In his 1981 paper “From Worlds to Possibilities”, Lloyd Humberstone shows a way to do modal logic without the apparatus of possible worlds. Instead of worlds he uses possibilities, which may, unlike worlds, be incomplete. The non-modal parts of the view are discussed again in section 6.44 of The Connectives, with some small presentational differences. In this paper I’ll set out this possibility frame approach, make some notes about its logic, and end with a survey of its possible applications.
Mathematically, possibilities are points in a model, like possible worlds are points in different kinds of models. But it helps to have a mental picture of what kind of thing they are. In “From Worlds to Possibilities”, Humberstone notes that one picture you could have is that they are sets of possible worlds. This isn’t a terrible picture, but it’s not perfect for a couple of reasons. For one thing, as Humberstone notes, part of the point of developing possibilities is to do without the machinery of possible worlds. Understanding possibilities as sets of possible worlds wouldn’t help with that project. For another, as Wesley Holliday (2025, 271–72) notes, the natural way to generate modal accessibility relations on sets of worlds from accessibility on the worlds themselves doesn’t always work the way Humberstone wants accessibility to work. So let’s start with a different picture.
Possibilities, as I’ll think of them, are stories. To make things concrete, I’ll start with a particular story: A Study in Scarlet (Conan Doyle (1995)), the story in which Sherlock Holmes was introduced. That story settles some questions, both explicitly, e.g., that Holmes is a detective, and implicitly, e.g., that Holmes has never set foot on the moon. But it leaves several other questions open, e.g., how many cousins Holmes has. It’s not just that A Study in Scarlet is a story. It has proper parts that are stories. The first chapter is a story, one that tells of the first meeting between Holmes and Watson. And arguably it is a proper part of a larger story, made up of all of Conan Doyle’s stories of Holmes and Watson. When a story \(x\) is a proper part of story \(y\), what that means is that everything settled in \(x\) is still true in \(y\), and more things besides are settled. When this happens, we’ll call \(y\) a proper refinement of \(x\). For most purposes it will be more convenient to use the more general notion of refinement, where each story counts as an improper refinement of itself.
Following Humberstone, I’ll write \(x \leqslant y\) to mean that \(y\) is a refinement of \(x\).1 As he notes, this notation can be confusing if one thinks of \(x\) and \(y\) as sets, because in that case the refinement will typically be smaller.2 But if we think of possibilities as stories, the notation becomes more intuitive. We have \(x \leqslant y\) when \(y\) is created by adding new content to \(x\). Keeping with this theme, I’ll typically model stories not as worlds, but as finite sequences. (In the main example in Section 2, they will be sequences of 0s and 1s.) In these models, \(x \leqslant y\) means that \(x\) is an initial segment of \(y\).
1 \(y \geqslant x\) will just mean the same thing as \(x \leqslant y\).
2 Holliday (2025) writes \(y \sqsubseteq x\) when \(y\) is a refinement of \(x\), mirroring this way of thinking about possibilities.
1 The Formal Structure
The Basic Language
To start with, assume we’re working in a simple language that has a countable set \(\mathcal{P}\) of propositional variables, and three connectives: \(\neg\), \(\wedge\), and \(\vee\). We have a set of possibilities \(W\), and a reflexive and transitive refinement relation \(\leqslant\) on them. The following rules show how to build what I’ll call a Humberstone possibility model on \(\langle W, \leqslant \rangle\). (I’ll call \(\langle W, \leqslant \rangle\) a possibility frame in most contexts, but a Humberstone frame when I’m comparing it to similar structures, especially in the context of discussing Holliday (2025).)
A Humberstone possibility model \(\mathcal{M}\) is a triple \(\langle W, \leqslant, V \rangle\), where \(V\) is a function from \(\mathcal{P}\) to \(\wp(W)\), intuitively saying where each atomic proposition is true, satisfying these two constraints:
- For all \(x\), if \(x \in V(p)\) and \(y \geqslant x\), then \(y \in V(p)\). Intuitively, truth for atomics is persistent across refinements.
- For all \(x\), if \(\forall y \geqslant x\;\, \exists z \geqslant y: z \in V(p)\), then \(x \in V(p)\). This is what Humberstone (2011, 900) calls refinability, and it means that \(p\) only fails to be true at \(x\) if there is some refinement of \(x\) where it is settled as being untrue.
Given these constraints, Humberstone suggests the following theory of truth at a possibility for all sentences in this language. (We’ll treat \(\rightarrow\) as a defined connective, with \(A \rightarrow B =_{df} \neg A \vee B\).)
\[\begin{align*} [\text{Vbls}] \quad & \mathcal{M} \models_x p_i \text{ iff } x \in V(p_i); \\ [\neg] \quad & \mathcal{M} \models_x \neg A \text{ iff } \forall y \geqslant x, \, \mathcal{M} \nmodels_y A; \\ [\wedge] \quad & \mathcal{M} \models_x A \wedge B \text{ iff } \mathcal{M} \models_x A \text{ and } \mathcal{M} \models_x B; \\ [\vee] \quad & \mathcal{M} \models_x A \vee B \text{ iff } \forall y \geqslant x \;\, \exists z \geqslant y : \mathcal{M} \models_z A \text{ or } \mathcal{M} \models_z B. \end{align*}\]
Given these definitions, it’s possible to prove three things. First, every sentence in the language is persistent. If \(\mathcal{M} \models_x A\) and \(x \leqslant y\), then \(\mathcal{M} \models_y A\). For any sentence, truth is always preserved when moving to a refinement. Second, refinability holds for all sentences in the language. This is, as Humberstone notes, easier to state using [\(\neg\)]. Once we’ve added negation to the language, refinement becomes the claim that, for any \(A\), if \(\mathcal{M} \nmodels_x A\), there is some refinement \(y\) of \(x\) such that \(\mathcal{M} \models_y \neg A\). Third, for any set of sentences \(\Gamma\) and sentence \(A\), the truth at a point of all sentences in \(\Gamma\) guarantees the truth of \(A\) iff the sequent \(\Gamma\) entails \(A\) in classical propositional logic.
In this paper, I’m going to discuss three extensions of this language. I’ll introduce them in reverse order of how much they are discussed in Humberstone, starting with one he does not discuss at all: infinitary disjunction.
