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Thread: Mathematical truths v. scientific truths (knowledge of)

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    Mathematical truths v. scientific truths (knowledge of)

    There is no difficulty in assuming that we don't actually know anything about the material world. We don't even need to claim to know that we know nothing about it since we may believe it is the case, or indeed believe it is not the case.

    It won't make any difference if we can only have beliefs about the material world and it won't make any difference if we can know the material world but are unable to tell the difference between belief and knowledge.

    Our possible claims to knowledge can be seen as just beliefs we believe are knowledge. Beliefs we think of as knowledge but are not can be regarded as beliefs free from doubt.

    And I don't see why they would have to be any substantial difference in behaviour between people who believe they know and people who believe they believe. I certainly seem myself to keep behaving as if I knew my way around even though I believe that I actually don't.

    We can think of human beings as acting according to whatever they happen to believe and we can think of scientific theories as particular beliefs. We don't even need to pretend that scientific theories are knowledge since it is good enough to believe they are true. And that we act accordingly.

    There is also no difficulty is ranking our beliefs in terms of the kind of justification we think we have to have them. And we can rank science as our most certain belief about the material world. I certainly do.

    And we can also rate our own perception of the material world as more reliable most of the time than anything other people might want to claim they know. And I certainly do.

    Is mathematics any different?

    Obviously, we think of mathematics as an abstraction, and apparently even most mathematicians do. However, there is also no difficulty if we accept that no mathematician knows any mathematical truth.

    As in the case of our beliefs about the material world, it is enough to think of mathematicians as acting according to their beliefs about mathematical truths. Here again, there is no substantial difference between believing you know and believing you believe as long as you act according to your belief, which seems to be what the notion of belief suggests we do anyway.

    Knowledge of the material world is similar to the notion of the infinite. It makes no practical difference as long as you don't claim to know that you know. Claiming to know that you know does expose you to the risk of being contradicted by the facts of the matter, or at least by something you may come to believe about them at some point in the future.

    Why take that risk? Well, we believe it can be very useful. We believe we can leverage our claim to have something we don't have to gain some substantial material advantages. And we all do it, anyway (I believe).

    British mathematician Andrew Wiles may believe he proved Fermat's Last Theorem and maybe he did, and that he did or did not would make a substantial difference, but that he merely believes he did, and that we all believe he did, instead of all believing that we know he did, doesn't make any substantial difference.

    Just like when I say "I love you". What matters is not that we should know it to be true, but that we should all believe it to be true, and then not even forever, but just on the moment.

    And act on it.

    Which explains gracefully why we seem to be here at all to tell the tale.

    There is no good reason to assume any substantial difference between our various senses of perception of the material world, on the one hand, and our mental capabilities, such as memory, logical sense, and even feelings and abstract thought, on the other. The latter capabilities are essentially, in all but name, perception of the material world, with the caveat that they are perception of that part of the material we call our body, and concerning our memory and our logic, that part of the material world we call our brain.

    Thus, mathematical abstractions behave in every way as do anything in the material world. We perceive them where they are, in this case in our own brain and they appear to us just as characteristically themselves as do the tree in my garden. Differences are just as significant as differences between vision and hearing, for example.

    Thus, mathematical abstractions may just be as unknowable as anything else. You know the one you have in mind, but then you can't hold in mind all mathematical abstractions at the same time. And there goes our claim to knowledge down the drain.

    Further, since physics is now suggesting a universe which many people like to think of as essentially mathematical--that is, somehow mathematical in nature, whatever that may mean exactly--there is a near perfect analogy between knowledge of the material world and knowledge of the Platonic world of mathematical abstractions.

    We are able to believe we know them.
    EB

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    Math and science are abstractions in that it is a result of our thoughts, metaphysics. Systems of thought.

    What sets science and math apart is injective quantification. Numbers. In sconce the definitions are the Systems International units. Math I believe reduces to counting.

    As to 'physics says' physics does not say anything. Physics is mathematical models that are predictive. Someone in physics may say 'the universe is mathematical, but that is philosophy, a perspective.

    Scientific knowledge is expressed mathematically. A chemical process. Calculating the trajectory of a space probe.

