant 'layer', as I guess would be obvious anyway.

    ↪Trestone


    In ordinary mathematical logic, contradictions are syntactical, not requiring assignment of truth values. Meanwhile, as far as I can tell, your layer logic is described primarily semantically in terms of truth values; I don't know the syntax of whatever proof system you have in mind, so I can't evaluate the means by which you would prevent (syntactical) contradictions. You could assert that provability entails soundness, but we need to prove that, not just assert it, and you can't prove it without first stating what the proof system is. — TonesInDeepFreeze

    Trestone: I do not fully understand, as I do not know the technical terms (syntax?, proof system?).
    For me a contradiction is, if the same statement is shown as true and not true.
    In layer logic the statement has to be in the same layer,
    as being true in one layer and being false in another layer is allowed and no contradiction.

    That stands without your response.

    I would guess that layer logic does disprove contradictions. That is, layer logic disproves all formulas of the form 'P & ~P' (where they "reside" (or\e whatever way you say it) in the same level [should be layer]). — TonesInDeepFreeze
    Trestone: No, only when there are different layers used.
    In most classical proofs that are indirect or by contradiction,
    different layers are used, if they are transferred to layer math,
    so many are disproved, but not those in the same layer.


    Is my guess correct?

    And does layer math prove the following?:

    ~0=1 Trestone: false in layer math

    and

    ~Ex (x is a natural number & x>x) Trestone: false in layer math

    And you admit that layer math does not prove the fundamental theorem of arithmetic. So layer math would not seem to offer much as a mathematical foundation anyway.
    Trestone: It is not so good for multiploikation and primes - b ut what if the real worl is so?

    what you say is that there are three truth values and that statements are evaluated at different levels [should be layers]. You haven't given even the starting point: description of evaluation of truth and falsehood for atomic sentences, compound sentences, and quantificational sentences. — TonesInDeepFreeze
    Trestone: In many cases it helps, that in layder 0 all sentences have truth value "undefined".
    That often can be used for starting. More I have not looked upon yet.


    That stands without your response.

    how I handle the proof of the halting problem — Trestone


    You begin with:

    with layer logic we have to add layers if a program has to give a value/result:
    A given program halts or not in layer k for given input data. — Trestone


    But no axioms or rules of inference by which to claim that.

    So to follow along with you in your layer math, one just has to accept the arbitrary lines in your arguments as given by you personally (there is no objective codification). You do not provide one with a way to check whether the lines you put forth are axioms or theorems of layer math but instead one must rely solely on your dicta as to what constitutes a valid line or inference in an argument.

    Trestone: Yes, I have not developed a full layer informatics, I just added layers to programms,
    that give a result. That was enough to abandon the Halting problem and
    create Non-Turing algorithms.


    my earlier handling of Cantor´s diagonalization and proof in layer logic — Trestone


    You begin with:

    (t marks the layers, W(x,t) ist the truth value of x in layer t, -w stands for „not true“ or „false“
    ther value „undefined“ I left out to make things easier).
    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 is defined with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w ) — Trestone


    In ordinary logic, truth values apply to sentences. It seems that had previously been the case in your discussion of layer logic too. Here you mention the truth value of x, So I take it that x ranges over sentences there. But then we find x ranging over prospective members of the set A. So which is it? x ranges over sentences or x ranges over prospective members of sets? So far, what you've given is pseudo-math or gibberish dressed up with undefined math/logic-sounding verbiage.

    Trestone: x is a member of a layer set and therefore itself a layer set.

    Also, you mention things (which I guess are sentence) as being true or false in layers, but now here we find that functions too are things in layers. But you've not stated what a layer sis or what kinds of things can be in layers or, as I mentioned earlier, how it is determined a given atomic, compound, or quantificational sentence is true or false in a layer.

    Trestone: Yes, I am not very precise. Everything where you can ask if it has a truth value
    (is it true, falser or undefined?) needs a layer in layer logic/math.


    F: M -> P(M) a bijection — Trestone


    Are you there asserting that there exists such an F? If you are, but without first proving the existence of such an F, it would seem to be question begging, since by supposedly refuting Cantor's theorem, you're claiming to prove that there does exist such an F.

    Trestone: like in the proof of Cantor, I asume herre that such a F exists.

    the proof about the power set can be similary be "unproofed" like the halting problem — Trestone


    Just to be clear, these are all distinct:

    (1) A proof of ~P in a given system..

