Suppose you actually did this, as an experiment, just looking for each subject's response.

Let's say you have 1200 subjects; one coin toss each, and let's say you get 600 heads and 600 tails. So 600 subjects get a single interview, and 600 get two. Total of 1800 interviews.

How many interviews are conducted, the toss having come up heads? 600.

Then for any given interview, the chances that it's one of the heads interviews are 1 in 3. I think that's all the reasoning you need.

****

If you're inclined to double-check, you might try something like this:

$\small pr(\mathsf{heads | interview}) = \cfrac{pr(\mathsf{heads}) \times pr(\mathsf{interview | heads})}{pr(\mathsf{interview})}$

Not perfectly clear what that means though, so let's explicitly take subjects one at a time, something like this:

$\small pr(\mathsf{hds}(x) | \mathsf{intv}(x)) = \cfrac{pr(\mathsf{hds}(x)) \times pr(\mathsf{intv}(x) | \mathsf{hds}(x))}{pr(\mathsf{intv}(x))}$

That looks a little more tractable. Certainly pr(heads(x)) is 1/2 for everyone. If in fact x gets heads, then x is one of the 1800 interviews, so pr(interview(x)) would be 1/1800, and for tails 2/1800. For an "average" subject x, then, the value should be their average (because halves), which is 1.5/1800, or 1/1200. Hey, that looks right! Maybe we needn't have bothered about the heads and tails...

(Is 1/1200 really the right value for pr(interview(x)) -- doesn't it make a difference whether their coin came up heads or tails? Yes and no: what we want here is an absolute probability, not a conditional one, so we deliberately average out the cases to get a baseline, and the result is just what you'd expect. Pick an interview at random, and the chances of it being an interview of a specific subject are, on average, 1 in 1200. Just as we leave pr(heads(x)) at the baseline of 1/2 for everyone.)

What about pr(interview(x) | heads(x))? What does this mean, and how do we assign a value to it? It's the chances that a given interview is of subject x, given that x's toss came up heads. That's the 1/1800 we just looked at.

Now we have values for everything on the right:

$\small \begin{align}pr(\mathsf{heads}(x) | \mathsf{interview}(x)) &= \cfrac{1/2 \times 1/1800}{1/1200}\\ &= 1/3 \end{align}$

That is, given any interview, the chances of the subject of that interview's toss having come up heads are 1 in 3. Exactly the same as above.

SB ought to reason that it's more likely the interview she's currently engaged in is one of the tails interviews, so her credence for heads should be 1/3.

This is a basic problem with Hume's approach. His proposed separation appears to between the senses, and reason. But "reason" in its proper definition is only the rational and logical activity of the mind. This leaves a vast amount of mental, or brain, activity which is obviously not reasoning, and obviously not activity of the senses, as unassailable, in an uncategorized grey area. — Metaphysician Undercover

So close, but isn't it the case that Hume is precisely discovering that a lot of mental activity cannot be attributed to the senses or to reason? Isn't that what we've been talking about for pages now? How on earth can you end up claiming that Hume overlooked this, when he's the one that drew our attention to it in the first place?

If we adhere to the principles then, we sense continuity, but the brain wants to break up the continuity into discrete, or distinct parts for the purpose of understanding. Therefore, individual, fixed and distinct objects is a creation of the brain, hence mind (even reason?) rather than senses. Now Hume says that this is an unjustified creation, an erroneous fiction. — Metaphysician Undercover

Obviously there's a problem with saying "brain, hence mind (even reason?)," and Hume is very clear that we cannot attribute the theory of external objects to reason. As for mind — well, perceiving (in the modern sense) sure looks like a mental activity, in addition to being a physical activity, but you'll have to look carefully to figure out what in your perceiving is down to the peripheral nervous system and what the central. No reason at all to think it's only one or the other. Hume doesn't talk about the brain much, so I don't think it's helpful to read his claims about the mind as just being 18th-century speak for 'brain'.

So the problem here is with the sense/reason division. As described above, there is vast area of activity which fits neither category, it lies between these two. We can find other ways of dividing, sense/brain, or body/mind, but each has its own problems of not being able to properly account for everything, sp we get aspects, parts of reality which have no category. This indicates that this sort of division is not the best way to go. — Metaphysician Undercover

And again it's Hume who takes enormous pains to insist that there is centrally important mental activity clearly not attributable either to the senses or to reason. Somehow you've talked yourself into accusing him of doing what he, quite remarkably for his time, did not do.

if someone represents our observations of change as seeing the way things are at one moment (a perception), then seeing them in a different way at the next moment (the next perception), and we conclude with the use of reason, that change has occurred between these two perceptions, this is really not the way that we actually sense change. Through the senses we are actually perceiving change directly. — Metaphysician Undercover

This is just assertion.

I'm not taking a position on whether you're right, but on what grounds will anyone agree or disagree?

Honestly, the only thing I can see here worth doing, is try to determine what our powers of discrimination are scientifically, to treat this as an empirical question.

No doubt the terms in which we investigate the question will change, but I think we'll have to allow that based on what the investigation shows. "Senses" will turn out to be far too coarse, as will "perception". We'll want to know which neurons are activated, what level of input it takes to do so, how long that takes, when they're ready to be activated again, at what point information is passed up to the central nervous system, how much information is enough to act on, all with or without conscious awareness, and then there are additional questions about what we become aware of, how, and when.

The problem though is that reason works best with static descriptions, predications with laws of logic, like non-contradiction, so it does not properly apprehend what the senses give to it, change. — Metaphysician Undercover

If the story you're telling is that reason distorts the true nature of the senses, or of their testimony, then that seems to me not a story worth telling. Better actually to go and look at how our nervous system works. That would include what the senses actually do and how, but also how our brains organize the information we have about the world. Reason comes in elsewhere, I suspect, and I think so far as all this goes, Hume's general approach is the right one, regardless of what particulars he may have gotten wrong.

In this case, it's the cow that he saw that establishes his conclusion that there's a cow in the field. He is mistaken about which cow he saw, but that doesn't undermine his conclusion. — Andrew M

One point about the analysis I was offering is that it is deductively more palatable. Whenever we've talked about Gettier on the forum, or introduction is a real sticking point, and thus existential generalization is. People accept it in math class, but they balk at someone in real life inferring something of the form A v B v C v ... from A. I went around that by treating the name as a description. Now, instead of or introduction, we have and elimination, which doesn't seem to bother anyone.

Or introduction is not a crucial element of Gettier cases; it's just the easiest way to construct them. All we need is a situation in which your reasons for believing some proposition are not the reasons it's true. Stated abstractly, I think it's obvious this happens, and that when it does we think of this exactly as being lucky, a little like this:



"It's Disembodied Reggae Space Voice, but that's just a coincidence, you didn't know that!" (If you let the clip play on, you'll also be treated to Phineas asking Baljeet to quit arguing with the soundtrack. That show ...)

Descriptivism as a theory of names is controversial, of course, but there are a couple of specifics here in its favor: first, we're not looking at reference in general, but at recognition of a particular we are familiar with (to continue RussellFest, a particular we know by acquaintance); the other point is that the part of the description we keep is the sortal.

