Metaphysical necessity: which others worlds are possible?
Position: Abstract necessitism
There may or may not be other worlds that belong to the concrete realm, as we’ve seen. But are those enough to cover all the modal variety of our discourse?
The metaphysical level of necessity has usually been invoked when mere physical necessity didn’t seem like it could match our imaginative power. For instance, Pegasus probably couldn’t exist under the classical laws of Newton. But the extension of physically possible situations has considerably ballooned since the advent of quantum theories. Any macroscopic state of affairs is governed by a probability distribution, hence the straightjacket of conservation laws and the likes is only extremely likely, but not certain, to hold. The probability that I would at this instant turn into a bird and fly away is so vanishingly small we lack the mathematical notation to make it intelligible: quantum field theory allows virtual particles to pop up from the vacuum for tiny fractions of a second; the likelihood of seeing a complex macroscopic structure emerge from the vacuum for a macro-duration is basically zero... but not quite zero, hence it is possible. Ideas about what alternate laws could look like (the "string theory" landscape, different dimensional configurations) do not tell us if complex objects such as ourselves could evolve in nomically possible worlds, but it’s not excluded that many "exotic" situations could be nomically realizable.
With such a tremendously broad and deep realm of possibilities at hand, do we even need metaphysical necessity? Are there even worlds that are metaphysically possible, but nomically impossible? One important way to answer that is to consider incomplete worlds.
Take a fictional character like Rastignac. He is defined by all the situations and characterizations that Balzac provided and nothing more. There are physically conceivable worlds where a concrete version of Rastignac exists, endowed with the personality and adventures imagined by Balzac. But in all of those worlds, Rastignac exemplifies plenty of other properties: he is of a certain height, he was born on a certain day, his DNA is in a certain way, etc. — any concrete human does. One possible answer is that we ought to consider the fictional Rastignac as a sort of equivalence class over all the concrete Rastignacs with their heights, their childhoods, their internal organs... It is an interesting way to dispel the issue, yet not entirely convincing to me. What if we move the needle to consider even more incomplete objects? For instance, those with a single, relatively simple, macroscopic property: "The color blue was wary of the color orange but felt kinship with the color green". Colors don’t have feelings and don’t talk, so this doesn’t represent a fully realized situation in any concrete possible worlds. The equivalence class approach is not very convincing here: abstracting from physical/nomic situation where "something blue is wary of something orange but not of something green" seems awfully contrived.
Luckily, we have just the right setup to represent incomplete, or in any way non-physical situations: our abstract realm and the encoding relation. Even better, we have developed just the right tools to describe mathematical theories! Let’s verify that it is a good fit...
We will start by defining a world as: \[\mathfrak{w} \text{ is a world} \hat{\longleftrightarrow} \mathbf{A}\mathfrak{w} \wedge (\forall P(\mathfrak{w}P \equiv \exists \phi (P=[\lambda x \phi])))\] Astute readers will have noticed this is exactly the same definition we gave for theories. Since we’ve used an equivalence sign in both cases, I’m equating the two: worlds are theories, and vice versa, we just use each term in different contexts. So, like a theory, a world is an abstract object that only encodes propositional properties. If we call \(\mathfrak{B}\) the world corresponding to Balzac’s fictional stories, then it will simply encode propositions that define that world to us, like "Rastignac was ambitious" or "Vautrin was a convict".
Similarly to theories, we can define the concept of truth at a world: \[\mathfrak{w}\vDash \phi \text{ ('$\phi$ is true at world $\mathfrak{w}$') } \hat{\longleftrightarrow} \mathfrak{w} [\lambda x \phi]\] A proposition is true at a world when the abstracted property is encoded by that world. If \(A\) was the property denoting ambition and \(r\) denoted Rastignac, we could write that since it is true, in Balzac’s fiction, that Rastignac is ambitious, then \(\mathfrak{B}[\lambda x (Ar)]\).
