What was the question? — Banno

" My understanding is that "classes" can include rules, but I don't understand how to do that formally while I do understand naive set theory at least. "

I suppose I was looking for reassurance of this distinction between sets and classes -- either in the right or wrong way. I've thought of sets as any collection of individuals whatsoever and classes as collections of individuals with inclusion rules. Is that bollox in my head that I'd need to defend or let go of, or something sensible from your perspective?

Why can't a car have 3 wheels and why wouldn't a broken car still be a car? — Hanover

Why wouldn't a car of any state or component missing be anything but a "piece of metal?" Because a knife is a piece of metal. You can't expect someone to buy or barter for a "piece of metal" without some sot of deeper and thorough designation, could you?

This is interesting, really. Is a pure metal shell of a car with no furnishing, engine, or internal infrastructure a car? Average person would say no (or would they?). Is a car with all those things but that doesn't start or function a car? Average person would say yes and of course call that a "broken car" or a "lemon" or a "clunker," But it's interesting because while one is considered a car that fails to perform the function of a car (yet can be made or altered to do so) the same is true of the shell of the car without any other parts. So explain that, eh?

A set is a single object. Elements are multiple objects. So a set is not identical to its elements. — litewave

A bit of care is needed here. A set is identical to its elements, and nothing more. No box. I hope we agree on that. So we can write that the set S = {a,b,c}; and say that S is identical to {a, b, c}; and by that we would mean that where we write "S" we might instead write {a, b, c}, and vice versa.

"object" is ontologically loaded. I'd include "property" there.

A set is a collection of individuals. They need not have anything related to one another, or share anything at all -- the individuals are the set and there's nothing else to it. The pebble on the ground and the sentence I say 5 miles away can form a set. — Moliere

Yes. So what, if anything, would we want to say about identifying such a set with some property? I take it you don't want "being in set X" to count as a property -- nor could it, on the OP's proposal.

Yes. So what, if anything, would we want to say about identifying such a set with some property? I take it you don't want "being in set X" to count as a property -- nor could it, on the OP's proposal. — J

I'm open to 'being in set X' because I think Russell's paradox is legitimate, and generally I like the paradoxes of self-reference as a point of thought -- stuff like the liar's paradox seem to sit here.

But, yes, it could not count on the OP's proposal which is why the paradoxes of self-reference came to mind.

I don't want to say anything about identifying a set with a property for this very reason :D

"Classes" is a more specialised term. If we stick to sets, we can start with individuals - a,b,c; and form sets of these - {a,b}, {b,c} - and then sets of these - {{a,b},{b,c}} - but avoid mixing them - {a, {a, b}}. Then we have a hierarchy that avoids Russell's little conundrum.

:up: Thanks.

That's all very rough, of course. More detail can be found at The IterativeConception

I mean that the property is the set.

Gotcha. I was thinking of "identifying" (verb), which made me think of the epistemological questions.

There may not be a universally agreed specification of justice, so different people may identify justice with different sets of acts. It's easier with redness, which can be specified with reference to a certain range of wavelengths of light, although the exact boundaries of this range may not be universally agreed either.

But doesn't this mean that there would be many different versions of the same property? So there would really be "justice(Tom), justice(Greg), justice(Sandra), etc. Given properties' roles in metaphysics, that seems problematic, particularly since the stipulation re modality would seem to indicate that we should be considering "everything that might possibly be considered x." A pretty popular idea in contemporary analytic metaphysics is the "bundle/pin cushion" view where things just are their properties (plus or less a sort of "bare particular substratum" or haecceity). Yet we seem to have opened the door on there potentially being as many properties as there are (potential) opinions. Wouldn't this risk making everything into everything else? This is why I personally prefer an intensional explanation of properties.

Even on deflationary accounts, this still seems like it could present problems because it would mean we are guilty of equivocation any time a different set is specified (and how, in practice, could we even determine that different sets had been specified?).

A similar sort of issue occurs with due to modality. If we're consequentialists, presumably any act might be "just" or "good" as well as "unjust" or "evil" given the right context. Does that mean good and evil represent the same set and thus the same thing? Likewise, wouldn't the set of contingent falsehoods end up being the same as contingent truths? But then are truth and falsehood the same thing unless they are necessary?

Also, what about the property of "being a property?"

, I supose that answers your question? THere is a difference between why we count an animal as having a heart, and that we cont an animal as having a heart.

