## "What is truth? said jesting Pilate; and would not stay for an answer."

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Weather forecaster is a good example. Does the weather forecaster actually know something, or just pretend to know what is actually not known?
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Does the weather forecaster actually know something

Does anyone actually know anything, according to you?
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By Andrew's definition, we can't honestly call anything knowledge, because we can't really know whether it actually is knowledge or not. I don't agree, that's why I argued against that.
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The issue is, who determines whether or not it is raining.

There's your problem. No one determines whether or not it is raining. :roll:
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By Andrew's definition, we can't honestly call anything knowledge, because we can't really know whether it actually is knowledge or not. I don't agree, that's why I argued against that.

Then, how do, or could, we know that something is knowledge, according to you? (A concise, short-winded answer will do just fine).
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What I said, is that your logic is not valid without a premise of temporal continuity. That a coin might disappear without one noticing, is just a simple example as to why such a premise is necessary.

If you mean that my argument is only valid in a world very much like ours, I agree. If you wanted to discuss jars of coins in a hypothetical world in which coins randomly appear and disappear, that's rather different from the discussion I believed we were having. I understood you to be making a point about the necessity of a free human judgment that assigns a number to the coins, but it appears I was mistaken.

To return to the issue at hand: I consider my arguments valid in worlds very much like this one. In worlds like this, if the number of coins in a jar can be determined by counting them, then you can know, without counting, that there is a specific number of coins in the jar.

Do you agree?
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What is "a number"? Are you taking a position of Platonic realism here?

No. A number is a value. It is the "propositional content" of one or more mathematical symbols. For example, $0.25$, $1\over4$, and $2\over8$ are different mathematical symbols that refer to the same mathematical value/number.

How do you honestly believe that there are objects called "triangles" which have never been called by that name?

Being called a triangle and being a triangle are two different things. Something can be a triangle even if it isn't called a triangle. The word "triangle" has a meaning, and objects can satisfy that meaning even if we do not talk about them. Something that satisfies the meaning of the word "triangle" is a triangle even if we do not call it a triangle.

Decapitation is going to kill me even if I call it a non-fatal injury. Saying something doesn't make it so, and not saying something doesn't make it not so.

The issue is, that the thing must be judged to be of that kind. because a "kind" is something artificial, created by human minds, a category. A natural object isn't just automatically of this kind or that kind, because it must fulfill a set list of criteria in order to be of any specific kind. And, whether or not something fulfills a list of criteria is a judgment.

This is where we disagree. Objects exist and have properties even if we are not aware of them. We define the word "triangle" such that an object is a triangle if it has such-and-such a property. If some object exists and has such-and-such a property then it is a triangle, even if we are not aware of this object and/or it having this property.

I think the relevant metaphysical dispute is regarding the claim that objects exist and have properties even if we are not aware of them. Your argument depends on this claim being false. Are you, then, assuming something like idealism?

This does not tell us whether "there are 66 coins" is the product of a judgement, or whether it's something independent from judgement. Nor does it tell us if there is 66, or 67 coins. It really tells us nothing. It is a useless statement.

It's not a useless statement. It's a sound argument.

1. There are only 66 coins iff "there are only 66 coins" is true
2. There are only 67 coins iff "there are only 67 coins" is true
3. There cannot be both only 66 and only 67 coins
4. Therefore, "there are only 66 coins" and "there are only 67 coins" cannot both be true

Do you disagree with one of the three premises, or do you disagree that the conclusion follows?
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No one determines whether or not it is raining.

Lots of people do. I do it every day before I go outside. Don't you? I do not see how you could be using "determine" in any way other than this here. So let's not regress back to the dishonesty.

Then, how do, or could, we know that something is knowledge, according to you? (A concise, short-winded answer will do just fine).

Your question is misleading. We do not judge if something is knowledge or not, because we do not see, or sense things which might be judged as knowledge. What I think is that "knowledge" is something which we infer the existence of, through people's actions.

As I said earlier. "knowledge" consists of principles used for willed actions. If a person acts intentionally then the person has knowledge. What is required is to judge actions, and if they are judged as intentional, then the person has knowledge.

If you mean that my argument is only valid in a world very much like ours, I agree.

No, that's not what I mean. I mean your logic is only valid if you state that premise of temporal continuity. You seem to have a habit of thinking that valid logic can rely on unstated premises. That is not acceptable. The premises required for inference must be stated.

To return to the issue at hand: I consider my arguments valid in worlds very much like this one. In worlds like this, if the number of coins in a jar can be determined by counting them, then you can know, without counting, that there is a specific number of coins in the jar.

You don't seem to understand the reality of "a world very much like ours". In our world, time passes, and things change as time passes. Change is primary, and change is what we take for granted, as we take for granted that time passes. Since time passing, and the associated "change", are what we take for granted in "a world very much like ours", the proposition that something stays the same as time passes, cannot be accepted without justification.

Because of the reality of change, we cannot count the coins in a jar at one time, and logically conclude that the number of coins in the jar was the same at an earlier time, unless we premise that there was no change in the quantity over that period of time.

Do you agree?

Obviously not, your argument is not valid because it is missing a very significant premise which is required to make the conclusion that you do.

