Modality

Possible worlds

There are many ways the world could be. I could have eaten a different breakfast this morning. Your parents could have never met. There could be no matter in the Universe. From that intuition, possibility became a puzzle for logicians very early on. Prima facie, propositions act as truth-bearer in the context of classical, bivalent logic: each one is either true or false. For instance, "Plato was a soldier" and "Plato was a married bachelor" are both false. Yet these propositions intuitively differ in an important way: Plato could possibly have been a soldier, but he could not have been a married man and a bachelor at the same time.

Statements such as those are called modal. To pursue our discussion of modality, which we have more than broached in our introduction to semantics, and start really getting into possible worlds (metaphorically), let me introduce some basic vocabulary and notation:

$A$ is possible ($\Diamond A$)
$A$ is necessary ($\Box A$) $\iff$It is not possible that $A$ be false ($\neg \Diamond \neg A$)
$A$ is impossible $\iff$ It is not possible that $A$ be true ($\Box \neg A$)
$A$ is contingent $\iff$ $A$ is neither necessary nor impossible ($\neg \Box A \land \neg \Box \neg A$)

Now, this could seem innocuous enough: we find here the type of syntax one is used to, practicing classical logic. But one key aspect is lost by introducing modal operators: extensionality.We’ve encountered that term a couple times in semantic or linguistic environments, so it’s not altogether a surprise to find it here. If you remember, extensionality implies that expressions sharing the same value can be freely substituted. Two true propositions are always equivalent, regardless of their internal structure, for instance: "Hillary Clinton is not the President" $\iff 1+1=2$. They can be used inter-changeably in classical, extensional contexts; for instance their negations are also equivalent to each other. Using modal operators, however, yields different results. "Possibly, Hillary Clinton could have been President" is true, while "Possibly, $1+1\neq2$" is false. Modal logic thus appears to be intensional.

With that being established, it is important to right away outline the de re/de dicto distinction. Consider the natural language sentence "Someone must be the President". Its meaning is ambiguous and could match two distinct propositions. The first one is: "Necessarily, there is a human who is the President". We note that the existence quantifier $\exists$ is inside the modal operator $\Box$. Such modal claims are called de dicto, because modality is exercised over a proposition. The second one is: "There is a human H, such that necessarily, H is the President". This is an entirely different idea. Here the modal operator is lodged inside the existence quantifier, and the claim is de re, because the modal claim is about a specific individual.

Now, an astute reader will have noticed that we’ve been liberally using our modal "intuition" without really explaining how we can decide that something is possible or necessary. Common-sense modal statements vary widely in that regard: "The sitting President must (necessarily) be Joe Biden" is false on anyone’s account. But what about those? "He must be a natural-born citizen" (because the Constitution say so). "He must be less than 120 year-old" (because no one alive is older). "He must be a carbon-based life-form" (because humans are). All of those appeal to different areas of knowledge, which can give rise to different types of modality. Yet they have in common that the modal claims are ultimately derived from empirical knowledge. This is called a posteriori epistemic necessity. Epistemic necessity can also be a priori if it solely relies on knowledge that’s independent of experience — since Kant it has been associated with "purely rational" thinking. "Three gallons of milk are heavier than two gallons of milk" is necessary a priori, but "Milk is white" is only necessary a posteriori. The distinction is far from obvious: don’t I need knowledge about what gallons are and about the additivity of mass to make that ostensibly a priori claim? It is generally assumed that definitions and "basic understanding" of terms by a proverbial "competent speaker" are an epistemic baseline (without which the concept of a priori would be untenable, since we do not know any word at birth, and claims need to be expressed with words). This type of subtlety can be seen as weakening the whole dichotomy, and indeed, the existence of a priori knowlege is far from universally admitted.

Since Kant, the notion of a priori epistemic necessity had almost universally been conflated with that of metaphysical necessity (Kant didn’t even distinguish between them, so the conflation was implicit). More recently, as we’ve seen, Saul Kripke built upon a series of semantic arguments to propose that these forms of necessity might not be entirely co-extensive after all, an idea that relies on his concept of rigid designators. Consider the following: "Carbon atoms have 6 protons". To Kripke, this is metaphysically necessary, because "Carbon" rigidly designates an elements that leads to cetain macroscopic properties of coal, graphite, diamond, etc. There could be a world where "Carbon" has 7 protons, but then it wouldn’t be Carbon, it would be Nitrogen, regardless of the name locals give it. Yet the number of protons in Carbon atoms is only knowable a posteriori, in particular no one could have known this three centuries ago. Kripke also pointed out the existence of a contingent a priori, for instance with indexical statements like "I am here now". Dualism between epistemic and metaphysical necessity is not a received view and is still undergoing intense debate.

It’s also worth mentioning the special case of nomic necessity which is based on laws of nature. That "massive objects travel slower than light" is epistemically a posteriori, but its metaphysical status depends on one’s concept of such laws. Necessitarians would say it is metaphysically necessary, contingentists would deny that.

It is finally time to introduce possible worlds proper. The intuition behind them was first put into words by Leibniz. The idea is simply to rephrase modal statements about necessary and possible truths as respectively: "in every possible word, $A$" and "there exists a possible world where $A$". The great advantage of this interpretation is that it not only provides a quite elegant reduction of modality, it also helps retrieve extensionality. Let’s go back to our example: "Hillary Clinton is not the President" $\iff 1+1=2$. In the possible world (PW) picture, we need to specify that these propositions are both true at our world. The fact that Hillary Clinton could have been President is now interpreted as: there is at least one world where she is. This has absolutely no bearing on the truth-value of $1+1=2$ at other worlds. Semantically speaking, this is nothing more than the fact that the truth of $\exists xFx$and $\forall x Fx$ are not in general determined by the truth value of $Fx$ for any particular value of $x$.