    Newtons laws of motion are a scientific truth, but not absolute. They work over a range of velocities but not all velocities.

    I do not know if there are mathematical truths analogous to scientific truths. Math is a collection of axiomatic systems that are shown to be logically consistent, no ambiguities.\\The OP uses the word truth in a general way. One has to define what a math and science trutrh is to be able to have a conversation.

    Is 1 + 1 = 2 a truth or the result of the definitions of counting and arithmetic? 1 + 1 = 2 is simply a shorthand notaion for counting. It is a model.

    Applying the model, 1 apple + 1 apple = 2 apples. The statement is true in the context of definition of integers and arithmetic.

    The idea of matemetics a kind of separate reality with independent truths IMO is a philosophy. Math is not out there, it is all in our heads. Describing perception of math is limited by language. Math as a reality is a manner of speaking.

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    Well written.

    There’s a lot I disagree with, of course, but it was a nice read.

    I believe there is traffic out on some of the roadways right now. If I merely believe it, I’m possibly mistaken. Often misunderstood is that if I know it, I’m also possibly mistaken. But, that needs to be taken in a specific kind of way to understand it. I’m not even going to try and explain it. It’ll derail focus from my point.

    So from another angle, if I merely believe it (that there is traffic out on some roadways right now), it may be untrue. What I believe may be untrue. The belief may be untrue; however, if I know it (if I know that there is traffic out on some roadways right now), then the belief is not untrue.

    Curious. I’ve been taught that knowing implies believing but not the inverse. Broken down, if I know P, then I believe P, but if I believe P, I don’t necessarily know P. If that holds up, then we can make fun of people who claim “I don’t believe it; I know it.” Weird isn’t it? That one knows something but also (strangely) doesn’t believe it? That kind of suggests there’s no good reason to act on what one knows but rather on what one believes, especially if knowing implies not believing. Kinda backwards, I would say.

    Of course, what’s actually going on is the person is meaning that she doesn’t merely believe but also in addition knows that. This brings me back full circle to my question. How is it that my belief may be untrue if I do indeed know that there is traffic on the roadways right now if knowing it implies that it is true?

    I suggest there is a underlying consistency even if not so obvious from a ground perspective. One is merely talking in probability. One is saying. “I could be wrong, but I’m confident that there is traffic on the roadways right now.” But, a person that thinks they know it doesn’t want to express that sentiment. Take me, for example. I’m willing to concede that I’m not so all-knowing that it’s logically impossible that I’m in error, but my confidence is far higher than is suggested by qualifying my statement with, “I could be wrong, but.”

    Does anyone realize how staggeringly extreme it is to deny knowledge on such a ridiculous basis that we “MIGHT” be brains in vats? That shows that it’s logically possible. LOGICALLY possible. That’s the BAR we should use for a demaracation between belief and knowledge? Are you kidding me?

    That’s NOT how normal people speak. That’s how some philosophers take things to ridiculous extremes. Hell, the entire scenario was made up by philosophers to push the bar further down the extremist line, so if anything, knowing it’s made up by philosophers to give thrust to their extremism run-amuck is actually a good reason to NOT believe we are brains in vats.

    This thread, however, is not quite like many that have come before it. What meaningful difference does it make whether we believe P versus whether we know P given that we’re gonna react the same regardless?

    But first, a little about the subject matter. It’s so easy in conversations like this to lose sight of the ball.

    P1 I believe that there is traffic on roadways right now.
    P2 I believe that I believe that there is traffic on the roadways right now.

    Does the truth of one affect the truth of the other? Just pause and wrestle with that. Oh, and whatever anyone does, please, for the love of blueberry muffins, don’t confuse them with:

    P3 there is traffic on the roadways right now

    “P2” has an air of self-awareness about it. With P1, I can ask myself, do I believe that? Do I believe that there is traffic on the roadways right now? Hmmm, yes, yes I do. I have that belief. The subject matter is belief. It’s about a belief of something else, but it’s a belief nonetheless. With P2, THE SUBJECT MATTER has changed.

    I wouldn’t ordinary go down that road (with belief), but a turn of phrase has been used in the OP that has been dealt with in these philosophical chambers before—regarding knowledge.