No. A number is a value. It is the "propositional content" of one or more mathematical symbols. For example, 0.250.25, 1414, and 2828 are different mathematical symbols that refer to the same mathematical value.

OK, I'll agree with "a number is a value", but I think that "propositional content" is somewhat vague or ambiguous, so I'll leave that for now. I understand "a value" as a principle which a human being holds within one's mind, concerning the desirability or utility of different types of things. Values often serve as principles by which we make judgements. So a value, as I understand "value", very clearly cannot exist independently of a human mind.

Being called a triangle and being a triangle are two different things.

This is something which needs to be justified. If "a number" is a value, then "a triangle" is also a value. So "triangle", as a concept is a simplified version (a representation) of an underlying complex concept, just like 2, as a number. is a simplified version (a representation) of an underlying complex concept.

So we have multiple layers of representation here. We have the word "triangle". We have the value 'triangle' (which is other than the word, like the number is other than the numeral). Then we have the underlying complex concept, three sided, straight lines, 180 degrees, different types, and all the associated mathematical principles.

Further, we now have the application of the value (the principle of action), which is the naming of a thing "a triangle". You seem to be asserting that a thing which a person might name as a triangle, has an independent property, which you call "being a triangle", which is separate from being named a triangle. How could you justify such a claim?

What you are saying, in effect, is that when you name something as a triangle, you are correct in an absolute sense, because the thing already has the property of "being a triangle" before you name it as such, therefore you cannot be wrong in your naming. And if you accept the reality, that you might be wrong in your naming, then if the thing does have that independent property, how would you ever know this? And if you cannot ever really know if the thing has this independent property or not, how is your assertion that it does, ever justified?

This is where we disagree. Objects exist and have properties even if we are not aware of them.

Yes, we disagree here. A "property" is a concept, usually quite complex, like the mathematical concept of "triangle" referred to above. We simplify the complex concept by naming it with one word, like "triangle", "large", "hot", "red", etc.. The word is supposed to represent "a concept" which in Platonist words is an intelligible object. The intelligible object represents the underlying complex concept. So a property is a complex concept. Objects do not have properties, as properties are concepts, and in application we assign the concept to the thing. It's called predication.
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You seem to be asserting that a thing which a person might name as a triangle, has an independent property, which you call "being a triangle", which is separate from being named a triangle. How could you justify such a claim?

The independent property is having three edges and vertices.

A "property" is a concept

Properties are something that objects have. Objects don't just exist as some property-less simple. They have a nature, including a mass, an extended position (i.e. a shape), and often a certain kind of movement.

That we decide which words refer to which properties isn't that the object only has these properties if we refer to it using these words. This is the fundamental mistake you keep making. If something has three edges and vertices then it is a triangle even if we do not call it a triangle.

Do you just not understand/disagree with how reference works, or the use-mention distinction?

This is how "truth" is most commonly used. When someone is asked to tell the truth, the person is asked to state what they honestly believe.

If I ask someone to tell me the truth of where my kidnapped wife has been hidden I'm not interested in where the person believes my wife has been hidden; I'm interested in where she's actually been hidden. The request to "tell the truth" is premised on the notion that things actually are as this person believes them to be. I have no interest in an honest belief if it's erroneous.
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I think the mathematical vocabulary is clearer: if they can be counted, then the cardinality of the set of coins in the jar exists and is unique, though we do not know its value until we count.

Yes, but it still assumes counterfactual definiteness. Which makes total sense for coins in jars (I'm not disagreeing with your argument with MU).

A person who has no lap has nothing in their lap. Russell's analysis of definite descriptions works just fine here, but physicists don't read Bertrand Russell. It's also tempting here to give a counterfactual analysis: if a standing person holding nothing were to sit, they would have an empty lap; if a standing person holding a child on their back and nothing else were to sit, they would have an empty lap, until another child scrambled onto it; if a standing person holding a child against their chest were to sit and loosen their grip upon the child even a little, they would have a child in their lap, and they would sigh with relief.

I think Strawson's presuppositional analysis is a closer fit. To make a different analogy, if a pointer is measured to be pointing North along the North-South axis, then what direction is it pointing along the West-East axis? A counterfactually-definite East or West direction presupposes that the pointer is also aligned along the West-East axis, but it isn't. Yet a measurement along that axis will give a definite result (in QM, West or East with equal probability).

Quantum mechanics may have some specific prohibitions on the use of counterfactual values in calculations, but it is, for me anyway, inconceivable (!) that we could get along without counterfactuals. They're hiding absolutely everywhere.

"You keep using that word. I do not think it means what you think it means." But, yes, it's difficult to imagine a world without counterfactuals.
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The issue is, who determines whether or not it is raining. Here, you are asserting "In the first scenario it is raining, in the second scenario it is not". Do you know whether or not it is raining in each scenario, in an absolute way? If so, I can give you an answer. If not, I cannot. This is because I cannot say whether Alice has knowledge or not unless I know infallibly whether or not it is raining. You have provided no justification for your assertions, therefore I cannot honestly give you an answer. So I do not believe that you know infallibly whether or not it is raining in each of those scenarios

They are hypothetical scenarios, and you know up front whether or not it is raining in each scenario. In the first scenario, it is raining (that's a given premise of the hypothetical). In the second scenario, it is not raining.

