Having the property red is not the same as the property red. — frank

Hm yes, the problem will be in the property of being red, which I equated with these two properties. It seems ok to equate being red with having the property red. But being red should not be equated with redness when redness is meant as general redness while being red is meant as particular redness.

Anyway, if there are genuinely different properties that have the same set of instances (for example, properties like general redness and particular redness), then my OP proposal of identifying a property with the set of its instances fails.

Membership in the red set entails having red as a property. Entailment doesn't get you to identity, though. Or if so, how? — frank

This way:

For example, let's take property red or redness (X = red): The property of "being in set red" is the same as the property of "having property red", which is the same as the property of "being red", which is the same as property red. So, the property of "being in set red" and property red are one and the same property. — litewave

So if I say the peony is red, I mean it's in the set of all red things. So did we change from the set is the property to being in the set is the property? — frank

These two properties have exactly the same instances and if I got it right, they are one and the same property, just described differently.

This is interesting but confusing. Is "Being in that set means having that property" different from "'Being in that set' is a property of the pebble"? I thought we didn't want set membership to count as a property. — J

On further thought, I find this confusing too. The property of "being in set X" may seem to be the property of members of set X, but perhaps it is actually the property of the set membership relation instead. (A member is in set X but "being in set X" is not the property of the member but of the set membership relation between the member and set X.) Similarly, the property of "having property X" may seem to be the property of instances of property X, but perhaps it is actually the property of the instantiation relation instead. (An instance has property X but "having property X" is not the property of the instance but of the instantiation relation between the instance and property X.)

Since I equate set with property, members of set are equated with instances of property, and set membership relation is equated with instantiation relation. — litewave

My reply above was a groaner, wasn't it. Perhaps the property of "being in set X" could be interpreted as a property of the set membership relation but it is clearly a property of elements of set X, first and foremost.

So, given that I propose identifying property X with the set of instances of property X, it seems that the elements of set X share two properties: property X and the property of "being in set X". And these two properties have the same extension - all elements of set X, so they are coextensive properties. However, I think that these two properties are not really different; they are one and the same property, just described differently. The property of "being in set X" is the same as the property of "having property X", which is the same as the property of "being X", which is the same as property X. So, the property of "being in set X" and property X are one and the same property.

For example, let's take property red or redness (X = red): The property of "being in set red" is the same as the property of "having property red", which is the same as the property of "being red", which is the same as property red. So, the property of "being in set red" and property red are one and the same property.

Can we agree that only one possible world actually exists (the actual world)?

In that case, your set includes "things" that do not exist, never have existed, and never will exist (they are non-actual possibilities). Let's focus on this subset of your big set. Does it have any members? Are the members things? If so, what is a thing? — Relativist

If only the actual world exists, then a property has instances only in the actual world, and the property is still a set of its instances (but the instances exist only in the actual world). The instances of a property are whatever has the property.

This is interesting but confusing. Is "Being in that set means having that property" different from "'Being in that set' is a property of the pebble"? I thought we didn't want set membership to count as a property. — J

On further thought, I find this confusing too. The property of "being in set X" may seem to be the property of members of set X, but perhaps it is actually the property of the set membership relation instead. (A member is in set X but "being in set X" is not the property of the member but of the set membership relation between the member and set X.) Similarly, the property of "having property X" may seem to be the property of instances of property X, but perhaps it is actually the property of the instantiation relation instead. (An instance has property X but "having property X" is not the property of the instance but of the instantiation relation between the instance and property X.)

Since I equate set with property, members of set are equated with instances of property, and set membership relation is equated with instantiation relation.

So you're saying that having a property is a matter of being a member of the set of all things that have that property. That's trivially true. — frank

The point of my OP is that the set actually is the property. That may not be obvious.

So the peony has the set of all red things. How does it have that set? — frank

By being an element of the set, thus having what all the other elements of the set have.

. The property of redness is the set of all red things.
2. A peony has the property of redness.
3. A peony has the set of all red things.

Help me out here. That doesn't make sense. — frank

It sounds weird if when you think of the set you think of all the red things. It makes you think that the peony somehow has all the red things, which is absurd. But the set is not all the red things. It is something else, which all the red things have in common.

So in a way what I'm asking here is to say "How does this notion of unification fit within a strict logical definition?" — Moliere

Well, in predicate logic you have individuals that have/satisfy a property/predicate. I propose that the property is the set of these individuals.

A red ball has the property of redness. A red ball is not the property of redness, though. They're two different things, so it's hard to see how a collection of red things would be equivalent to redness. — frank

Because a collection is something different than its elements, yet it is also something that is common to the elements.

