Identification of properties with sets

litewave

I am proposing that we could plausibly identify a property with the set of all things that have this property. This set would be the property, and the elements of this set would be the instances of the property. For example, the property of redness would be identified with the set of all red things, or the property of being a car would be identified with the set of all cars.

At first, one might find such an identification counterintuitive because redness and carness (being a car) seem like something singular that is shared by its instances rather than a multiplicity that encompasses the instances. But a set is a single thing too and its elements can be said to participate in or share the character of this thing.

Another objection might be that visually a set of red things just doesn’t look like redness or a set of cars just doesn’t look like a car. But what does redness or carness look like? Whenever we try to visualize redness as something that is red or carness as a car we always visualize a particular instance of redness or carness, not the redness itself or the carness itself. Redness and carness seem to elude visualization because they are somehow dispersed across space, time and possible worlds.

Yet another objection might be that a property may be instantiated not only in things existing now but also in things that existed in the past or will exist in the future and that a property would exist even if it is never instantiated. 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). There cannot be a property that is never instantiated because it must be at least possible for a property to be instantiated (a property that is impossible to instantiate would contradict the very notion of property as something that can be possessed) and if it is possible for a property to be instantiated then its instance exists in some possible world. Not sure if this requires modal realism (modal fictions might be regarded as elements of a set too).

Finally, there may be a concern that necessarily coextensive properties, for example “equilateral triangle” and “equiangular triangle”, collapse into one property because their instances constitute one and the same set. However, I would say that such properties are indeed one and the same property, just described in different words.

RussellA

I am proposing that we could plausibly identify a property with the set of all things that have this property. — litewave

This seems a similar concept to using set theory to define the natural numbers.

Frege and Russell proposed defining a natural number n as the collection of all sets with n elements.
Wikipedia - Set-theoretic definition of natural numbers

litewave

↪RussellA

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".

RussellA

I am proposing that we could plausibly identify a property with the set of all things that have this property. — litewave

I am just wondering:

Suppose only three things in the world have the property of redness.

Consider the set {red car, red apple, red book}

It seems that a set is not the same thing as the elements within the set, as a box is not the same as the things inside the box (an analogy given by Copilot). For example, even though all the things inside the box are red, the box itself could be black.

So to say that we can identify a property with the set of all things that have this property may be like saying we can identify the property of redness with something that is black.

Is this valid?

litewave

↪RussellA

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.

Count Timothy von Icarus

↪litewave

I am proposing that we could plausibly identify a property with the set of all things that have this property.

What exactly do you mean by "identify" here? I suppose one issue might be circularity. How do you know what belongs in a set? Suppose we take "justice" or "just acts." In virtue of what are all "just acts" properly members? If we identify the property of "being just" as simply "being a member of the set of 'just acts'" we don't seem to get anywhere.

Further, even if I had the set of all past just acts to refer to, how exactly does this help me determine which acts would be just in the future, or to identify new just acts? If all I know about justice is that "just acts belong to the set of just acts," then future identification seems impossible. I would have to instead look at the set and reason out a sort of definition, or at least an observable pattern, by which to identify future or potential members. But in this case, it seems to me that it is really the essence of justice, signified by the definition, that is actually what "membership is in virtue of."

litewave

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.

Hanover

I am proposing that we could plausibly identify a property with the set of all things that have this property. This set would be the property, and the elements of this set would be the instances of the property. For example, the property of redness would be identified with the set of all red things, or the property of being a car would be identified with the set of all cars. — litewave

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. 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.

Searle, Wittgenstein.

litewave

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.

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. — litewave

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.

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. — litewave

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

Hanover

It depends on how "car" is specified. — litewave

Maybe replace "specified" with "used." Otherwise, you just have a purely prescriptive language, and not one that really exists.

litewave

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.

Banno

↪litewave

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

There are problems, though. Perhaps not the ones identified by Hanover and Tim.

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.

What we can say is that the extension of a property is just those things to which the property applies,

You are right to puzzle over the notion of a "property".

Banno

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

Identity can be defined extensionally using substitution, and without circularity. That's how it is done in modern logic.

Banno

↪litewave

Russel's "box" metaphor doesn't work becasue a set just is it's elements. Asking if redness is red is a category error.

Moliere

↪Banno

That's my understanding, at least given Russell's paradox. (which the OP reminds me of)

It feels plausible that a set can be identified by a property or even a set of properties.

But, no, they're different -- a set is its' elements, rather than a property which all the elements share.