In the first scenario, Alice has a justified, true belief that it is raining, i.e., she knows that it is raining. In the second, Alice's belief is false, so she does not know that it is raining.

That is, according to your representation of "knowledge", which requires infallibility.

No, as demonstrated by the first scenario, Alice knows that it is raining not because she is infallible (or because she had ruled out all other possibilities such as Bob hosing the window), but because she had a justified, true belief.
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which would seem to suggest equating the empirical object with the ding an sich, if not the noumenon?

Close enough. To get closer, change “if not” to “but not”.
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if a pointer is measured to be pointing North along the North-South axis, then what direction is it pointing along the West-East axis?

I feel like I'm doing something wrong because I keep wanting to refute the examples. (Also, it reminds of my first my earliest experiences in philosophy, when I kept thinking that old-timey philosophers just didn't know enough math.) I'll try to think of an example after I do this one.

In this example, since you're only interested in direction from a point, defining that relative to a pair of orthogonal axes is at best an intermediate step (if you defined a location first and then converted it). What you ought to be saying is that the pointer is 0° off North. For jollies, you can throw in that it's 270° off East and 90° off West, but why bother? The extra axis adds nothing.

You didn't even have to align your direction right on the North-South axis to get here: if it were pointing exactly Northeast (45° off North), or, you know, almost anywhere, it's not aligned on either of your canonical axes! Oh my god! Its direction is undefined!

The only measurement always available is how far off a given axis it is. So just start there, and only use the half-axis from origin to North. Or take that direction as the default, define it as 0° and do other directions relative to that, whatever, but why would you define more than one axis in the first place? (Put this way, East-West is, to begin with, defined as passing through 90° off North and 270° off North, or 90° off South, defined as 180° from North.)

I think it's presented as pointing exactly North to support the illusion than some measurements could be made and some couldn't. But that's not what's happening here. We have a system that is useless for measuring anything but one or maybe two directions, which means we're not measuring at all, we're classifying directions as "North" (and maybe as "South") and "not North". That's not measuring.

I'm doing all this because it looks like this was a purely verbal conundrum. It seems to present a genuine problem (like the lap) but does not, and one way you know it doesn't is that it doesn't even do properly what it was pretending to do. The suggestion seems to be that directions generally have a North-South component and an East-West component, except for the degenerate case where you're actually on one of the axes, and then the other value doesn't go to zero but is suddenly undefined and maybe can have any value at all! Horrors! But the system supposedly breaking down only works for the case of pointing exactly North or, I guess, exactly South. This wasn't a genuine question but an intuition pump.
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To make a different analogy, if a pointer is measured to be pointing North along the North-South axis, then what direction is it pointing along the West-East axis?

Isn't this like asking for the z coordinate of a point plotted on a plane?
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In our world, time passes, and things change as time passes.

For instance, if there were so many coins in the jar that I would die before I could finish counting them, then I would have to pass this sacred duty on to my son, and no doubt him to his daughter, and now we're writing a Kafka short story, not doing philosophy.

The issue here is not all of metaphysics but a simple conditional: if they can be counted -- if -- then there must be a specific number of coins in the jar right now. All of these other issues are different ways of saying that as a matter of fact they can't be counted. (And that doesn't tell us whether the jar has a specific number of coins or not.)

I say the conditional is true. Do you say it is false?
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For those of you losing patience with all this, I'll jump to the end. What I provided was a sketch of an algorithm, an algorithm that could be instantiated in a machine, and at no point in the machine's operation is human judgment required to "assign" a number to anything. Coin counters are quite real and there's probably one at the front of your local supermarket. They claim, correctly, to represent the value of the coins in your jar before you dumped them in.

But didn't a human being have to design the machine, so isn't it just an embodiment of human judgment? Since we designed the coins and what values they represent, we have to design the machine to, you might say, take that into account; but you could also say that we design the machine to factor out (not in) complications we have added to the process of counting, to keep them from interfering. We tell the machine that objects of roughly the same size and weight are to be counted as the same thing so that it can count without the need for it to make such a judgment. (The machine, for instance, tallies only the nominal value of the coins, and won't notice if a rare coin worth a thousand dollars was mixed in with the dimes.)

I count money using a machine every day I go to work; the machine is easily fooled, and its mistakes are sometimes interesting. (A roll of nickels that is a little over, IIRC, is very close in weight to a roll of dollar coins, but a \$23 difference in value. This has caused some head-scratching in the cash room now and then.) But it is easily fooled because all it does is count, and counting doesn't require -- so the machine doesn't offer -- judgment.
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Then, how do, or could, we know that something is knowledge, according to you? (A concise, short-winded answer will do just fine). — Janus

Your question is misleading. We do not judge if something is knowledge or not, because we do not see, or sense things which might be judged as knowledge. What I think is that "knowledge" is something which we infer the existence of, through people's actions.

As I said earlier. "knowledge" consists of principles used for willed actions. If a person acts intentionally then the person has knowledge. What is required is to judge actions, and if they are judged as intentional, then the person has knowledge.

