Since Frodo is not real, he could not be a member of the non-empty class of those who walk into Mordor. — Banno

You mean, since Mordor is not a real place, the class of people who've been there is empty.

I'll go you one better.

Marianne Moore published two versions — I think 'published', maybe she only contemplated doing this — of a poem called 'Poetry'. The short version goes like this:

I too, dislike it.

The longer version, with the indentation butchered by our software:

I too, dislike it: there are things that are important beyond all this fiddle.
Reading it, however, with a perfect contempt for it, one discovers that there is in
it after all, a place for the genuine.
Hands that can grasp, eyes
that can dilate, hair that can rise
if it must, these things are important not because a

high-sounding interpretation can be put upon them but because they are
useful; when they become so derivative as to become unintelligible, the
same thing may be said for all of us—that we
do not admire what
we cannot understand. The bat,
holding on upside down or in quest of something to

eat, elephants pushing, a wild horse taking a roll, a tireless wolf under
a tree, the immovable critic twinkling his skin like a horse that feels a flea, the base—
ball fan, the statistician—case after case
could be cited did
one wish it; nor is it valid
to discriminate against “business documents and

school-books”; all these phenomena are important. One must make a distinction
however: when dragged into prominence by half poets, the result is not poetry,
nor till the autocrats among us can be
“literalists of
the imagination”—above
insolence and triviality and can present

for inspection, imaginary gardens with real toads in them, shall we have
it. In the meantime, if you demand on the one hand, in defiance of their opinion—
the raw material of poetry in
all its rawness, and
that which is on the other hand,
genuine, then you are interested in poetry. — poets.org, first published 1919

Imaginary gardens with real toads in them.

What shall we say about that?

Frodo is a hobbit, therefore the class of hobbits is not empty - they are fictional creatures. — Banno

Hobbits, then, form a subclass of the class of fictional creatures, right?

So Boromir could well have argued, at the council of Elrond, that Frodo could not possibly carry the One Ring to Mordor because he was a hobbit, and thus fictional. Why do you suppose he didn't? But then, maybe no one at the council knew that hobbits are fictional, perhaps through some mischief of Saruman's. Still, you'd think Gandalf would have known, as much time as he spent with them. (Like curling up with his favorite book, I guess.) Luckily, it all worked out. Being fictional didn't stop Sam and Frodo from carrying out their task, so maybe it's not as strong an argument as it seems.

If the real is so elusive, so difficult to establish — Tom Storm

I only said that some empirical questions are hard to answer. It took a long time and a lot of money to observe the Higgs, and for a long time it was a thing we just could not do. Some questions about the past we likely will never be able to answer. The existence of books about a guy from Nazareth named 'Jesus' (or something like that) suggest he was a real person, but it's hard to know for sure for a great number of reasons. We know surprisingly little, as I recall, about the personal life of Shakespeare, but he too was probably a real person.

If you're interested in my opinions, I'll give you one: I find the stories about Bigfoot hoaxes persuasive, the guys that made the footprints for fun, running along behind a pickup, the guy that dressed in the costume for a wannabe filmmaker, plus I'm convinced by the argument that a breeding population of bigfeet would have to be big enough that we're likely to have had incontrovertible proof by now, if they existed. So I think there's no Bigfoot, and I will be very surprised if it turns out there is.

We're in 'prove a negative' territory, but I'm pretty confident the class of bigfeet is empty. That's harder to determine than whether I'm out of Pop Tarts but not as hard to determine as whether there are gravitons.

It's you who are in need of an account of how we can talk rationally about fictional or imagined characters. — Banno

I'd love to. Fiction is interesting because pretending is really interesting.

No idea why it should change how I think about logic though.

Sounds like we're fucked then and to a large extent doomed to be the playthings of the likes of Osama bin Laden and Trump. — Tom Storm

?

How do we determine what counts as fictional and what does not? Is Allah fictional... Jesus? — Tom Storm

Who knows? There are arguments, there's evidence, and some empirical questions are hard to answer.

