When Everything Becomes One: Identity Universalism and the Paradox Puzzle

Alright, let’s dive into some pretty deep philosophical waters. We’re talking about a view that’s a bit mind-bending, combining two powerful ideas about how reality is put together. It’s called Identity Universalism, and it’s the lovechild of two other concepts: Unrestricted Composition and Composition as Identity.

Think of it this way: Unrestricted Composition (UC) says that any bunch of things, no matter how random – say, your left sock, the moon, and the concept of justice – compose something. Yep, a single entity made of those things. Pretty wild stuff, right? Composition always happens, everywhere, all the time.

Now, add Composition as Identity (CAI) to the mix. This is where it gets really interesting. CAI says that when some things compose something, they *are* that something. Literally. Your body isn’t just *made of* your head, arms, and legs; it *is* them. They *are* it. It’s not that the whole is identical to each part separately (that would be bonkers!), but that the one composite thing is identical to the many things that compose it, taken together. As one philosopher put it, “the many and the one are the same portion of Reality.”

So, Identity Universalism (IU) is the view that says (i) things compose an object *exactly when* they’re identical to it, and (ii) composition *always* happens. Any collection of things? They compose something, and that something *is* that collection of things.

This sounds incredibly bold, maybe even a little scary. And for good reason! It seems to lead straight into some classic philosophical headaches, particularly those involving paradoxes like Russell’s and the tricky business of counting things (cardinality) that Cantor explored.

The Building Blocks: UC and CAI

Let’s unpack these two ideas a bit more. Unrestricted Composition is appealing partly because trying to *restrict* composition seems to lead to vagueness. When exactly do things compose something? Is there a composite object made of the atoms in my chair? How about the atoms in my chair *plus* one atom from the moon? If you try to draw a line, it seems arbitrary or vague. And some philosophers, like David Lewis, argued that composition simply *can’t* be vague. So, the easiest way out? Just say everything composes something.

Composition as Identity, on the other hand, gets a lot of mileage from explaining how incredibly close the relationship between a whole and its parts feels. Think about that land example: if a guy sells off all six parcels of land but claims he still owns the whole plot, it feels wrong. Why? Because the land *is* the parcels. The connection is super intimate, almost like identity itself. CAI takes that intuition and runs with it, saying the connection *is* identity. It offers an explanation for this “intimacy” that other views might just see as a coincidence.

There are arguments suggesting that if you accept CAI, you might be pushed towards accepting UC anyway. And conversely, if you accept UC, CAI makes it seem less ontologically extravagant. If the composite object *is* just the parts, then accepting that a composite exists for *any* collection of parts isn’t adding extra stuff to the world; you’re just describing the same stuff in a different way. They seem to have a bit of a symbiotic relationship.

But put them together, and you get Identity Universalism. And that’s where the real fun (and potential trouble) begins.

The Fregean Parallel and the Paradox Threat

Now, hang tight, because we’re going to take a quick detour to visit the philosopher Gottlob Frege. Frege had a system involving objects and concepts. Concepts were like properties, and they had “extensions” – objects that somehow represented the collection of things the concept applied to. Frege had two key principles:

• Unrestricted Second-Order Comprehension: For any property you can describe, there’s a concept for it.
• Basic Law V: The extensions of two concepts are identical *if and only if* exactly the same objects fall under those concepts.

Basic Law V is an “ontological explosion” principle. It basically guarantees a distinct object (an extension) for every distinct concept (every distinct way of grouping objects). But this is where Russell’s paradox crashed the party. Russell found a concept – the concept of being the extension of a concept that doesn’t apply to itself. Does the extension of *this* concept fall under the concept itself? If it does, it doesn’t. If it doesn’t, it does. Contradiction!

The deeper lesson, often attributed to Cantor, is that you can’t have a one-to-one mapping from all possible ways of grouping things in a domain to the things *in* that domain. There are always strictly more ways to group things than there are things themselves.

