• PL Olcott
    626
    I forgot to say this: L is the formal language of a formal system

    My True(L,x) predicate is defined to return true or false for every
    finite string x on the basis of the existence of a sequence of truth
    preserving operations that derive x from

    A set of finite string semantic meanings that form an accurate
    verbal model of the general knowledge of the actual world that
    form a finite set of finite strings that are stipulated to have the
    semantic value of Boolean true.

    False(L,x) is defined as True(L,~x).
    Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))

    Finite string expressions that are not truth-bearers are rejected
    as a type mismatch error for every formal system of bivalent logic.

    Truthbearer(English, "This sentence is not true") is false.
    Truthbearer(English, "This sentence is true") is false.
    Truthbearer(English, "a fish") is false.
    Truthbearer(English, "some fish are alive") is true.

    Truthmaker Maximalism (is what the above ideas are anchored in)
    https://plato.stanford.edu/entries/truthmakers/#Max
  • Lionino
    1.8k
    that derive x fromPL Olcott

    From what?

    string semantic meaningsPL Olcott

    Semantic and meaning mean the same thing. Is the quote above supposed to mean "meaningful strings" as for example "the dog bites the ball" instead of "gorbyr dortug equerxi"?

    verbal model of the general knowledge of the actual worldPL Olcott

    What's "general knowledge" supposed to mean as opposed to just "knowledge"? Also, if the answer to the question in the previous quote is "Yes", "string semantic meanings" and "verbal model" approximately mean the same thing, if they mean anything at all.

    that
    form a finite set of finite strings that are stipulated to have the
    semantic value of Boolean true
    PL Olcott

    ?
    You are writing things from a train of thought but the purpose of writing is communication, you need to include the train of thought, not just its destination.

    A set of finite string semantic meanings that form an accurate
    verbal model of the general knowledge of the actual world
    that form a finite set of finite strings that are stipulated to have the semantic value of Boolean true.
    PL Olcott

    We have an X that forms a Y that forms a Z, but X and Z seem awfully similar, as if they mean the same thing.

    False(L,x) is defined as True(L,x)PL Olcott

    Do you mean False(L,x) is defined as True(L,¬x)?

    Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))PL Olcott

    Yes, something is a truth-bearer if it is true or false.

    Finite string expressions that are not truth-bearers are rejected
    as a type mismatch error for every formal system of bivalent logic.

    Truthbearer(English, "This sentence is not true") is false.
    Truthbearer(English, "This sentence is true") is false.
    Truthbearer(English, "a fish") is false.
    Truthbearer(English, "some fish are alive") is true.
    PL Olcott

    That is the naïve reply to sentences such as "This sentence is a lie". Claiming that it is not a truth-bearer is alike hand-waving, you must give some account as to how it is not a truth bearer.

    Another further issue is that by the law of non-contradiction, something is X or it is not-X. Something is true or it is not true. Replying that "a fish" is neither true or false while nevertheless defining "false" as not-true violates the LNC. The "type mismatch" is encompassed in "not true" just like "green" is encompassed in "not salty" when you ask the equally nonsensical question "Is the colour green salty or not salty?" — do we really need to come up with a concept of "salty-bearer" or can't we just say things that taste salty are salty and everything else is not salty?
  • PL Olcott
    626
    That is the naïve reply to sentences such as "This sentence is a lie". Claiming that it is not a truth-bearer is alike hand-waving, you must give some account as to how it is not a truth bearer.Lionino

    It is far too much to unpack all oat once.
    I don't say: "This sentence is a lie",
    I refer to the strengthened Liar Paradox: "This sentence is not true."

    ?- LP = not(true(LP)).
    LP = not(true(LP)).

    ?- unify_with_occurs_check(LP, not(true(LP))).
    false.

    The means LP is rejected as not a truth bearer in Prolog because
    it has an infinite cycle in its evaluation graph.

    This sentence is not true.
    What it is not true about?
    It is not true about being not true.
    What is it not true about being not true about?
  • Lionino
    1.8k
    I don't say: "This sentence is a lie",
    I refer to the strengthened Liar Paradox: "This sentence is not true."
    PL Olcott

    Those two mean the same.

    ?- LP = not(true(LP)).
    LP = not(true(LP)).
    PL Olcott

    No clue what that means.

    The means LP is rejected as not a truth bearer in Prolog because
    it has an infinite cycle in its evaluation graph.