A specific remark needs to be made in that interpretation about de re claims. Let’s take an example: "The French PM could be a trained pianist". The de dicto interpretation is that there are worlds where there is a person who happens to be the French PM and is also a trained pianist. De re, we have instead that: there are worlds such that the French PM Jean Castex is a trained pianist. The major difference is that in the latter, we’re talking about a specific, rigidly-designated individual across worlds — in order to make de re statements, we are thereby committing to some concept of transworld identity.

I hopefully have managed to convey in which way possible worlds are useful in a philosophical context. But what hasn’t transpired is exactly what they are, and what should be their place in our bestiary of things.

Concretism

Concretism, or possible-worlds realism, was first developed in details by D.K. Lewis. His answer about what worlds are is pretty simple: they are maximal spatiotemporally connected objects. Beyond the barbarous phrase lies the simple idea that something is part of our world if and only if there is a continous "patch" of spacetime linking us. To exist in a world would simply mean to physically be a part of it. This definition is rife with problems: as mentioned many times, the intricacies of modern physics and uncertainty about the fine structure of spacetime makes the connectedness conceit dangerous to use. A workable physical theory of the multiverse may yet be developed, but for now let’s consider this a purely philosophical hypothesis. We’ll sidestep the details of the worlds’ definition and just retain the main features: worlds are causally isolated from each other, and objects within are worldbound (they can’t travel to other worlds nor exist in different worlds).

So far we’ve talked about Lewis’s worlds, but have said nothing of our world. To Lewis, our world is just one among others — one that we happen to be part of. The immediate consequence would be that in an ontological sense, things in possible worlds exist in the exact same way as things around us. Since things in our world obviously have special status in our eyes, Lewis calls them actual. But this is strictly to be understood as an indexical rather than an existentially charged distinction. The actual president Joe Biden exists in the same way as all possible presidents, just like this room (that I’m here now) exists in the same way as all other rooms on Earth. If we picked out one specific possible world and called it w, then "actual" and "in w" would play the exact same role. Another way to say this is that Sherlock Holmes is not an actual human being, but he exists, and is actual relative to a world where Conan Doyle’s stories happened.

One major problem of this approach is the hurdle of transworld identity: we would like to make de re statements and talk about "all that I could’ve done" or "all the ways Federer’s career could have turned out". But in Lewis’s picture, that would suggest than an object could exist across worlds, which we posited is not the case. His response to that objection was to develop a theory of counterparts. Broadly speaking, your counterpart would be "the closest thing" there is to you in a different world. This offers ample theoretical maneuverability, but the inherent vagueness makes it difficult to generalise. A version of Aristotle who ate a different dinner one day is intuitively a valid counterpart. An Aristotle who never taught Alexander too. But what of an Aristotle who dies as a child? Would he still be an Aristotle counterpart, or just "some person in that world"? What if Aristotle’s parents had a different son, also named Aristotle but with different DNA, and that person wrote and taught the same things "our Aristotle" did? Would he be "an Aristotle" despite originating in different gametes?

Fara has aimed to develop this intuition towards a more rigorous model, using sortal sameness instead of similarity. Remember that in discussing ordinary objects, we’ve talked at length about such relations: a potential counterpart of Aristotle would then be someone whom we’d feel comfortable calling "the same person as Aristotle" if he suddenly popped up into our world. The trouble with going that route is that the relation is context-dependent. Someone can be "the same person as Aristotle" without being "the same philosopher"; which one applies in a modal context depends on the sentence and the perception of a specific rigid designator.

One consequence of Lewis’s possible worlds explaining all of our modal claims, be they written, said or thought, is that there must be lots of them. This is expressed by a principle of plenitude: for any way the world could be, there is a world that is this way. It goes without saying that his ontology violates quantitative parsimony, and then some, but Lewis rejects that frequent criticism by making a sharp distinction with qualitative parsimony (many objects, but one type), a response that I largely share.

Abstractionism

Abstractionism (which Lewisian realists sometimes call, a tad pejoratively, ersatzism) takes the opposite stance. Lewis started by positing possible worlds and endorsed the modal semantics that construct intensional entities like propositions or predicates out of them. Abstractionists, on the other hand, take propositions and properties to be ontological primitives, which we have pre-theoretic intuitions of, but cannot be set up in a constructive fashion. In that picture, possible worlds are derived from these entities.

As one would expect, possible-worlds abstractionists take PWs to be abstract entities, and most offer a precise characterization based on some already familiar entities that play the role of "world-surrogates". What exactly that would be is the differentiation factor between different versions of abstractionism. Carnap famously proposed that worlds be maximum sets of consistent sentences. Others gave the project a semantic tinge, using sets of propositions. One interesting avenue in that vein defines worlds as states of affairs, which are the truth-makers to propositions’ truth-bearers. States of affairs can define "anything", like "France being a kingdom in 2021" or "all squares being circular". We know those examples are not consistent with our world, yet they exist (as abstract entities). However they are not actualized; in the prevalent jargon, we’ll say they don’t obtain. If they did, that would lead to the existence of facts: it is a fact that France was a republic in 2021. The vision of worlds as states of affairs is reminescent of Wittgenstein’s idea that we don’t live in a world of things, but of facts.