    P1: I know that there is traffic on roadways right now.
    P2: I know that I know that there is traffic on roadways right now.

    P3: there is traffic on roadways right now

    Let it be known that P1 is not P2. Readers, keep your eye on the ball. Don’t be duped into thinking that a response regarding one stands toe to toe as a reasonable response for the other. P2 is so screwed up that I can think of three varying interpretations of it off the top of my head—talk about ambiguity!

    And with that, I go back to the question. What difference (well, what meaningful difference) does SAYING one believes and SAYING one knows —beyond whatever actual difference there is between ACTUALLY believing or ACTUALLY knowing?

    His point, I guess, is that it really doesn’t matter if I know or merely believe I do when I pull out on the roadway. If I believe there is traffic (and think I know there is traffic), then I’m going to act no different when I pull out whether I think I know and am correct or think I know and am incorrect)—I’m gonna act on my belief.

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    Quote Originally Posted by fast View Post
    Well written.

    There’s a lot I disagree with, of course, but it was a nice read.
    You're a true gentleman.

    Quote Originally Posted by fast View Post
    Does anyone realize how staggeringly extreme it is to deny knowledge on such a ridiculous basis that we “MIGHT” be brains in vats? That shows that it’s logically possible. LOGICALLY possible. That’s the BAR we should use for a demaracation between belief and knowledge? Are you kidding me?
    The brain in a vat is not any basis to deny knowledge. It's only a sort of quaint and somewhat absurd story to help you understand what is meant.

    The basis for claiming that knowledge of the material world doesn't exist is, and has always been, common sense. Read Greek philosophers. Read the empiricists. It is now exactly what science implies. Whatever goes on inside your brain is only at best a probabilistic model of your environment, nothing like the 100% true required by knowledge. Assuming you even know the information that's inside your brain, what you know is then a model. You know the model, at best. And the concept of knowledge is itself part of the model. What does knowledge even mean, when meaning itself is part of the model? We are the model. Knowledge of the real world outside the model is a nonsensical concept. Knowledge is whatever the model decides is knowledge and what it is knowledge of is itself part of the model. If the model works properly, then we believe we know the world. And what it means for the model to work properly is also part of the model. Knowledge doesn't exist outside cognitive systems. There is no knowledge in a tree. Interactions and regularities, sure. Knowledge is a technical relation between different parts of the model. There are no atoms as such. There are no electrons as such. There is no spacetime as such. These things are what the model uses to represent the world. The model doesn't even represent. Representation is itself a technical relation between different parts of the model, which, therefore, is not a model. Think of Plato's shadows on the wall of the cave. But we're not the men looking at the shadows. We are the shadows. If we know anything, then we are shadows among shadows knowing shadows among shadows. And if we know anything in this sense, then, sure, we know something and we know that something through and through. But, the real world outside hat the shadows are the shadows of?

    Quote Originally Posted by fast View Post
    That’s NOT how normal people speak.
    It is. Which is why we have the distinction between knowledge and belief.

    We say we believe that p whenever we believe that we don't know that it is true that p. In which case, we mean that p is possibly false.

    We say we know that p whenever we believe that we know that p is true. In which case, we obviously and literally mean that p is not possibly false.

    But since it happens routinely that p turns out to be false, it is clear that when we say we know that p, we in fact only believe that we know that p.

    Nice trick, but sometimes, it just doesn't work and the charade is revealed. Except for the True Believers.

    Quote Originally Posted by fast View Post
    That’s how some philosophers take things to ridiculous extremes. Hell, the entire scenario was made up by philosophers to push the bar further down the extremist line, so if anything, knowing it’s made up by philosophers to give thrust to their extremism run-amuck is actually a good reason to NOT believe we are brains in vats.
    No one ever seriously suggested that you might be a brain in a vat. It is an Edgar Poe story for the little children who haven't got the necessary concepts to understand the situation like grown-ups.

    If you take that story to be an actual claim about the real world, then you are in serious need to grow up.
    EB

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    Boolean algebra is axiomized classical logic. It is called algebra because it uses varibales and equations.

    Variables are Boolean, a binary true or false. Same basic logic as formal logic.