"unification" -- I'd say this is an extra-logical notion. We may posit the set consisting of ununified elements, for instance -- is this then not a set because the elements are ununified? Is it possible to posit such a set? — Moliere

Well, I'm trying to describe the concept of set in some intuitive terms. You may say that the concept of set is extra-logical but I wouldn't be able to make sense of logic without it. Like, why are the conclusions in syllogisms necessarily true if the premises are true?

The set is an object that somehow unifies different objects without negating their different identities. One over many.

Red has the property of redness? That doesn't sound right. — frank

I said instances of redness have the property of redness. The property of redness itself doesn't seem to be red, hence it doesn't instantiate itself.

There is a branch of mathematics that deals with these kinds of issues, called fuzzy logic, as there's certainly nothing stopping us trying to make rigorous treatments of our pretty vague concepts about the real world, which I haven't looked into all that closely but maybe of interest to you. — boethius

Ok, I haven't studied fuzzy logic, it may be a useful way of dealing with uncertainties, but ontologically I regard every set as completely specified, just like in set theory.

Of course, doesn't stop us talking about a set of red things, and that can be useful to do, but if you want a rigorous definition you'd need to solve all these problems; otherwise, the definition becomes the set of red things which I will decide on a case by case basis as I get to them to resolve all edge cases in a way I'm confident won't result in any contradictions whatsoever; which is not how a set is usually defined in formal logic. — boethius

We can agree on many things that should be included as elements in the set of red things (instances of redness), for example ripe tomatoes and their various parts, pools of blood and their various parts etc. If you also include individual atoms, so be it - that will be how you specify the property of redness and thus the set of its instances. If you don't include individual atoms, so be it - you will specify a somewhat different property and set and you will call it redness. Properties and sets objectively exist as completely specified but what you call them is your choice.

Am I right in agreeing with you that the property of redness is the set of all red things? — RussellA

That's what I am saying the property of redness is - the set of all red things (the set of all instances of redness).

What is this extra "unspoken property" doing for us in understanding what a set is? — Moliere

The extra property (the set) is a thing that is the result of the unification of the elements into one thing (while keeping the elements distinct from each other), which is thereby shared by the elements.

It's just seems like you're mixing categories if you say redness is the set of red things. It's closer the set of all shades and hues of red. — frank

Shades and hues of red are instances of redness, so they all have the property of redness.

That does not make it invalid to talk about sets of "everything red" for example, but we can know ahead of time that such a concept cannot be developed into something rigorous without axiomatization. — boethius

When I say that a property is identical to the set of all objects that have this property, I mean that the property is completely specified and thus the set is completely specified. In practice we usually don't have such complete specifications and we talk about approximately specified properties like "redness", but that doesn't refute my claim that a property (completely specified) is identical to the set of all objects that have this property.

Does this not mean that saying the box can only be black if it contains instances of blackness violates the Zermelo-Frankei set theory, in that the singleton set must be distinct from the element it contains? — RussellA

Aha, I think I see what you mean. The singleton set is distinct from the element it contains and so it is something additional to the element. The element is red and the singleton set is something else (though probably not black, because that would require some more complex structure that can absorb light). I propose that the set of all red objects is the property "redness" but this property probably does not look red, in fact it probably does not look like anything that could be visualized because it is not an object that is contiguous in space or time.

Properties, qualities, characteristics, and so on, are mental or linguistic abstractions of the things described, or even the descriptions themselves. Your morphological derivations “redness” and “carness” indicate this. They are derivations, not sets or properties. — NOS4A2

Collections or sets are not just mental or linguistic abstractions though. As I am typing this I am actually holding a collection called "smartphone".

First point. ↪RussellA
might be understood as saying that in addition to the set consisting of {book, car, apple} there is a fourth item, grouping these together, the box the set comes in, as it where. That's not right. There is nothing in addition to the elements. — Banno

But that's exactly what I am arguing - there is a fourth object and this fourth object is identical to the set of the three objects. The set as a single object cannot be identical to three objects, so it is identical to a fourth object. The identity of the fourth object is fixed by the three objects because there can only be one set of the three objects, but the set itself is not identical to the three objects.

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. — J

Even the extravagant set that @Moliere has mentioned above is something in addition to the pebble and the sentence, and this something is a property that the pebble and the sentence share. It is an unimportant property for which we have no word, and being in that set means having that property.

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.

"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.

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".

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.

Russel's "box" metaphor doesn't work becasue a set just is it's elements. — Banno

A set is a single object. Elements are multiple objects. So a set is not identical to its elements. Even in set theory, a set is an object in its own right: if set A is an element of set B, it doesn't mean that elements of set A are elements of set B.

A fine piece of work. Nice anticipation of objections, especially the modal objection. Are you studying logic? — Banno

Thanks. Not on an academic level.