I think it is plausible to think of recognition as inherently a descriptive enterprise, involving a list of criteria. And sortals always play a special role. That post looks a little odd:

(1) I believe that's Daisy out there.
(2) I know Daisy to be a cow. — Srap Tasmaner

It's like that because what I thought I was going to do was have the farmer form a belief regarding the particular, Daisy, non-descriptively, and then infer further beliefs from his beliefs about Daisy; doing that would jam an and in between the recognition and the other inferences, creating two new scopes and allowing us to screw around with the reasons they're true. (I had a sort of Twin-Earthy idea that Daisy might turn out to have been a schmow all along, but with all the other things the farmer knew about Daisy still true.)

But non-descriptive recognition is so implausible, and implausible in particular if you give up the sortal. I think our beliefs are in almost every case centered on sortals; Daisy, for instance, is not just a particular, she is a particular cow. Certainly for the task of recognition, the list of Daisy's properties is going to begin with cow, and then include features (the nick in her ear, her weight, etc.) that distinguish her not from the farmhouse or the farmer's wife or the Milky Way or the tractor or democracy, but from other cows.

So all of that is to bolster the sense that you do have knowledge if you infer that you've seen a cow from your belief that you've seen Daisy the cow, even if you actually saw Clarabelle the cow.

Back to Gettier. Must there always be a false lemma when your reasons for believing a proposition are not the reasons it's true? No, obviously. It was an act of the Kansas state legislature (I'm guessing) that made Topeka the capital; you believe it because you learned it in school. There is, doubtless, a causal chain between that session of the Kansas state legislature and what your teacher told you or you read in a textbook, but it's a causal chain you cannot possibly be familiar with from beginning to end.

Now suppose you're in grade school and your teacher — because he's a bit of a prankster, or because he wants to make some point about remembering, or whatever — writes a list of cities on the board and a list of states. His intention is that he'll catch out some of the students wanting to match up Wichita to Kansas, even though Wichita is not the capital, and both Wichita and Kansas should remain unmatched. But, because he's also vulnerable to accessibility bias, he actually writes Topeka on the board. When he asks the payoff question, "What about Kansas?" a bunch of kids say "Topeka!" and he responds, "Oh-ho! But Topeka" — here he looks at the board — "is right there. Shoot." He intended there to be a conflict between the list the students had memorized and the list in front of them, to see if they could be tricked into taking what's in front of them instead of relying on what they remember, but he inadvertently made the lists the same. Now he has no experiment, because some of the students may have done exactly what he hoped, chosen Topeka remembering only that it's in Kansas. That's necessary but not sufficient for being the capital of Kansas, so it's not wrong, but it's still a mistake. But because of his mistake, it's a mistake that's undetectable. Of course, little kids tend to be pretty candid, so if he just asks, "How many of you remembered that Topeka is the capital of Kansas?" and "How many of you chose it because it's only city in Kansas on the board?" he'll probably get some hands up for each, and some embarrassed giggles.

And there's the other part of the Gettier case. Our epistemic agent always has explicit knowledge of what reasons they're relying on, what they're inferring from. To defeat no-false-lemmas, we have to construct a case in which those reasons are true and do provide strong enough support for the conclusion, but we can't do that counterfactually — that is, with reasons that the conclusion might have been true but isn't — so what we need are independent reasons, as "I learned it in school" is, epistemically if not causally, independent of "The Kansas state legislature said so."

My classroom wasn't intended to be a Gettier case, only a neighbor that illustrates the issues. But it's close, because it has elements of getting the right answer for the wrong reasons, only it adds a twist that the wrong reasons are coincidentally the right reasons. That's a funny thing, because it's almost as if "Wichita" is misspelled "Topeka" on the board, but in the teacher's mind is still the word "Wichita". So there's a false lemma here, but pushed back from the kids to the teacher. Never even realized on the board, but it's there in the teacher's beliefs. It's similar to my thing about Russell's clock, having the worker set the clock correctly from a watch that only happened to be right. Pushing the false lemma out over the agent's epistemic horizon leaves us in an uncertain position I think: depending on how the details are presented, the agent might strike us genuinely lucky to acquire knowledge, or too lucky for his belief to count as knowledge. It's like the conflicting intuitions among philosophers about the fake barn cases. What's interesting here is that most of the kids can probably report whether they had knowledge because they know whether they remembered, and it's the remembering that would be factive, but even some of them might not be sure they would have stuck with Topeka had Wichita been written on the board. And some might not know whether they would have remembered without being prompted. "I know it when I see it" is a real thing.

I'm going to take a break, but I really think we should be able to construct a clear case, roughly along the lines above, of Gettier case without a false lemma.

Charles Mingus, Blues and Roots

Story behind this one, I believe, is that one of the Ertegun brothers suggested he do a whole record in the vein of Haitian Fight Song, from The Clown.

So here's that:

Duke Ellington — Jamal

As a pianist, Ellington is thought of as having a percussive style, as opposed to say the fluidity of Art Tatum, the great pianist of the swing era. So there's a tradition that runs from this



through Monk to Cecil. Watch Monk play, oh my god:

possible worlds — Banno

I have very strong doubts that stories count as possible worlds.

semantics — Banno

I don't see how to avoid semantic issues: the truth conditions of "Frodo carried the ring to Mordor" look nothing like the truth conditions for "Washington crossed the Delaware." That ought to be obvious.

But the problem with my position is apparently that 'within the story', or from an 'in-world perspective', Frodo going to Mordor has exactly the same sort of truth conditions as Washington crossing the Delaware has (in our world, if that needs to be said). We can carry out such an analysis by pretending that Frodo is a person, Mordor is a place, the one ring is a thing, and so on.

But we are also aware of the book as a textual artifact and must analyze it as such. Whatever happens in the story happens because the author says it did. So one way to frame the issue here is to ask how these two frames of analysis are related. Is one dependent on the other? Are they dependent on each other? Independent of each other?

(Incidentally, I wanted to refresh my memory so I checked the wiki for "willing suspension of disbelief" — it's nearly Coleridge's phrase, as I thought, but he didn't mean what I learned in school. The wiki article is interesting and notes a sort of response from one J. R. R. Tolkien!)

I'm initially inclined to think that the 'in-world' analysis is parasitic on the textual analysis, precisely because whether something counts, within the story, as having happened, depends entirely on whether the storyteller says it did. Arguments about what did or didn't happen in Tolkien's story are settled — or at least, attempted to be settled — by reference to the text. Thus to ask whether Pippin accompanied Frodo and Sam to Mount Doom is elliptical for asking what it says in the book.

But now 'what it says in the book' is going to be from the in-world perspective, so indeed we have to understand the sentences in the story by taking them as pretend. If we could not carry out such an in-world analysis much as we would analyze sentences like "Washington crossed the Delaware," then we could not answer any question of the form, "What does the book say?"

So it appears the two sorts of analysis are interdependent. There is an extensional layer, what the book does or doesn't say; and an intensional layer, what what it says means within the story.

Having at least scratched the surface of the sort of work I imagine is necessary, has it become any clearer whether Frodo is real? If by Frodo we mean a hobbit person, then in-world, of course he is; in our world certainly not. In our world, Frodo is a fictional character, which is a real thing just as stories are a real thing. We seem to need a definition for "fictional character" and the obvious one is that a character is whatever counts as a person within the story, from that in-world perspective. Especially in fantasy literature, this may present some problems, because the characters in the story may not all share a perspective on what is a person, and the storyteller has a perspective on this too, again perhaps shared and perhaps not. Ghost stories are the obvious example.