When I wrote "something" I did not have sense objects in mind; I think that should have been obvious. So your objection that "we do not see, or sense things which might be judged as knowledge" is irrelevant.

My question was concerning how to distinguish between belief and knowledge. Beliefs can be understood to be "principles used for willed actions". So "being intentional" cannot be a sufficient criterion for saying that someone has knowledge as opposed to merely having belief.

Bear in mind I am not concerned with "know-how" but with 'knowing-that' (knowing how to do anything does not seem to have anything to do with justified true belief). So, do you have a way to distinguish between knowledge and belief, or do you reject the distinction?

Close enough. To get closer, change “if not” to “but not”.Mww

OK, if I understand you correctly, then you would say the ding an sich, being the empirical object, is empirically real? The usual interpretation seems to be that it, like the noumenon, is thought by Kant to be transcendentally ideal.

It has occurred to me in the past that there seems to be a sense in which the empirical object, from our point of view, understood to be a whole and unified entity, and since it is not known as such by us, but is known only as sensorially acquired images and impressions (themselves empirically real), is transcendentally ideal. The flip side being that the noumenon would be transcendentally real (in itself, but not to us, obviously).
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you would say the ding an sich, being the empirical object, is empirically real?

Yes, it is an ontological given, real in the sense of being necessary for our perceptions. But to say it is empirically real is to say we can know something about it, contradicting the predicates of the philosophy to which it belongs. Space and time are attributed empirical reality because we can say something is known about them, to wit: we can know how and why they relate to the possibility of experience.

Ooooo....transcendental ideality. If noumena are tough, this one is damn near incomprehensible. Transcendental anything is the mode of pure reason from which synthetic a priori cognitions are given necessarily. Transcendental this or that simply means a priori conditions are necessary for judgements on them. A concept is transcendental merely from the very restrictive mode of how we think about it.

Given all that, we cannot arrive at a priori cognitions with respect to the ding an sich, insofar as any knowledge whatsoever about them is itself impossible. Therefore, they cannot be attributed transcendental ideality. Same with noumena, which can be thought a priori, so are knowable merely as a transcendental conception, as are all the categories, but still cannot be considered as have the attribute of transcendental ideality.

In keeping with the text, there are only two transcendental idealities, our ol’ pals, space and time. Some, in particular Schopenhauer, say causality too, but Kant does not.

Anyway....this is far too complex to get into here, because the concept is spread out over so much stuff. And sorry this doesn’t help much.
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Anyway....this is far too complex to get into here, because the concept is spread out over so much stuff. And sorry this doesn’t help much.Mww

Thanks, grist for the mill; and I don't expect anything to be cut and dried when it comes to Kant. It seems to me the transcendental/ empirical dichotomy opens up paths for whole suites of different ways of traversing the territory. What more could we ask of good philosophy than such fertile ambiguity? Unless we are one of those seeking a sterile clarity.
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The independent property is having three edges and vertices.

That something has "three edges and vertices" is a judgement. Who makes that judgement?

They have a nature, including a mass, an extended position (i.e. a shape), and often a certain kind of movement.

As I said, these are all things which we say about objects. And all you are doing is confirming this by saying it. How are you going to justify your assertions? What makes you think that mass, extended position, and movement, are anything other than concepts?

That we decide which words refer to which properties isn't that the object only has these properties if we refer to it using these words. This is the fundamental mistake you keep making. If something has three edges and vertices then it is a triangle even if we do not call it a triangle.

All these shapes and things which you say are the real properties of objects, are just products of our perceptual apparatus. These images, like edges and vertices, are created within the mind, There is no reason to believe that they are part of the objects themselves. The images created in the mind are just representations, like symbols, and there is no reason to believe that the symbol bears a likeness to the thing it represents. We can learn this from language. Words generally are not similar to whatever they represent. So whenever the mind creates an image, like a taste, a sound, or a visual image, we ought to believe that the image is a representation, like a symbol, and there is no reason to believe that the thing represented is anything at all like the symbol.

Science has been very good to demonstrate the reality of this to us. A taste, or smell, consists of molecules, which have no similarity to the smell or taste. Sound consists of waves, which is nothing like the image we hear. And, since the light which reflects to our eyes is the result of an interaction between electrons and photons, the boundaries between objects are nothing like the edges of a triangle (these are presumed to be straight lines).

If I ask someone to tell me the truth of where my kidnapped wife has been hidden I'm not interested in where the person believes my wife has been hidden; I'm interested in where she's actually been hidden.

Sure, you are interested in where your wife is hidden. That's obvious. But you are asking the person to provide you with what they honestly believe when you ask them for the truth. Remember, the person honestly might not know where your wife is. This is the way communication works, you cannot demand of others, to give you what you want. Such demands get you nowhere. So you must ask them to give you what they are capable of giving you, rather than demanding that they give you what you want. You want to know where your wife is hidden. The best that the other person can provide you with, is their honest belief, whether or not they know where she is. That's a simple fact. Therefore it is a mistaken approach for you to demand that the person give you the information you want, when their honesty provides you with the best that they can give you anyway. This way you respect the fact that the person might not be capable of giving you what you want. So the proper approach is to encourage them to give you their honesty. And that is to encourage them to tell the truth (honesty), rather than demanding The Truth (absolute, what is the case).