Banno

↪Moliere

"Property" is a problematic notion for our Thomist friends, who puzzle over "what properties really are". Less so for more recent logic, which recognises that a property is a single-place predication, and not all that special. The Thomist might ask "But what IS redness, really?" and that whole structure of essences that ensues, while the modern logician might respond "It's just what 'x is red' expresses - nothing more mysterious than that."

That might well be what

↪litewave

is leaning in to.

Moliere

↪Banno

I've expressed it as "a set is a collection of objects -- where objects are logical objects (any name whatsoever) -- that need not share anything in common other than being in that collection of objects"

Not sure how right I am as I still think on these things.

↪litewave

Good OP!

Identity can be defined extensionally using substitution, and without circularity. That's how it is done in modern logic. — Banno

This -- and the earlier queries of @Count Timothy von Icarus and @Hanover -- is where my attention is drawn as well.

The extension, let's say, of all beings with kidneys is the same as that of all beings with hearts. (I don't know if that's true, but no matter). We can also say, The extension of all triangles with equal angles is the same as that of all triangles with equal sides. But if we were to decide, as the OP recommends, that we have identified the properties of having a heart and being equilateral by pointing to their sets, how do we deal with the problem that the reason why the respective sets are co-extensive is different in each case?

In the case of the triangles, it's not implausible, as you point out, to declare that being equilateral and being equiangular are two ways of describing (or should it be "naming"?) the same property. That's because the two descriptors are logical equivalents -- to assert the first is to assert the second, a priori. But hearts and kidneys are different. A posteriori, it turns out that there are biological reasons (again, as we're supposing) for beings with kidneys to have hearts, but that is not a conceptual or logical equivalence.

Does this matter, on the question of whether we're zeroing in on a property, in each case? I'm not sure, because I'm not sure how you understand "property". Are you recommending a new use for the term "property," or a new, improved analysis of what "property" has always meant?

Also, @Count Timothy von Icarus's query is a fair one: Yes, we can define identity in the usual way of modern logic, but the OP is asking us to stretch. Is that kind of identity really intended to be the same? Your answer was: "I mean that the property is the set." That "is" deserves expansion. If all you mean is what @Banno means -- the "is" of logical identity -- fair enough, but we want to be sure.

Banno

↪Moliere

Yep, although "objects" is ontologically loaded - I'd use "individuals". And an individual is just what we give proper names to - "a", "b", "c" or "Moliere", Banno"

litewave

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.

I've expressed it as "a set is a collection of objects -- where objects are logical objects (any name whatsoever) -- that need not share anything in common other than being in that collection of objects" — Moliere

Would that mean that "being in that collection of objects [or individuals, per @Banno]" is a shared property? Can an object "wander in," so to speak, and partake of that property? This may not be a question about your definition so much as an expression of uncertainty about "property".

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. — litewave

I think this replies, with a cross-post, to part of my question, thanks.

Moliere

↪J

I think

↪Banno

covered this well in that "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

Relations are different from sets in that they are somehow connecting one set to another, but sets have no "rules" for inclusion.

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.

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. — litewave

Yes, you are basically correct.

Lets take a closer look at what might happen were we to come across an organism with what appeared to be kidneys and yet no heart. We have a choice. We can decide that there are organisms with kidneys and yet no hearts, and say that the extension of "animal with kidneys" is different to the extension of "animal with a heart". Or alternately we could maintain that all animals with hearts also have kidneys, and simply say that although these organs do much the same thing as kidneys do in animals with hearts, they do not count as kidneys.

Now kidneys filter blood, which requires a circulatory system, which typically requires a heart. So the latter is probably the biologist's best option.

The Thomists amongst us see this as somehow nominalist and arbitrary. Btu that's how words work.

litewave

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.

Moliere

A set is a single object. — litewave

That single object is the collection, but the thought here is that there's nothing more to that than being the collection of the individuals in the set.

We can name a set, so it can be an element -- and is an element of the set that is itself -- but a set need not be a single object (or name) at all. The empty set comes to mind here.

Banno

sets have no "rules" for inclusion. — Moliere

Best acknowledge early that sets are best formed in stages, so as to avoid Russel's paradox. Start with individuals, then sets of individuals, then sets of sets of individuals, and never the twain.

Moliere

↪Banno

Does that avoid Russel's paradox?

If so, does it do so by delaying the question? :D

I'm good with forming sets in stages either way. Defining sets in a technical manner is something I still think on and think I don't understand, really.

Banno

Does that avoid Russel's paradox? — Moliere

In a very rough form its how Z works, so yes.

What was the question?

Identification of properties with sets

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