But in our example it's clear enough that within The Lord of the Rings Frodo is a real person and a hobbit, and so for us he would count as a fictional character.

Can we spell this out as truth conditions for "X is a fictional character"? Can we just say "X is a fictional character if and only if there is a fictional story within which X is considered a person"? What sort of X do imagine filling in here? I mean, there's a temptation just to plug in a name there and call it a day. But it's the semantic value of that name that is exactly our problem.

On the right-hand side, we want to take the in-world perspective, and leverage that to define a term in our world, on the left-hand side. In Middle Earth, we want to say, Frodo is a person; in our world, he's a fictional character. Is this the same 'entity' we're talking about? Has it a dual existence, in one 'world' as one sort of thing and in ours as another? Is this no different from saying that chocolate can exist as something yummy for one person and something repulsive for another?

The pretending that matters here is done in our world, and I think this might provide a start on a solution. I don't think we really want to say that Tolkien pretends — and we pretend with him as readers — Frodo is real and a hobbit. That sounds right, but there is no Frodo for him to pretend is real, as there is real chocolate for one person to like and another dislike. (It's no good to say that Tolkien pretends his fictional character is a real person because (a) there is no such character until he does the pretending, and (b) we were trying to rely on the pretended reality in order to define the character, not the other way around.) Instead, I think we say that Tolkien pretends to be telling a true story — at least in some sense. (We still don't have an account of pretending to hand.) Among other things, Tolkien pretends to be telling a story about Frodo. He isn't actually, because there is no Frodo, but he can tell a story about the Frodo who doesn't exist as if he did. But he's never actually talking about Frodo, only pretending to.

You can tell a story about a real person, and within that story the semantic value of that person's name is the person. You can also pretend to tell a story about a person who doesn't exist, and the name of the person you pretend to be talking about has no semantic value, but you pretend it does. (Deja vu. I think I've written that on this forum before, but I had forgotten until just now.) The important thing is to see that the pretending is precisely that the story is about someone; it's not.

That's still not quite right because I think we need to make an even stronger claim to make sense of this. What is telling a story, telling a story about something that really happened, the sorts of stories we tell all the time? It's a recounting of events you know to have happened.

Fictional storytelling is pretending you're doing that, when you're not, and your audience knows you're not. It's next-door to lying but without the intent to deceive. When you lie, you maintain a pretense that you're telling the truth, but when telling a fictional story that's not it exactly. You pretend to be recounting. In the course of recounting, you pretend to narrate events that happened, as you would real events; you pretend to talk about people and places, as you would talk about people and places when recounting. But you're not doing any of those things, you're pretending to. The pretense sweeps in everything, beginning with the idea that you're in a position to tell the story because you know what happened, when and where and who did it and to whom. You don't. You don't know any of those things, but you pretend you do. (It is not true, for instance, that you're the only person who knows, since you're the storyteller; you don't know things you're making up.) You might even pretend you translated the story you're telling from an old manuscript bound in red leather. But that's not true either.

I think that does still leave 'fictional character' as someone you pretended to be talking about but weren't really — it's just that we want to read that holistically. It's the whole story that carries the pretense of being a recounting of events, not atomistically a matter of an entity in the story not being real. If I thought it would fly, I'd just say that fictional storytelling is pretending to tell a story, that it's a flow of speech meant to sound like a story but isn't really. But I doubt anyone will plump for that.

That's the best I can do tonight. Some of the analysis near the beginning of this post might still be okay, but the whole in-world/our-world analysis might be kind of a blind alley. Worth exploring though, and maybe it's salvageable. But it does seem to me now that the right starting point is where I've ended up: fictional storytelling is parasitic on the sort of true narratives we trade in all the time, and the primary pretense is that it is this sort of speech one is engaged in. (Not for nothing, but early novels overwhelmingly presented themselves as diaries or letters to establish this pretense of being a recounting of actual events, a tradition Tolkien keeps to.)

This does not seem like a good way of describing reasoning, the act of passing from one perception to another. — Metaphysician Undercover

The insight which captivated and shook him, is that those laws don't look much like reason. — Srap Tasmaner

If the cow that he doesn’t know about is the one that establishes the truth, then he didn’t know there was a cow in the field. — Ludwig V

Not sure I agree. I already worried over this a bit too, so I get the concern here.

I think the essence of this objection is to deny that the existential generalization actually takes place. If we leave aside justification for a moment, the idea is something like this:

(1) I believe that's Daisy out there.
(2) I know Daisy to be a cow.

In fact, I know a number of things about Daisy, so the presence of Daisy in the field entails the presence of a creature with any such feature. That is, the inference I'm prepared to make is that wherever Daisy is, an instance of Daisy's features is, including something being a cow. -- That is, I also accept that sometimes Daisy features are present without her, singly or in bunches, because other things are cows, other things are placid, other things ruminate, etc. (If our farmer prefers tropes to properties, all bets are off.)

I'm not immediately going to claim this is enough to justify the EG. I think first we take a detour through some obvious counterfactuals. We're trying to give due weight to the idea that the farmer only believes there's a cow in the field because he believes Daisy is, and he knows Daisy is a cow. That's to say, if he did not believe Daisy was in the field, he would not believe a cow was in the field.

But that sounds far too strong, at least because if it turns out to have been Clarabelle, the farmer will retreat to: I knew it was a cow and I thought it was Daisy. He might even be genuinely surprised and wonder how such a mistake happened: I could have sworn it was Daisy.

What I want to note here is that on discovering his mistake, the farmer will quite naturally itemize the Daisy features he correctly identified in search of the one he was mistaken about. All of which suggests his Daisy belief was -- contra my sympathies for a casual account of names -- in fact a sort of compound belief regarding those many descriptive features of Daisy. Maybe this is specific to cases of recognition, but the farmer's ready recasting of his belief as a compound suggests there's a list of criteria for recognizing Daisy and he was right about some of them, but not all. (Such lists look easy to Gettier-ize.)

And if all of that is right, then the EG was compound to begin with, even with only Daisy criteria in mind: there is something in the field that is a cow, and is placid, and ruminates, and has a nick on her left ear, and is pretty fat for this time of year, etc. And if that's right -- and even if that list is somehow taken as open-ended -- it can be split: there's something in the field that's a cow; there's something in the field that's placid; and so on.

Which, again, is why the farmer won't feel nearly such a fool if it turns out to be Clarabelle, even if he's very surprised, because he will still have gotten a lot right.

(I was going to head in a completely different direction, so I'll wait to see what people think of this before trotting out alternative analyses.)

Well if a valid deduction is enough to be deductively justified — neomac

I think the general view is that inference confers justification no more than it confers truth; rather, valid deduction is often expected to conserve justification and to conserve knowledge, just as it conserves truth. Since people may not make inferences they're entitled to, you have to add some clause, such as Gettier does, that the agent makes the valid inference. (Similarly, you don't automatically know everything entailed by what you know, but if you made all the inferences you're entitled to, you would.)

Some nice points in this thread, which I'll reread.