More precisely, worlds can be defined defined as total possible states of affairs. A total state of affairs t is one that either includes or precludes every other state of affairs (with $s \downarrow$ denoting that $s$ obtains, inclusion means that $\forall s (\Box t \downarrow \rightarrow s\downarrow)$, and preclusion $\forall s (\Box t \downarrow \rightarrow \neg s \downarrow)$. For $t$ to be a possible state of affairs is simply for t to possibly obtain: $\Diamond t \downarrow$. We can see that possibility is back to being a primitive concept: the abstractionist approach is not a reducive analysis of modality. As a result, extensionality is lost once again: modal claims can be explained in terms of possible worlds indeed, but we need modal operators to define them. One interesting feature of those models is that objects are not worldbound, thus facilitating transworld discourse. For $a$ to exist at a world $w$ is simply for the proposition "$a$ exists" to be included in the state of affairs representing $w$.

Of course, within that theory of worlds as states of affairs can be nested a theory of states of affairs themselves. For example, some metaphysicians represent all particulars as bundles of properties, thus making a world a giant cloud in the phase space of properties. Another one, fleshed out by David Armstrong, called combinatorialism, relies on the principles of logical atomism, as developed by Russell and Wittgenstein. The basic idea is that any recombination of individuals and relations from our world constitutes a possible state of affairs.

Two sides of the same coin?

Ultimately, picking a side in that debate could seem like a pure matter of ontological preferrence. Neither option is prescriptive nor predictive about what is possible. They just pick a different ontological primitive (possible worlds for concretism, propositions for abstractionism).

That formal symmetry is broken by the nature of the respective ontological commitments. We’ve briefly noted there were different types of modality, and shall come back to that in due time. Each flavor of modality yields a different batch of possible worlds. Physically possible worlds need to be consistent with all fundamental laws from our universe. If we restrain ourselves to those, Lewisian realism is a very attractive position, since some aspects of our best available theories do entice us to consider the multiverse as a serious option.

But we generally expect more from metaphysical possibility. A claim like "There could have been a 20-foot tall woman" is biologically false, but many would consider it to be metaphysically true, like most fictional accounts. The issue is that this biological state of affairs is not made impossible by a "wrong turn" in evolutionary history or some other contingency, it is truly grounded in fundamental physics (presumably, a body that large with the same structure as that of humans could not display the same range of movement, which our possible giantess presumably does, and could likely not survive for long). To argue that such a being can live and thrive while all else in the universe stays equal, is to say that fundamental laws need to be different, perhaps in incredibly subtle ways, to accommodate this one remarkable addition. If one compounds such examples, the only outcome is that some of these possible universes must have no laws at all.

This weakens the concretist conceit of worlds with the same "degree" of existence as ours: if only "some worlds" have a set of fundamental laws comparable to ours in their scope and complexity, is our world only special by chance? And what would it even be like to live in one of the others? Abstractionism offers more flexibility in that regard. Representing a world that is "like the actual world, plus Pegasus" is mostly a matter of appending the right facts or property-bundles to it. Having Sherlock Holmes exist in a world where Conan Doyle’s stories have happened doesn’t imply there is a universe where the famous detective is as existent as myself, where he was born, dies, has internal organs and a subjective life during every single second of his time on earth. It simply involves a state of affairs that includes everything Conan Doyle described about his character.

Some combination of concretism and abstractionism depending on which type of modality we are considering might then be then best option.

Impossible worlds

To consider impossible situations is understandably a more questionable step. Possible worlds have been introduced to explain modal claims; indeed, there are ways the world could have been: Alain Juppé could be the president of France. But there are also ways the world could not have been: Alain Juppé could not have won the election and lost it at the same time. The gist of the idea behind impossible worlds is that we need to ontologize such situations in order to talk and think about them, or use them in counterfactual reasoning.

Unsurprisingly, what qualifies as impossible depends on what is deemed possible, being its exact complementary. We’ve seen above that possibility "in general" does not exist and there are various levels of modality. A world that is nomically impossible in the sense it violates our laws of physics could still be metaphysically possible. The more pointed examples of impossible worlds we shall then consider are the ones that violate metaphysical posits (e.g. in which Alice can be born from different parents and still be Alice), or logical ones (in which even the rule of non-contradiction may be broken, and something called a "round square" can end up being round and not round at the same time). Clearly logical possibility is entailed by metaphysical possibility.

The nature of these impossible worlds is generally considered to match that of their possible peers: concretists and abstractionists will simply have impossible worlds inherit their favourite modal metaphysics. Indeed, there doesn’t seem to be any cogent reason to have much ontological difference there. Obviously, this doesn’t mean that we can simply import the work done on possible worlds and call it a day. They may be ontologically similar, but impossible worlds, by definition, break metaphysical or logical principles, which inevitably makes them structurally distinct. They display varying degrees of anarchy, which anyone arguing for their existence will need to contend with. All these developments are key to the prospects of rigorously defining impossible objects.

Meinong’s jungle

Nonexistents and indeterminates

Nonexistent entities are particulars that don’t exist, or lack being. This concept arises naturally when considering some ordinary assertions: if I say that there is no current king of France (negative existential statement), that James Bond is a spy (claim about fiction), or that the first female president will have executive experience (statement about future events), to what extent am I conjuring these "persons" into some form of being?

Alexius Meinong, who pioneered the type of approach we will describe, thought that such objects carried considerable explanatory force, and deplored his fellow philosophers’ prejudice in favor of the actual. For instance, to a Russellian about names, one cannot make any true statement about objects that lack existence (Russell denied they are anything else than disguised descriptions). The consequences can be counter-intuitive: a statement like "the current king of France is a king" would be false.