    This sentence is not true.
    What it is not true about?
    It is not true about being not true.
    What is it not true about being not true about?
    PL Olcott

    It is the criticism of the liar paradox refering to nothing. It was discussed in the thread I linked.
  • PL Olcott
    626
    It is the criticism of the liar paradox refering to nothing. It was discussed in the thread I linked.Lionino

    I have spent two decades on this. It <is> a truth predicate that would
    work because Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x)) screens out
    epistemological antinomies that Tarski get stuck on.
  • Lionino
    1.8k
    I have spent two decades on this.PL Olcott

    That's rad.

    It <is> a truth predicate that would work because Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x)) screens out epistemological antinomies that Tarski get stuck on.PL Olcott

    @jgill@fishfry
  • PL Olcott
    626
    What's "general knowledge" supposed to mean as opposed to just "knowledge"?Lionino

    General knowledge can be expressed in a finite set of finite strings.
    Specific knowledge of everything is unmanageably large and infinite.
  • fishfry
    3k
    It <is> a truth predicate that would work because Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x)) screens out epistemological antinomies that Tarski get stuck on.
    — PL Olcott

    @jgill@fishfry
    Lionino

    I'm familiar with Pete's work from other forums.
  • tim wood
    8.9k
    True(L,x)PL Olcott
    Tell us, how do you know True(L,x) is true?
  • PL Olcott
    626
    True(L,x)
    — PL Olcott
    Tell us, how do you know True(L,x) is true?
    tim wood

    Expressions that are {true on the basis of meaning} are ONLY
    (a) A set of finite string semantic meanings that form an accurate
    model of the general knowledge of the actual world.

    (b) Expressions derived by applying truth preserving operations to (a).

    True(English, "a cat is an animal") is true.
  • tim wood
    8.9k
    (a) A set of finite string semantic meanings that form an accurate
    model of the general knowledge of the actual world.
    PL Olcott
    Ok, but how exactly do you decide what is, or is not, a member of this set?
  • PL Olcott
    626
    (a) A set of finite string semantic meanings that form an accurate
    model of the general knowledge of the actual world.
    — PL Olcott
    Ok, but how exactly do you decide what is, or is not, a member of this set?
    tim wood

    It is simply all of the details of every fact of the world. General knowledge is a finite set of axioms.
  • tim wood
    8.9k
    The problem is that you have a set. But it is by no means clear how you create that set.
  • PL Olcott
    626
    ↪PL Olcott The problem is that you have a set. But it is by no means clear how you create that set.tim wood

    That is not the point. The set of verified facts of the world is defined to exist, it is merely
    not written down all in one place yet. LLM AI models might be able to achieve this
    within a few years.

    The point is that when we know all of the general knowledge facts of the world then
    we can easily screen out every epistemological antinomy (as a type mismatch error
    non-truth-bearer) that many of the undecidability proofs depend on. Tarski undefinability
    proof depends on this.

    https://liarparadox.org/Tarski_247_248.pdf
    https://liarparadox.org/Tarski_275_276.pdf
  • tim wood
    8.9k
    That is not the point.PL Olcott
    That is the point. Not only have you not got it; it may not be achievable - that depending on the exact details and definitions. You invoke an oracle, but give no account of it other than some hand-waving. And what do you mean by "verified fact"? Is a verified fact different from just a fact? How do you verify it - what does verified mean? Do you even know what a fact is? Do you know the difference between fact and true?
    Of course you can do what you want and claim what you want. But when your claims cross over into nonsense, they amount to an illegitimate appropriation. Make it legitimate.
  • PL Olcott
    626
    That is the point. Not only have you not got it; it may not be achievable - that depending on the exact details and definitions. You invoke an oracle, but give no account of it other than some hand-waving. And what do you mean by "verified fact"? Is a verified fact different from just a fact? How do you verify it - what does verified mean? Do you even know what a fact is? Do you know the difference between fact and true?tim wood

    If we merely encoded all of the rules of algorithms, logic, and programming in a single
    formal system then when when no sequence of truth preserving operations from these
    basic axioms derives x or ~x then x can be rejected as a type mismatch error on the basis
    that all formal systems of bivalent logic require every expression to be a truth-bearer.
  • tim wood
    8.9k
    If we merely encoded all of the rules of algorithms, logic, and programming in a single formal systemPL Olcott
    All right, a programming language.
    truth preserving operations (TPOS)PL Olcott
    An example or two, please?