I'm only surprised no one has yet used the phrase "instrumental rationality," which could be defined something like, the rational selection of a course of action to achieve a given goal -- the kicker being that this means any goal, however arbitrary. Sometimes "reasonableness" is contrasted specifically with instrumental rationality in submitting to judgment also the worthwhileness of the goal and the acceptability of the means of achieving it, so a broader decision-making process.

He and George Adams (ts) were in Mingus's last quartet, and then carried on as a band.

Pullen had a unique technique that involved rolling his hand over the keys to get clusters of notes (and some otherworldly effects). There's a cute video on YouTube of his band appearing on a show Ramsey Lewis hosted, and Ramsey tells him, I tried it, tried to play like you do, and I ended up with bandaids all over my hands.

Here's a good place to start:

My starting point was maybe his first record as a leader. He does a killer version of Bemsha Swing. Gets how Monk had already broken into, let's say, tactical atonality. Monk understands what can be done with a piano as a physical thing, not just as a manifestation of music theory.

For a sort of point between Monk and where Cecil ends up, don't miss the incredible Don Pullen.

If you've listened to some other earlyish Ornette but not to Free Jazz, just spin it. There's just more players, but it's very listenable. I only finally got around to it in the past year, and it's nothing to be afraid of. (It used to be said there were two routes into free jazz (my music theory is almost non-existent, so grain of salt here): Ornette just passes right by the theory of harmony and frees melody from it; Cecil layers in more, augmenting traditional harmony, broadening it. Free Jazz the record is definitely still on Ornette's end of the spectrum.)

I'll certainly revisit late Coltrane, so thanks for your impressions.

Relations are different, distinct from the things which are related. Yet the relations are necessarily present within the mind, and are part of the mind. — Metaphysician Undercover

I think this is exactly the point of difficulty. I know I'm going to have trouble explaining it, because I can't quite see it clearly myself, but here goes.

I think these relations are, let's say, of the mind, but not present to mind; that distinction belongs exclusively to perceptions, and the relations among perceptions are not themselves perceptions.

What are they then? I think they are something like laws. You do not directly observe the law of gravity, you do not observe the force of gravity causing the tree limb to fall onto your car, you only observe the limb first there and then here, your car first fine and then smashed. Something, we believe, caused this passage from one state of affairs to another, but what it was is not something we can observe, but only postulate. (The passing itself, the atomist Hume might even say, we do not observe, but only the limb first attached to the tree, then at many points between the tree and your car, then on your car. I'd have to go back to see whether Hume thinks we actually observe motion.)

The laws of nature are there in somewhat the same way the laws of inference are in an argument. We have our premises, we pass from one formula to another, reaching a conclusion, but if we rely on modus ponens or conjunction elimination, they are not there in the argument as premises, but as the laws that carry us from one formula to the next. We're used now to axiomatic deduction systems in which the rules of inference are explicitly chosen — and thus part of the system though part of no argument — but in olden times, modus ponens would be present only implicitly, and perhaps postulated, or discovered, as a legitimate way of getting from some claims to others.

With those analogies in mind — and I think they're close to Hume's intentions and world-view — most of the book is an exploration of the mechanisms by which we pass from certain perceptions, be they impressions or ideas, to other perceptions, generally (but not always) to new ideas. He says something like this on almost every page of the book — we pass smoothly from this one perception to this other one because of the resemblance between them, that sort of stuff. It's everywhere, because it's the whole point of the book. But those resemblances, for instance, they're something we can reflect on and have ideas about, as he has done, but they are not themselves perceptions present to the mind. (There's a regress argument here, but I'm not sure it's Humean. It's the same problem you would have if you had nothing with the status of an inference rule, and had to take modus ponens as a premise. That doesn't work.)

Our subject here, the belief in body, I believe is something like one of these laws of thought, not an idea we have but how we pass from the lamp impression to the lamp object belief, from the book impression to the book object belief, and so on. The double existence theory is an idea we have about this habitual passage from one perception to another, something like our ideas about causality, an attempt to justify to reason our expectation that one perception will follow another because it has in the past. (A rule of inference in psychology is more naturally seen as a habit or custom of inference.)

But aren't I loading up the mind with stuff, when Hume says it's simple? I don't think so, but I'd have to look closely at the text. Certainly the spirit of the thing is as I've said: he's looking to understand how we pass from one perception to another, so he's going to postulate laws of the mind somewhat as Newton postulated laws of the physical universe. The insight which captivated and shook him, is that those laws don't look much like reason.

Couple things about Elvin Jones: he told some interviewer that part of the secret of his style, the polyrhythmic thing, is that he always hits something on the beat, just not always the same thing. Also, when Mingus was forming a group in the late fifties, the only drummer he wanted was Elvin Jones, but Elvin was playing with someone else at the moment, so Mingus taught saxophonist Danny Richmond how to play drums, and Danny was his drummer for the rest of his life.

I think it might be the liner notes to the Coltrane I recommended where Trane says of Elvin, "Sometimes he's too much even for me."

I have limited understanding and appreciation of free jazz. I mostly know Pharaoh from later albums, not his Impulse! stuff. Coltrane's last couple years, I don't do. I've listened to lots of Cecil Taylor, and find him really interesting, but I there's a lot I don't really get. Ornette is easy compared to a lot of stuff out there. Understanding and loving the many varieties of free jazz (and fusion, for that matter) remains on my to-do list.

Hume seems to rely here on two important premises, or principles. The first one is stated clearly and explicitly, that we cannot doubt the existence of body, that to do so would be unreasonable. — Metaphysician Undercover

"Unreasonable" cannot be the right word here.

The second premise appears to be a bit more obscure, but it has to do with what is present to the mind. Simply stated, the principle seems to be that the only thing present to a mind, is perceptions. This is made very evident from his description of mind as a simple unity of perceptions. — Metaphysician Undercover

There's no "seems to be" about it. He says it in so many words. I quoted him saying it on page 1 of the thread.

However, to maintain this conclusion, it is necessary that identity could not have been truthfully, accurately, or reasonably derived from a source other than perception. — Metaphysician Undercover

I think your disagreement is with Book I Part I Section I, where Hume claims that all our ideas are derived from impressions. Thus, no innate ideas.

Then, when he considers identity, and continued existence, and finds these not to be supported by perceptions, he wrongfully concludes continuous identity to be imaginary, fictitious, even erroneous. But this is only because he doesn't consider the other category of influences in the mind, the causes which come directly from one's own body, instinct, intuition, desire, intention, and the good. — Metaphysician Undercover

All the perceptions of the human mind resolve themselves into two distinct kinds, which I shall call Impressions and Ideas. The difference betwixt these consists in the degrees of force and liveliness, with which they strike upon the mind, and make their way into our thought or consciousness. Those perceptions, which enter with most force and violence, we may name impressions; and under this name I comprehend all our sensations, passions and emotions, as they make their first appearance in the soul. By ideas I mean the faint images of these in thinking and reasoning; such as, for instance, are all the perceptions excited by the present discourse, excepting only, those which arise from the sight and touch, and excepting the immediate pleasure or uneasiness it may occasion. — The very first words of the book (after the Introduction)

So, no, Hume is not ignoring other causes that arise from within your own body: they are all impressions.