Following this introduction, it may seem I’m just retracing my steps here: aren’t non-actual possible objects and impossible objects archetypes of these nonexistent entities? Indeed they are. But in (im)possible-worlds modal metaphysics, the worlds are members of our ontology, and the objects their denizens. There is no such intermediation in traditional versions of Meinongianism. Meinong introduced more precisely incomplete objects, as we’ll see, whose modal nature is essentially indeterminate.

But let’s come back to our first point: how committed shall our ontology be to objects we’ve defined as nonexistent? Meinong denies there is a contradiction: everything is an object, even though some of these objects lack any form of being. This is formalised with the concept of so-being (Sosein): the various ways that an object can be. Meinong’s key principle is that of indifference: an object is characterized by its so-being (we might say, its properties, its essence...) and its being or non-being is indifferent to that nature. The Sosein of the square rectangle is being a rectangle with the property of being square; that of the square cirle is being a circle with the property of being square. We know that the former exists (has being) while the latter doesn’t (has non-being); yet to Meinong the Sosein is not affected by the Sein — they are independent. Unlike Russell, Meinong is able to state that the square circle is square, or that the current king of France is a king.

His typology actually goes deeper than a simple binary:

Real objects are defined as having existence and thus also being. Meinong includes physical and psychological objects there, so we might say that conrresponds to our first and second realms.
Ideal objects lack existence but they subsist and according to Meinong also have being. That would include for instance numbers, true propositions or states of affairs that obtain. Many objects from our third realm of abstract entities would thus be in that category.
Non-being objects are neither real nor ideal: they do not exist, nor do they subsist. They include what Meinong called "non-contradictory" objects (the golden mountain, Pegasus...) and "contradictory" ones (the round square, the largest prime number...). With hindsight we might prefer to use "possible" and "impossible" to reap the benefits of modal notions.
Indeterminate objects do not have being, but they also don’t have non-being. This category essentially matches the class of incomplete objects. An object with only one property, like that of being blue, cannot exist nor subsist. Meinongians insist that incomplete objects are distinct from properties: the object that is only blue (which is blue) is not the same as blueness (which is not blue). Meinong also includes here objects like a triangle. An indeterminate triangle is not an ideal object, since it is incomplete. It could be a complete ideal object if we knew its three angles and the length of one side.

In my view, the main flaw in Meinong’s original typology is the lack of clarity as to what separates ideal objects from the non-being realm. Even Russell showed signs of the utmost confusion on these matters, in a harmful fashion for Meinong’s legacy. At times he seemed to believe the golden mountain or false propositions to be subsistent entities. To be fair, the distinction between susbsistence and "non-being" does seem fairly nebulous to the modern Meinong reader.

It is best to consider Meinong’s ideas, not as a consistent body of ontological doctrine (which would "explain" being), but rather as the origin of an influential principle: that it is possible to de-ontologize large swathes of things whilst still referring to, quantifying over, or thinking about them. Alongside that founding principle, neo-Meinongians, starting with his student Mally, have proposed different ways to fine-tune the resulting metaphysical positions and dispel paradoxes. The first one, possibilism reverts to a possible/impossible-worlds model in an attempt to port the Meinongian nomenclature there. The second one, the dual-property strategy, distinguishes between nuclear and extra-nuclear properties. Extra-nuclear attributes can be ontological ("is existent"), modal ("is possible"), intentional ("is believed by me") or technical ("is complete").

Lastly, veering away even further from Meinongian orthodoxy is the dual-copula strategy. According to proponents like Edward Zalta, properties can be exemplified, but also encoded. For instance, a real dog in my street can exemplify the property "being a dog". But White Fang only encodes the property "being a dog", as well as "being a wolfdog", "being born wild", etc. But White Fang exemplifies many others: "being a fictional character", "having been created by Jack London", "being famous", "being nonexistent", etc. Similarly, the round square encodes exactly two properties ("being round" and "being square") but "is" not round nor square (if we assign that English-language copula to exemplification). The round square, however, is impossible, nonexistent, non-subsistent... Objects can thus be seen as correlates of sets of properties.

Fictions

Our discourse is saturdated with a type of entities whose existence is questionable: fictional ones. They can be literary characters like Rastignac, the ambitious young man from la Comédie humaine, mythical figures like Pegasus, the flying horse, places like Rivendell, the refuge of Elves in the Tolkienian Third Age, institutions like the Ministry of Magic from J.K. Rowling’s lore, and much more. Such entities have sometimes appeared in sections above as examples of possible, impossible, or inexistent objects. Indeed, should one entertain a realist position about such ficta, it is tempting to simply consider them a subclass of those broader categories, according to one’s preferred modal doctrine. I’ll take the opportunity to recapitulate those: first, possibilist approaches including Lewisian modal realism and diverse forms of ersatzism (chiefly, propositionalism and combinatorialism); second, Meinongian theories, either orthodox or reconstructed (with dual properties or dual copula).

What all these models cannot account for, though, is the act of creation. They all tend to build a Platonic-style picture of objects not located in time and independent from everything but their constituents (e.g. their properties). But it’s a platitude to say that fictional objects were created when the fictional work they belong to was first recited, written, sung...The contention that fictions are "seized" or "discovered" rather than created or imagined is a hard one to accept. Hence the appearance of a new class of realist models, called creationism or artefactualism, which aim to describe fictional objects with a beginning in time, as well as an explicit dependence upon a creator. We may note this approach’s defects are complementary to those of possibilism and Meinongianism. These traditional stances offer clear identity criteria for fictional objects but ignore their origin in the human mind, while creationism accounts for this fact, but struggles to provide clear identity criteria.