Infinitary Disjunction
We’ll add to the language a new symbol \(\bigvee\), which forms a new sentence out of any countable set of sentences not containing \(\bigvee\). Intuitively, it is true when one of the sentences in the set is true. More formally, its truth at a possibility is defined as follows:
\[\begin{align*} [\bigvee] \quad & \mathcal{M} \models_x \bigvee ({A_1, A_2, \dots}) \text{ iff } \forall y \geqslant x \;\, \exists z \geqslant y :\text{ for some } i \, \mathcal{M} \models_z A_i. \end{align*}\]
Again, it’s fairly simple to show that this addition to the language will preserve persistence and refinability. But while this is simple, it is significant, because things could easily have been otherwise.
Quantifiers
The second extension will be to add quantifiers, following a suggestion in Humberstone (1981, 331). Assume, as usual, that the language has a stock of names \(c_1, \dots\), and for each \(n\), a stock of \(n\)-place predicates \(F^n_1, F^n_2, \dots\). A first-order (Humberstone) possibility model is a structure \(\langle W, \leqslant, D, V \rangle\), where \(D\) assigns a non-empty domain of objects to each point, and \(V\) interprets the non-logical vocabulary. More precisely:
- \(D\) is a function assigning to each \(x \in W\) a non-empty set \(D(x)\), the domain at \(x\).
- \(V\) assigns to each name \(c_i\) and each \(x \in W\) either a designated element \(V(c_i, x) \in D(x)\), or is undefined at \(x\).
- \(V\) assigns to each \(n\)-place predicate \(F^n_j\) and each \(x \in W\) a set \(V(F^n_j, x) \subseteq D(x)^n\), the extension of \(F^n_j\) at \(x\).
These must satisfy the following constraints:
- Domain monotonicity
- If \(x \leqslant y\), then \(D(x) \subseteq D(y)\).
- Name coverage
- For each name \(c_i\) and each \(x \in W\), there exists some \(y \geqslant x\) such that \(V(c_i, y)\) is defined.
- Persistence for names
- If \(V(c_i, x)\) is defined and \(x \leqslant y\), then \(V(c_i, y)\) is defined and \(V(c_i, y) = V(c_i, x)\).
- Persistence for predicate extensions
- If \(\langle o_1, \dots, o_n \rangle \in V(F^n_j, x)\) and \(x \leqslant y\), then \(\langle o_1, \dots, o_n \rangle \in V(F^n_j, y)\).
- Refinability for predicate extensions
- If \(\langle o_1, \dots, o_n \rangle \notin V(F^n_j, x)\), then there exists some \(y \geqslant x\) such that for all \(z \geqslant y\), \(\langle o_1, \dots, o_n \rangle \notin V(F^n_j, z)\).
Given a model and a variable assignment \(g\) mapping variables to objects, truth at a point is defined as follows. Write \(g[v/o]\) for the assignment that maps variable \(v\) to object \(o\) and otherwise agrees with \(g\). For a term \(t\), write \(\llbracket t \rrbracket^{g,x}\) for the denotation of \(t\) under \(g\) at \(x\). This will not always be defined, e.g., when the term is a name that only gets defined at subsequent refinements.
\[\begin{align*} [=] \quad & \mathcal{M}, g \models_x t_1 = t_2 \text{ iff } \forall y \geqslant x \;\, \exists z \geqslant y : \llbracket t_1 \rrbracket^{g,z} \text{ and } \llbracket t_2 \rrbracket^{g,z} \text{ are both defined and equal}; \\ [F^n] \quad & \mathcal{M}, g \models_x F^n_j(t_1, \dots, t_n) \text{ iff } \forall y \geqslant x \;\, \exists z \geqslant y : \langle \llbracket t_1 \rrbracket^{g,z}, \dots, \llbracket t_n \rrbracket^{g,z} \rangle \in V(F^n_j, z); \\ [\forall] \quad & \mathcal{M}, g \models_x \forall v \, A \text{ iff } \forall y \geqslant x \, \forall o \in D(y) : \mathcal{M}, g[v/o] \models_y A; \\ [\exists] \quad & \mathcal{M}, g \models_x \exists v \, A \text{ iff } \forall y \geqslant x \;\, \exists z \geqslant y \, \exists o \in D(z) : \mathcal{M}, g[v/o] \models_z A. \end{align*}\]
The propositional connectives are handled exactly as before.
The surprising thing here is the \(\forall\exists\) pattern in clauses for identity and for non-logical predicates. We need to write the clause that way, rather than just saying for example that \(Fa\) is true at \(x\) if \(\llbracket a \rrbracket^{g,x} \in \llbracket F \rrbracket^{g,x}\), for two related reasons. First, we’re in classical logic, so for any name \(c\) we want \(c=c\) to come out true, even when \(c\) lacks a denotation at \(x\). Second, we want \(\forall y: Fy\) to entail \(Fc\). So if \(\forall y: Fy\) is true at \(x\), we need \(Fc\) to be true at \(x\) even when \(c\) lacks a denotation.
The underlying problem here is that it’s impossible to have an equivalent of the refinability condition for names. It seems reasonable to have a name, say ‘Holmes’, which picks out a particular individual, say Holmes, at all refinements where Holmes is in the domain. But we don’t want that to imply that Holmes is already in the domain. So even if ‘Holmes’ will never denote something other than Holmes, it’s not already true that it denotes Holmes. The complications in the above clauses are downstream of the fact that denotation does not satisfy refinability in just that way.
In the special case where every name already has a denotation at \(x\), the atomic clauses simplify in just the way you would hope. If \(t_1, \dots, t_n\) are variables, or names that already have denotations at \(x\), then by persistence of names and predicate extensions the \(\forall\exists\) quantifier prefix collapses: \(\mathcal{M}, g \models_x t_1 = t_2\) iff \(\llbracket t_1 \rrbracket^{g,x} = \llbracket t_2 \rrbracket^{g,x}\), and \(\mathcal{M}, g \models_x F^n_j(t_1, \dots, t_n)\) iff \(\langle \llbracket t_1 \rrbracket^{g,x}, \dots, \llbracket t_n \rrbracket^{g,x} \rangle \in V(F^n_j, x)\). The more complex clauses above are needed only to handle the case where some name occurring in the formula lacks a denotation at \(x\) but is guaranteed to acquire one.
This is a possibilist treatment of the universal quantifier, in contrast to the actualist quantifiers discussed in Harrison-Trainor (2019). I’ll return in Section 3 to the reasons we are best off using possibilist quantifiers, and the difficulties this raises for talking about just what’s true in a possibility.