I think fiction is a pretty subtle thing, and there are simpler cases to consider. Lots of things used to exist and don't anymore. The class of Tokyo hotels designed by Frank Lloyd Wright used to have one member, the Imperial, but now it's empty. We may have evidence, from Audubon or something, that there was once a bird called the Whiffle-Breasted Woodpecker, now believed extinct; we would say that class used to have members and now it doesn't. But someone may spot one someday, and then it will turn out that class is not empty after all.

Does the Higgs boson exist? We had the class, defined theoretically, for years before we could manage observations that showed that class to have members. The Michelson-Morley experiment was widely taken as showing that the class of luminiferous aether is empty.

So I gather you are saying that Sheldon cannot be a unicorn - that the class "Unicorn" is empty?

That seems to me to be an unneeded step to far. — Banno

I have absolutely no idea why you think so. Of course the class of unicorns is empty. For all x, x is not a unicorn.

But Frodo, of course, is fictional, and not real. If being member of a class is the same as being real, then Frodo cannot be a member of a class, and so not a member of the class "hobbits". If we followed that rout, we would not be in a position to talk rationally about fictional or imaginative characters. That's the step too far. — Banno

There are no hobbits. The class of critters that are hobbits is empty. It's pretty clear to me.

What you need is an account of how talking about fiction works. Not only would I be disinclined to monkey with logic just for that, I'd assume you'll need logic to keep working in the usual way to carry out such an analysis.

That is, one might set up a domain by ejecting imaginary and fictional stuff. — Banno

Really? I would have thought imaginary entities don't exist and so don't need to be 'ejected' from the domain of discourse. There are no unicorns or hobbits for me to eject, are there?

Frodo, being a member of the class "Hobbit", is real. — Banno

Nope. We pretend there is such a person and that he is a hobbit. There isn't, and he isn't.

Is Sheldon a horse or a unicorn? — Banno

I was offering an example of a real horse named Sheldon disguised as a unicorn. His not being a unicorn doesn't make him not real. He's a real horse.

there is the class of things that are not real. We don't want to treat that as empty, while still saying it has members. — Banno

It exists, it is empty, and it has no members.

If Sheldon is a unicorn, the by p(a)⊃∃(x)p(x) Sheldon exists. Are you happy to say that? — Banno

If Sheldon is a unicorn, the class of unicorns is non-empty, yeah. (And I have no issue with existential generalization.) If your argument concludes that an empty class has a member, that's a contradiction, so one of your premises is false, for instance, "Sheldon is a unicorn." That can be false even when Sheldon is quite real, because a horse.

A better approach might be to suppose that being member of a class is not the same as being real. — Banno

I'm saying that's exactly what it is.

Existential quantification is not about what is real and what isn't. — Banno

Sure it is. Says so right on the tin.

if 'real' is 'member of a non-empty class', then Sheldon proves that unicorns are real. That doesn't look right — Banno

You mean like my example in which Sheldon is a horse? Sheldon's being a member of the class <horse> means Sheldon is real; doesn't make the class <unicorn> non-empty.

Another way to put what I'm saying: makes no difference to your mind what the source of the perception is. All, as Hume says, are 'on equal footing'.

Here's a choice line from Part II Section VI:

To hate, to love, to think, to feel, to see; all this is nothing but to perceive.

This is not a unique situation: logic is concerned with the validity of arguments; whether they be sound is someone else's problem.

It may be there is no purely mental difference between a veridical seeing and an optical illusion: the same predictions of your future states are generated. The difference is out in the future, when your expectation is confirmed or must be revised.

As logic is incomplete without some means for determining the truth of premises, so beliefs (expectations, inferences, whatever you like there) would be incomplete without some means of testing and revising them -- so, action.

Nature may find the simplest way of making things work — Manuel

The key exemplar of course is evolution by natural selection, a relatively simple mechanism which yields 'endless forms most beautiful'.