The thing I dislike about chess is the lack of creativity — TiredThinker

No.

and the ability to beat someone in like 2 moves — TiredThinker

They have to help.

It sounds like you don't understand the game, don't play very well, and are frustrated.

Please stop. Learn about the game or don't, but quit mouthing off about it.

Some of us find the game beautiful. You don't, at least not yet. Let's just leave it at that.

You might as well try to convince me I don't really like jazz because it's "just noise".

This thread shouldn't exist.

It is really about what happens in the story whether it is fictitious or not. — Amity

Whether I went to Atlanta is not a matter of whether anyone said it. Whether Pippin went to Mordor is exactly a matter of whether Tolkien said it.

I would say that sentences in fiction communicate truths — Amity

Another kind of truth about reality can be found in the likes of Charles Dickens.
In contrast to a fictional truth, this might be termed a 'genuine' truth. His stories present moral or political truths some readers can relate to. — Amity

Leaving aside generalities like "All happy families are alike," which might be literally true, the truths you get from fiction are not stated. You read a story by Chekhov or Raymond Carver -- if you're a teenager in a crappy English class, you just say wtf? But if you're an adult with some experience, some curiosity about the world, some sensitivity maybe?, you might reflect on the story and on the lives we lead and have something like insight. The truth you find is not stated in what you read.

Imaginary gardens with real toads in them.

What shall we say about that?
— Srap Tasmaner

I bolded your question to which I attempted a response. Still waiting for your reply — Amity

Some of this is just Moore thumbing her nose at the poetry-reading public, oddly, because she was famous for including snippets of newspaper and magazine articles in her poems. She was the oddest of ducks.

Otherwise, I think she's saying something like what I suggested above. It's the spirit in which Seamus Heaney said poetry shows us "a glimpsed alternative," and Geoffrey Hill said a poem must be "a fortress of the imagination." Creative, imaginative thinking ought to be contagious, just as rigorous thought ought to be, as when Wittgenstein said, "I should not wish to have spared anyone the trouble of thinking."

Again, why we can't use sentences as truthbearers. — frank

And again, I'm not saying they are.

Here's one way "Washington has crossed the Delaware" can depend on context: if the context is the Revolutionary War, "Washington" may be used to refer, by metonymy I guess, not to the person Washington but to the army he commands or some part of it; Washington the person may not have crossed at all. But the logical form changes little: it is true if some concrete persons used to be on one side of the river and are now on the other.

Now consider what makes "Frodo went to Mordor" true, if it is: there is a sentence or sentences in the book written by Tolkien which say or imply that Frodo went to Mordor. Whether those sentences are part of the book determines its truth-value, if it has one, not the sequential locations of any person. Note that we are not concerned with whether what Tolkien said is true, because it isn't, but only with whether he said it.

See the difference?

No, I'm saying that the form of the proposition expressed by "Frodo went to Mordor" is different from the form of the proposition expressed by "George Washington crossed the Delaware." The semantic value of "George Washington" is a person. If you want to say the semantic value of "Frodo Baggins" is also a person but he happens to be fictional, I'll just say this is not something persons can be. You can only be one of those.

They're both real in their respective frameworks — frank

I get the impulse to say this, I do, but I think it's more complicated than that.

You would somehow distinguish between a person who read LOTR as a novel, and someone who thought it was true. What is that distinction and how does it affect the truth-value of statements like "Frodo carried the ring to Mordor"?

Consider this too: even if it is true that Frodo went to Mordor, it is not made true in the same sort of way that "George Washington crossed the Delaware" was, by the person George making such a transit; it became LOTR-true by Tolkien writing words to that effect.

Normally you cannot deduce P from someone's saying that P. You might count it as evidence, but that's it. Here we have an act with illocutionary force: Frodo went to Mordor if and only if Tolkien said he did.

Insofar as there is some framework within which "Frodo went to Mordor" is true, it consists exactly of what Tolkien did or did not say, including things like this very sentence. The 'framework' comes into being along with the statements it makes true.

If asked, whether Pippin accompanied Frodo into Mordor, you might answer, no, Pippin was in Gondor. I contend that we do not mean that as a statement of historical fact, but as elliptical for "Tolkien says he was in Gondor" or "The book says he was in Gondor," something like that. Within context, we understand that, but it means such sentences actually have a different logical form than "Washington crossed the Delaware."

There's a lot more to this. Fiction does not respect the law of bivalence: if the book doesn't mention in what order the members of the fellowship left Elrond's house, then there is no order that truly or falsely describes their group. Fictional worlds are fundamentally incomplete.

And I do love Elvin Jones's drumming — Jamal

If you haven't heard it listen to Out of This World, the opening track of Coltrane's self-titled release on Impulse!

Mine for today is

Pharaoh Sanders, Heart is a Melody
Pharaoh, ts; William Henderson, p; John Heard, b; Idris Muhammad, d

and it's very special to me. (Got to to see Pharaoh in Atlanta a couple years ago.)

Many years ago, I used to live in the DC area and would sometimes listen to Jazz 90, knowing nothing about jazz but a little curious. One late night driving up Connecticut, I heard the opening track of this record and became a lifelong jazz fan. I was blown away.

But I didn't hear who the performer was, only that it was a performance of John Coltrane's composition Olé.

Next day I went to Tower Records and bought a copy of that album, my first jazz record. As I learned about jazz and branched out - Eric Dolphy is on the record, under a pseudonym, and he led me to Mingus - I learned there was tenor player Coltrane knew (the west coast expert on mouthpieces, when Trane was having trouble with his) and later played with, in his free period and who also recorded on Impulse! in the 60s and 70s, so eventually I found my way back around to Heart is a Melody.

It's a beautiful record and the performance of Olé has one of the most jaw-dropping moments in recorded jazz, far as I'm concerned.

he was the first to seriously open the box out of which his successors would step. — Mww

Two ways to look at this. (1) When Hume say 'Nature' deemed this matter, the belief in body, so important that it did not leave it up to our fallible reasoning, he might well have said 'God' but didn't. So that's a pretty daring step, and we have to remember he's writing more than a hundred years before Wallace and Darwin, though 'anticipating' them here and there in the Treatise. (2) On the other hand, you could say that the structure of the argument is the same, just substituting 'Nature' for 'God', so it's not such an advance as it might seem.

*

Hume is quite clear that the belief in body does not arise either from the senses or from reason, but from a sort of instinct, and much of this chapter is in some ways a description of how we adapt ourselves to having this instinct — thus the 'double existence' theory.

But more than that, the senses alone or reason acting upon the deliverances of the senses, would seem to support the opposite conclusion, that there are no bodies distinct from us and continuous over time. The senses and reason are not just unable to deliver the belief in body; they support disbelief in body.

If we want to say, as @unenlightened suggests, that Nature is trustworthy here, that there are bodies, then not only is looking for reasonable grounds for that belief a mistake, because it won't deliver them, but the use of reason will actually lead us astray, so far as it is able. Hume is just as clear that reason is impotent to overcome the instinctive belief in body, but it is trying.

That ought to bother us. The short chapter before this, on scepticism with regard to reason, was all about our failings as reasoners in fact, our simple fallibility. But here, someone is lying to us. If it's Nature, and there's nothing we can do about it, that's troubling. If Nature tells us truly that there is body, but our senses and reason tell us there isn't, then our senses and our reason are mistaken.