    And the Truths these TPOs are expected to preserve, whence them - your having only a language? And, "no sequence"? How do you define "no sequence"?

    Let's imagine you generate a listing of all possible propositions/statements of 100 or fewer symbols in length - a very long list. Let's further imagine you can test for and eliminate all nonsense strings, leaving only those that are syntactically "healthy." Still a long list. Now to test each for truth with your TPOs. The result for each item on the list being either T for true, or F for everything else. What is your specification for both the TPO and truth itself that the TPO can distinguish what is true from what is not?

    And what about strings of longer than 100 symbols? There are lots of true propositions/statements longer than 100 symbols that your TPOs will record as F.

    Unless you have already created an encyclopedia of sorts. Then you could test against that, but that would be very far indeed from being either conclusive or exhaustive or in itself interesting.
  • PL Olcott
    626
    If we merely encoded all of the rules of algorithms, logic, and programming in a single formal system
    — PL Olcott
    All right, a programming language.
    truth preserving operations (TPOS)
    — PL Olcott
    An example or two, please?

    And the Truths these TPOs are expected to preserve, whence them - your having only a language? And, "no sequence"? How do you define "no sequence"?
    tim wood

    {All cats are animals}
    {All animals are living things}
    therefore {All cats are living things}

    The principle of explosion is not truth preserving.
    {All cats are animals} // axiom
    {No cats are animals} // false assumption
    therefore FALSE
  • tim wood
    8.9k
    {All cats are animals}PL Olcott
    You have a programming language - where does a statement about cats come from? How do you know "no cats are animals" is false and not itself an axiom?
  • PL Olcott
    626
    {All cats are animals}
    — PL Olcott
    You have a programming language - where does a statement about cats come from? How do you know "no cats are animals" is false and not itself an axiom?
    tim wood

    Expressions that are {true on the basis of meaning} are ONLY
    (a) A set of finite string semantic meanings that form an accurate
    model of the general knowledge of the actual world.

    (b) Expressions derived by applying truth preserving operations to (a).

    The above algorithm specifies True(L,x) and False(L,x) defined
    as True(L, ~x).
  • tim wood
    8.9k
    Expressions that are {true on the basis of meaningPL Olcott
    Ah, meaning. What is that? How does your program assess or even recognize meaning? I am asking the simplest and most basic questions because it seems to me you must have both asked and answered them. But so far I have no evidence of that in this thread, or seen it in your other threads.

    Above your syllogism as an example of a TPO
    {All animals are living things}
    {All cats are animals}
    therefore {All cats are living things}
    PL Olcott

    There are many more possible premises/conclusions. The questions here are how does your PTO or your computer know what is true or meaningful? How does it construct the right syllogism from the possible premises?
  • PL Olcott
    626
    Ah, meaning. What is that? How does your program assess or even recognize meaning? I am asking the simplest and most basic questions because it seems to me you must have both asked and answered them. But so far I have no evidence of that in this thread, or seen it in your other threads.tim wood

    https://en.wikipedia.org/wiki/Ontology_(information_science)
    of Rudolf Carnap / Richard Montague {meaning postulates} that stipulate relations
    between finite strings as providing the semantic meanings that form an accurate
    model of the general knowledge of the actual world.

    Rudolf Carnap told Willard Van Orman Quine that the otherwise totally
    meaningless finite string of "Bachelor(x)" is defined as the otherwise totally
    meaningless finite string "~Married(x)" and Quine just could not get it.

    The full definition of "Married(x)" entails (at least) billions of other meaning
    postulates defining "Human(x)".
  • tim wood
    8.9k
    of Rudolf Carnap / Richard Montague {meaning postulates} that stipulate relations between finite strings as providing the semantic meanings that form an accurate
    model of the general knowledge of the actual world.
    PL Olcott
    English, please. Simple sentences are good.

    I'll try to make it simpler. Given some string, call it Σ, we can start by supposing that Σ is/is not meaningful, is/is not true. How do you know/decide? Because I infer you have your program do it, the question is really, how does your program decide?
  • PL Olcott
    626
    I'll try to make it simpler. Given some string, call it Σ, we can start by supposing that Σ is/is not meaningful, is/is not true. How do you know/decide? Because I infer you have your program do it, the question is really, how does your program decide?tim wood

    Facts are sentences that are defined as true. Cats <are> Animals is defined as true.
    The otherwise totally meaningless sequence of letters of "cats" and "animals" are defined to have the <is a type of> relation to each other.