    (2) A meta-proof that P is not a theorem of given system.

    (3) Pointing out a line in a purported proof of a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).

    (4) A meta-proof that P is false in a given model of a given theory.

    So, letting P = Cantor's theorem, do you you claim either (1) or (2) regarding layer math? (I take it that you do claim (4) or something like it.)

    Trestone: (5) I do not disprove the original P of Cantor,
    but a transferred P2 in a new model, layer math.


    His new world is pure nonsense and fantasy for the Cave people. — Trestone


    That's question begging. One can just as well say you've not left your own cave, as you are not familiar with the logic and mathematics that has been explored by generations of logicians and mathematicians who have themselves studied alternatives including types, orders, levels in set theory, quantification over theories themselves, modalities, possible world semantics, topological semantics, and even para-consistency.
    TonesInDeepFreeze

    Trestone:
    "You will know them by their fruits" (Matthew 7:15-20)
    Even if Matthew warns here of bad people,
    I am astonished what new fruits (not only in mathematics)
    are in range with layer logic.


    Yours
    Trestone
  • TonesInDeepFreeze
    265
    I am sorry that I can not answer most your questions to formal details.Trestone

    You're lacking not just all the formal details, but even a coherent outline.

    F: M -> P(M) a bijection between themTrestone

    Again, you can't assume that there is bijection from M onto PM. You are purporting to prove there is such a bijection, so you can't do that by assuming there is one. Unless layer logic is so meaningless that it allows proof by question begging.

    The problem is not just a lack of familiarity with the technicalities of mathematical logic, but that you don't understand even the very basics of even informal logical reasoning.

    in layer math, the existence of F does not lead to a contradictionTrestone

    You don't know that unless you've proven the consistency of layer math. But, of course, you can't prove the consistency of something that is not formed with sufficient determinateness to tell what is a theorem and what is not.

    hope you have a little understanding for a "Columbus"Trestone

    That would be the Columbus who brought widespread fatal disease, subjugation, and genocide. Meanwhile, you bring confusion, ignorance, and misinformation. Not as bad, so I wouldn't insist on the comparison.
  • TonesInDeepFreeze
    265
    I do not know the technical terms (syntax?, proof system?).Trestone

    Those are utterly basic to the subject. If you don't even know what proof is, then you can't very well explain whatever alternative system you have in mind.

    For me a contradiction is, if the same statement is shown as true and not true.Trestone

    That's kind of okay in an everyday sense. But a contradiction is actually a formula of the form

    P & ~P

    (Or, more generally, a set of formulas that implies a formula of the form P & ~P.)

    indirect or by contradictionTrestone

    I've informed you twice already that you are misusing that terminology. The particular proofs we are talking about in ordinary mathematics are not indirect (and 'proof by contradiction' is another term for 'indirect').

    That is, layer logic disproves all formulas of the form 'P & ~P' [?']TonesInDeepFreeze

    No, only when there are different layers used.Trestone

    So, for example, within a single layer you don't prove "It is not the case that both 2 is even and 2 is not even".

    ~0=1 Trestone: false in layer mathTonesInDeepFreeze

    Really, you think it is false that 0 does not equal 1?

    ~Ex (x is a natural number & x>x) Trestone: false in layer mathTonesInDeepFreeze

    Really, you think there is a natural number that is greater than itself?

    In many cases it helps, that in layder 0 all sentences have truth value "undefined".Trestone

    You haven't said how, in general, one evaluates the truth of atomic sentences, compound sentences, or quantificational sentences in any layer.

    Non-Turing algorithmsTrestone

    Pray tell, what class of algorithms do you have in mind that are not Turing machine computable? Actually, to reduce even more confusion and misinformation than you've already posted, pray don't tell.

    You begin with:

    (t marks the layers, W(x,t) ist the truth value of x in layer t, -w stands for „not true“ or „false“
    ther value „undefined“ I left out to make things easier).
    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 is defined with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w )
    TonesInDeepFreeze

    like in the proof of Cantor, I asume herre that such a F exists.Trestone

    You are completely confused.

    Cantor proves that such a bijection does not exist. You purport to prove that such a bijection does exist.

    To prove that such a bijection does not exist, we may assume one does exits, then derive a contradiction, then conclude that such a bijection does not exist (and that is not indirect proof).

    But to prove that such a bijection does exist, you can't start by assuming that such a bijection does exist. That is question begging.