Modal Operators
The third extension will be the introduction of modal operators. Here I’ll follow Humberstone (1981) very closely, save that I’ll have a plurality of modal operators rather than just one. So I’ll use these structures to define (as Holliday (2025) does) multi-modal logics. But I’ll follow Humberstone, and not Holliday, in defining modal operators in terms of accessibility relations \(R_i\) satisfying these three conditions3:
3 I’m using the names for these that Holliday uses, which are more evocative than Humberstone’s original names.
- UpR:
- If \(x \leqslant x'\) and \(x' R_i y\), then \(x R_i y\).
- RDown:
- If \(x R_i y\) and \(y \leqslant y'\), then \(x R_i y'\).
- RRef++:
- If \(x R_i y\), then there exists \(x' \geqslant x\) such that for all \(x'' \geqslant x'\), \(x'' R_i y\).
UpR says that if a refinement of \(x\) can access \(y\), then \(x\) itself can already access \(y\): accessibility is not something that can be gained by adding detail to the source. RDown is a converse of this; it says that accessibility cannot be lost by adding detail to the target. RRef++ says that if \(x\) can access \(y\), there is some refinement \(x'\) of \(x\) where it is settled that \(x'\) can access y. This last access can’t be overturned by further refinement of \(x'\).
Given these constraints, the truth condition for the box operator is:
\[\begin{align*} [\Box_i] \quad & \mathcal{M} \models_x \Box_i A \text{ iff } \forall y \, (x R_i y \Rightarrow \mathcal{M} \models_y A); \\ \end{align*}\]
This should be familiar: \(\Box_i A\) is true at \(x\) iff \(A\) is true at every \(R_i\)-accessible possibility.
Humberstone treats \(\Diamond\) as a defined connective, \(\Diamond_i A\) just means \(\neg \Box_i \neg A\), and I’ll follow suit. If we spell out what it means for \(\neg \Box_i \neg A\) to be true, we get the rule [\(\Diamond_i\)]Official. But if we impose the above three constraints on \(R_i\), we can see that this is equivalent to the more familiar [\(\Diamond_i\)]Simple.
\[\begin{align*} [\Diamond_i]_{\text{Official}} \quad & \mathcal{M} \models_x \Diamond_i A \text{ iff } \forall y \geqslant x \;\, \exists z \geqslant y \, \exists w \, (z R_i w \text{ and } \mathcal{M} \models_w A). \\ [\Diamond_i]_{\text{Simple}} \quad & \mathcal{M} \models_x \Diamond_i A \text{ iff } \exists y \, (x R_i y \text{ and } \mathcal{M} \models_y A); \end{align*}\]
If \(R_i\) obeys UpR, then [\(\Diamond_i\)]Official will imply [\(\Diamond_i\)]Simple. For \(\Diamond_i A\) to be true at \(x\) according to [\(\Diamond_i\)]Official, there must be some refinement which can access a point where \(A\) is true, and so by UpR, \(x\) itself can access that point. If \(R_i\) obeys RRef++, then [\(\Diamond_i\)]Simple will imply [\(\Diamond_i\)]Official. If there is some \(y\) such that \(xR_iy\) and \(A\) is true at \(y\), then by RRef++, there is some refinement of \(x\) such that every refinement of it can access \(y\), and hence can access a point where \(A\) is true. So these are equivalent.
From now on, I’ll use [\(\Diamond_i\)]Simple when working out what’s true at points in particular models. But when we are proving general facts about the language, it will help to remember that \(\Diamond_i\) is a defined connective, so we don’t need an extra part of inductive arguments to cover it.
Modal Constraints
Why should we impose the three constraints Humberstone proposes? It is not hard to show that they guarantee that persistence and refinability hold for sentences generated using these new modal connectives. At least, it isn’t hard as long as we remember that \(\Diamond\) is being treated as defined, so the only new step in the inductive proofs involves \(\Box\). And UpR guarantees persistence for \(\Box\) sentences, while RRef++ guarantees refinability.
But this is overkill. As Humberstone points out, we haven’t used RDown in the proof, so this doesn’t explain why we’d impose RDown. Holliday (2025, 62) further points out UpR is stronger than we need for persistence. We could weaken it by making greater use of the fact that \(A\) is persistent. All we need is that if \(x \leqslant x'\) and \(xRy\) then there is some \(z \geqslant y\) such that \(x'Rz\). That will guarantee the key fact if \(x'\) can access a world where \(A\), then so can \(x\).
So we need other arguments for these constraints other than their role in ensuring persistence and refinability. Humberstone offers two other arguments here. One anticipates the applications of possibilities to multi-modal logics. If we want to use possibilities for tense logic, then we want the converse of any accessibility relation to satisfy the constraints on an accessibility relation. So if all acceptable accessibility relations satisfy UpR, they should also satisfy RDown. This isn’t convincing on its own though. For one thing, as already noted, we might not need UpR. For another, by these lights we should worry that the system is incomplete because we haven’t put in a converse of RRef++.
The argument that Humberstone spends more time on, and which I think is more compelling, comes from rethinking the relationship between \(R_i\) and \(\Box_i\). It’s very tempting to read the truth condition for \(\Box_i\) as being explanatory from right-to-left. On this way of thinking, facts about which \(\Box_i\) sentences are true at a point are grounded in which non-modal sentences are true and which \(R_i\) relations obtain. But while this is tempting, it isn’t compulsory.
We could instead take the modal facts as given, and ask what accessibility relations must obtain to be consistent with them. The process here is going to resemble the construction of canonical models. We take the sets of consistent sentences as given, and say \(s_1R_is_2\) iff whenever \(\Box_i A \in s_1\), then \(A \in s_2\). Humberstone’s defence of the constraints takes a similar path. Imagine that instead of using the recursive clauses for the definition of modal truths, we simply have a frame \(\langle W, \leqslant \rangle\) and a stipulated assignment of truths at each point in \(W\). These assignments will be presumed to satisfy persistence and refinability. Now say that \(xR_iy\) iff \(\mathcal{M} \models_y A\) whenever \(\mathcal{M} \models_x \Box_i A\). Given those assumptions, what conditions must \(R_i\) satisfy?