On the other hand, traditional antirealist approaches such as Russellianism quickly get into trouble with fictional entities. The fact they don’t refer to anything real leads to a statement like "Rastignac is French" to be prima facie false, which is impractical. It’s always possible to use a definite description, but even outside of fictional contexts, descriptions of proper names have suffered significant blows. One significant development is called fictionalism. It essentially introduces a fictional operator that allows to distinguish between internal and external metafictional sentences. "According to J.R.R. Tolkien’s legendarium, Gandalf is a wizard" is an internal sentence, that can be true because according to these works of fiction, the name "Gandalf" refers. "Gandalf is an icon in fantasy literature" is however an external sentence, and brings us back to square one: Russellians think that in our world, Gandalf does not refer. Various strategies have been devised, including one that construe realist positions as fictional universes, interpreting external sentences as: "According to fictional realists, Gandalf is an iconic character" (that seems slightly contrived to me). Lastly, one may note the additional difficulty of sentences that seemingly mix both modes, like "the Flash runs faster than Usain Bolt" or "Sherlock Holmes is smarter than any actual detective".

Developments about fictional entities specifically are justifiable by the importance of fiction in our culture. In addition, it’s interesting to note how the dedicated approaches described above have outgrown their intial scope. Artefactualism has been applied beyond its creationist origins (focusing on individual creations of fiction at a single point in time) to all types of human-originated references (institutions, laws, nations...). Meanwhile fictionalist conceits have been generalized to apply to all sort of entities, including, fruitfully, mathematical objects. Purported truths about apparently abstract entities can then be deemed true in the modal context of useful discourse (equivalent to a fictional context). Therefore, "4 is even" would be false but "according to mathematical discourse, 4 is even" would be true. This line of thinking has also allowed to set up the foundations of modal fictionalism, an anti-realist take on possible worlds.

Modality

Our construction of the world is progressing nicely, with the nature of ordinary objects and properties elucidated. Complex sentences involving these elements seemingly make up most of our thoughts and discourse: "I am thirsty", "Buck the dog has a tail", etc. All those sentences participate in a local state of affairs, but there is an understanding that the situations pertaining to our direct environment are part of larger and larger states of affairs, up to a maximal one, "the world".

Yet, this is far from covering the whole extent of possible discourse. Consider:

"I should have taken an umbrella this morning."
"If Nadal had become a pro soccer player, Federer may have won 29 majors."
"I didn’t think this could ever happen."

None of those describe a state of affairs in the actual world. Instead they relate to possible situations. Modality is the conventional term to encompass notions of possibility and necessity. What is true in no possible world is impossible; what is true in at least one possible world is possible; what is true in every possible world is necessary; what is not necessary is contingent.

The metaphysical nature of possible worlds has already been pondered above (in the "abstract world" section). But prior to taking a metaphysical stance, we need to distinguish between all the different types of modality. Consider now these sentences:

It is necessary for the President of the United States to be at least 35 years of age.
It is impossible to travel at more than 300,000 km/s
If $A\rightarrow B$ and $A$ is false, it is still possible that $B$ be true.

They are all couched in the language of modality, yet they clearly appeal to very different flavors. Ultimately, in a formally rigorous framework, modal logic should be reducible to the understanding of a set of fundamental laws: something is possible if it wouldn’t violate any of those laws. Or, equivalently: a possible world is a way in which our world could have been if things had been different, while still being subject to the fundamental laws.

Our three modal sentences above are true by virtue of, respectively, the U.S. Constitution, Einstein’s general relativity, and classical logic. Let’s outline the following varieties of modality, and the corresponding laws:

Physical necessity: what is necessary under the exact physical laws of our Universe.
Nomic necessity: what is necessary under "some" physical laws.
Metaphysical necessity: what is necessary under the laws of metaphysics.
Logical necessity: what is necessary under the laws of logic.

There is no reason why the metaphysical nature of the possible worlds implied by these modalities should be the same. You may remember that we observed how most metaphysical conceptions of possible worlds simply fall under "concretism" or "abstractionism": either there are other universes, metaphysically indistinguishable from the "actual" one (whose only differentiating factor is that I live in it), or possible worlds are just another abstraction. As I’ve outlined, the two approachs, save for the ultimate ontological truth, are conceptually equivalent for the most part. Concretists typically favor their approach because modal statements stem from a simple fact of the matter: something is possible because it is really true somewhere. Abstractionists dispense with concrete worlds but are left to contend with defining what’s possible. The systematic approach to building a formalized ontology that we’re engaging in right now gives us the best of both worlds: we don’t need concrete worlds to guide us because a clear set of laws would be enough to build the modal abstraction of possible worlds, if that was our only option.

Nomic necessity: are there other worlds than ours?

Position: Multiverse concretism

It’s a good thing that our theory is neutral towards concrete "other worlds". There are developments in modern physics that suggest these may exist (and perhaps provably so one day), so we don’t want to preclude them; at the same time, assuming their existence as a mandatory feature of our model seems like one step too far. Let’s just mention, with an open mind, the relevant scientific avenues.

Physically possible worlds in cosmology

The universe could be finite, or infinite. The impact on "other worlds" does not stem from any complex physical modeling, but from a simple mathematical fact: infinity is big. Intellectually, this is a truism, but not one that human minds are very good at handling. Yet the implacable fact remains that if (1) the universe is infinite, and (2) it is homogeneous and isotropic (that is, if the physical conditions that we observe locally around the Earth are not exceptional), then everything that could exist, does. It may be one gazillion bowjillion light years away, but under such conditions, there must be another Earth, with another Jonathan that’s exactly identical to myself and living the same life, but is right-handed. Or if there’s any physically viable way they may exist, one with unicorns.