The request to "tell the truth" is premised on the notion that things actually are as this person believes them to be.

This is incorrect. The request to "tell the truth" is clearly a request for honesty. This is evident because it is most commonly used to determine whether or not the person knows the information which is wanted. You do not necessarily know whether the person has correct information concerning the whereabouts of your wife, so you need honesty to determine this. In relation to the subject you are interested in, your wife's location, you must get people to speak honestly, before you can even determine who has the beliefs which you are interested in. And that is what "tell the truth" is premised on, the attempt to determine whether the person has beliefs which are relevant to your interest.

They are hypothetical scenarios, and you know up front whether or not it is raining in each scenario. In the first scenario, it is raining (that's a given premise of the hypothetical). In the second scenario, it is not raining.

You're missing the point. Unless you explain how one could "know up front" whether or not it's raining (someone might be hosing the window), you are just begging the question.

No, as demonstrated by the first scenario, Alice knows that it is raining not because she is infallible (or because she had ruled out all other possibilities such as Bob hosing the window), but because she had a justified, true belief.

It's only justified by your begging the question, which is not justification at all.

The issue here is not all of metaphysics but a simple conditional: if they can be counted -- if -- then there must be a specific number of coins in the jar right now.

I've agreed to this already. We see a quantity of coins and we assume that they can be counted. If they can be counted, there is a specific number, as you say. So we are inspired to count them, assuming that there is a specific number, and therefore they can be counted. Then we do count them. And after we do, we need to rely on a premise of temporal continuity to say that there was the same number at the earlier time as there was at the time of the count.

Do you still have trouble with this?

But it is easily fooled because all it does is count, and counting doesn't require -- so the machine doesn't offer -- judgment.

Your counting machine does make judgement. That's what an algorithm is, instructions for making judgement. As human beings, we have created machines designed to make these simple judgements for us. But machines are now making more and more complex judgements for us. The AI is designed to be adaptable in its judgement capacity.

My question was concerning how to distinguish between belief and knowledge. Beliefs can be understood to be "principles used for willed actions". So "being intentional" cannot be a sufficient criterion for saying that someone has knowledge as opposed to merely having belief.

I follow the traditional formula, knowledge is a particular type of belief, justified and true. Justified is having been proven, and true is honest (that's my difference, how I define "true). Generally, being intentional shows knowledge, because we do things in set ways (justified beliefs), and we honestly believe in what we are doing.

Bear in mind I am not concerned with "know-how" but with 'knowing-that' (knowing how to do anything does not seem to have anything to do with justified true belief). So, do you have a way to distinguish between knowledge and belief, or do you reject the distinction?

Knowing -that is a type of knowing-how, just like knowledge is a type of belief.
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You didn't even have to align your direction right on the North-South axis to get here: if it were pointing exactly Northeast (45° off North), or, you know, almost anywhere, it's not aligned on either of your canonical axes! Oh my god! Its direction is undefined!

That's almost exactly the point. Suppose that you live in a grid world where you can only move and measure things along the North-South or West-East axes. Now an unobserved arrow might be mathematically represented as North-East in grid world (i.e., a linear combination of North and East arrows). But an arrow is only ever observed pointing along one of the grid lines. Thus raising the question of which direction the arrow is actually pointing (if it has a definite direction at all) when not observed.

I'll give a real-world example now. Suppose that you have an interferometer (see Figure 3) and a photon travelling East hits the first beam splitter. The photon could reflect and travel North, or continue East. Since we don't know which way the photon went, let's represent it with a North-East arrow. But, assuming counterfactual-definiteness, it's definitely travelling the North path or definitely travelling the East path. In fact, if we place detectors on those two paths, we will indeed measure the photon on one or the other of those paths.

So far so good. Now suppose we don't measure which path the photon takes. In this case, the photon will arrive at a second beam splitter where it will again either reflect or continue in the same direction. The classical prediction is that the photon will end up at either detector 1 or detector 2 with equal probability (i.e., a North path photon will either reflect or continue in the same direction; same with the East path photon). But what actually happens is that the photon is only ever measured at detector 1, as predicted by QM.

QM represents the photon as being in a linear combination of travelling both paths which results in interference at the beam splitter. One could say that the linear combination (the North-East arrow) is just a mathematical representation, and that the photon took one and only one definite path (a hidden variable that is definitely North or definitely East). But there are various no-go theorems that say, in effect, that that purported solution creates more problems than it solves.

Isn't this like asking for the z coordinate of a point plotted on a plane?

Not quite, since one can conceive of a z coordinate even if it is not plotted. For example, a bird (at z altitude) that casts a shadow (point) on the ground (plane). Or if there is no z-dimension, z is always 0.

Whereas in the analogy, there is only one arrow pointing North. So there is no sense in which that arrow points definitely either West or East.
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They are hypothetical scenarios, and you know up front whether or not it is raining in each scenario. In the first scenario, it is raining (that's a given premise of the hypothetical). In the second scenario, it is not raining.
— Andrew M

You're missing the point. Unless you explain how one could "know up front" whether or not it's raining (someone might be hosing the window), you are just begging the question.