The answer is that then \(R_i\) will satisfy these three constraints: UpR, RDown and RRef++. If \(\mathcal{M} \models_y A\) whenever \(\mathcal{M} \models_x' \Box_i A\), then by the persistence of \(\Box_i\), we know that \(\mathcal{M} \models_y A\) whenever \(\mathcal{M} \models_x \Box_i A\), so UpR holds. A similar argument, using the persistence of \(A\), justifies RDown. Finally, RRef++ follows from the fact that \(\Box_i A\) is refinable. If \(\Box_i A\) is not true at \(x\), that means that it is determinately not true at some refinement. So if \(\mathcal{M} \nmodels_x A\), there must be some refinement \(x'\) such that for all further refinements \(x''\), \(\mathcal{M} \nmodels_x'' A\). If \(\mathcal{M} \nmodels_x \Box_i A\), then there must be some possibility \(y\) such that \(xR_iy\) and \(\mathcal{M} \nmodels_y A\). So as long as \(x''R_iy\), refinability will be satisfied. I don’t see how to prove there isn’t a weaker condition that would also work, it’s possible we could use the refinability of \(A\) to find some weaker condition, but I don’t quite see how that would work. So I think RRef++ also follows from this way of thinking about accessibility.
In the next section I’ll discuss what logics can be defined using frames that satisfy all of these conditions.
2 Logics Determinable on Humberstone Frames
Holliday (2025, sec. 8.2) raises an interesting question. As well as the familiar Kripke frames most commonly used as a semantics for modal logic, and the Humberstone frames defined above, he introduces a class of ‘full possibility’ frames, which weaken some of Humberstone’s constraints. It won’t matter here exactly what these weakenings are, but what does matter is that using these weakened frames, Holliday shows how to determine logics that are not determinable on any class of Kripke frames. To state this more precisely, for any class of frames \(\mathsf{F}\), let \(\mathrm{L}(\mathsf{F})\) be the set of sentences true at all points in all models definable on some member of \(\mathsf{F}\). Then let \(\mathrm{ML}(\mathsf{F})\) be the set \(\{\mathrm{L}(\mathsf{X}) : \mathsf{X} \subseteq \mathsf{F}\}\). That is, \(\mathrm{ML}(\mathsf{F})\) is the class of logics that can be determined using just \(\mathsf{F}\).
If we let \(\mathsf{K}\) denote the class of Kripke frames, and \(\mathsf{FP}\) denote the class of full possibility frames, Holliday (2025, sec. 2.5) constructs a very clever argument to show that \(\mathrm{ML}(\mathsf{K}) \subsetneq \mathrm{ML}(\mathsf{FP})\). If we let \(\mathsf{H}\) denote the class of Humberstone frames, it follows from the fact that every Kripke frame is a Humberstone frame and every Humberstone frame is a full possibility frame that \(\mathrm{ML}(\mathsf{K}) \subseteq \mathrm{ML}(\mathsf{H}) \subseteq \mathrm{ML}(\mathsf{FP})\). The fact that \(\mathrm{ML}(\mathsf{K}) \subsetneq \mathrm{ML}(\mathsf{FP})\) implies that at least one of those inclusions is strict, it isn’t clear which one. He leaves the question of which one it is as an open question.
I don’t have an answer to that question as asked, since it is asked about languages whose sentences have finite length. I do have a proof that if we allow infinite disjunction, as discussed above, then \(\mathrm{ML}(\mathsf{H}) \neq \mathrm{ML}(\mathsf{K})\). If we expand the language by allowing infinite disjunctions, at least the first inclusion is strict. I will show this by constructing a single Humberstone frame that, in the infinitary language, defines a logic with no Kripke equivalent. The construction will follow Holliday’s very closely, but differ just enough to ensure compliance with Humberstone’s conditions.
The Frame
The frame is built from two copies of the set of finite binary sequences, i.e., sequences of 0s and 1s of any finite length, including the empty sequence. Call one copy the left-handed sequences and the other the right-handed sequences. The refinement relation is: \(x \leqslant y\) iff \(x\) and \(y\) have the same handedness and \(x\) is an initial segment of \(y\). So within each copy the frame is just the binary tree ordered by extension, and no left-handed sequence refines a right-handed sequence or vice versa. It will help to have some notation for referring to points in this model. When \(s\) is a finite binary sequence, I’ll write \(s^L\) for the left-handed version of \(s\), and \(s^R\) for the right-handed version.
The Accessibility Relations
I’ll define first a single accessibility relation and then a separate infinite family of accessibility relations. The single relation, which I’ll write \(R^{\rightarrow}\), is such that \(xR^{\rightarrow}y\) iff \(x\) is left-handed and \(y\) is right-handed. The family of relations, each written \(R^{\leftarrow}_i\) for \(i \in \mathbb{N}\), is such that \(xR^{\leftarrow}_iy\) iff \(x\) is right-handed, \(x\) does not have a \(0\) in its \(i\)-th position (either because \(x\) has length less than \(i\), or because it has a \(1\) in position \(i\)), and \(y\) is left-handed.
That \(R^{\rightarrow}\) satisfies UpR, RDown, and RRef++ is obvious. It is also obvious that for each \(i\), \(R^{\leftarrow}_i\) satisfies UpR and RDown. It’s only a little harder to show that it satisfies RRef++. Assume \(xR^{\leftarrow}_iy\). So \(x\) is right-handed and \(y\) is left-handed. If \(x\) is of length at least \(i\), then \(x\) itself can serve as the refinement such that every further refinement can access \(y\). If \(x\)’s length is less than \(i\), extend \(x\) with enough 1s to create a sequence of length \(i\). The result will be a refinement such that every further refinement can access \(y\), as required.
The Example
Now consider the proposition (\(\ref{Splitting}\)), a minor variant on Holliday’s example (). (I’m using \(\mathsf{T}\) for an arbitrary tautology.)
\[\begin{equation} \neg \Diamond^{\rightarrow} p \vee \bigvee_{i \in \mathbb{N}}( \Diamond^{\rightarrow} ( p \wedge \Diamond_i^{\leftarrow} \mathsf{T} ) \wedge \Diamond^{\rightarrow} ( p \wedge \neg \Diamond_i^{\leftarrow} \mathsf{T} ) ) \tag{\textsc{Splitting}} \label{Splitting} \end{equation}\]
I’m going to make three claims about (\(\ref{Splitting}\)). First, it is true throughout the frame I just described. Second, \(\neg \Diamond^{\rightarrow} p\) is not true on all models on that frame. Third, there is no class of Kripke frames throughout which (\(\ref{Splitting}\)) is always true and \(\neg \Diamond^{\rightarrow} p\) is not always true. From this it follows that \(\mathrm{ML}(\mathsf{K}) \neq \mathrm{ML}(\mathsf{H})\).