    Moreover, we don't even need to assume such a bijection exists and derive a contradiction. Rather, we argue from universal generalization: Let f be any function from S to PS. Then we show that S is not a surjection.

    x is a member of a layer set and therefore itself a layer set.Trestone

    You skipped the problem I mentioned. First you use 'x' to stand for a sentence, then 'x' to stand for a set. That, like virtually everything else you say, is nonsense.

    the proof about the power set can be similary be "unproofed"TonesInDeepFreeze

    "unproved" is nonsense terminology unless you specify which of these you mean:

    1) A proof of ~P in a given system..

    (2) A meta-proof that P is not a theorem of given system.

    (3) Pointing out a line in a purported proof in a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).

    What you seem to think is that you have accomplished a variation of (1) by showing that in your own system it is not the case that there is not a bijection between S and PS. But you fail from the very start by assuming that there is such a bijection.

    "You will know them by their fruits" (Matthew 7:15-20)Trestone

    For crying out loud, you compare yourself to Lenin, the hero of the Cave, Christopher Columbus, and Matthew the Apostle. Maybe get a grip and cut back on the self-aggrandizement.

    Anyway, the fruitfulness of set theory is in having a recursively axiomatized theory for the mathematics of the sciences, conceptualization and formalization of computability theory that enables, among other things, the computer technology we all use to type things on the Internet, a treasure of peer-reviewed articles that provide reading about fascinating intellectual accomplishments, and a starting point for approaches that differ from classical mathematics, including multi-valued logics, constructive mathematics, paraconsistent logic, and others that can be quite exotic. Meanwhile, the fruit of your postings is from the poisoned tree of ignorance.
  • Trestone
    51
    Hello TonesInDeepFreeze,

    thank you for still answering me!

    I do not think that we will in near time agree on the main points of Layer Logic,
    but to have a look from different sides helps me to understand,
    where open questions and points might be.

    With the two examples with the “~” I made a mistake, as you noticed:
    I thought the “~” refered only to the next sign, and not to the whole term.
    ~0=1 Trestone: true in layer math and
    ~Ex (x is a natural number & x>x) Trestone: true in layer math

    Perhaps this is exemplary: If we are not used to the language of the other,
    even small misundestandings can create totally wrong interpretations.
    (And probably most times I am wrong).

    Looked upon only as a theory, it is not nice, that natural numbers might have
    different prime decompositions in different layers.

    But it could help my theory without being fully formalized:
    If (perhaps in 100 years) we handled big enough numbers (with computers),
    the following could happen:
    On one day we get for a number n the prime decomposition P1.
    One week later we get on the same computer with the same program for n
    another prime decomposition P2 (and similar disturbing results with other computers).

    (After a speculation of me, there are layers in the real (physical) world,
    and they increase with every physical interaction (except gravity).
    So to increase layers we only have to wait a little time.)

    Most probably this will be handled as computer bugs
    and nobody will remember Layer Logic.

    But by chance somebody might check even old articles -
    and Layer Logic could be one of the candidates to explain the surprising effect.

    As I can not tell how large n and the layers have to be,
    I can not say how long we will have to wait for an Arthur Stanley Eddington.
    (Yes, now I have compared myself with Albert Einstein,
    but he is already part of my nickname)

    So “sit and wait” is maybe not the worst strategy to develop Layer Logic ...

    Yours
    Trestone
  • TonesInDeepFreeze
    265
    ~0=1 Trestone: true in layer mathTrestone

    What is the proof in layer math?

    ~Ex (x is a natural number & x>x) Trestone: true in layer mathTrestone

    What is the proof in layer math?

    In general, without stating axioms and a proof system, how do know what is true in layer math? Is there a layer math phone hotline you call and they tell you what's true or false?
  • Trestone
    51
    Hello TonesInDeepFreeze,

    in the German link to layer logic there are some more definitions.
    Layer Logic in German at ask1.org

    For example this to the natural numbers (here in English):

    N1: Definition of successor function M+ for level set M
    (for the construction of natural numbers):

    Vt> 0: W (x e M+, t + 1): = W (x e M, t + 1) v W (x = M, 1)
    Let us consider the 0: W (x e 0, t) = f for t> 0.
    “Zero” is therefore empty from t = 1 (independent of the level).

    1 = 0 +: W (x e 0+, t + 1) = W (x e 0, t + 1) v W (x = 0.1) = W (x = 0.1)
    From t = 1, "one" contains exactly the one element "zero".