Now, bringing our actual theories back into play: is the universe, in fact, infinite? There is a common misconception that the Big Bang models imply a finite universe (a tiny ball that grew up quickly to become a large object). That is however not the case: following the Big Bang, spacetime expanded, which means that density quickly decreased: what was concentrated into a tiny ball was all the matter in the currently observable universe. But it’s perfectly possible to imagine an infinite, isotropic (and extremely dense) universe, expanding into an infinite, isotropic (and less and less dense) universe.

What really determines its finitude is spacetime curvature. Most recent observations seem to point to a negative curvature, which implies... an infinite universe!

If that were true, what would be the impact on our model? At first sight, it would be quite devastating. Remember that we engineered the concept of ordinary objects to enforce a clear countability. I today am the same person I was in 1992, and the same person as myself-without-hair or myself-minus-one-finger. All these Jonathans add up to one single human being. If the universe is indeed infinite and isotropic, there are alternate-world Jonathans which will be captured as "the same human being" under the relative identity relation we’ve defined. But since lots of them are living a different life, the careful construction of ordinary properties falls apart: I cannot say anymore that Jonathan lives in London, since some alternate-world members of the equivalence class live in Tokyo or on Mars.

The solution isn’t conceptually difficult: we can reframe our model using world-indexing. For that, we need to first assume that the concrete world can be partitioned: $\aleph = \bigcup \aleph_w$. Our ’actual world’ is only a distinguished index $w_\mathfrak{a}$. Then we can define $w$-restricted sets of macroscopic objects $\aleph ^{\bigodot} _w$, the fusions or divisions of primitive objects from $\aleph_w$ only. Lastly we can systematically rework our definition of relative identity conditions by imposing that to be "the same" implies that there is some $w$ such that the objects in relation are both members of $\aleph ^{\bigodot} _w$. The effect is double:

this precludes any trans-world object to be "the same" as anything else.
this ensures that two things can be the same only if they’re from the same world.

So Tokyo-Jonathan and Mars-Jonathan are not the same as me, and the ordinary object architecture can again serve its purpose.

That said, we are not done with difficulties. The fact the concrete world can be "naturally" partitioned was a leap of faith, and is far from obvious. Yet, general relativity helps us here. After all, when we commonly talk of "the universe" in astronomy, we typically mean "the observable universe". From our vantage point, it is estimated at around a 100 billion light-years radius. But obviously it’s the 4D picture that counts. We know that more and more of the universe becomes observable, as invisible galaxies will progressively come into view. Yet even if we waited for an arbitrarily long time, we wouldn’t be able to observe at an arbitrary far distance, due to the combination of the reduced rate of expansion and the density of dark energy. Therefore, if the universe as a whole is infinite, it must be possible to partition it into an infinity of spatio-temporally disjointed parts, which we’ll call worlds. To refine what I said earlier: if Tokyo-Jonathan exists in one part of the universe that’s wholly disjointed from ours, it will not be the same as me; but if he happens to exist in the same part, albeith 90 billion light years away, he is the same.

We can remedy that by either reframing the relative identity relation from common discourse with an implicit circumstantial restriction (e.g. to "planet Earth, around our day and age"), or by considering that the odds are so long that it would happen (even with trillions of galaxies in the observable universe!) that it’s an acceptable deviation from our initial intent.

Physically possible worlds in quantum physics

A more contemporary development that could tie in with a concretist approach to modality is the so-called Everettian, or ’many-worlds’ interpretation of quantum theory. There is little space here to attempt a full exposé of the matter, so allow me to propose a brief summary instead.

Quantum mechanics is often seen as an intrinsically probabilistic, indeterministic, nonlocal theory, but what it really does is to merge two things:

Some kind of fundamental wave equation. In non-relativistic Quantum Mechanics (QM), that is Schrödinger’s equation. In relativistic QM that is the Klein-Gordon equation or the Dirac equation. In the standard model of Quantum Field Theory (QFT), that is a more complex system of equations that’s governed by the relevant interactions’ symmetry groups. Point is, none of those is qualitatively different from classical equations by Fresnel or Maxwell. They are differential field equations governing the evolution over time of a wavefunction. That evolution is unitary, and the resulting model is entirely deterministic and local.
A concept of ’wavefunction collapse’. Indeed, the spoiler is that, unlike electromagnetics or optics, quantum mechanics cannot stop there, for we know that electrons, atoms and many more bodies don’t behave like waves at all when we observe them. In comes the key postulate, often called Born rule: it states that
- the outcome of any measurement (for instance of position, momentum, energy...) is a random function whose probability distribution is given by the wavefunction.
- after such a measurement, the wavefunction has collapsed onto a pure state that matches the outcome.
Let’s summon Schrödinger’s always serviceable cat to try and picture things. Pre-measurement, the cat is assumed to be in a superposition of quantum states: \[\lvert\psi_{cat}\rangle = \alpha \lvert \textrm{live}\rangle + \beta \lvert \textrm{dead}\rangle\] The ’live’ and ’dead’ wavefunctions are called "pure states": they are eigenfunctions derived from the wave equation and correspond to a physical state where we have perfect information about the cat’s status. When we make a measurement by looking at the cat, he could be either live or dead. If he is alive, then the wavefunction collapses, and \[\lvert\psi_{cat}\rangle = \lvert \textrm{live}\rangle\] That process is by definition non-deterministic; it’s also non-unitary and nonlocal.