Here's perhaps the classic reply. Having a kidney is not the very same as having a heart, and yet all animals with kidney also have hearts. We can say that the extension of "Having kidneys" and the extension of "Having a heart" are the very same. — Banno

As long as it is possible (logically consistent) for an organism to have a heart without a kidney, or vice versa, then the set of all possible instances of having a heart is different than the set of all possible instances of having a kidney, and thus these two properties are differentiated. You don't even need to conceive of some extraterrestrial organism, just take some ordinary animal whose kidneys have been removed while the heart stayed in place. However, as I mentioned in OP, if two properties are necessarily coextensive, for example "equilateral triangle" and "equiangular triangle", then they are one and the same property, just described in different words.

So when you say everything must have a referent, you're speaking modally, meaning it has a hypothetical referent in a possible world? I didn't get that from your OP. — Hanover

Yes, all instances of a property, in all possible worlds, constitute the set that I indentify with the property.

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

It can. And with such a specification it would be a different property, identical with a different set.

There are properties that exist that are not of a referent, like the property of being the King of France attaches to no object, yet being the King of France is a property nontheless. — Hanover

Is it possible (logically consistent) for the property of being the king of France to be instantiated? If yes, then it is instantiated in some possible world. If not, then it would be self-contradictory.

There are also no essences of objects that would dictate which set all examples belong, like whether a particular car belongs in the set of cars is contextually dependent. — Hanover

It depends on how "car" is specified. Usually it is specified as "self-propelled vehicle on four wheels". In that case, the property of being a car is the set of all self-propelled vehicles on four wheels.

What exactly do you mean by "identify" here? — Count Timothy von Icarus

I mean that the property is the set. But knowing only that justice is the set of all just acts will not help you know which acts belong to this set or specify what justice means. 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.

A set is a different object than any of its elements. But if the box is black then it also contains instances of blackness, not just redness. For example the walls of the box may be black. Your example looks like the property of redness contained in a black box.

Numbers are properties (universals/general entities) too, so defining a number n as a collection of all things that instantiate the number n would be another example of identifying a property with a set (collection). However, a thing need not instantiate a number n only by having n elements. For example, here is a set that has only 1 element but it instantiates number 2 by having 2 embeddings of elements:

{ { { } } }

This is actually how Zermelo defined number 2. Von Neumann defined number 2 as the following set of 2 elements:

{ { }, { { } } }

Both von Neumann's and Zermelo's definitions of number 2 are actually just particular instances of number 2.

Frege and Russell's definition of a natural number n as the collection of all sets with n elements is closer to capturing the general/universal nature of number n, but a complete definition of a natural number n would be a collection of all instances of number n. The property that Frege and Russell defined is not "number n" but "set of n elements" or "set of cardinality n".

Consciousness is a weird thing. I wouldn't be so surprised if it experienced a static structure as moving, especially if the structure is a smooth sequence. As the ontologist Dua Lipa sings, "Illusion, I really like the way you're movin".

If fundamental reality wasn't inherently consistent, life couldn't exist. — RussellA

Which means that life would exist, but it wouldn't.

One cookie that first was part of the letter E, suddenly becomes part of the letter F. (And later becomes a period according to T Clark). It is clear that that only happens in our mind. — Carlo Roosen

Why would it only happen in our mind? The cookie is out there, it is a part of collection E, then a part of collection F. Collections E and F are out there too.

Fundamental reality must be the particles, the cookies AND the letters, somehow. — Carlo Roosen

In other words, reality consists of collections of collections of collections... Welcome to set theory, the instantiation of abstract mathematics in concrete forms. Every set (collection) is an object in its own right, not identical to any of its elements but something in addition to them.

It is logically possible for there to be a present King of France. That is, in some possible world there is a Present King of France. — Banno

It may be logically possible in some possible world but not in ours.

The trouble with "The present King of France is bald" is that given there isn't a present King of France, It's unclear what truth value the sentence has. If there is no present King of France, then he is neither bald nor hirsute. — Banno

Since "the present King of France" has a logically inconsistent definition (in our world), it is not an entity but nothing. Nothing has no properties, so I would say that "The present King of France is bald" is false because it attributes the property of baldness to nothing. I am not sure about negative properties though. "The present King of France is NOT bald" rephrased as "Nothing is NOT bald" seems true because it denies the property of baldness to nothing. But does that mean that nothing has a negative property of "non-baldness"?

There's nothing logically inconsistent about the present King of France, no contradiction that follows from the very idea. — Banno

It would be logically inconsistent for an entity to exist at a place and time where it doesn't exist. The present king of France doesn't exist on our planet, therefore it would be logically inconsistent for him to exist on our planet.