    (Therefore 0 is not the same as 1).

    General: n + contains exactly the elements n, n-1,…, 1 in level t> 0

    The addition can now also be defined in the same way as the classic procedure:
    W (xen + m+, t + 1): = W (xe (n + m)+, t + 1) = W (xe (n + m), t) v W (x = (n + m), 1 )

    As natural numbers are layer sets in layer set theory,
    I have to define what n>m means (and to be more exact, what it means in layer k).

    I propose, that a natural number set n is greater than a number set m in layer k,
    if n has more elements than m in layer k.
    As x always has the same elements as x in layer k>0
    we can say, that in all layers k>0 there is no x that is a layer natural number
    and that fulfills x>x.

    Yours
    Trestone
  • TonesInDeepFreeze
    265
    You seem to expect others to understand your notations without your having explained them from the bottom up. It borders on solipsism.

    Do you understand the notion of either using commonly known notation or explaining our own personal notation starting at its most simple?
  • jgill
    1.2k
    On one day we get for a number n the prime decomposition P1.
    One week later we get on the same computer with the same program for n
    another prime decomposition P2 (and similar disturbing results with other computers).
    Trestone

    Why would another prime decomposition, P2, arise if not for a computer implementing this layer logic? Thus, avoid disturbing results by ignoring layer logic.

    Sorry. I see things from weary old eyes. :roll:
  • Trestone
    51
    Hello TonesInDeepFreeze,

    there are two problems:

    First, even having studied mathematics 30 years ago,
    I never liked the formalisms.
    And when studying philosophy, my feeling was,
    that the (formal) approach to logic was not good.

    In oppsition to all this I developed my new logic
    using and developing my own notations over almost twenty years –
    in (mostly German) discussions and not in a very systematic way.

    Over the years less and less reactions came to my writings,
    so now I am used to being “a voice crying in the wilderness”.

    The second problem is, that meanwhile I can hardly take
    the position of someone to whom Layer Logic is new.

    To me all looks easy and clear – as I have lived with it for so long time.
    Here questions can help, but point one is a problem.

    And maybe unconsciously I want to be the only one
    who understands Layer Logic,
    so that I am the only one who can play with it ...

    Yours
    Trestone
  • Trestone
    51
    Hello jgill,

    in my eyes you do not have to install Layer Logic on a computer:
    It`s already there.
    (Of course this is only a daring hypothesis of me.)

    It is the same as with the General Theory of Relativity,
    Eddington had not to install it to the sun andf the light rays,
    it was already part of the world (if it was true).

    In everyday life we do not notice that we have Layer Logic instead of classic logic,
    as most propositions are not layer dependent.
    But prime decompositions of very large numbers could be.

    I assume that the the change of layers is very easy in everyday life
    (and in computers):
    we just have to wait for some time.

    (More exact: We have to wait for the next physical interaction
    (not by gravity) in our environment.
    And with every evaluation a computer has to use a new layer)

    As usually propositions do not change with layers,
    we do not notice this change of layers in our surroundings.

    But the layers could be there all the time ...

    Yours
    Trestone
  • jgill
    1.2k
    As usually propositions do not change with layers,
    we do not notice this change of layers in our surroundings.

    But the layers could be there all the time ...
    Trestone

    You have developed a personal philosophy based on LL. Congratulations.
  • EricH
    338

    Perhaps a naive notion here, but whether a statements/procedure can take a value or true or false depends on the context. I apologize in advance if I am going over stuff that you are familiar with.

    1) Factual statements
    Are we making a statement about the real physical world (AKA Kant's nuomenal world, existence, the universe, reality, everything that is the case, etc)? Not sure what country you're from, but in the USA if you give testimony in a court of law you swear to tell the truth - which basically means that if a statement accurately (or as accurately as possible) describes an event in the real world, then it is true. This is the correspondence theory of truth.

    With this in mind, the sentence/proposition "This sentence is false" does not describe any event in the real physical world, and as such it cannot be assigned a truth value.

    2) Logical/Mathematical Statements/propositions
    Logic/math statements do not refer to any event (real or hypothetical) in the physical universe, but are only true or false depending on the rules within the particular mathematical/logical system framework being used. While there are many such frameworks, at the risk of over simplifying each of these logical frameworks works in a similar fashion. You start off with axioms which are defined to be true within the particular logical framework you are working in - and then you have rules for generating other statements. For example, here are the axioms for Peano Arithmetic. I am not an expert in this, but there are many people on this forum who are highly knowledgeable and might provide you with more details.

