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.