You and I know up front because I created the hypotheticals that way. The question is not about what you and I know, which is a given, but about what Alice knows.
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I follow the traditional formula, knowledge is a particular type of belief, justified and true. Justified is having been proven, and true is honest (that's my difference, how I define "true). Generally, being intentional shows knowledge, because we do things in set ways (justified beliefs), and we honestly believe in what we are doing.

I don't think you can claim to follow the traditional formulation, because your understanding of what constitutes justification and truth is not in accord with the usual understanding. The usual understanding does not demand "proof" to underpin justification, and does not consider truth to be dependent on human intentions, honest or dishonest.

Knowing -that is a type of knowing-how, just like knowledge is a type of belief.

JTB is a definition of propositional knowledge, not know-how. Even if propositional knowledge could be, at a stretch, considered to be a kind of know-how; there are many other kinds of know-how which have nothing to do with truth or justification.
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Preamble

Well, this is humbling. I wrote a rambling, exploratory post last night that I thought ended in a pretty good place, a really interesting place, but with a problem, one I've been interested in for a long time. Then this morning it occurred to me that there might be a sort of solution suggested by how I arrived at the problem, so I wrote an addendum to last night's post. And not until I was actually writing the words this morning did it occur to me what I've been talking about for days.

TL;DR

What I have been claiming about the number of coins in a jar is simply that we can know a priori that if they can be counted then there is already a specific number of coins in the jar; we can only know a posteriori what that number is.

I do not think I have ever had occasion to make a claim to knowledge that so clearly fits the definition of a priori. Whaddya know.

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But an arrow is only ever observed pointing along one of the grid lines. Thus raising the question of which direction the arrow is actually pointing (if it has a definite direction at all) when not observed.

Does it? Your QM example gets there, I guess, but I've got nothing to say about that.

What isn't clear in your grid world example is what would motivate this question. If you sometimes observe an arrow pointing North and never observe anything else, what would make you think that it exists the whole time but the rest of the time it's pointing somewhere you can't observe? As you say, we don't seem to be able to distinguish pointing somewhere else from not pointing at all, or, as I put it before, we're really talking not about measuring but about two classes, North and not-North, which would also include just not pointing at all.

You must have some reason for positing that the arrow is pointing non-northwards when unobserved, right? But by stipulation, you don't. So I'm still at a loss. If the point is just exactly this, that if you, in essence, only imagine a situation, then you can't make measurements, that seems indisputable. You had a pithy quote to that effect.

---- Enough of that. I think I have better answers below, toward the end, or part of an answer anyway. ----

My claim, as you know, was not that I could figure out how many coins are in a jar by imagining counting them. That's clearly false. It was a claim about the nature of counting, that it does not "create" the cardinality of the set, that the cardinality of a set does not fail to exist until its members are counted, but that counting (to borrow a phrase from the wiki you linked) reveals a pre-existing unknown value.

What I have imagined happening here is, roughly, the mathematization of a physical problem: counting in the real world is a physical process, taking time, consuming energy and so on, but the result -- well, I suppose I can't really finish that sentence the way I want, because clearly what we're talking about now is information. I want to say that there is an aspect of what's going on that it is mathematical, and thus non-physical and non-temporal, but information is after all physical. Yuck. But there is also a mathematics of information, so maybe I come out okay. Gonna leave that alone for the moment.

What I'm trying to say is that if the math didn't work the way it does, then the physical process of counting could not work the way it does. It's not that the mathematics constrains your actions, but it does constrain the results. Performing a physical task such as counting or measuring or dividing, all this business and much more, in a way that doesn't respect the mathematics won't reliably produce the right result. (Hence engineering.) And therefore the mathematics can give you some insight into what the right procedure must be.

And that seems right. Philosophy and mathematics are old friends. Plato will refer to this cluster of disciplines -- philosophy, mathematics, music, astronomy -- as if it's perfectly obvious why they go together, and indeed it is, if you think this way. The impulse to mathematize a problem is sound. It's what we do.

To come back to our issue -- I suppose I think of the physical counting of the coins as counterfactual, but mathematics, after all, is what it is at all possible worlds, and is never counterfactual. That's why it seems so clear to me that I am entitled before counting to make only the claims about an unperformed count that mathematics would entitle me to make, that the result I will get exists and is unique, though I do not know its value. If I follow an incorrect procedure, that's not true. If I cannot follow the correct procedure, that's not true. But I can know what a correct procedure is and what result it must produce if it can be followed. And that claim is based on the mathematics, so not counterfactual.

What remains -- and it's too big for me -- is some explanation of how mathematics (non-physical, non-temporal) is implicated in the performance of a physical task in the actual world.

Does this make any sense? I could go back and edit, but maybe it's clearer if you can watch me stumbling toward figuring out what I want to say...

+++

The last problem mentioned --- roughly, idealization, the function of ideals in our thinking, and so on --- does have a possible solution here, of a sort.

I suggested that I can know some things about counting a set of objects without counting them because there is mathematics that constrains how counting works, and I can know the mathematics because, unlike the counting itself, it is never counterfactual.