    So now the question here is, within what logical framework are we asserting the proposition "This sentence is false"? You said it is a "paradox of classical logic". The term "classical logic" is a bit vague, but here is the SEP article on Classical Logic Perhaps there is a way to translate the sentence into classical logic syntax (it's beyond my capabilities) but I'm reasonably confident that even if the sentence could be formulated it would have a value of false


    Put differently, the sentence "This sentence is false" does not express a coherent thought and thus cannot take a truth value. There are many examples of nonsense sentences in the philosophical literature. "Quadruplicity drinks procrastination". "Colorless green ideas sleep furiously." We recognize that these sentences are grammatically correct and we also recognize that they are nonsense sentences. So there is no need to construct an elaborate logic system to handle such statements.

    But maybe I'm missing your point altogether.
  • TonesInDeepFreeze
    265
    now I am used to being “a voice crying in the wilderness”.Trestone

    To get out of the wilderness, you could you formulate your notions in a way that other people can follow them, step by step, from basic to more involved.

    maybe unconsciously I want to be the only one
    who understands Layer Logic,
    Trestone

    You're doing a great job to make sure that you are.
  • Trestone
    51
    Hello EricH,

    originally the Layer Logic was only a theory and a new logic system
    like others.
    So I handled truth values with Layer Logic but I did not bother what truth really was.

    Later I noticed, that I could descripe cause and effect with layers,
    and now all measured properties are described by me with layers.
    Here I am near to Factual Statements.
    But why would we need the real physical world for "This sentence is false"?
    It is a logical proposition to me and so part of the logical world
    (but I can not decide wether it is true or false).

    Transferred to Layer Logic the liar LL has different events/objects in different layers:
    „For all k=0,1,2,3,...: This proposition LL is true in layer k+1, if LL is not true in layer k
    and LL is false in layer k+1 else.“

    The definition of the truth value of LL in layer k+1 only depends on values of layer k.
    This values are like events/objects in layer k (here logical and not physical).

    As I showed this definitions lead to alternating truth values u,t,f,t,f,t,...,
    and there are no problemds with the truth v alues any more).

    The idea with Layer Logic is to use a classic logic (like L3 of Łukasiewicz)
    and add layers 0,1,2,3,4,... to it – for every time a truth value is determined.

    Using the layers hierarchical when defining propositions,
    most paradoxes and antinomies can be avoided.
    And proofs that use contradictions are mostly valid no more,
    as true in one layer and false in annother layer are allowed truth values.
    So the Layer Logic helps at much more problems than only the liar.

    The use of self references is explicitely allowed in Layer Logic,
    it is only not allowed within a layer.

    LL is a sentence that makes perfect sense in Layer Logic,
    it is an example of a sentence with changing truth values with different layers.
    Classic logical sentences would have a constant truth value in all layers,
    except in layer 0 where all layerr statements have the value “undefined”.

    But there is a more factual use / interpretation of Layer Logic:

    if an object in physics has a measured proposition,
    whe can apply a layer to this measurement:Objekt O has proposition p in layer k
    (= when measured in layer k).

    Now I think that if there is a physical interaction (not gravity) around O,
    the layer in the sourrounding will increase (say to k+1).

    As there are physical interactions all the time,
    we only have to wait a very short time to get a new and higer layer
    (for example in a computer).

    So if we have a property, that changes from layer k to layer k+1,
    we just have to wait untill both layers are reached one after the other.
    Such a property could be the prime decompensation of a large number n.

    And so we could see in a physical experiment that there are layers
    (or something other strange).

    Yours
    Trestone
  • TonesInDeepFreeze
    265
    Logic/math statements do not refer to any event (real or hypothetical) in the physical universe, but are only true or false depending on the rules within the particular mathematical/logical system framework being used.EricH

    That is not universally accepted by all philosophers of mathematics.

    The term "classical logic" is a bit vague,EricH

    Classical logic is exactly formalized. It's not vague.

    Perhaps there is a way to translate into classical logic syntax (it's beyond my capabilities) but I'm reasonably confident that even if the sentence could be formulated it would have a value of falseEricH

    The subject is explicated by Tarski's Theorem. If a theory has a truth-predicate for itself, then the theory is inconsistent, thus it has no model, thus there is not a model in which to evaluate the truth of falsehood of 'this sentence is false'.