Now, back to Identity Universalism. Pluralities (like “my head, arms, and legs”) are kind of like concepts (ways of grouping things). Fusions (like “my body”) are kind of like extensions (objects representing those groupings). And guess what? IU entails a principle that’s a dead ringer for Frege’s Basic Law V. Let’s call it V*:

• V*: The fusion of some things (the Xs) is identical to the fusion of some other things (the Ys) *if and only if* exactly the same things are one of the Xs as are one of the Ys.

Just like Basic Law V, V* is an ontological explosion principle. Given UC (everything has a fusion) and CAI (the fusion *is* the plurality), V* guarantees a distinct object (a fusion) for every distinct plurality. This looks *exactly* like the setup that led Frege into Russell’s paradox. It seems like IU should face a parallel contradiction.

Dodging the Bullet: Why IU Avoids Russell’s Paradox

So, how does Identity Universalism, despite having this V* principle that looks so dangerous, manage to avoid the paradox? Here’s the scoop: it has a very peculiar, non-standard view of pluralities.

Normally, we’d think that if you have a composite object, say, my body, it can be composed of different pluralities. My body is composed of my top and bottom halves. It’s also composed of my left and right halves. And surely, the top and bottom halves aren’t the *same* plurality as the left and right halves, right? Something (like my top half) is one of the first group but not one of the second.

But if CAI is true, my body *is* the top and bottom halves, and my body *is* the left and right halves. By the logic of identity, the top and bottom halves *must* be identical to the left and right halves as pluralities. This means that under CAI, something is one of the top and bottom halves *if and only if* it is one of the left and right halves. This feels super weird! It suggests that something can be “one of” a plurality without being identical to any *specific* member you might list. My top half is one of the “left and right halves” plurality, even though it’s not identical to the left half or the right half.

This is what philosopher Ted Sider calls a “collapse.” Under CAI, being one of some things collapses into being a part of their fusion. If something is one of the Xs, it’s a part of the fusion of the Xs. And if it’s a part of the fusion of the Xs, it’s one of the Xs. This isn’t how we usually think about pluralities or sets.

This “Collapse” principle is key. It means that CAI (and thus IU) has a very restricted view of which pluralities actually exist. It rejects what’s called Unrestricted Plural Comprehension – the idea that for any property you can describe, there’s a plurality of exactly the things that have that property.

For example, under CAI, there can’t be a plurality of all and only the fundamental particles in the universe. Why? Because the universe is composed of fundamental particles, but the universe itself is a part of the universe’s fusion (which is the universe), and the universe isn’t a fundamental particle. The “Collapse” principle means that if the fundamental particles formed a plurality, the universe would have to be one of them (since it’s a part of their fusion), which it isn’t. So, no such plurality exists.

This restriction on pluralities is precisely what saves IU from Russell’s paradox. The problematic pluralities needed to generate the contradiction (like the plurality of all things that are proper parts of something) simply don’t exist under CAI’s peculiar conception of pluralities. The “ontological explosion” happens, but it’s not catastrophic because the domain of *pluralities* is smaller than we might have thought.

The Cantorian Clash

Okay, so IU dodges Russell’s paradox. That’s a pretty neat trick! But avoiding the paradox isn’t the same as avoiding the *insight* behind it, which is the Cantorian idea about cardinality. Remember, Cantor’s theorem tells us there are always strictly more subsets of a set than elements in the set. The plural version says there are strictly more pluralities of things than there are things themselves.

Identity Universalism, with its V* principle and the resulting one-to-one correspondence between objects (fusions) and pluralities, is committed to the idea that:

• (I) There are at least as many things as there are pluralities of them.

This directly conflicts with the Cantorian insight:

• (II) There are strictly more pluralities of things than there are things themselves.

So, while IU avoids the contradiction, it remains squarely anti-Cantorian when it comes to the relative sizes of the domain of objects and the domain of pluralities.