This combination of two disjointed processes characterizes much of quantum theories’ strangeness, and is at the core of all resulting metaphysical puzzles. Cue the interpretations of quantum mechanics: what they all seek to interpret, above all else, is that process of wavefunction collapse.

I believe it is important to dispel an oft-held misconception that conflates the collapse of the wavefunction with quantum decoherence. Like wavefunction collapse, quantum decoherence has a part in explaining how the appearance of classical physics emerges, at macroscopic scale, from the undeniably quantum foundations of our world. Yet, quantum decoherence is part and parcel of the unitary, deterministic world of wave equations. What it does explain is why, at macroscopic scales, objects behave in particle-like fashion. The result of countless interactions and stimuli on a macroscopic body outside of the lab is that the density matrix does not feature any mixed terms — the kind that generate interference patterns in a double-slit experiment —, and thus that the system’s quantum state converges towards a probabilistic mix of pure states. In other words, wave-like quantum effects are supplanted by a distribution that matches the epistemic uncertainty of an ideal observer.

That is why we can observe interference in a double-slit experiment easily with electrons, less easily with atoms, and not at all with small dust particles. However, no wavefunction collapse has taken place. Decoherence tells us why we can’t observe cat-interferences, but doesn’t tell us why and how we observe a live cat or a dead cat, rather than a superposition. Formally, the density matrix may not have near-zero terms like \[\lvert\textrm{live}\rangle \langle \textrm{dead} \lvert\] but it still looks somewhat like \[\frac{1}{2} ( \lvert\textrm{live}\rangle \langle \textrm{live} \lvert + \lvert\textrm{dead}\rangle \langle \textrm{dead} \lvert)\]

From that point on there are three possible attitudes towards wavefunction collapse:

Ignore the problem altogether. Most Copenhagen-like interpretations don’t treat wavefunction collapse as a physical process at all, because they don’t treat wavefunctions as physical either, instead considering them as a measure of the information available to an observer. A number of other approaches plainly accept the co-existence of the unitary, continuous process and the non-unitary, discontinuous collapse.
Treat wavefunction collapse as a physical process that can be explained by a revamped wave equation. Usually that entails giving up on the unitary nature of the quantum time evolution, while retrieving it as an approximation at small scales. An objective collapse (independent of any measurement or observer) is then part of the physical theory, rather than a primitive that needs to be uncomfortably tacked on. Penrose has expressed a particularly elegant proposal that involves the interface between the quantum superposition principle and general relativity’s equivalence principle.
The last group plainly denies that wavefunction collapse takes place at all. Hugh Everett criticized the usual conception of the observed object as a quantum system and the observer as a classical one. The truth is that upon observing a quantum state, the observer becomes entangled with it. According to Everett, there is no discontinuity at all: the only physical process is the unitary wave equation, that keeps moving the quantum state through time.

Post-measurement, the universal quantum state involves a superposition of several entangled states. Coming back to the cat, there would be an entangled state of "Jonathan-who-saw-a-live-cat" with "live-cat" and "Jonathan-who-saw-a-dead-cat" with "dead-cat". The fact that I saw a live cat is not, in that picture, the outcome of some obscure rolling of the dice. It simply stems from the fact that "I" am described by the former quantum state, not the latter.... but I could have been the latter as well. Of course, this observation is not a punctual event: what I have seen will influence my behavior, and then the whole universe going forward. Each measurement event is then re-framed as the branching out into a "different timeline" although the "real world" can still be described as a superposition of all those timelines.

Here are our "possible worlds"... We have used the term "measurements" but won’t endorse a psychologist or subjective approach: objective events can cause wavefunction collapse (and thus ’timeline forking’) in the absence of consciousness, human or otherwise. Therefore, at every turn, new timelines appear, covering all the possible ways the world could be observed at that particular moment. This is exactly our definition of physically possible worlds.

The metaphysical implications are not easy to parse for the Everettian enthusiast. The same issues as for the infinite-universe case arise, whereby we need to somehow partition the world to avoid utter confusion in our modal logic. The physical world may be equal to the universal quantum wavefunction, a superposition of mixed states, but we want to be able to define a world in the metaphysical sense as one where all past observations are indeed certain.

If the characterization of objective measurement "by the Universe" can be rigorously developed, there is a chance to define those worlds properly. Instead of carving the world into spacetime bits like in the previous proposal, we would thus be carving a Hilbert space (or some more complex superstring/quantum gravity avatar) into sectors.

Nomically possible worlds

Those are also conceivable in the context of our current physical theorizing. This term designates worlds that are possible under "some" laws of physics that are different from ours.

One example of multiverse model harboring universes with different laws is given by eternal inflation. By way of brief reminder, inflationary cosmology was borne out of Guth’s work, which proved fruitful in explaining away major issues with standard Friedmann/Lemaître cosmology (essentially, general relativity plus a Big Bang concept with sui generis expansion of the spacetime metric).