It is not impossible that some mechanism just as simple yields the complexity of mind, something like Friston's free energy principle, maybe.

On the inevitability of 'idealism from the inside', I left out the other bit, which Hume doesn't, which is that the organism will believe there are external objects and all that, just as we would studying such a creature in its environment, but the idealism comes in at the explanation stage: that, strangely, in analyzing the behavior of organism, we are driven to imagine that it must behave as if there were only mind, even if, as with our own case, we refuse to believe any such thing. Objects fairly hurl themselves against the mind, but to the mind it's just impressions, from somewhere beyond the Markov blanket.

Perhaps it's that we believe in objects, but our minds do not!

a complex mental framework — Manuel

I do think in some ways the question is, how complex? The ongoing debate in linguistics is between those who think some specialized faculty is necessary, and those who think quite general faculties get you language.

I've quoted Herbert Simon's suggestion before, that our mental lives are complex not because our minds are complex *in themselves*, in their machinery, but because our environments are complex, and culture only increases that complexity.

The other major issue seems to be something like this: we know that we are creatures embedded in an environment, all of our science begins with that understanding; but just as surely, we know that *from the point of view* of such a creature, there is only mind. On this, broadly, Hume, Kant, the Tractatus, and modern psychology are agreed. It is not so, but it *must* appear so, from the point of view of the organism.

That's interesting. And Hume was on the right track, broadly, in thinking that what you can learn from this recognition is not what's in the world -- whether there be objects, for insurance -- but something about how minds work.

But then he goes on to say: "But as no beings are ever present to the mind but perceptions; it follows that we may observe a conjunction or a relation of cause and effect between different perceptions, but can never observe. it between perceptions and objects." (p.212)

I think this last quote is problematic, a stimulus is needed. — Manuel

This is the point I've been trying to make that Hume recognizes the need for laws that govern the relations between perceptions, as Newton gave laws governing the relations between objects. That is, I think he conceived the project this way, to do for thoughts what Newton did for bodies.

Hume says you cannot argue your way out of a paper bag, but fortunately you don't have to, because the world is already present and available to be made sense of. — unenlightened

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

As to those impressions, which arise from the senses, their ultimate cause is, in my opinion, perfectly inexplicable by human reason, and 'twill always be impossible to decide with certainty, whether they arise immediately from the object, or are produc'd by the creative power of the mind, or are deriv'd from the author of our being. — Part III, Section V, p. 84

my chamber — Manuel

Indeed. It's why I was thinking we'd need to graph out the arguments, because they are sometimes presented in terms that other arguments will undermine.

I haven't spotted a similarly straightforward example in the Treatise, but there's this in the Enquiry:

This table ... preserves its existence uniform and entire, independent of the situation of intelligent beings, who perceive or contemplate it.

But this universal and primary opinion of all men is soon destroyed by the slightest philosophy, which teaches us, that nothing can ever be present to the mind but an image or perception, and that the senses are only the inlets, through which these images are conveyed, without being able to produce any immediate intercourse between the mind and the object. The table, which we see, seems to diminish, as we remove farther from it: but the real table, which exists independent of us, suffers no alteration: it was, therefore, nothing but its image, which was present to the mind. — Section XII, Part I

How does this argument work? Hume demonstrates that only perceptions are present to the mind, not objects, by showing that perceptions change when objects don't; but then he will later use the fact that only perceptions are present to the mind to argue that the hypothesis of double existence is insupportable, that we have no grounds for a belief in an object separate from our perceptions as their cause. — But then where does that leave this argument which originally established that only perceptions not objects are present to the mind? If we can't contrast the apparent extension of the table with its 'real' extension, then we have no argument at all. We have something vaguely of the form P → Q → ~P. Yikes.

And it happens all over the place, his description of his chamber being another example, and his simple reliance on his own identity.

(1) P → Q
"If the object of a knowing is an appearance, then the knowing is filtered."