For the first claim, I’ll show something slightly stronger, namely that at each point one or other disjunct in (\(\ref{Splitting}\)) is true. If the point is right-handed, then the first disjunct is true, since each right-handed point is a dead-end with respect to \(R^{\rightarrow}\). So we just have to look at the left-handed points. Let \(x\) be an arbitrary left-handed point. If there is no \(y\) such that \(xR^{\rightarrow}y\) and \(p\) is true at \(y\), then again the first disjunct is true.
Now consider the case where \(x\) is left-handed, and there are right-handed points such that \(p\) is true at \(y\). Here it will be helpful to denote a right-handed point \(y\) as \(s_y^R\). Among those \(s_y^R\) at which \(p\) is true, consider the ones where \(|s_y|\), the length of \(s_y\), is minimal. There must be some such points, since sequence lengths are non-negative integers and \(p\) is true at some accessible right-handed point. Let \(i = |s_y| + 1\). Let \(s_y^R \oplus \langle 0 \rangle^R\) and \(s_y^R \oplus \langle 1 \rangle^R\) be the right-handed sequences generated by adding either a 0 or a 1, respectively, to \(s_y^R\). Since \(p\) is true at \(s_y^R\) and truth is persistent, \(p\) will be true at both \(s_y^R \oplus \langle 0 \rangle^R\) and \(s_y^R \oplus \langle 1 \rangle^R\). Since \(s_y^R \oplus \langle 0 \rangle^R\) has a 0 at its \(i\)th position, it is a dead-end with respect to \(R^{\leftarrow}_i\). So \(\neg \Diamond_i^{\leftarrow} \mathsf{T}\) is true there. So since \(x\) is left-handed, and \(s_y^R\oplus \langle 0 \rangle^R\) is right-handed, \(\Diamond^{\rightarrow} p \wedge \neg \Diamond_i^{\leftarrow} \mathsf{T}\) is true at \(x\). A similar argument shows that \(p \wedge \Diamond_i^{\leftarrow} \mathsf{T}\) is true at \(s_y^R\oplus \langle 1 \rangle^R\), so \(\Diamond^{\rightarrow}(p \wedge \Diamond_i^{\leftarrow} \mathsf{T})\) is true at \(x\). And that implies that the \(i\)th disjunct of the right-hand disjunction of (\(\ref{Splitting}\)) is true at \(x\), as required.
The second claim, that \(\neg \Diamond^{\rightarrow} p\) is not true on all models on the frame, is trivial, since it will fail at some left-handed points whenever \(p\) is true at some right-handed point.
For the third claim, we follow Holliday’s argument particularly closely. In particular, we’ll appeal to his insight that “Worlds cannot split, but possibilities can” (Holliday 2025, 95). Consider any class of Kripke frames such that \(\neg \Diamond^{\rightarrow} p\) is not valid on that class. Look at the class of models on those frames where \(p\) is true at exactly one world. For any disjunct of the right-hand disjunction of (\(\ref{Splitting}\)) to be true at a point, that point must access a world where \(p\) is true that is a dead-end with respect to \(R^{\leftarrow}_i\), and also a world where \(p\) is true that is not a dead-end with respect to \(R^{\leftarrow}_i\). That’s impossible if \(p\) is true at just one world. So throughout this class of models, the right-hand disjunction of (\(\ref{Splitting}\)) will be false. But if \(\neg \Diamond^{\rightarrow} p\) is not valid on the frame, there will also be points in this class of models where \(\Diamond^{\rightarrow} p\) is true. So the whole disjunction will be false at those points, and hence (\(\ref{Splitting}\)) is not valid on the frame.
Putting these together, there is no class of Kripke frames that validates exactly the sentences valid on this particular Humberstone frame. So \(\mathrm{ML}(\mathsf{K}) \neq \mathrm{ML}(\mathsf{H})\), and hence \(\mathrm{ML}(\mathsf{K}) \subsetneq \mathrm{ML}(\mathsf{H})\).
3 Quantifiers and Necessitism
There are two distinctive features of the way that I introduced quantifiers in Section 1. First, I introduced the universal quantifier as basic, and simply defined \(\exists\) as \(\neg \forall \neg\). Second, I introduced the universal quantifier as possibilist, at least in the sense that it ranges over all objects in a possibility and all its refinements. As I noted, there is a passing remark in “From Worlds to Possibilities” where Humberstone suggests that is how he would do it. It’s worth going over why this ends up being the best way to do things.
The aim here, as it was in Section 1, was to keep classical logic while still using things, like incomplete possibilities, which are usually taken to be incompatible with classical logic. But in this section the primary contrast to classical logic is not some non-classical propositional logic, but a free logic. In classical logic, this argument looks fairly compelling, as long as the modal logic we’re using is normal.
\[\begin{align*} 1. && a = a && \text{=Intro} \\ 2. && \exists x(x = a) && \exists\text{-Introduction} \\ 3. && \Box \exists x(x = a) && \text{Necessitation} \\ 4. && \forall y \Box \exists x(x = y) && \forall\text{-Introduction} \\ 5. && \Box \forall y \Box \exists x(x = y) && \text{Necessitation} \end{align*}\]
That is, necessarily every object is such that necessarily, something is identical to it. If we are working in a possible worlds framework and our logic is at least as strong as KTB, this implies that any two worlds connected by the ancestral of \(R\) must share the same domain. So all worlds that could be relevant to the truth of any sentence at \(w\) have the same domain as \(w\).
In the possibilities framework, we have a bit more flexibility while still making claims like \(\Box \forall y \Box \exists x(x = y)\) come out true. We could go for a constant domain semantics, as in the possible worlds picture. The upside of the constant domain semantics is that our quantifiers could be ‘actualist’. We could say that \(\forall x: \Phi x\) is true at a point, relative to an assignment \(g\), iff \(\Phi a\) is true at \(g[x/o]\) for any \(o\) in the domain. That is more intuitive than the possibilist clause I’ve endorsed for \(\forall\).
The reason I’ve endorsed this less intuitive clause is that having constant domains does not feel like it matches the spirit of the possibilities framework. Some stories have more characters than others. In the story that consists of one proposition, Romeo loves Juliet, it feels like the domain should just consist of Romeo and Juliet. But now we face a challenge. If the quantifiers range just over the things in that possibility, we can’t have \(\exists x(x = a)\) be a logical truth in that to be a logical truth, it seems like the domain of the quantifier can’t range just over Romeo and Juliet.