Now I do not think that there is general answer to the question of why we group some things together. And I think Thomistic talk of essences tries to paper over that difference, by pretending that it's essence all the way down, while never quite telling us what an essence is.

Added: case in point, as it seems.

I supose that ↪this answers your question? — Banno

Yes, you and @litewave both crossed posts with me. But I still have questions, above, about the identification of property with set, for litewave to consider.

Now I do not think that there is general answer to the question of why we group some things together. — Banno

Right. The "bleen people" group as they do (choosing bizarre intersections of "green" and "blue"), and while they are doing something we find impractical and hard to parse, they may have their reasons.

But doesn't this mean that there would be many different versions of the same property? So there would really be "justice(Tom), justice(Greg), justice(Sandra), etc. — Count Timothy von Icarus

Yes. I know you probably don't care for that conclusion, but I think it's exactly what happens. There are indeed different construals and attempts at definition for an abstraction like "justice." But, to anticipate your objection, that doesn't mean that anything goes, that some nonsense from Tom deserves to be taken as seriously as "justice(Rawls)." The fact that we cannot define something doesn't mean we can't know anything about it, or can't tell a promising clarification or interpretation from one that isn't. Look at the Republic. Justice never gets a satisfactory definition, but it would be hard to read the book carefully and not believe you've learned something about the subject.

But I still have questions, above, about the identification of property with set, for litewave to consider. — J

Yep. The answer might be to drop the notion of "property", which is somewhat anachronistic anyway. It reifies a semantic difference.

Yep.

The more obvious objection, to my mind, ties into the modal caveat in the OP:

This problem can be fixed by clarifying that a property is the set of not only its presently existing instances but also of its past and future instances and of all its possible instances (existing in possible worlds)

Isn't it possible that people might consider properties all sorts of ridiculous ways? I don't see a mechanism here for dismissing Tom's opinion on the grounds that it is "nonsense" when we have already opened things up to every possible set configuration. Yet this would seem to make "everything to be everything else."

I don't think the "opinion based flexibility" works with the modal expansion. And something like "all possible opinions that aren't 'nonsense,'" seems to ignore that there are many possible opinions about what constitutes "nonsense." This is made more acute by the modal expansion, but I would say it applies just as well for what you've said, since there is the question: "who decides what is nonsense?"

Look at the Republic. Justice never gets a satisfactory definition, but it would be hard to read the book carefully and not believe you've learned something about the subject.

Sure, but lacking a definition seems to me to be much different issue. On this account, we don't have many different claims about what justice is, but many different justices. It's a positive metaphysical claim to say that justice just is the set of things each individual considers to be just. So, we'd be justifying a positive metaphysical claim using our own profession of ignorance.

I'm inclined to agree that defining "properties" in such broad terms is fraught with difficulties. An answer might be to drop "property" rather than extensionality.

Unless, perhaps, you can offer a definition of "property" that we might use? I suspect this will bring us back to your circular definitions of "essence".

If there are no properties, in virtue of what would some things be members of "the set of red things" but not others?

Or in virtue of what would different individual things he discernible?

If there are no properties, in virtue of what would some things be members of "the set of red things" but not others? — Count Timothy von Icarus

Yep.

What's your answer? That red things are exactly those that have the property "red"?

And you think this helpful?

If I ask you what it is to have the property of red, will you say that it is to have redness?

Do you like merry-go-rounds?

A set is identical to its elements, and nothing more. No box. I hope we agree on that. So we can write that the set S = {a,b,c}; and say that S is identical to {a, b, c}; and by that we would mean that where we write "S" we might instead write {a, b, c}, and vice versa. — Banno

I really don't think that a set is identical to its elements. A single object is not identical to many objects. We can write that the set S = {a,b,c} but we cannot write that the set S = a,b,c. Also, in set theory, the set {{a,b,c}} is different than the set {a,b,c}; the first one is a set with one element, while the second one is a set with three elements. I know that in our everyday life this distinction is unimportant because for example a set of three apples weighs the same as the total of the weights of the individual three apples and so the set in itself doesn't add any additional weight to the weights of the apples. Well, that's how our particular world works - forces like gravity act on elementary particles and can be added up. But in a different possible world a force might act only on certain sets and not on their elements; you might then get a set of three apples that weighs a pound while each apple alone is weightless.