(2) R → ~Q
"If the object of a knowing is an action, then the knowing is unfiltered."

(3) R → ~P
"If the object X of a knowing is an action, then X is not an appearance."

You were on the right track: like a modus tollens but there's a condition hanging over it in (2).

We can use "real" to differentiate in particular explicit cases - a real painting, a real foot, by understanding what the contrary is - a counterfeit painting, an artificial foot.

But some folk wish to contend that there is a way of using "real" that somehow goes beyond that, having no contrary. — Banno

Why 'having no contrary'? Or do you only mean in the 'pants' sense, deriving it's meaning from the contrary?

I mean, it's true that we're never going to predicate of some object 'imaginary', not in earnest, but only as a manner of speaking. The logical form of such a claim is just going to be '~∃xFx' which doesn't commit us to anything. We can comfortably say something like 'Unicorns aren't real but imaginary'; no one's attributing a property to something that also has the property of being a unicorn.

I think we would like to be able to say something like, "If something is a unicorn, then it doesn't exist," or maybe if you have a name, like from a story, "If Sheldon is a unicorn, then he doesn't exist." I guess we can stuff that directly into classical logic, but I don't think it's a very comfortable fit. It is, however, pretty straightforward to say that if something (or Sheldon) is a unicorn, then it (or he) is a member of class known to be empty, so that's a contradiction — and the conclusion is just that (say) "Sheldon is a unicorn" is false; we'll only need to go for "Sheldon is not a unicorn" if "Sheldon" is known to refer — if, say, Sheldon is a horse with a horn affixed to his forehead.

The class of unicorns can be as real as you (whoever you are) generally take classes to be; it just happens to be empty, but that doesn't mean there's any particular problem talking about it. And if we define 'imaginary' as 'member of an empty class', it ought to serve pretty well as an opposite for 'real' in that most general sense, and show up in arguments about where we'd want it to.

Oh yeah, and then 'real' in this general sense is 'member of a non-empty class'. Which is fine.

Bonus anecdote:

Story Robert Creeley tells — didn't happen to him but another poet, I forget who — that after a reading someone from the audience came up to ask our poet about something he read, "Was that a real poem, or did you make it up yourself?"

Here's another way: there can be, I think Hume thinks, nothing in the perception itself that would tip off the mind as to its origin or nature. Thus we have no surefire way of distinguishing veridical observations from hallucinations or dreams or optical illusions. Hume accepts the usual argument as a step toward considering perceptions only, however they appear to the mind.

So we still do not get at the source of individuation — Metaphysician Undercover

But Hume explicitly doesn't care.

Same page is where he says all these mental phenomena (perceptions, feelings, ideas, what have you) are 'on the same footing.' And he assumes they are presented to the mind as discrete, already individuated packets.

He is absolutely *not* going to say they are shaped by the mind, because that suggests there is something to be shaped, something that already has a distinct existence outside the mind. But he explicitly wants only to look at perceptions etc. insofar as they are dependent on the mind: for Hume they exist at the moment we are conscious of them, and that's it.

I mean, they're different in quantity, not quality. They're both cardinal numbers, just of different sizes.

Now there are transfinite ordinals, but you'd have to ask someone else about those.

There is such a thing as equivocation between two or more meanings or usages of a term, right? I repeatedly described countability in its non-mathematical sense of “able to be counted — javra

Except (a) you want specifically to talk about mathematical infinities, and there's prior art there you might as well become familiar with; and (b) the mathematical usage of 'countable' is actually something a lot like 'able to be counted', because listable.

I think what's throwing the discussion off is that we don't normally talk about the cardinality of a line except when we're considering it as a collection of points, the continuum, which is not countable. But that's not really measuring its length, different deal. If you have an infinite ruler marked off in centimeters, you'll be counting again.

Are the infinities of natural numbers and of real numbers two different infinities? — javra

Yes. The cardinality of the set of natural numbers is aleph-0; the cardinality of the set of real numbers is aleph-1, aleph-0 raised to the aleph-0 power. It is not known whether there is a size in between, but I think most mathematicians think not. Could be wrong.