What we don't have from Hume — not to my memory, maybe someone else who's been in the Treatise more than I lately could say — is an extended discussion of how the senses or reason subtly go wrong in this matter (or in any of the others in the Treatise, for the matter, such as causality). We might have. If he thought Nature had gotten everything right for us beforehand, then he would assume, I think, that the reflections which lead to the opposite conclusion must have a flaw somewhere, and we would have hundreds of pages devoted to finding those quite subtle flaws. This would not be a matter of reason and experience grounding the belief in body, say, but of them at least not kicking against it. Then, at least, while there would remain important beliefs beyond the reach of reason and experience, we could continue to trust them regarding such beliefs as they do ground.

As it is, Hume of course accepts that our senses can deceive us, but he's very specific: our senses tell us that the table grows smaller as we withdraw from it, though it does not; but our senses are completely honest and trustworthy about the table appearing to grow smaller. They may lie about the objects that occasion them, but not about themselves. In the end, he will not lay this charge of deception at their feet, because the senses are not responsible for our belief in the table as an external object in the first place! That belief is implanted in us by nature. (But not by being implanted in the senses or in our reason.) But if nature has done right by us, the senses deceive us by their very nature, by providing the mind with changing and interrupted perceptions rather than the distinct and continuous bodies that we should be perceiving. But if that is the nature of our senses, and they are blameless in doing only what they can do, then it must be down to reason to find the necessary connections among those scattered perceptions and assemble them into the proper wholes nature has rightly told us there are; but reason cannot do so. Instead we must feign such unity in our perceptions, using not reason but fancy.

Our senses, then, lie, by telling the only truth they can; our reason lies by preserving only such truth as it is given by the senses; and imagination tells the truth by lying to us ('double existence') about the incompatibility of our instinctive beliefs with the beliefs we derive from experience by reason.

I am terrible at chess playing a computer at an easy level and I am killed off pretty quickly. Isn't chess basically an OCD game of perfectionism? — TiredThinker

Did you lose again right before posting this?

Does chess even exercise useful parts of the brain? — TiredThinker

Not

always.

*

Decide if you want to be a better player. If you do, that is an achievable goal.

Have a look at Levy Rozman's YouTube channel. There's material there (and elsewhere) to help you play better if you want, or you can just watch it for chess appreciation, to understand a little what people love about the game.

*

For the record, I quit tournament chess without ever making master, but I had a couple master-level performance ratings. I miss it.

And his point was that the referent (if any) of "Fido" is the dog so named, whereas people (and at least half of philosophers) think this can't be right: reference, being logical after all, must be from word to other word. — bongo fury

Like who?

the causal connection may not always be sufficient for knowledge. Consider the fake barn scenario. — Andrew M

I agree this has to be done carefully, and I tried to cover some of the obvious issues. It looks like some of the "not the right way" issues just get kicked down to "not caused the right way", so causal connections are not just a 'win card' we can play.

If I get a handle on Goldman's approach, I'll report back, particularly on how he deals with barns. I will note, in passing, that it's my impression there is even less unanimity on the barn cases — that is, in a lot of Gettier cases there is little conflict among philosophers' intuitions, but with the barn cases I believe there is.

Which is interesting because it means the conflict there is a new data point to explain.

Even in the concept of "actual" or complete or whole infinity, can every open possibility be actualized?

I'm very open to learning otherwise, by what I currently understand by infinite length is that actualizing every open possibility would entail a limit/boundary/end of open possibilities ... thereby negating its affirmed infinitude. Am I misinterpreting something in the terminology? — javra

Well there's a formal out, if you want to take it, and then there are new questions.

The formal out is that in modern logic (Frege's logic, which he developed specifically for formalizing mathematics), "every" is of course no sort of number at all. "Every" indicates a conditional: "Every sperm is holy" says "If something is a sperm, then it is holy." This veers somewhat sharply away from the old treatment of universal generality (from Aristotle and medieval logicians, the square of opposition) in that universals are no longer taken to have 'existential import'; in this case, the existence of sperm is not entailed, and the claim is vacuously true if there are no sperm to be holy or otherwise. (Frank Ramsey was even of the opinion that universal generalities were exactly this, habits or rules of inference, nothing more, and not really quantification in the way people think.)

For our case, "Every colinear point is included" says "If a point is colinear with any two points already included, it's also included." Now that doesn't say, "If a point is colinear with any two points already included, add it"; it looks like a rule for adding points, but instead it claims directly that they are all already there. The rule is the line. You don't really construct the line at all, and then know what you have constructed, but by knowing the rule, know the line.

This is how mathematics makes the infinite comprehensible. No human being will ever have the opportunity to observe a one-dimensional line of any length, much less of infinite length; but any human being is capable of understanding the rule that defines such a line.

Of course, one can say, that's not really infinity; or one can say, that really is infinity and thus no one really understands such a rule, they only know how to work with it formally, as a bit of symbolism. (I think I've now alluded to all the principle schools of the philosophy of mathematics: realism, intuitionism, and formalism, for what that's worth.)

Not sure how this fits your thing, but there it is.

Honestly, @apokrisis is the only guy I know around here who's comfortable with this sort of metaphysics, and I learned the habit of looking for constraints from him. He'll mainly tell you that whatever system you're cooking up is a partial reconstruction of his own, but he'll understand what you're up to. You know the drill.

I do think it might be worth thinking a little more about how dimensions work, because they are so explicitly a matter of adding degrees of freedom, each of which is constrained by what was previously an added degree of freedom. That's a curious pattern. There are weirdnesses we're passing by, like fractals and space-filling curves, but gotta walk before you can run.

Hope some of this has been helpful.

And fundamentally I think all of this is to one side of issues in logic and ontology.
— Srap Tasmaner

I don't understand what you mean. Please explain, thanks. — Amity

I only mean that modern logic of the sort we typically use these days in philosophy is Frege's logic: there are objects — so ontology — and functions, literal functions like you learned about in math class that map objects or sets of objects onto the set {0, 1}, truth-values. That's it.

When we talk about Middle Earth, we're only doing logic very indirectly: we're talking about what Tolkien did and did not put in the books, so there are truth-values to be had here, and there are objects, but the objects are, at bottom, words. Did Pippin accompany Frodo and Sam to Mount Doom? That question is not about any persons or places or travel anyone undertook, not really. It's a question about what sentences are in the book, and what the logical relations among them are, or can be worked out to be. The rules of inference are already a kind of pretend; they work as if they treat of known objects we can quantify over and apply known predicates to, but they are only truth-preserving not truth-engendering. There is no truth to the sentences in fiction, so there is no truth to preserve, but the sentences can still be related to one another logically.

Because fiction seems to be about persons, places, and events, it's in one sense a handy showcase for how logic works, but only if we pretend. If we want to say things that are genuinely true and false about fiction in the same way we say them about objects we do find in the world, then we must do this complicated double analysis, that works out the logical connections among sentences on the pretend level, but in the end only quantifies over the words and sentences that make up this textual artifact.

Unless you find reason to disagree with this generalization regarding the determinacy of such infinities, I think I'm good to go. — javra

There is one other little hitch though: a line, for example, not only can or may contain all the points in a plane colinear with it (that is, with any two of the points on the line), but it must and does.