    A true if light on false if light off
    B true if daylight false if dark
    C true if room is lit false if not
    A OR B = C C is true or false depending on the true or false of expression on the left.

    In Boolean/math logic there are no syllogisms. Only true or false propositions processed by linear logic.

    To the OP in science or technology for that matter truth is a demonstration. The BB Theory is not truth, it is the best fit model to available data.

    The last 300 years of science shows science is open to revision at any time. There are no absolutes.

    I go by Popper, for something to be science it must be testable. The only objective knowledge is the result of an experiment. Everything else is subjective debate and at some point it becomes philosophy. Such as multi universe derived from quantum physics.

    Not that it is not a useful discussion, you seem to be debating language and semantics.

    I believe that I believe makes no sense to me logically. It makes sense as an emphasis in a debate.

    From a science view I would sample traffic over several days and devolve a probability of traffic at any time in the day. What I believe is irrelevant, the numbers speak.

    I have seen heated debate in engineering groups over an issue. Even when something is demonstrated some still argue.

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    Quote Originally Posted by fast View Post
    This thread, however, is not quite like many that have come before it. What meaningful difference does it make whether we believe P versus whether we know P given that we’re gonna react the same regardless?
    But if we actually knew the world, we would not have to change what we say our knowledge of it is, and we do that all the time. The fact that we are forced to accept that we were wrong in claiming to know shows that we didn't know even though we thought we knew. If we can be deceived again and again, or if we can deceive ourselves so easily, why should anyone take anybody's claim to knowledge seriously? Instead, we behave as if people had said "we believe that". When scientists say the Big Bang was 13.7 billion years ago, everybody will take it as evidence that scientists *believe* the Big Bang was 13.7 billion years ago, not that they know it. We also take what they say as evidence that the Big Bang was indeed 13.7 billion years ago, but evidence doesn't imply truth and doesn't imply knowledge, only plausible probability and therefore belief.

    Quote Originally Posted by fast View Post
    But first, a little about the subject matter. It’s so easy in conversations like this to lose sight of the ball.

    P1 I believe that there is traffic on roadways right now.
    P2 I believe that I believe that there is traffic on the roadways right now.

    Does the truth of one affect the truth of the other? Just pause and wrestle with that.
    I believe there is a tree in my garden. Wait... I may either know that I believe there is a tree in my garden, or only believe that I believe there is a tree in my garden. Since beliefs are entirely self-evident subjective facts of my own mind (to me at least), I know that I believe there is a tree in my garden (at least whenever I actually believe there is a tree in my garden).

    But wait, again... Do I know there is a tree in my garden? I may either know that I know there is a tree in my garden, or only believe that I know there is a tree in my garden. Since knowledge of the material world is not at all any self-evident subjective fact of my mind (to me at least), I don't know that I know there is a tree in my garden. Now, clearly, I could have believed that I know but I don't, and obviously I can only believe that other people don't know but believe they have a tree in their garden (even when there is no tree at all).

    So, where is the difficulty here? I guess there is one but it is so difficult I can't even think of it.

    Ah, yes, does the fact that I know or believe that I believe p affects my belief that p? Not really, but the two "I believe" don't have the same status. One is a proposition, i.e. a putative fact of the world, the other is my claim about this putative fact. When I say I believe that p, the putative fact is that I believe that p, while the claim is that this putative fact is real. When I say I believe that I believe that p, the putative fact is that I believe that that I believe that p, while the claim is that this putative fact is real. The first "I believe" is a claim. The second "I believe" is the putative fact object of the first "I believe". Not at all the same status. And the two beliefs remain independent of each other. The second belief may be about any fact of the material world, while the first belief is about a particular kind of facts, my beliefs, which are subjective facts.

    But no problem, I think.

    Quote Originally Posted by fast View Post
    Oh, and whatever anyone does, please, for the love of blueberry muffins, don’t confuse them with:

    P3 there is traffic on the roadways right now

    “P2” has an air of self-awareness about it. With P1, I can ask myself, do I believe that? Do I believe that there is traffic on the roadways right now? Hmmm, yes, yes I do. I have that belief. The subject matter is belief. It’s about a belief of something else, but it’s a belief nonetheless. With P2, THE SUBJECT MATTER has changed.
    Sure, no problem.