That doesn’t quite settle things though. In the propositional case, we made the truth value for conjunctions depend only on what was true at that possibility, while the truth value for disjunctions depended on later refinements. Perhaps we try a parallel move here, and have the truth of a universally quantified statement range over only things in that possibility, while existentially quantified statements range over more things.
That won’t work though. If the domain of \(\langle \textit{Romeo loves Juliet}\rangle\) is just Romeo and Juliet, and universal quantifiers only range over the domain of that possibility, then the sentence \(\forall x(x = \mathit{Romeo} \vee x = \mathit{Juliet})\) would come out true. But by $, we would then get that Mercutio is identical to Romeo or Mercutio is identical to Juliet. We don’t want that, because there are clearly refinements of our simple story where Mercutio is neither Romeo nor Juliet. Shakespeare’s play is one such refinement.
So it looks like we are forced to say that quantifiers range over future refinements as well as the actual possibility. Still, we want a way to be able to say, within a possibility, that all the \(F\)’s in that possibility are \(G\). We can’t say that with \(\forall x(Fx \rightarrow Gx)\), since there might be a counterexample of an \(F\) in a later refinement which is not \(G\). Is there any way to talk about what’s just true in the possibility?
Harrison-Trainor (2019) suggests that we can do this by adding a special predicate \(E\) for “exists” which is satisfied at a world by all and only the things in the world’s domain. Then we could say what we wanted as \(\forall x ((Fx \wedge Ex) \rightarrow Gx)\).
That’s on the right track, but it can’t be right as it stands. The problem is that we want all our predicates to be persistent and refinable, or the logic won’t work. A special predicate \(E\) is satisfied only by Romeo and Juliet in \(\langle \textit{Romeo loves Juliet}\rangle\) would not comply with refinability. For any object (other than Romeo and Juliet), we want them to not be \(E\). But there is no later world where there is some name \(c\) for them, and \(\neg Ec\) is true. So by refinability, all objects are \(E\), defeating the point of the proposal.
I think this is just a terminological issue though. The key thing is to not use a predicate here, but a demonstrative. There are two ways we could do this.
First, we might introduce a new plural referring term \(\mathbf{U}\), short for Us, which picks out everything in our possibility. Imagine introducing \(\mathbf{U}\) while waving your arms around and saying that it refers to us. The thought is that \(\mathbf{U}x\) does not mean that \(x\) exists, but rather that \(x\) is one of us. If Romeo or Juliet were to introduce \(\mathbf{U}\) that way, then the thing they say by uttering “Holmes is not one of us” would be true. Plausibly, though this depends what the ontology of fictional characters is, we also say something true by saying “Holmes is not one of us.”
There is a slightly more complicated, but I suspect ultimately more natural, way to get the same result. Let’s introduce a new name @ for the actual possibility. As with \(\mathbf{U}\), it is important that this is a directly referring expression. It is indexical, in the way that ‘actually’ is in two-dimensional semantics (Davies and Humberstone (1980)). Then we introduce a special new two-place predicate \(I\), meaning ‘in’. Using those we can say that all actual \(F\)’s are \(G\)’s by saying \(\forall x((Fx \wedge Ix@) \rightarrow Gx)\). The move here is parallel to what modal realists have to do to understand ordinary, world-bound, quantification.
There is one odd thing about either of these moves. They both involve indexicals. So one might expect that there will be a distinction between the proposition that these utterances express, and the ‘diagonal’ proposition consisting of those possibilities that make true the proposition the utterance would express there. In the possibilities framework, there can’t be these diagonal propositions. The problem is just the one we ran into when we tried to do actualist quantification. We can certainly find a set of possibilities where the content the utterance would have if uttered there is true there. But that set will typically not be a proposition because it won’t satisfy persistence or refinability. In general, one complication of the possibilities framework is that we cannot find a proposition for arbitrary sets of possibilities, and that is what causes the non-existence of the diagonal proposition here.
Once we understand world-bound quantification using one or other of these indexicals, the possibilities framework turns out to have some interesting interactions with recent debates about necessitism. Following Williamson (2013), there has been a resurgence of attention in the idea that everything exists necessarily. Here is a very quick argument for necessitism.
- The right logic for quantified modal logic is a normal modal logic with classical first-order logic.
- So it is a logical truth that \(\Box \forall y \Box \exists x(x = y)\).
- The right modal logic is not just normal, it is at least as strong as KTB.
- In the possible worlds framework, the only way to have \(\Box \forall y \Box \exists x(x = y)\) and for the modal logic to be as strong as KTB is for the domain to be constant across all worlds connected by the ancestral of \(R\).
- So the domain of all these \(R\)-connected worlds is constant, i.e., necessitism is true.
Here’s a tempting, but I suspect ultimately mistaken, way to respond to this argument. The possibilities framework shows that the move from step 4 to step 5 is wrong. In the the possible worlds framework, it’s true that we need a constant domain to validate these logical truths. But we don’t need a constant domain in the possibilities framework to make \(\Box \forall y \Box \exists x(x = y)\) come out true. So if that’s the master argument for necessitism, the possibilities framework provides one with tools to resist it.
This seems too quick to me for three reasons. One is that a theory which separates domains of possibilities from the truth conditions for the quantifiers as dramatically as this theory does is not likely to be particularly convincing to necessitists. That’s not worrying on its own; lots of good arguments aren’t convincing to theorists committed to the other side. The other two concerns, however, are more pressing.
So the second worry is that this is far from the only argument for necessitism. Several of the more pressing arguments involve second-order modal logic. I haven’t even tried to formulate second-order logic in the possibilities framework, and until that’s done, it’s impossible to respond to these arguments. And the third worry is that plenty of necessitists do not think that that necessitism has anything to do with constant domains. Here is how Williamson introduces the view.
Call the proposition that it is necessary what there is necessitism, and its negation contingentism. In slightly less compressed form, necessitism says that necessarily everything is necessarily something; still more long-windedly: it is necessary that everything is such that it is necessary that something is identical with it. In a slogan: ontology is necessary. Contingentism denies that necessarily everything is necessarily something. In a slogan: ontology is contingent. (Williamson 2013, 2)
On Williamson’s own account, necessitism looks like what we get to in step 2 of the little argument. The distinction between the worlds framework and the possibilities framework only becomes relevant after the game is already over.
A better use of the possibilities framework might not be to reject necessitism, but to explain away some of its counterintuitive consequences. Surely, say the contingentists, some things might not have been, and other things that are might not have been. In the possibilities framework, we have a way of accommodating these intuitions, even while endorsing necessitism.