The first puzzle was linked with the isotropy of the observable universe: the energy density in any direction we look is remarkably constant. Even the Cosmic Microwave Background radiation, representing the residual photons that were first able to travel freely when the universe cooled after the Big Bang, is isotropic, with variations of $10^{-5}$ at the most. It’s not a shocking fact in itself. The early universe must have had high degrees of anisotropy because of quantum fluctuations, but fluids like a cannister of gas or a pool of water reach thermal equilibrium when they cool down, and the universe has had billions of years to do so... But the difference lies in the rate of expansion necessary to explain the universe’s size in traditional Big Bang models. It is so high that many regions would find themselves part of distinct causal patches: in other words, they would "travel away from each other" so fast that they would find themselves causally isolated before thermal equilibrium could have taken place. After that point, since communication is impossible, there is no reason the same temperature would be observed in all regions. Inflation can explain that since, as the name indicates, it describes a mechanism (usually involving a scalar quantum field called inflaton) whereby expansion would be exponential for a brief moment (in some models, expansion by a factor of $10^{78}$ for a duration of $10^{-32}\textrm{s}$), before resuming a ’moderate’ pace.

A second enigma is the flatness of the universe: previous cosmological models of course admit a near-flat universe as part of their solutions, but that’s a highly unlikely outcome that can be best described by fine-tuning via a cosmological constant. Inflation automatically explains this with its scale and suddennness. The universe right after the Big Bang was still in a high-energy state where spacetime curvature was a direct result of quantum fluctuations. Inflation was so intense and fast that our whole observable universe expanded from a spacetime volume tiny enough, that even quantum fluctuations created no major homogeneity.

Lastly, this superfast dilution could explain the complete absence of magnetic monopoles or other heavy particles predicted by most candidate GUTs (Grand Unification Theories): since their abundant production would have preceded the inflation phase, their density would be so low that it’s unlikely to find even one in our observable volume.

So, the basic principles of inflationary cosmology seem quite promising. But its initial version was not fit for purpose: it couldn’t explain how inflation would stop and result in a re-heated universe that matches observations. I won’t get into details about all rival models that attempt to remedy this. Suffice it to say here that many of them, and chiefly eternal inflation, describe an overall inflationary universe with no beginning nor end, but where inflation can stop locally, basically creating, hot "bubbles", each of which is like an independent universe with its own Big Bang. New bubble are constantly appearing, then expanding, though of course by definition at a much slower pace than the inflation multiverse (if we are to consider individual bubbles as universes, this is what we should then call the super-structure). This gives the multiverse a fractal structure with in principle an infinity of bubbles. Bubble-universe are endowed with different initial conditions, since those arise from tiny inhomogeneities from quantum fluctuations (still visible in the material structures of a universe, such as our galaxy filaments). But they can also feature different physical laws. Even though we might assume there would be irreducible "multiversal" laws, the way symmetries break (with an element of randomness) could lead to differences in the way extra dimensions would compactify, or different fundamental interactions. Since an infinity of worlds would exist with physical laws that are distinct from our "universal laws", this would be a realization of the nomically possible worlds concept.

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.

Logical necessity and beyond: are there impossible worlds?

Position: Hyperintensional sententialism

The concept of impossible worlds is a common one in philosophy. Of course, "impossible" is not precise enough; what precedes has taught us that we should ask "relative to which modality?" As we’ve seen, for instance, there are worlds that are metaphysically possible but physically impossible. But the term "impossible world" has generally been used with the idea of metaphysical impossibility in mind. Even within the metaphysically impossible, there are still distinctions to make, for logical possibility encompasses metaphysical possibility (because the laws of logic are prior).

Let’s first consider the metaphysically impossible, but logically possible. We already know how to represent such worlds: via the same construction as above, but with closure under the logical entailment $\vdash^{log{}}$ (if we attempted to use metaphysical consequence instead, some contradiction would have us land on the universal object that encodes everything). Here are some situations which would fall in this category:

An object that is both abstract and concrete (this violates the definition of these concepts).
A concrete object that encodes a property (this violates one of the metaphysical axioms).
A property that doesn’t have a Form (this goes against the comprehension principle).
Two existing objects whose mereological fusion does not exist (this goes against the rules of our mereology).
An intentional mental state that’s independent of any physical object (in contradiction with our buildup of mental states).
A version of you that wasn’t born from the same parents (this one requires more development: by using the term "you" in our language, I implicitly refer to a human being. As we’ve seen, in our metaphysical construction, that is an ordinary object which is individuated by some criteria. In the case of human beings, that normally involves some core genetic identity: Nixon could still have been Nixon if he had never been President, but he couldn’t have been Nixon if his mother hadn’t been Hannah Milhous. Therefore for you to have other parents than yours is not logically impossible at all, but metaphysically impossible).

Logically impossible worlds are beyond any form of necessity. They can be represented as state of affairs without any rule of closure. Basically, "anything-goes" world. Situations can include:

An object that exemplifies $P$ and not-$P$ (which is against the propositional logic that our ontology is built on. That can be a truth however in the theory of paraconsistent logic, which plays the role of an impossible world where statements like "$P$ and non-$P$" can be true at that world).
$1=0$ (at the most basic level, these are just some inkblots on a page. But implicitly, when humans use these symbols, they refer to some version of arithmetic that’s consistent with Peano Number Theory, so we’re really talking here about $0{_\mathfrak{pnt}}$ and $1_{\mathfrak{pnt}}$. We know that equality to be false in Peano arithmetic, which is closed under logical consequence; therefore it can only be true in a logically impossible world).
An object in Euclidean space that is both round and square.

All in all, the Zalta-inspired formalism we’ve described above has proved its worth once again. We use the term of sententialism to characterize the present section in order to distinguish more sharply between metaphysically possible and impossible worlds. Although the abstractionist formalism is similar on either side, it is not the same to define objects in the context of our metaphysical theory that enforce the laws of metaphysic in their internal semantics, or to define objects that are metaphysically inconsistent, and thus live in contradiction with the language they’re couched in. The latter are best characterized as sentences that we can talk or think about, but hardly deserve the world moniker.