    Quote Originally Posted by fast View Post
    I wouldn’t ordinary go down that road (with belief), but a turn of phrase has been used in the OP that has been dealt with in these philosophical chambers before—regarding knowledge.

    P1: I know that there is traffic on roadways right now.
    P2: I know that I know that there is traffic on roadways right now.

    P3: there is traffic on roadways right now

    Let it be known that P1 is not P2. Readers, keep your eye on the ball. Don’t be duped into thinking that a response regarding one stands toe to toe as a reasonable response for the other. P2 is so screwed up that I can think of three varying interpretations of it off the top of my head—talk about ambiguity!
    LOL. You're not going to convince anyone with half-baked arguments. What are your three interpretations?

    Me, I see just one, along the same line as my interpretation about belief of belief, apparently the same as yours. Where's the problem suddenly?

    I could well believe that I know. And indeed many people seem to believe they know a lot of things, only to be contradicted by their own experience at some point.

    I definitely know that I know pain whenever I am in pain. But I clearly don't know that I know that there is a tree in my garden. That there is a tree in my garden would be a fact of the material world, not a fact of my mind. I fail to see any good reason implying that I would know such a fact. And that makes the putative fact that I know that there is a tree in my garden also a fact of the material world, and therefore not one that I could possibly know that I know it.

    Quote Originally Posted by fast View Post
    And with that, I go back to the question. What difference (well, what meaningful difference) does SAYING one believes and SAYING one knows —beyond whatever actual difference there is between ACTUALLY believing or ACTUALLY knowing?

    His point, I guess, is that it really doesn’t matter if I know or merely believe I do when I pull out on the roadway. If I believe there is traffic (and think I know there is traffic), then I’m going to act no different when I pull out whether I think I know and am correct or think I know and am incorrect)—I’m gonna act on my belief.
    Exactly. There is no difference between how we act on our beliefs and how we act on our knowledge, IF, AS I SAID, we "are unable to tell the difference between belief and knowledge".

    And the mere fact that we can't should be enough. Anyone prepared to bet their own life and those of the people they love on the age of the universe being 13.7 billion years old?

    Not me, sorry.
    EB

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    Quote Originally Posted by Speakpigeon View Post

    Anyone prepared to bet their own life and those of the people they love on the age of the universe being 13.7 billion years old?

    Not me, sorry.
    EB
    Wow. Same theory that produces that prediction also produced the bomb. You willing to sacrifice your friends and family over doubt about the bomb.

    That would be a "be my guest".

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    There is something off kilter in our varied usage of language that sustains my drive to resist acquiescing to how certain scientifically-backed purported truths are reported. There is the appearance of denial on my part, but it has more to do with language than it does with substance. It’s not the discovered scientifically based truths I deny but how those truths are articulated that drives my resistance.

    There is an intermediary sensing process between the brain and the outside world. That’s why the brain can at best only indirectly sense the outside world and cannot directly sense the outside world.

    When a person walks into a room, the brain (the brain, I say—that all important necessary organ that plays a huge roll in making us who we are) does not directly sense anything in the room itself (but perhaps rather directly senses a representative reflective and possibly distorted percept model of the room), but (and I submit) despite that truth, the person (person, this time—that thing that walks, talks, and weighs a hell of a lot more than a brain) walking into the room can directly sense many of the things in the room. Science might deny my purported claim, much like how they might deny that we can directly see what we look at, but that’s because there’s an unreconciled confusion between two different intermediary processes.

    To begin an explanation, let’s first take a quick gander at this direct/indirect distinction i’ve been using. First, a fuller example followed by a quick recap snippet:

    To illustrate, inmate Albert looks directly at a prison guard coming down the long hallway and informs fellow inmate Barry that the prison guard is approaching. Interested in determining how far away the prison guard is, inmate Barry sticks out a mirror to see if the prison guard out of direct sight is close or far away.

    Albert directly sees the guard while Barry also sees the guard but indirectly.