Maybe the distinction between a set and its elements is important for the emergence of consciousness from unconscious parts - the whole is conscious while its parts are not, because the whole has an additional property to the total of the properties of its parts.

I really don't think that a set is identical to its elements. — litewave

Sure. We both need to keep track of what is being said here. We are talking at cross purposes.

"Identical" is defined extensionally by substitution. I hope we agree that there is nothing more to the set {a, b, c} than a and b and c, no additional "setness" in the way @RussellA supposed by adding his box.

If redness is all things that are red in all possible worlds, then that set is infinite as is the set of of all things we're not sure are red. If there is infinitely uncertainty as to redness, then what value is our redness set in telling us what is red?

Mine might just be a vagueness objection that implicates infinite vagueness, but isn't the purpose of the extensionalism exercise to eliminate just that?

But doesn't this mean that there would be many different versions of the same property? So there would really be "justice(Tom), justice(Greg), justice(Sandra), etc.). — Count Timothy von Icarus

Different versions of the same property are actually different properties (although they are similar in some way significantly enough to call them "versions"). Tom may call property X "justice" while Greg may call property Y "justice". Properties X and Y are objective parts of reality and we can all agree what are their instances, but we may not agree which of the properties should be called "justice".

If redness is all things that are red in all possible worlds, then that set is infinite as is the set of of all things we're not sure are red. If there is infinitely uncertainty as to redness, then what value is our redness set in telling us what is red? — Hanover

Why would being infinite make it uncertain? There are infinite odd numbers, but no uncertainty here. Infinity does not lead automatically to vagueness.

Different versions of the same property are actually different properties — litewave

Yep. {a, b, c} is different to {a, b, d}. It would only amount to equivocation if we were to say that they were the same. Tim's objection is unclear.

"Identical" is defined extensionally by substitution. I hope we agree that there is nothing more to the set {a, b, c} than a and b and c, no additional "setness" in the way RussellA supposed by adding his box. — Banno

When we identify some thing extensionally/by substitution, it doesn't mean that we identify the thing with its extension. It means that we identify the thing in relation to other things. For example, we can identify a set extensionally in relation to its elements, which are different things than the set itself. At least that's how I understand extension because I don't think that one thing can be identical to many things.

I don't disagree with that, with some caution. So {a,b,c}= {a,c,b}. The care is that {a,b,c} is not other than [a,c,b}. So some caution with "It means that we identify the thing in relation to other things".

So again, when we say a set is identical to it's elements, we just mean that for example S = {a,b,c} and (extensionally) that where we can speak of S we can also speak of {a,b,c} and substitute one for the other.

I don't think we have a substantive difference in our opinions here.

Order of elements of a set doesn't matter, I agree. But I think it is important to emphasize the identity of a set as a single thing, distinct from its elements, because I propose to identify a set with a property, which is supposed to be a single thing too rather than multiple things.

Why would being infinite make it uncertain? There are infinite odd numbers, but no uncertainty here. Infinity does not lead automatically to vagueness. — Banno

That's an interesting point, but doesn't this reference a distinction in categories between analytically defined and empirically defined?

If sorting infinite root vegetables, some will be rutabegas, some we're not sure, and some will be Swedes.

But every other integer after 1 is odd no matter how high we count.

So, the rutebega set is infinite across all possible worlds as is the not-sure-if-rutebega set, but we have zero not-sure-if-odd set.

That is, I feel your odd number counter example was not applicable. It's of a different sort.

I also think my rutebega/Swede distinction raises another sort of problem along the lines of your kidney talk.

What's your answer? That red things are exactly those that have the property "red"?

I think it's pretty easy to identify red things. Color can be explained in various ways. Likewise, triangular things are those things with three sides, etc.

But even if we were left with properties as some sort of inexplicable metaphysical primitive, that still seems better than "nothing has the property of being triangular" which would seem to imply that nothing is triangular. Or more broadly, "nothing has any properties," which seems absurd. One will just end up reinventing the basic idea of a property under another name in order to say that anything is anything.

But I think it is important to emphase the identity of a set as a single thing, distinct from its elements, — litewave

I don't see what "distinct from" does here. S is different from a, but is it different from a, b and c? Extensionally, no.

Perhaps you are trying to capture the unity of the set. I'd see that as what we do in deciding to talk about a, b, and c together, rather than something in addition to a, b, and c.

What I want to be clear about is that there is no "box" - those curly brackets mark the set but do not add something to it like the box would.

I'll leave you to it.