There is some good sense in which possible beings, like Holmes, are not actual. They are not in the actual world. At least, necessitism does not entail that they are in the actual world. Holmes is not one of us. Conversely, the Holmes stories may well be possible (provided we fix where Watson’s war wound is), and I’m not in them. It’s true that the sentence \(\exists x (x = \textit{Brian})\) is true in the Holmes stories, but surprisingly that can be true without me being in the Holmes stories.
So once we understand quantifiers and indexicals in the possibilities framework the right way, we can say things that are very close to contingentism without actually giving up necessitism. I suspect the best way to interpret that is as good news for a kind of necessitism, one that is willing to concede that there is something to contingentist intuitions. Whether that kind of slightly qualified necessitism is viable will depend on further questions, in particular questions concerning how to do second order quantification in the possibilities framework, but it seems worth investigating.
4 Conditionals
The only discussion of possibilities (as opposed to worlds) in The Connectives is in the chapter on disjunction. But there are several potential connections to conditionals, and in this section I’ll go over a couple of them.
One connection concerns Conditional Excluded Middle (as discussed on pages 1008–1013 of The Connectives), and more generally the relationship between \(A \boxright (B \vee C)\) and \((A \boxright B) \vee (A \boxright C)\). On a Stalnaker–Lewis style approach to conditionals, these are equivalent iff there is a nearest possible world in which \(A\) is true, for any possible \(A\). It is natural to think about whether that is true in largely metaphysical terms, asking whether there really is guaranteed to be a single nearest world where \(A\) is true. And as Lewis (1973) argued, it is natural to answer that question negatively.
To take one striking example of that, consider the discussion by Jeremy Goodman (2018) of the example, originally due to Max Black (1952), of the two spheres alone in space. Black says that the spheres are really two, so this is a counterexample to the Principle of Identity of Indiscernibles. Let’s assume, for now, that Black is right, and we can call one sphere \(a\) and the other sphere \(b\). Goodman asks what we should say about the counterfactual possibility that one of the spheres is heavier. On Lewis’s picture, rejecting Conditional Excluded Middle, both (1) and (2) are false.
- If one of the spheres were heavier, it would be \(a\).
- If one of the spheres were heavier, it would be \(b\).
A common thought at this point is that this verdict really does follow from Lewis’s ‘nearest possible world’ semantics for conditionals, but that data about the inferential role of conditionals shows that Conditional Excluded Middle must be correct.4 This is, many think, a problem for the Lewisian view.
4 For a recent statement of this last view, with many more citations to similar statements, see Cariani and Goldstein (2020).
One move here, discussed by Humberstone (2011, 1011), is to use supervaluations. Perhaps it is in some sense indeterminate whether the world where \(a\) is heavier or the world where \(b\) is heavier is more like actuality. A better move is to analyse conditionals not in terms of possible worlds, but in terms of possibilities.
Here is one possible way to analyse conditionals, mixing Stalnaker’s approach with Humberstone’s possibilities.5 Extend a possibility model \(\langle W, \leqslant, V \rangle\) to a conditional possibility model by adding a selection function \(f\). This is a function \(\mathcal{P}(W) \times W \rightarrow \mathcal{P}(W)\), intuitively picking out the ‘nearest’ possibilities to a world where a particular proposition is true, satisfying these constraints.6
5 The particular formulation I’m going to use draws heavily on the theory presented by Andrew Bacon (2023, 382). Three of the four conditions are directly quoted from his paper, though I mean something different by them since on my version the variables pick out possibilities, not worlds. Bacon does not endorse AB in the form shown here; he only endorses a version of it restricted to the case where \(f(B, x) = \emptyset\). I’ll have more to say about Bacon’s paper presently.
6 Humberstone uses \(R\) rather than \(f\) for the same idea; I’m going to follow Bacon, who in turn follows Stalnaker, to highlight the connection to theories that validate Conditional Excluded Middle.
- MP
- \(x \in f(A, x)\) whenever \(x \in A\)
- ID
- \(f(A, x) \subseteq A\)
- CEM
- \(|f(A, x)| \leq 1\)
- AB
- If \(f(A, x) \subseteq B\) and \(f(B, x) \subseteq A\), then \(f(A, x) = f(B, x)\)
The cardinality constraint CEM guarantees that Conditional Excluded Middle will hold. But we don’t have to make invidious choices about whether the nearest possibility where one of the spheres is heavier makes \(a\) or \(b\) heavier. Rather, we just say that the nearest possibility is an incompletely refined possibility that makes the disjunctive proposition One of them is heavier true, without making either disjunct true. It will have refinements where each is true, but the nearest possibility will not validate either disjunct. This seems like an intuitive treatment of the case.
Goodman uses this example to argue that Black was incorrect, and the spheres are in fact discernible. His argument is that one but not the other will have the property of being the one that would be heavier if they were different. I don’t think this argument goes through on the possibilities framework, but settling that would require saying more about how higher-order quantification works in the possibilities framework. As noted in Section 3, I’m leaving questions about higher-order logic for another day. Instead I’ll turn to a puzzle that Bacon aimed to solve in the paper I just mentioned. That puzzle starts with an example from José Bernadete (1964), which Kit Fine (2012a, 2012b) developed into a deep challenge to several theories of counterfactuals.
There is a room that is very dangerous to cross. A man is thinking of crossing it, but he is warned off when he learns that it contains an infinity of gods. God1 will kill him if he makes it half-way across the room. God2 will kill him if he makes it a quarter of the way across. God3 will kill him if he makes it one-eighth of the way across. More generally, Godn will kill him if he makes it (1/2)n of the way across the room. Does he enter? Of course not; he’d be killed! But who would kill him? Presumably not God1; how would he make it that far? This generalises. Godn can’t kill him, because Godn+1 would already have done the job. So he would be killed by the gods, but not by any particular god. This doesn’t sound very plausible.
The case looks like the kind that motivated Lewis to reject what he called the Limit Assumption. This says that if \(A\) is possible, then relative to any world \(w\) there are some closest worlds where \(A\) is true. Humberstone (2011, 1014–15) discusses Lewis’s rejection of the Limit Assumption, and adopts the position that we shouldn’t impose it in general, but can freely talk as if it is true, because it doesn’t make a difference to the logic. This is right in the context Humberstone is writing in, but potentially misleading. The Limit Assumption does make a big difference to the logic if we have either quantifiers or infinitary connectives in the language.7 This fact is what Fine’s puzzle turns on.