The definitions can of course be questioned, but they are commonly established — javra

I warned you this would be trouble.

The usual way of using these words in mathematics is pretty straightforward. 'Countable' means there is a one-to-one correspondence between the set you have and a subset of the natural numbers, maybe all of them. So either finite, or 'countably infinite' like the natural numbers. We're talking about sets where you can write down the members in a list, even if that list goes on forever. 'Uncountable' is for bigger infinite sets. The real numbers, to start with, cannot be written down in a list that goes on forever, no matter how clever you are.

Obviously countable is nicer to deal with, because you can use algorithms that iterate (or recurse) their way through a list and you know that will get you not to the end but as far as you'd like to go.

(Also: Zeus could write out all the natural numbers in a finite amount of time just by doing the next one faster each step; not even Zeus could write out the real numbers in a finite amount of time. Lists are friendlier, even when they don't terminate.)

There is an issue around the individuation of perceptions.

There's that passage where Hume claims perceptions are exactly what they appear to consciousness as, etc etc, so he's basically claiming they are self-individuating.

* Here it is, p. 190

For since all actions and sensations of the mind are known to us by consciousness, they must necessarily appear in every particular what they are, and be what they appear.

It's easy to pass over that bit as just the usual empiricist sense-data talk, but without it he has no basis for claiming that our perceptions are interrupted.

Once you have a line, whether any other point in the plane is on it or not can be determined; it becomes an absolute yes/no question. Within a plane, every point is either on the line, above it or below, so the line perfectly bifurcates the plane. (Not for nothing, but given a line and a point, you figure out its relation to the line using a mathematical construction called a 'determinant'.)

There's also a sense in which a line, like any other function, gives a perfectly clear answer to how a segment of it can be extended: go on exactly like this.

It's altogether very well-behaved, and as sharply defined as, say, a triangle or some other sort of figure.

Hmmm. I was hoping you'd say you were okay with this example so we could compare it to another that you feel differently about.

Does it help at all to look at how mathematics handles this? Vaguely similar questions do arise in mathematics.

So, for instance, we say any two points in a plane determine a unique line. But if we go up a dimension, thus allowing that third coordinate to vary without bound, two points are not enough to pick out a single plane, and there are infinitely many planes that contain the line they determine. You need one more point, not on the line, to uniquely determine a plane.

Just an example. Mathematics does sometimes directly address how determinate its objects are, at least in this sort of sense, whether there's a unique solution, finitely many, infinitely many, etc.

Is this sort of determinateness any use to you?

Is there any property in taxicab geometry analogous to curvature?

Maybe the average distance of the intersections at which you turn from the impossible direct route. Not sure what the point would be, but it's interesting to think of taxicab routes as approximations of the direct routes that are unavailable. (Or vice versa.)

So for your question about the determinateness of mathematical infinities, you would say here that a line is I guess 'determinate enough' that we can pick it out as an object?

Ah, I see, you meant countable as a unit, as a line. Sure.

countable — javra

Just don't say that. It has a specific meaning in mathematics, and the length of a line is not countable in that sense.

Doesn't matter to whatever you're saying. Carry on.

I know so little about math, but I'm always eager to learn. — Real Gone Cat

Uh huh

we have indications of the existence of external objects, — Manuel

Do we?

I think it is true that, broadly, we take beliefs, our individual beliefs and the beliefs of others, as indicative of how things are, but we know that we cannot deduce P from anyone's belief that P. We can take a middle course and count the beliefs of others as evidence, but such a procedure is rarely available when considering your own beliefs. There are very specific circumstances where that's reasonable, but in general there's something illicit in counting your own beliefs as evidence of their truth.

What Hume says quite definitely is that we do embrace the principle of the existence of body. Can we count that as evidence of the truth of this principle? Hume presents arguments that the principle cannot be supported either by our senses or by reason. Does that mean he leaves open the possibility of an 'argument from instinct' or some such thing?