Do we still call it freedom, absence of constraint, if you must actualize every open possibility?

I've been speaking of a line as embedded in a plane, because it's simpler to visualize that way, and you can contrast a line to the other possible figures in a plane, but a line is, by itself, simply a dimension. It is in one sense a result of constraining a plane, but in another sense a constituent of an infinite number of planes, whether seen as an infinite collection of zero-dimensional points, or — more importantly here, I think — seen as a formal constituent of the plane, as representing one of its dimensions. And here's the kicker: any line can itself be considered a constraint that partially determines a plane, as can any point.

and that it differs from types of infinity that can be quantified and thereby numerated — javra

Sure.

So a line. On the one hand, there's a sort of procedure, which is repeatable, by which you can keep extending a line; there may be more than one way to do that — physically different techniques, for example — but they're all equivalent in the long run, because there's the usual asymmetry here: there is exactly one way to extend a line as a line and an infinite number of ways to extend it otherwise, with curves, angles, gaps, and so on. So we have a pretty strict formal constraint. On the other hand, we want to extend it forever, which requires the procedure to be repeated forever, without constraint.

If you think of the possible figures you could draw in a plane, you're constrained to the plane, but otherwise have complete freedom. If you compress and channel that freedom in a particular way, you can get a line: completely constrained in one dimension, but completely unconstrained in the other.

Is this the sort of thing you had in mind?

Which scores a stupendous predictive hit for Hume, even if I got it wrong. — unenlightened

Yes, I think that's right. The gist of the color constancy effect is that your brain prepares an interpretation of your visual environment and part of that is that objects have distinct and continuous colors (just along Humean lines) and it is this idealization of the objects in your environment that you are conscious of, not a faithful recreation of the color patches that make up your putative visual field.

nonquantifiable infinity — javra

Pleroma?

Well, it's clear that Frodo is not someone who we might meet at the shops, nor an historical figure, but a fictional character. And that is what one is claiming in saying he doesn't exist. — Banno

We agree he doesn't exist. But you want to still be able truly to predicate "is a hobbit" of him; I don't.

Honestly, I could meet you halfway, and allow a sentence like "Frodo is a hobbit" to be true under somewhat constrained conditions, and those conditions would involve spelling out some of what's involved in Frodo's hobbithood. But for an apples to apples comparison, "Frodo is a hobbit" cannot be true in the same way that "Seabiscuit was a horse" is true. And I don't mean there are different sorts of truth, but that the presuppositions of those statements are so different that I think the statements themselves don't even have the same logical form. Consider the difference between "It says in the book that Frodo is Bilbo's nephew" and "It says in the book that Seabiscuit beat War Admiral by four lengths": the books here are doing very different things.

Everything we say about Frodo is a sort of shorthand for referring to the literary work of Tolkien, and other work derived from it. (And so far as that goes, I don't in fact have a problem with "Frodo is a hobbit." It says so in the book.) It's what allows that strange slipping in out of the text that people fall into when talking about literature: "But on page 74, Frodo tells Sam that ..." Seabiscuit never did anything on a page. And in an obvious sense neither did Frodo. But page 74 says something about Frodo and Sam, and we treat that in a certain sophisticated way.

What I've been trying to get you to see is the shocking incompatibility between "Frodo is a hobbit" and "Frodo is a fictional character." (If "Frodo is a fictional character" is true, how come no one in the books seems to know that? Why would it change the book into some avant-garde foolery if some character in the book said this true thing?) No entity can be both those things. Nothing can be anything and also be non-existent. Flicka cannot be a horse and a fictional character; neither could Seabiscuit, who settled for just being a horse.

"Frodo is a fictional character" and "Flicka is a fictional character" do not predicate anything of entities named 'Frodo' or 'Flicka'. A first pass at parsing "Flicka is a fictional character" might be: "'Flicka' is the name of a character in a book." But 'is the name of' can't mean what it usually means because Flicka doesn't exist, so that's not right. We might as well say "'Flicka' is the name of a horse in a book by Mary O'Hara." Really? How did a horse manage to live in a book? Whatever Flicka is, and however the name 'Flicka' attaches to that, Flicka is not the sort of entity that in real life has a name in the usual way.

I think it's actually pretty hard to give a good account of how we talk about fiction. Most such talk hangs suspended from a counterfactual conditional like "If the story in Mary O'Hara's book were a true story ..." But that's not all of it, because as I noted, we freely pass back and forth between pretending Flicka exists as described and treating Flicka as a textual artifact — "Remember in Chapter 3 when Flicka was out in the thunderstorm?"

There's some pretty sophisticated stuff going on when we talk about fiction, but it's all obscured by familiarity. I remember reading a story somewhere about some Europeans traveling maybe to Japan, some place in the East with a very different theatrical tradition — I may have the details all wrong — and the point was made that the local audience was absolutely mystified by the idea of actors, people pretending to be the characters in stories and speaking their words. They were used to a sort of elaborated story-telling with music and so on, but basically one guy reciting. Acting was incomprehensible to them at first. This is the kind of sophistication we have with fiction that I think is hard to notice, and why I can't just reel off an account of how we think about and talk about these things.

And fundamentally I think all of this is to one side of issues in logic and ontology.

Right, that part is brilliant. Not only is there a double existence, but the perceptions an object occasions exactly resemble it, and of course vice versa. Why? No reason at all. No conceivable reason. It's just the sort of assumption we typically make, with no justification whatsoever. That bit is pretty humbling.

Nor do I personally take established concepts in mathematics to the foundational cornerstone of what "infinity" at large can signify. — javra

I hate to forestall this thread's death, but I am curious about this.

I looked back at the OP yet again, the centerpiece of which is this question:

Can mathematical infinities (e.g., geometric lines, infinite sets, and so forth) be ontically determinate? — javra

You're talking specifically about the mathematical versions of concepts in wider (and vaguer) use — and that wider use is what some of us assume lies at the foundation of mathematics, our intuitions about shapes, collections, counting, patterns, all that. I gather it's something like those intuitive, pre-theoretical ideas you really wanted to address, not their mathematical axiomatization.

Which is fine, and I can imagine doing a phenomenology of boundedness and unboundedness, that sort of thing. No doubt that would be interesting.

But there is still something odd about your decision — though maybe I've misunderstood you — to exclude mathematics. After all, we've had a few thousand years now of people thinking about just these things, and some of that thinking is what we call mathematics. The history of mathematics is far messier and various than your grade school textbook led you to believe, precisely because it's the history of people thinking about the sorts of things you've expressed interest in. Mathematics as it is now may strike you as somewhat rigid and narrow, and therefore of no use to you, but it is still a body of serious, rigorous thought about things like the infinite, so even if there's more to say than you can get out of established mathematics, it is surely the natural starting point, not the natural body of work to be excluded.

Maybe this thread would have gone differently if I had asked you directly to explain this:

And my stumbling block is that by defining determinacy as I did in the OP (i.e., having limits or boundaries set by one or more determinants), I run into this stubborn paradox of having to differentate semi-determinacy from what I've so far termed "mathematical infinities" ... which are, again, only partly infinite in some respect while yet being finite in others. — javra

Looking back at our exchange, I realize I hoped that what you're talking about here would become clear as we worked through some examples, but it didn't.