The little puzzle here, of what this mathematics is and how it connects to physical processes like counting coins, could be dissolved by reversing my description above: suppose instead we say first that there are things I can know about counting objects, without doing any counting, because they must be so (and thus are not counterfactual). And this sort of knowledge --- of just those aspects of a situation or process that must be so --- is more or less what we call mathematics.

If that's defensible, then we may be able to find our way back around to questions about truth, because truth appears to come in varieties, which is slightly disconcerting, and I've been presenting an analysis that relies precisely on a distinction between a priori and a posteriori knowledge, and have offered a half-baked suggestion for how you might get the former out of the latter (thus perhaps re-linking some sorts of truth, if not quite re-unifying them).
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You and I know up front because I created the hypotheticals that way.

What type of knowledge do you assume that a "hypothetical" gives someone? It's not true knowledge. When you assume hypothetically that it is raining, this does not mean that you have knowledge that it is raining.

I don't think you can claim to follow the traditional formulation, because your understanding of what constitutes justification and truth is not in accord with the usual understanding. The usual understanding does not demand "proof" to underpin justification, and does not consider truth to be dependent on human intentions, honest or dishonest.

I don't see that you have a point. Justified, in general does mean proven. To justify means to demonstrate the correctness of, and that is to prove. And the meaning of "true" is very problematic, as demonstrated by this thread. Some posting here want to reduce truth to a special form of justification, but that leaves knowledge as simply justified belief. And others want to consult common usage. That's what I did, and common usage of "truth" is grounded in honesty. If we tell the truth when making a proposition, we propose what we honestly believe. A proposition which does not present what the person honestly believes is not a truthful one.

JTB is a definition of propositional knowledge, not know-how.

As I said, I do not respect this separation. Knowing-that, or propositional knowledge is just a special form of knowing-how. Using language and logic is a type of acting, so this is a type of know-how.

Even if propositional knowledge could be, at a stretch, considered to be a kind of know-how; there are many other kinds of know-how which have nothing to do with truth or justification.

In categorization, if there is a category with sub-groups, then all the sub-groups have something in common which makes them all members of the broader category. So if all types of know-how are all types of "knowledge", then they all have something in common. To say what "knowledge" is, we need to determine what they have in common. I think that JTB, if understood in the right way, is a good proposal. It has been around for a long time, and stood the test of time. The most difficult issue is to determine what "true" means. As the title of the op suggests, we often ask, "what is truth?", without sticking around to determine the answer. And so JTB is rather useless if we do not understand what T means.
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The way I'd put it is that the thing-in-itself is a noumenon, i.e. something that can be thought but cannot be empirically encountered, but noumena is the general category while thing-in-itself is one particular noumenon.

I believe it was mostly invented to make a distinction between transcendental idealism and the pure idealism that Kant is concerned with criticizing -- it's absurd to think that the moon stops existing when no one looks at it, but we'll never encounter the moon-in-itself either. So thing-in-itself is more like a place-holder concept to guard against treating metaphysical (non-empirical, and unbounded by the categories) judgments about objects as knowledge -- such as objects are material/ideal, which cannot be determined through collective empirical judgment.

This isn't to disagree with anything you've written, which I've agreed with, but to complement it.
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the transcendental/ empirical dichotomy opens up paths for whole suites of different ways of traversing the territory.

Funny, innit? Dude spends 700-odd pages telling us how there is but one way to traverse the territory toward knowledge, but his one way requires an abundance of cautions about what we’re not supposed to do in order to get there. Which makes sense in its own way, for what we’re not supposed to do is what the philosophers before him told us to do.

Try this on, see how it fits, as to why neither the ding an sich nor noumena can be transcendental idealities.

Just take as accepted we cannot know anything of noumena because they require a non-sensuous intuition, yet ours is always and only possible from perception, which makes our intuition necessarily sensuous. So....regarding the path to knowledge, scratch noumena.

Now, objects of perception are given, so no need to look at those. But those objects are said to affect us, but they really only affect our sensing apparatus. Sounds objects make affects our ears, odor of objects affects our nose and so on, and we call these sensations. Each one of us has his own sensing apparatus; I can’t see with your eyes, so we can say that which affects the senses changes only the condition of the subject to whom the apparatus belongs. I hear something you don’t, my subjective condition is changed relative to yours.

But you could hear what I heard, everything is in place to make it possible, except the occasion for it. All this is physically determinable in its entirety, as any medical doctor will tell you, so this part ends here. Nonetheless, your subjective condition is changeable, even if it doesn’t change, so there is that which makes changes in your subjective condition possible, whether or not there is an occasion for it, and therefore this cannot be counted in the physical part.

Just take as accepted, anything not counted as physical is not counted as empirical, and anything not counted as empirical in some way is counted as a priori, and anything not counted as empirical in any way whatsoever is counted as pure a priori. It follows that whatever is there that makes changes in one’s subjective condition merely possible, is pure a priori. But it must be something, and thus is established and justified, a precursory condition.

The sound a lead ball makes is different than the sound a rubber ball makes, and the sound a ball makes is different than the sound a trash compactor makes. That all these make a sound is determined by the the matter of each, but the matter of these, while affecting the senses with sound, do not carry the information of what form the matter has. It is impossible for us to get “ball” out of the sound an object makes when it hits something solid. Without antecedent experience, you cannot get “telephone” out of some arbitrary ringing/clanking/buzzing sound.