    Put differently, the sentence "This sentence is false" does not express a coherent thoughtEricH

    'Express a coherent thought' is an informal notion. Without formalization, there is no effective procedure by which in general people may objectively and definitively determine whether or not something "expresses a coherent thought". Mathematical logic though deals with 'this sentence is false' in a formal way that is clear and objective.

    even if the sentence could be formulated it would have a value of falseEricH

    This bears repeating: If a theory has a truth-predicate for itself, then the theory is inconsistent, thus it has no model, thus there is not a model in which to evaluate the truth of falsehood of 'this sentence is false'.
  • synthesis
    915
    On the other hand I am interested to learn,more about the liar, extensions and the solutions?Trestone

    "All speech is lying."

    Where it is true, it is also false (because there is an equal amount of everything in everything).

    So whereas you might believe that some things are patently true or false, they are all true and false. It is due to the limitations of language (language's relationship with Reality) and all things intellectual that makes this the case.
  • EricH
    338

    Suggest you try to follow what @TonesInDeepFreeze is saying - they have deep knowledge of this topic.
  • EricH
    338

    Thank you for the intelligent response. It appears to me that we're more or less on the same side of this particular discussion - namely that the OP does not make much sense. My analysis was more informal - yours is clearly grounded in a deeper knowledge of math.

    The term "classical logic" is a bit vague, — EricH
    Classical logic is exactly formalized. It's not vague.
    TonesInDeepFreeze

    One question for you - when I said "the term classical logic is a bit vague" - I was referring to the way it was used in the OP. In my response, I pointed to the SEP article on Classical Logic Is this your understanding of the term classical logic? If not, could you point me in the right direction?
  • TonesInDeepFreeze
    265
    the OP does not make much senseEricH

    The poster refers to 'classical logic', but doesn't know what classical logic is.

    the SEP article on Classical Logic Is this your understanding of the term classical logic?EricH

    I only glanced through that article just now, but it looks good to me. I have found that Stanford Encyclopedia of Philosophy articles on logic are excellent and beautiful to read.
  • Trestone
    51
    Hello TonesInDeepFreeze,

    you remind me of the border guard
    who demanded the TAO-TE-KING from Laotse.
    (Again a comparison to a famous person ...)

    Unfortunately, I am unable to write down my ideas in 1000 lines
    of beautiful Classical Chinese.

    Therefore, here a short poem by me in German and English
    and quotations from the beginning of the TAO-TE-KING.

    Logeric
    Es pendelte ein Philosoph mit der Bahn
    nahm sich dabei der Logiklücken an
    verhedderte sich in Stufen
    denn die Geister die er gerufen
    verlachen Logik als Wahn.

    A philosopher commuted by train
    took care of logic gaps there in vain
    tangled up in layers
    called spirits in his prayers
    that laught at logic in wane.

    TAO TE KING (Laotse, Geman by Richard Wilhelm)
    Der Sinn, der sich aussprechen läßt,
    ist nicht der ewige Sinn.
    Der Name, der sich nennen läßt,
    ist nicht der ewige Name.
    "Nichtsein" nenne ich den
    Anfang von Himmel und Erde.
    "Sein" nenne ich die Mutter der Einzelwesen.
    ...
    Des Geheimnisses noch tieferes Geheimnis
    ist das Tor, durch das alle Wunder hervortreten.

    TAO TE CHING (Laotse, English by Stephen Mitchell)
    The tao that can be told
    is not the eternal Tao.
    The name that can be named
    is not the eternal Name.
    The unnamable is the eternally real
    Naming is the origin
    of all particular things.
    ...
    Darkness within darkness.
    The gateway to all understanding.

    Yours,
    Trestone
  • Trestone
    51
    Hello,

    maybe it helps if we try imagination:

    First we add a layer (0,1,2,3,...) to every statement with a truth value.
    For example “The liar statement is false – in layer 2” and
    “The liar statement is true – in layer 3”.

    All statements in layer 0 will have the value “undefinded”,
    and in all higher layers every statement has exactly one of the values
    “true”, “false” or “undefined”.

    Then we define truth values of statements by using higher layers
    for the defined values and lower layers for the defining ones.
    For example “Statement LL is true in layer k+1, if LL is not true in layer k
    and LL is false in layer k+1 else.“

    And fourth we use classic 2-valued logic and statements as meta logic
    if we talk about layer logic, especially if we talk about layers or truth values
    (like in the rules above).