    All of the facts about the world work this same way. You wanted it simple and provided a complex example. In one case "Σ" is a Greek letter. Even this begs the question: What is Greek? and What is a letter?

    https://www.mathsisfun.com/algebra/sigma-notation.html requires a whole mathematical infrastructure.
  • tim wood
    8.9k
    Facts are sentences that are defined as true.PL Olcott
    That's a pretty good definition! But you're missing the whole point. Who or what defines, and on what basis or by what criteria? If it's humans all the way down, I'll take that as an answer, but that will leave the question as to how your whole program will work, in as much as it will have to be preloaded with that which it is supposed to produce.
  • PL Olcott
    626
    That's a pretty good definition! But you're missing the whole point. Who or what defines, and on what basis or by what criteria? If it's humans all the way down, I'll take that as an answer, but that will leave the question as to how your whole program will work, in as much as it will have to be preloaded with that which it is supposed to produce.tim wood

    Yes it is the case that only humans have a complex language, however, apes have learned a symbolic language known as Yerkish. https://en.wikipedia.org/wiki/Yerkish

    There is a data structure known as a knowledge ontology that is based on a directed graph.
    https://en.wikipedia.org/wiki/Ontology_(information_science)

    A knowledge ontology has a unique integer (such as the CYC project's use of the 128-bit GUIDs) for each sense meaning of every word. https://en.wikipedia.org/wiki/Cyc

    A word is a string (AKA sequence) of characters such as "dog". The first sense meaning is the most common one: https://www.merriam-webster.com/dictionary/dog# These differing sense meanings have an integer index in dictionaries. A knowledge ontology might require an ISO standard dictionary so that its unique sense meanings expressed as 128-bit integers can correspond to their sequence of characters in this IOS standard dictionary.

    A knowledge ontology is an inheritance hierarchy of these sense meanings. This means that the sense meaning of {dog} (the animal) gets most of its meaning from {animal} and only adds details that distinguish a {dog} from other {animals}.
  • tim wood
    8.9k
    Ok, I understand your answer that meaning is assigned by people, the strings of which translated by the machine into a kind of index and score, and that the machine then associates strings on the basis of their index and scores. But these associations are probabilistic only, and neither in themselves truth bearing or producing, Assuming the program is creating some master list of true statements, how is the truth of any particular statement judged, if not by a person?
  • PL Olcott
    626
    But these associations are probabilistic only, and neither in themselves truth bearing or producingtim wood

    In other words you seem to believe that "a cat is probably an animal" and "a cat is probably not a fifteen story office building". I disagree.
  • tim wood
    8.9k
    Not at all - for a person. But all the machine has got is probability and some kind of heuristics.

    You seem to be in the position of a carpenter with a hammer who insists that his hammer drives nails, the problem being that the hammer can be used to drive nails, but that there never was nor will be a hammer that itself drives any nails. You have claimed seemingly endlessly and relentlessly claimed that your program/machine does something. I am just attempting to find out from you what exactly it does, and if relevant, how.

    So far I think your machine just generates strings of symbols as candidates for inclusion in a list, but that apparently require the judgment of a person for that inclusion.
  • PL Olcott
    626
    So far I think your machine just generates strings of symbols as candidates for inclusion in a list, but that apparently require the judgment of a person for that inclusion.tim wood

    We simply correctly encode all of the true facts of the world. When the discussion
    devolves into "facts according to who" I lose interest because the discussion has
    devolved away from actual truth.
  • tim wood
    8.9k
    We simply correctly encode all of the true facts of the world.PL Olcott
    Then why bother with a machine/program? You have simply gone to the trouble of creating a data-base - in theory because there are significant problems im creating one for real.

    Further, as you have defined a fact as a proposition defined as true, then being dismissive of "facts according to whom" is at the least disingenuous. And this in turn implies that while you in part seem to know what a fact is, you do not in fact know what one is. That is, that facts are never true. This a formal distinction that has not much use in the ordinary world, but is critical here. The problem is that in defining it true, you then take it to be true, and that only possible through either a deliberate or ignorant disregard of a distinction between instrumental truth, e.g., gravity is a force, the atom is like a small solar system, and a priori truth. The one being conditionally true and perhaps otherwise untrue, the other being always universally and necessarily true.

    So you have created in theory a listing of a lot of your or someone's opinions. Can you demonstrate having done any better than that?
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