You represent the conceptions of external objects as being dependent on, or necessarily caused by, perceptions. This denies the possibility that a representation of external objects could be entirely fictitious, imaginary, created completely by the mind. — Metaphysician Undercover

Not what I intended at all. I was, I thought, following Hume's usage in using the word 'perception' to cover both impressions and ideas; so a perception is something present to the mind, of whatever source, a perceiving that is done 'by the mind's eye' we might say.

It's best that you give proper context to "continu'd existence". This is what is expressed by Newton's first law, the law of inertia. — Metaphysician Undercover

Here's what I think Hume is saying.

As I watch the birds flitting about in my front yard, I have an impression of a bird first here, then there, then there, as it moves from tree to tree. On a naive view, we might say there is a body, some force is applied to that body, and thus the body is caused to move through the air. We can see in the bird's flight inertia, which carries it along a certain path (modified by gravity and drag) until it flaps its wings again, adding a new vector which modifies its course, and so on.

Now consider me watching this: I have an impression of the bird; that impression is replaced by a different one with the bird elsewhere. Did my impression of the bird move? Of course not; I'm sitting on my porch and all my impressions of the bird are in the same place. Did my impression of the bird acquire an inertial force carrying it from one place to another? Perforce, no: my impression did not move; my impression had no such force applied to it.

Whatever connects my impression of the bird in one place to my impression of the bird in another place cannot be the same sort of physical law (inertia) that connects the bird being first in one place to its a moment later being in another. Newton's laws may apply to external objects that occasion my impressions (if there be such things) but they clearly do not apply to my impressions themselves.

I think it's a line of thought something like this that lies behind Hume's arguments. Habit, custom, instinct, imagination, association, all these sorts of things will fill, within the mental realm, the role that Newton's laws fill in the physical realm, laws that connect one event to another.

And of course once started down this path, it'll become clear that even belief in the existence of distinct and persistent external bodies is also down to the operation of such mental laws. Hume recognizes that it is the science of human nature that must underwrite all the other sciences, including Newton's physics.

Of course you can disagree with Hume; but can you make a case against his view?

For instance, Hume's view as I'm presenting it (not sure I'm getting him right) could be tweaked to more closely resemble relatively mainstream psychology: to claim some physical law governs the behavior of external objects is to describe our expectations about their behavior, which will for us just be particular sensory experiences and that's what we're actually predicting.

There's (not coincidentally) a similar perspective flip in the subjective account of probability (and thus of statistics).

So what's the critique of such views beyond mere disagreement?

Are you remaining within the chapter? — Manuel

Yeah, except for going back to Part II Section VI for the existence stuff.

I think that some of this may be alleviated once you get to the part in which he discusses the imagination. — Manuel

Some of what?

So far as I can see, he's still talking about our perceptions of the object, and then the problem is how do these perceptions tell us something about the existence and continuity of these objects ("body"), which "we must take for granted." — Manuel

Two points:

1. Something else conspicuous by its absence is the word "representation"; when he covers this material in the Enquiry he uses phrases like "perception or representation", but there's no such suggestion here. The word might be in there somewhere, but there doesn't seem to be much use made of the idea; the whole flavor of the account is causal, mechanical. Whatever 'occasions' (Hume's word) our perceptions, they appear from mind's point-of-view as sui generis.

2. Even if there are principles connecting objects to each other 'out there', beyond our minds, those principles apply to objects, not to our perceptions of them — thus we must have our own mental principles, which will apply to our perceptions, in order to conceive something like causality. Similarly, even if there are principles connecting the presentations of an external object to mind — namely, the distinctness and continuity of the object — it will do us no good: we need principles that will relate certain perceptions to each other.

Hume thinks we can analyze the principles governing the behavior of perceptions, but if there are analogous principles governing the behavior of external objects, we cannot know what they are and cannot analyze them. We might as well presume there are no such principles, it makes no difference.