    This is where science-backed science guru Sam comes in and says, “nope, Albert doesn’t really and truly see any guard directly.” Don’t get me wrong; Sam knows his stuff; nice guy; well respected! Plus, I have no problem with the science that supports the notion that there is an intermediary process between the brain and the outside guardsman. I have no problem with that—and I have no problem with accepting that the brain cannot directly sense the guardsman since at best, the brain’s direct connection is limited to its immediate vacinity and thus can only directly tap into these formed models you speak of.

    There’s a subtlety that I refuse to let go of. If we speak of the brain (that thing that’s approximately 3 pounds), that’s one thing, but when we speak of a person (like the guardsmen that weighed more than 3 pounds when born), that’s something entirely different.

    Sam is correct in what he means; he’s just incorrect in what he says.

    To grossly simplify for illustrative purposes, and someone somewhere cognizantly recognize what I’m attempting to convey:

    There is one intermediary process between the brain and the outside world, but there is no intermediary process between the person and the outside world. It’s the intermediary process that serves as the divide between direct and indirect. Don’t allow the fact of one (the brain can’t see) alter your understanding of the other (the person can see).

    Alberts brain to guardsman: 1 (bodily functions)
    Albert to guardsman: 0 (nothing much)

    Barry’s brain to guardsman: 2 (bodying functions and mirror)
    Barry to Guardsman: 1 (mirror)

    In essence, science shows us that there are inner workings but conflates the parts with the whole before reporting. We shouldn’t accept a scientific claim for what I cannot do because an investigation of how things work yields insight. The insight sometimes shows us we were mistaken about what we once thought, but damn it guys, leave some room for we were right all along but now because of science know more now than we did.

    PS; I am not the trickster. The truth condition was never met to begin with if it’s later discovered that the claim of knowledge was false.

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    Notwithstanding non-gettier type issues, it’s often the case, especially in cases of strong justification, people know just what it is they say they know when they sincerely say they know it.

    It’s quite a simple checklist actually:
    1) do you believe it? Do you believe that there is traffic on the roadways right now? If yes, place a checkmark and proceed to number 2.

    2) is the belief justified? Is there good reason to think so that isn’t corroded by countervening reasons to think otherwise? If you have sufficient justification for your belief, place a checkmark and go to number 3.

    3) You’re done, congratulations. No question for this part. Sorry, nope, not gonna ask if the belief is true. I’m not gonna ask if the belief (what is believed) is ACTUALLY true. For one, I don’t want to hear any crap about whether you know or don’t know if the belief is true. The belief might be true or it might not be. I don’t care about that anyhow. In fact, your answer isn’t even needed. The truth is independent of your answer. If the justified belief is true, booyow, end of story.

    Don’t believe you know? That’s fine; you know it whether you believe you know it or not. Don’t get side-tracked. If your belief is false, then you don’t know it, no matter how justified your belief —your belief that P is true I’m talking about, not your belief regarding the truth condition.

    See, one interpretation of “I know that I know” P is having a justified true belief that you have a justified true belief.
    If you don’t believe you know that P is true despite having a justified belief that is true, that’s fine; that’s just means you don’t know that you know, but as I just demonstrated, that has no bearing on the truth condition of P that you have a justified belief for.

    PART 2

    Do you know if the clothes are in the dryer?
    Yes.
    Are you sure?
    Yes, I heard it just cut off.
    But, do you know that they’re still in there?
    Yes. I do know.
    But, do you reaaaaaly knowwww they’re still there
    Yes
    I could have snook in.
    No, i’d of heard you
    I could have been sneaky
    Possibility, but I know they’re there
    Aliens could have got them.
    There aren’t any aliens
    There might be

    Cartesian Certainty. Often times, when someone tags on “really” in front of “know”, that’s a sign that a higher degree of certainty is being sought after. On philosophy forums, “I know that I know” is usually a sure fire tale tell that extremism is amongst us. “I know that I know” popping up would be an unwitting substitute for Cartesian certainly being conflated.

    Part 3

    This one is infrequent but reminds me of something else. One may actually say, “I don’t believe it” when in fact they do believe it. Why? Well, think of it as an expression of disbelief following a tragedy.

    Seldom, but “I know I know” may be uttered somberly whereby it’s indicative of neither above. It’s like self-affirmation or some such: I know I put her medicine in here. I know I did. I know I know. I’m not losing my mind.

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