So I still don't have the faintest idea how what you call "mathematical infinities" inserted themselves into whatever you were working on, and why their arrival was such a problem.

If actual mathematics is no use in solving your problem, then presumably these "mathematical infinities" obtruded for non-mathematical reasons; but I can't figure out what sort of non-mathematical problem would drag in a bunch of — as a matter of fact, somewhat recondite, even for math — mathematical concepts.

If you're of a mind, and not burned out on the topic, take another swing at it. It is, after all, a philosophy forum not a math forum. Maybe if you could explain a little more clearly how your problem relates to mathematics without being a mathematical problem, we could make some progress.

I'm still going back through the section on and off, but we end up with three 'theories', right?

There's (1) the instinctive view that we directly see objects. Then there's (2) the sceptical, philosophical view that only perceptions can be present to the mind, and perceptions don't have the key properties of being distinct from us and constant over time. Then imagination gives us (3) the 'double existence' theory, which posits a constant object of which we have changing perceptions, giving both instinct and reflection whatever they want, without actually justifying this move.

Is that the overall structure as you see it?

On the other hand if one took a regular checkerboard, and reconstructed the whole scene, there would be no illusion because the eye would correctly identify the squares that were the same colour, despite the variation in lighting due to shadows. — unenlightened

It's a nice thought, but demonstrably false.

Here's another video (a bit tech-bro, but that's what you get) about illusions related to color constancy, mostly done with real-life models.

One may be justified in believing that p even if p is false. This opens the door to Gettier cases, no matter how stingy or generous the criteria are. The problems actually arise when S believes the right thing for the wrong, but justifiable, reasons.

How to respond? Well, my response to your farmer is 1) he thought he saw a cow, 2) he didn’t see a cow, 3) there was a cow. I observe that a) 1) and 3) are reasons for saying that he knew and that b) 2) is a reason for saying that he didn’t. I conclude that it is not proven that he knew, and that it is not proven that he didn’t, so I classify the case as unclassifiable. — Ludwig V

I'll tell you what I think is the obvious thing to say here: the problem with the farmer's belief that there's a cow in the field is that it was not caused by seeing any of the cows in the field. Or: there was no causal connection between the farmer and a cow that contributed to the farmer's belief.

This is more or less a typical Gettier case because the conclusion is an existential claim that is true in virtue of the existence of some particular: it is true that there is a cow in the field because this particular cow, let's call her Alice, is in the field. In a sense, we don't even have to talk about what's wrong with a cow-belief being caused by a bit of cloth, about whether the interpretation of blurry light spots is a reliable method of cow detection; all we have to say is that the farmer has what looks to be a belief about Alice despite that belief not being causally connected to Alice.

This was more or less Alvin Goldman's response to Gettier, and it does seem to get something right. (Only just started looking at Goldman, so don't ask me about his views.)

My way of putting this raises some issues though: in what sense is the farmer's belief about Alice? This doesn't look good at all. Since Alice played no role in the farmer's belief formation, it's pretty clear Alice is no part of the content of the farmer's belief. Alice does play a part in the existential claim; Alice is what makes that claim true.

We can get to Alice, as a matter of content, with the obvious counterfactual claim: had the farmer seen Alice instead of the bit of cloth, and seen that Alice is a cow, then in that case he would of course know that there was a cow in the field. But he might have seen Alice and mistaken Alice for a bit of cloth flapping in the breeze — so not seen that Alice is a cow — and formed the mistaken belief that there's a bit of cloth in the field, which might also be Gettierly true. That's a little uncomfortable for the causal account, as it stands so far, because it's just requiring the seeing itself to be a factive mental state. But at least now Alice, under some interpretation, is part of the content of the farmer's belief.

And that seems a reasonable starting point: Alice ought to play a causal role in beliefs about Alice. I don't think there is a remaining problem with the existential generalization after all because we can just enumerate it: if Alice, Bobbie, Clarabelle, and Dixie are the cows in the field, then the truth of such an existential claim as we're concerned with is a truth about at least one of those: one of those four ought to play a causal role in the farmer's belief, expressed as an existential generality.

Are we any better off though? Suppose the farmer thinks the cow he's seeing is Clarabelle, when it's Alice, even though Clarabelle is out there in the dark. There is some lingering oddness about the existential generalization; it feels a little unreal, like the content of his belief still involves Clarabelle, though expressed with reference to "a cow", and so his basis for believing that the generalization is warranted is suspect.

There are obvious cases in which the farmer would reach for the general, disjunctive claim because he sees a cow and doesn't know which one. What about in this case, where he believes he does know which cow makes the disjunction true? Now that's a funny thing, because it's very natural to have different degrees of confidence here: I for sure saw a cow, and I'm pretty sure it was Clarabelle — if it wasn't Clarabelle, I assume it was Alice or Bobbie or Dixie. That last clause can fail if a neighbor's cow has gotten into his field, but even that won't affect his high level of confidence that he saw a cow, some cow. We might even plump for him knowing it was a cow, while denying that he knows which one.

And that's a reminder that you absolutely can know a disjunction is true without knowing that one of the disjuncts is true. The law of the excluded middle is a clear enough example, but we might forget in these more mundane, probabilistic cases.

The farmer, then, could be in a state of disjunctive knowledge, connected causally to some truth-making cow, the actual content of which is a belief he holds only partially and could even be wrong about. (Something still weird about that formulation.)

In the original version of the story, it's a bit of cloth that is causally related to the farmer's implicitly disjunctive belief and another disjunct is true. What's different here from the case above where one cow is mistaken for another — and so there's acceptable disjunctive knowledge — is that you cannot see that a bit of cloth is a cow, because it isn't. We are relying on the seeing being factive, and that's already expressed as predication; what causation gives us is an explanation for the acceptability of the predication: you can see that something is a cow only if it is a cow. Which I hope is another way of saying that some cow ought to be causally involved in your formation of cow beliefs.

@creativesoul, I think some of your concerns are addressed above. @Andrew M, any thoughts?

Except it's not a world of objects but of perceptions; objects are mere prejudice. Empiricism slides into idealism.
— Srap Tasmaner

No. it's not the objects that he denies, it's the reasoning. Of course there are objects; of course they aren't in the mind, and of course they are not the product of reason. When you follow strict reasoning you end up with 'Yikes!'. Natural impulses are a better guide. — unenlightened

You're absolutely partly right.

Of course, he does not deny that there are objects, because he claims that we cannot. I'm happy with the word 'prejudice' there.

On the other hand, he makes no 'argument from instinct' that I can see. He might have, but he doesn't.

And you're right that the intellectual context matters, as you noted before. Descartes does give something like an argument from irremediable prejudice: that which we cannot doubt must be true. Hume (and Kant after) seems to me unmoved by this argument. Why could there not be some falsehood we cannot help but believe?

There are optical illusions like this, that work even when you know they're illusions (the Ames window, the checkerboard illusion and other color constancy shenanigans), because they depend on deepish features of our visual processing. Empiricists love their optical illusions, so Hume, were he aware of these examples, would no doubt consider such things slam-dunk counterexample to any proposed 'argument from instinct'.