Just take as accepted, there is now what we call phenomenon, which is only a representation of a change in subjective condition caused by the affect of an object on sensory apparatus. OK, so...eventually we get to know what these objects are, but there still needs be the matter arranged in a certain form such that the present phenomenon subsequently becomes a specific experienced, known....named....object. But don’t forget...we’re still in the early stages, just past having been affected by an object of perception. In Platonic fashion, we know that there is a sensation, but we do not know how the sensation is to be represented because as yet It hasn’t been. It happens that just as your subjective condition can be changed, so too can the matter of objects be arranged into a certain form, which must be the case, otherwise we’d never be able to distinguish one from another. Thus, all matter is arrangeable, which makes explicit there is that which makes the matter of an object arrangeable in its particular form, again, even if there no object present to affect the senses, which makes whatever that is, a pure a priori whatever. And this whatever must cover everything perceived, from the matter of the object of the moon arranged as a mere simple circle, all the way to, e.g. a pine cone, the matter of which is arranged in the form of a complex Fibonacci sequence.

But there are virtually innumerable objects, any one of them distinguishable from any other and any one of them possibly an experience, which suggests there is something common to the arrangement of matter, common to all objects without exception. So it is that the pure a priori whatever can be given a certain name, can be thought as a certain conception, can pertain to nothing else at all, and has no other purpose, except the possibility of arranging the matter of every single object of a possible experience in accordance with the manner in which we are affected by them.

Because we have constructed this entire scenario in a speculative, or intellectual, fashion, it is pure a priori. Because we have constructed it with absolutely singular purpose, that is with respect to our subjective condition alone, it is ideal. And due to the mode of its construction, from pure reason alone, it is transcendental.

That conception which meets these criteria is space; space, therefore is a transcendental ideality. And at the same time, because it has to do with empirical conditions of real physical objects, logically space has empirical validity. But there’s still something further along to consider, because all that’s been accomplished so far, is the exposition of the relation of an object to us, which says nothing of the relation of objects to each other, for which account must be made insofar as we actually can be simultaneously conscious of more than one object. And, while we always sense an object as it is in one space, we can also sense the same object in a different space. Something lurks in the shadows of the mind.....

Neither the thing-in-itself nor noumena, while being transcendental conceptions a priori, never affect our subjective condition sufficient to change it, their matter is never subjected to the ideality of space such that representation as phenomena are given necessarily, hence neither can ever be a possible experience, which thereby makes them unknowable in its most exact sense.

Cut and dried. Obvious to even the most casual observer. Yeah, right.

Now....about that rational part of the system. No? Maybe in another life, then. With an endless supply of gin and tonic. Or maybe some serious Matanuska Thunderfuck ganja maan. Play Black Sabbath at 78, talk to Lord Immanuel Himself. (Sigh)
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This isn't to disagree (...), but to complement....

This....

So thing-in-itself is more like a place-holder concept to guard against treating metaphysical (non-empirical, and unbounded by the categories) judgments about objects as knowledge.....

.....complements rather well, I must say.

The way I'd put it is that the thing-in-itself is a noumenon

While I can’t refute that, as people are certainly entitled to think whatever they wish, but I’m reluctant to agree with it. Standing prejudices, doncha know.
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I don't see that you have a point. Justified, in general does mean proven.

Justified cannot mean proven. When it comes to empirical beliefs, nothing we consider ourselves justified in believing can be proven. The provenance of proof is in logic and mathematics, not in inductive reasoning.

As I said, I do not respect this separation. Knowing-that, or propositional knowledge is just a special form of knowing-how. Using language and logic is a type of acting, so this is a type of know-how.

I haven't disputed that, but it does not follow that all kinds of know-how are forms of knowing-that, which is why I have been trying to point out to you that there are kinds of know-how that have nothing to do with justification, truth or even belief.

I cannot find anything to disagree with there, but I still cannot say that I'm entirely clear on your view of just what the distinction is, according to Kant, between noumenon and ding an sich. Maybe I'll have to go back to reading the CPR again (when I can find the time).

Regarding the rejection of the idea of intellectual intuition, would you say that is on account of the impossibility of inter-subjective and cross-sensory corroboration?
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Justified cannot mean proven. When it comes to empirical beliefs, nothing we consider ourselves justified in believing can be proven. The provenance of proof is in logic and mathematics, not in inductive reasoning.

It seems you have a misunderstanding of justification. Empirical evidence, along with logic comprise justification. All logic requires premises, and most are grounded in empirical evidence. Justification is not limited to empirical evidence alone. I don't even know how empirical evidence without some form of inference would work as justification for a belief. You just observe evidence with no inference?

I haven't disputed that, but it does not follow that all kinds of know-how are forms of knowing-that, which is why I have been trying to point out to you that there are kinds of know-how that have nothing to do with justification, truth or even belief.

I fully understand that, but I don't see the relevance. As I said what I was looking for is what is common to all knowledge. Depending on how one defines "justified", justification may be conceived of as what sets knowing-that apart from other forms of knowing how.
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