    And last we imagine a world where we all live in the same invisible layer
    that grows with time.

    What consequences would all this have for logic and math?

    I believe there would be fewer contradictions
    and there could be a rather simple set theory.

    Yours
    Trestone
  • TonesInDeepFreeze
    265
    you remind me of the border guard
    who demanded the TAO-TE-KING from Laotse.
    Trestone

    A better comparison is with the boy who declared, "The Emperor has no clothes."
  • Trestone
    51
    Hello TonesInDeepFreeze,

    about a month ago I wrote a fairytale about layer logic,
    based on this story with The Emperor`s Clothes.

    The role of the Emperor and the boy/child is played by other people in my version ...

    Link to German fairytale:
    https://www.leselupe.de/beitrag/der-logik-neue-kleider-146395/

    In English:

    The Logic´s New Clothes

    Once upon a time there was an emperor who loved science.
    He called many bright minds around him who were eagerly researching.

    One day two logicians came and made their discipline palatable to him
    with the following words: Our logic is not only two thousand years old,
    it is also the basis of all science and only those who are stupid
    and not suitable for true science cannot understand it, because it is very easy.

    The Emperor did not understand why he needed this logic now,
    and so he asked deserved statesmen and ministers what they thought of it.
    They did not want to show any nakedness and emphasized the universal validity
    and unquestionable truth of the new logic.

    So the emperor dared to go to the streets with the new old logic.
    Even when the root of 2 turned out to be irrational by means of logic,
    one could suddenly find uncountable infinite sets and the arithmetic showed
    that most true sentences could not be proven, this was seen and believed
    as an indication of the sophistication of this logic.

    "But the logic has no clothes on - it doesn't work!" finally called a little child.
    Then they quickly threw logic a few layers over and ended the procession.

    There is also a story where I compare myself with Kassandra,
    the prophetess that nobody listens to.

    German link:
    https://www.leselupe.de/beitrag/die-logik-von-troja-146428/

    In English:

    The logic of Troy

    Troy had withstood the siege of the Greeks for ten years.
    The voices calling for Agamemnon to withdraw grew louder and louder.
    Odysseus called for the carpenters and started one last trick.
    The Greeks withdrew with all their ships and left behind on the beach
    just a huge wooden horse.
    It was supposed to grant Athena's blessing on the journey home
    and was dedicated to her Logos.

    With the Trojans, Laocoon and Kassandra warned against bringing
    so calamity into the city.

    But Laocoon and his sons were killed by a snake sent by Athena -
    whoever kills is right, that is the logic of the Greeks and Trojans!

    And although Kassandra always prophesied the truth, no one believed her anymore,
    because Apollo, who had been rejected by her, had cursed her gift of vision this way.

    "Do you really want to break down the walls of common sense for this Danaer gift?" -
    They did it with joy and had a feast.

    The Greeks who crawled out of the horse's belly late at night,
    then celebrated an orgy of a completely different kind,
    opened the gates for the returning warriors and slaughtered Troy with fire and sword.

    And since nobody listens to Kassandra's calls for alternatives even today,
    we still use the captivatingly simple logic of the Greeks and Trojans ...

    Yours
    Trestone
  • TonesInDeepFreeze
    265
    Our logic is not only two thousand years old,
    it is also the basis of all science and only those who are stupid
    and not suitable for true science cannot understand it, because it is very easy.
    Trestone

    That's not an argument logicians would give for the worthiness of logic.

    the arithmetic showed
    that most true sentences could not be proven
    Trestone

    What logician ever said that?

    There is no sentence that can't be proven in some system. However for any system (of a certain kind), there are sentences not proven in that system.

    Why don't you at least learn to properly identify that which you falsely comment on?

    They did not want to show any nakedness and emphasized the universal validity and unquestionable truth of the new logic.Trestone

    Fear of not being accepted is not a basis on which logicians endorse work in the field of study. Or, if you claim it is, then please point to a logician of whom it is true.

    it doesn't work!Trestone

    It's question begging just to claim that it doesn't work. Actually, logicians prove it does work in the sense of (1) proving soundness, (3) proving completeness, (3) displaying the development of mathematics from axioms.
  • TonesInDeepFreeze
    265
    The logic of TroyTrestone

    An analogy so attenuated in connection with logic that it's ridiculous.
  • Trestone
    51
    Hello TonesInDeepFreeze,

    Is it good or bad to be ridiculous?

    Yours
    Trestone
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