So far, a world sounds like just a state of affairs, as generic as can be. In the case of mathematic theories, the key point in the definition was the rule of closure. Similarly, worlds must abide by some rule of closure. \[(\mathfrak{w} \vDash \phi) \wedge (\phi \vdash \psi )\rightarrow \mathfrak{w} \vDash \psi\] In the context of mathematical theory, this rule was simple: it ensured that all the the logical consequences of encoded propositions were themselves encoded. It also excluded inconsistent theories as distinct entities, since those would encode every property, and thus be identical to each other.
The same idea can be generalised here. Let’s say that \(\vdash^{\mathrm{phys}}\) represents some form of consequence under the laws of physics. Then by enforcing rules of closure with that version of consequence, we should end up with a physically possible world. Since \(\mathfrak{B}\) encodes that Rastignac is an ambitious man, it physically entails , for instance, that Rastignac had two parents and a unique DNA genome. Balzac didn’t write about DNA (how could he?), so such a propositional property is certainly not part of the "axioms" of that world.
This is why fictions and many other situations that describe incomplete worlds are best construed as metaphysically possible worlds. As regards the formalism we’ve introduced, this simply means that we will use metaphysical consequence \(\vdash^{met{}}\) in our rules of closure. By analogy with what precedes: any statement that is entailed by the axioms of a world following the law of metaphysics is also true at the world in question.
The concept of laws of metaphysics has often proved vexing in the past. Luckily, dear reader, we’ve just been toiling hard to establish them. They cover every rule described since the logical section: about being and non-being, about concrete and abstract, about essence and accident, about mereology, etc.
Still re-tracing our steps from the mathematical section, we can define abstract objects as such: \[r_{\mathfrak{B}}=\imath y ( \mathbf{A}y \wedge \forall P ( yP \equiv \mathfrak{B} \vDash P r_{\mathfrak{B}} )\] In plain English: "Rastignac-at-Balzac-world is an abstract object that encodes all properties that Rastignac exemplifies at that world". It’s important to note that, especially in fictional discourse, the objects within propositional properties are typically ordinary objects, so statements whereby they exemplify some property are in fact simplified notations to sum up statements about some equivalence class.
That’s another incompleteness factor: Balzac tells us lots about "ordinary Rastignac", but nothing about any "concrete Rastignac". Unlike ordinary objects in the actual world (i.e. ours) or other physically realised universes, there is nothing "beneath" the ordinary object, which we just have to take at face value.
I would call this position about metaphysically possible worlds a form of abstract necessitism. Indeed we have it that possible worlds are in general abstract (notwithstanding the potential existence of a concrete multiverse). And because of the way abstract objects are allowed to freely combine with each other in the fashion we’ve described at level 3c about abstract combinatorialism, we also land on a necessitist position, which we can loosely translate as "everything that could exist, exists".
Coming back for a second to the potential multiverse: should it truly exist, there would be no contradiction with abstract necessitism. Remember that we defined the abstraction of a concrete object that encodes every predicate exemplified by that concrete object. Any individual universe in a concrete multiverse would then have its own abstract equivalent. Give or take some isomorphism (to convert an abstract object that encodes predicates to one that encodes propositional properties reflecting the same truth), we can map any concrete universe onto a physically or nomically possible world from the abstract realm.
Lastly, we need to come back to our original introduction of modality in this formalized ontology. Remember that I introduced \(\Box\), the necessity operator as a primitive one, and \(S5\) modal logic is part of the fundamental axioms that we’ve adopted. At the same time, we’ve been explaining there is no absolute modality and that it needs be relative to a specific set of laws. It is the case for the \(\Box\) and \(\Diamond\) symbols we’ve been using: they refer to metaphysical modality. Indeed we’ve only seen them used in the definition of metaphysical laws. When we say that an abstract object encodes a property necessarily, this ought to mean at least metaphysical necessity, and cannot mean logical necessity since this is not a logical principle (logically there’s nothing wrong about an object just ceasing to encode something). We ought to be careful about circularity, since the necessity operator is used in laws that purportedly ground the definition of metaphysical modality. But by building things up axiomatically as we have, we can likely avoid such pitfalls.