Ultimately, Hume is trying to convince his perceptions that they are perception. — Richard B

As the above makes clear, I don't think so. I think Hume's idea is that it's irrelevant what the source of our impressions is; however they come to mind, they are now mental phenomena, and whatever principles govern the relations among mental phenomena must be mental principles, not such principles as we imagine govern the behavior of external objects.

He doesn't really need the tyranny of Nature here, does he? The sceptic is not admitting anything of any consequence. — unenlightened

I'm still unclear on this. All I've come up with so far is that Hume believes he must provide some explanation for our universal assumption that there are distinct, continuing, external objects. Whether it's true is just not at issue; we have no reason to believe it (he says) but we do, so he believes he must explain why we hold this unjustified view.

The idea of existence conjoined with the idea of any object makes no addition to it, but existence conjoined with the idea of any object adds everything to it. — unenlightened

I mean, it's clear that there's a difference between imagining there's a chocolate cake in the kitchen and believing there is. Hume has already dealt with that difference, earlier in the book. What else is there to say?

I've always thought the logical argument against existence being a predicate was convincing: an object existing and not existing must have the same properties, else existing is not what distinguishes them.

I'm not sure what good it is, but Hume's point that to imagine something is to imagine it existing — that's pretty interesting. It actually sounds plausible, but of course you can't tell the difference! It's a strange thought that undermines itself.

I am unsure whether the intention is to put the above passage into first order logic as it stands independently of Kant or into first order logic such that it agrees with Kant's philosophy. — RussellA

I understood the issue in the OP to be the logic of the quote, whether it's Kant or not. Everyone else seemed to assume it was Kant, so I did too. Maybe a mistake, but not relevant to the logical question.

I still find the quote pretty straightforward, but only if you're comfortable with the logic.

I cognize something x. Cognition is a higher level function of the brain. I can cognize about x both as an appearance and a noumenon. — RussellA

That may be, but in the quote given, there's no as-ing of the object of cognition.

(doesn't deny the conjunction of P and Q). Rather, it takes that denial for granted: — bongo fury

I did longer versions of this where I included it as a premise: 'If x is an appearance, then x is not a noumenon' — I think the conditional was all that was needed, not a biconditional, for what it's worth.

restatements of the original position -- if all cognition is of appearances, in that very case there can be no cognition of noumena (since noumena aren't appearances) — Moliere

Anyway, I think @Moliere has the right idea: there's no argument per se going on here, but an explanation of terms. Kant isn't establishing a result, just clarifying.

We often say things like this just to make clear what would and what wouldn't count as a counterexample, for instance: Either all cars are Toyotas, or there is something that is a car and is not a Toyota. It's not obvious to people who do don't do FOL all day that the opposite of a conditional is a conjunction, but it is. It sounds right, sure, but is it logically valid? Yeah, it is.

Is that what Kant is doing? Am I on the wrong track?

Either all cognition is cognition of appearance, in which case there can be no cognition of noumena, or there can be cognition of the noumenon, in which case cognition is not essentially cognition of appearance — KantDane21

Is he, for instance, arguing that cognition is essentially cognition of appearance? How could he be, since one of the disjuncts says that it isn't?

As I said, we could do this up more completely with predicates and quantifiers, we could even throw in some modal operators to cover 'essentially', but I think all that's overkill. It's a simple passage, and @KantDane21 was just a little confused about its basic logical form.

Now if we wanted to try to handle @RussellA's suggestion that there is 'cognition of x as F', and that's what's at stake here, that looks like the sort of thing classical logic is really crappy at. There's no 'as' operator because that suggests predicate interpretation, which suggests fiddling with the domain in the middle of an argument, which I guess you can do without going all dialethic, but what's the point? (I'm thinking of simple stuff like 'x is short for a basketball player but tall for a man'. You can give a semantics for 'short' and 'tall' relative to a population and thus allow someone to be short in one sense but tall in another, if that's something you really need.)