Standing Your Ground: Why the Conflict Might Not Be Fatal

Does this conflict with Cantor sink Identity Universalism? Not necessarily, argues the paper. The standard proofs of the plural version of Cantor’s theorem rely on some form of plural comprehension principle – essentially, the ability to form a plurality for certain described conditions. But, as we’ve seen, CAI (and thus IU) *rejects* unrestricted plural comprehension.

The identity universalist can argue that the same features of their view that blocked Russell’s paradox (namely, the restricted view of pluralities dictated by Collapse) also block the *proof* of the plural Cantor theorem. The pluralities needed for the proof simply don’t exist in their system.

They can then reasonably push back against the Cantorian, asking for independent reasons to accept the necessary plural comprehension principles or the metaphysical necessity of the plural Cantor theorem itself, *given* the identity universalist’s starting assumptions about composition and identity. It’s a bold stance, but they can stand their ground.

The Curious Case of Pluralities

Still, there’s something intuitively a bit wonky about rejecting unrestricted plural comprehension. It means accepting that there’s a thing x1, a thing x2, etc., which are all the fundamental particles, but denying that there are *some things*, the Xs, such that they are exactly those fundamental particles. It feels like denying that the existence of a collection of things follows from the existence of those very things. A plurality is supposed to *be* those things themselves. This might feel less like a different philosophical view and more like… well, talking about something else entirely. It seems to challenge the very idea of plural quantification.

Two Ways to See Reality

Can the identity universalist say more to make this less strange? Perhaps. The paper suggests borrowing an idea from recent metaphysics about “joint-carving” – which concepts or ways of describing reality best capture its fundamental structure.

We could distinguish two versions of Identity Universalism:

• Singularist IU: The fundamental reality is best described using singular terms and mereological notions (part, whole, composition). Pluralities are understood in terms of these singular concepts.
• Pluralist IU: The fundamental reality is best described using plural terms and notions (plural quantification, being one of). Singular objects (especially composites) are understood in terms of these plural concepts.

Both views agree that a portion of reality can be seen as both one (a composite object) and many (the plurality of its parts). But they disagree on which perspective is the “joint-carving” one, which is more fundamental.

The neat trick here is that in the “joint-carving” language of either view, the correspondence between objects and pluralities becomes trivial. For the singularist, talking about a plurality of parts gets cashed out in terms of the composite object they form. So, saying there’s a unique plurality for every composite object becomes saying there’s a unique composite object for every composite object (itself) – a trivial truth. For the pluralist, talking about a composite object gets cashed out in terms of the plurality it’s identical to. So, saying there’s a unique composite object for every plurality becomes saying there’s a unique plurality for every plurality (itself) – also a trivial truth.

From either the singularist or the pluralist perspective, when described in their preferred fundamental language, the statement “there are at least as many things as pluralities” (I) becomes trivially true, and “there are strictly more pluralities than things” (II) becomes trivially false. This gives the identity universalist a story about why their anti-Cantorian stance isn’t just stubbornness, but a consequence of how they see the fundamental structure of reality.

Conclusion

So, Identity Universalism – the view that any things compose something and are identical to it – faces some serious initial worries about paradox and cardinality, largely because it entails a principle (V*) that mirrors Frege’s problematic Basic Law V. However, its proponents can argue that CAI’s unique conception of pluralities, which restricts which pluralities exist, prevents the derivation of Russell’s paradox. While this view remains in conflict with the plural version of Cantor’s theorem, the same restrictions on pluralities also block the standard proofs of that theorem within the IU framework.

The biggest challenge remaining seems to be the intuitive oddity of rejecting unrestricted plural comprehension. But even here, the identity universalist might appeal to a distinction between singularist and pluralist ways of understanding their view, arguing that in the most fundamental description of reality, the object-plurality correspondence is a triviality. While there are still details to fill in, Identity Universalism seems more resilient against these specific paradox and cardinality concerns than one might initially expect.

Source: Springer