Another route would be to note that Kant is apparently making a point about cognition, rather than about which objects fall into the class 'noumenon' and which 'appearance'. So we could instead define type of cognition (again, awkward for FOL but doable), something like this:

If x is a cognizing, then x is an a-cognizing (that is, cognizing something as an appearance).

We could, again, do it all up with quantifiers, but in essence all we want to say is this:

Either all cognizing is a-cognizing, or some cognizing is n-cognizing.

And the intent there is to leave room for the same 'object' to be a-cognized and n-cognized. But it is still implied here that a-cognizing and n-cognizing are disjunct 'mental events' or types of cognizing.

And again I take it that the point of passages like this is just to clarify what the denial of the conditional would amount to logically, what we would need to show as a counterexample.

A = x is cognizable, B = x is a noumenon (and ~B = x is an appearance)

1. (A → ~B) v (A & B)
2. (A → ~B) v ~(~A v ~B)
3. (A → ~B) v ~(A → ~B)
True

Not sure what everyone else is doing, or what your P and Q lead to, but the argument is obviously right and I think this is the simplest way to formalize it. (We could do quantifiers and stuff, but it's really simple.)

My way is cleaner than yours because my sentential variables are really pretend predicates, just leaving off the variables. That means we don't stuff quantifiers (like 'all') into the variables.

The main thing is to render 'All F are G' as 'F → G'; that is, read it as 'If something is an F, then it's a G.'

The only other rules used were de Morgan's law to get from (1) to (2), and then the equivalence of P → Q to ~P v Q to get from (2) to (3). Or you can just recognize that if something is both A and B, that means being A doesn't imply not being B.

And then we're done, because (3) is a tautology, so (1) is valid.

2.6 is the section about existence, and it's only a couple pages. I quoted some bits.

One minor point maybe worth going back to is his use of the word 'specifically', also used in our section when recalling 2.6, when he denies that external existence can be taken as something 'specifically different' from our perceptions. I haven't checked, but I strongly suspect that word here means 'different in kind', 'specific' from 'species'. That's how I read his point, but I don't think I mentioned I was relying in part on this particular word.

So the argument begins

(1) Existence, external or not, is not something we have any separate conception of.

But then we get the other part:

(2) The notion of external existence cannot be taken as specifically different from perception.

That sounds like it's telling us what sort of thing something that doesn't exist is.

I think that means (1) is at least a little misleading as I've phrased it.

(2) is part of the larger claim that it's only perceptions in our mind, nothing else.

So if we have an idea of existence, it's that sort of thing, a perception; but (1) had already shown that existence itself is not a perception, but just something part of every perception, so in a way nothing.

(Some of the confusion here is just rhetorical. It's a pretty common move to say something doesn't exist, at least not as the kind of thing you think it is, not if we take it to be what you think it is, but it does exist, just as something completely different. We're keeping the name, but changing the meaning. Like that.)

But there is something else going on here, because Hume says our idea of external existence turns out to be different relations, connections, and durations that we *attribute* to perceptions.

So these are ideas *about* perceptions.

Thus instead of thinking some idea we have is of an object that we also think exists, we will have the perception or idea of the object and then *attribute* to that perception the properties that will be refuted in 4.2, distinct existence and continuity over time.

But attributing, like relating or associating -- these don't sound like perceptions but ways of handling or working with or acting upon perceptions. We can, in addition, have ideas about what we're doing when do this sort of thing, and Hume bundles some of these mental behaviors together and calls them our notion of external existence.

The status of these mental behaviors might be clearer if we look back at Part I where most of the basic machinery is laid out.

On the whole, I find your reading of him to be quite accurate — Manuel

I'm leaning on his actual words too much though.

I'm not absolutely certain I've gotten to the bottom of the arguments in 2.6

I should be clearer. I was only asking if my reading of Hume was plausible, not whether what he says was.

I do want to evaluate the arguments as we come to them, but only once we know what they are!