7 If we assume a constant domain, then whether \(A \boxright \forall x: \phi(x)\) and \(\forall x: A \boxright \phi(x)\) are equivalent depends on whether we adopt the Limit Assumption.
Stated without the Limit Assumption, Lewis’s view is that \(A \boxright B\) is true at \(w\) if there is some world where \(A\) is true such that there is no closer world where \(A \wedge \neg B\) is true. If we assume that for any n, the world where the man enters and Godn+1 kills him is closer than the world where he enters and Godn kills him, then Lewis is committed to both (3) and (4).
- If the man were to enter the room, he would be killed by either God1 or God2 or \(\dots\).
- For each n, it is not the case that if the man were to enter the room, he would be killed by Godn.
As Fine notes, Lewis’s theory of counterfactuals is committed to denying a principle he calls Infinite Conjunction.
- Infinite Conjunction
- If \(A \boxright C_i\) is true for each \(i\), then \(A \boxright (C_1 \wedge C_2 \wedge \dots)\) is true.
With the Limit Assumption, Lewis’s semantics would endorse Infinite Conjunction. But it would also have a problem. Which of the gods would kill the man? Any choice seems not only arbitrary, but mistaken. Let’s spell this out a bit more carefully. Fine’s way of spelling out the paradox makes heavy use of a principle he calls Disjunction. (This is called Subj. Dilemma by Humberstone (2011, 1015).)
- Disjunction
- If \(A \boxright C\) and \(B \boxright C\) are true, so is \((A \vee B) \boxright C\).
Then both (5) and (6) seem like they should be true.
- If the man had entered the room, and been killed by one of God1 through God2k, he wouldn’t have been killed by Godk (because God2k would have killed him first).
- If the man had entered the room, and not been killed by one of God1 through God2k, he wouldn’t have been killed by Godk.
Putting these together using Disjunction, we get the following sentence. It’s a mouthful, but it’s important to spell it out for what comes next.
- If either the man had entered the room, and been killed by one of God1 through God2k, or he had entered the room, and not been killed by one of God1 through God2k, he wouldn’t have been killed by Godk.
To finish off the paradox, let’s add a new principle, Antecedent Substitution.
- Antecedent Substitution
- If \(A\) and \(B\) are provably equivalent in classical logic, and \(A \boxright C\) is true, so is \(B \boxright C\).
Then using Antecedent Substitution we can get from (7) to (8).
- If the man had entered the room, he would not have been killed by Godk.
And since k is arbitrary in (8), we can derive (4), without any appeal to the metaphysics of counterfactuals. It looks like our only options, short of abandoning classical logic, are to give up one of Infinite Conjunction, Disjunction, or Antecedent Substitution. In Fine’s original discussion of the paradox he introduces several more principles that could in theory be given up, but Brian Embry (2014) convincingly argues that really it has to be one of these three, and I’m following his lead.
If these are the three options, it isn’t obvious which one to take. All three paths forward have their proponents, or at least are consequences of otherwise plausible views. I’ve already noted that Lewis (1973) is committed to rejecting Infinite Conjunction, because he does not endorse the Limit Assumption. This does not look great; it’s crucial to reasoning with counterfactuals that if some things would each be true were \(A\) the case, then were \(A\) the case they would each be true.
Fine (2012b) recommends giving up Antecedent Substitution. He develops a theory of conditionals that doesn’t use possible worlds, but instead uses incomplete states. These are not entirely unlike Humberstone’s possibilities, but the resulting theory is quite different. I suspect the key distinction, the one that drives all of the rest of the results, is that Fine takes disjunctions to be true at a state only if a disjunct is true at that state. Anyway, Fine thinks that the misstep in the trilemma above is the derivation of (8) from (7). That step requires substituting \(A\) for \((A \wedge B) \vee (A \wedge \neg B)\) in an antecedent, which Fine takes to be illegitimate.
There are a couple of reasons to be unhappy with this way of getting out of the problem. One is that this did not feel like the most controversial step when we were developing the problem. But a bigger one is that Fine’s resolution of the trilemma ends up endorsing not just Disjunction, but also its converse. This is the principle that Humberstone (2011, 1016) calls Conv. Subj. Dilemma, the key being that from \((A \vee B) \boxright C\) one can infer \(A \boxright C\). The criticisms of Fine in Embry (2014) largely centre on this aspect of Fine’s view, and its consequences. But the key problems with the principle are already pretty clear in Humberstone (2011, 1016–22). So I won’t go that way.
That leaves Disjunction. This is the step that Andrew Bacon (2023) rejects, and it’s what I’ll reject as well. I think there are two key reasons to worry about giving up Disjunction, and they are pretty hard worries to address in the possible worlds framework. But they both seem more or less manageable in the possibilities framework.
The first worry is that without Disjunction, we have to give up the idea that the selection function \(f\) is in any sense a measure of similarity, or really any kind of nearness. Here is how Bacon puts it:
The second consideration in favour of Disjunction is that its validity is predicted by the now dominant account of counterfactuals, prominently defended by Lewis and Stalnaker, based on similarity semantics. For roughly, if the closest \(A\) worlds are \(C\) worlds, and the closest \(B\) worlds are \(C\) worlds, then the closest \(A \vee B\) worlds are \(C\) worlds. (Bacon 2023, 374)
Now for Bacon, this isn’t a worry, because he thinks there are independent reasons to reject the picture of similarity or nearness as being foundational to counterfactuals. I don’t find those reasons convincing, and basically agree with the response to them that Fine (2023) gives. But it’s not just critics of Disjunction who connect it to the similarity picture. The same connection is drawn by Humberstone, who does endorse Disjunction. He first notes that, by analogy with other maximal relations, we should expect that the nearest \(A \vee B\)-world will be either the nearest \(A\)-world or the nearest \(B\)-world (Humberstone 2011, 1015). He then argues more formally that if the set of nearest \(A\)-worlds is generated by an underlying three-place similarity relation \(S_wxy\), meaning \(x\) is at least as similar to \(w\) as \(y\) is, and if for any \(w\), \(S_w\) is a total preorder on worlds, then Disjunction is guaranteed to hold (Humberstone 2011, 1025–26).
That argument seems irrefutable if, but only if, we’re working in a possible worlds framework. If we’re in a possibilities framework, it doesn’t look right. It could be that the nearest possibility in which \(A \vee B\) is true is not identical to either the nearest possibility in which \(A\) is true or the nearest possibility in which \(B\) is true, but is instead a coarsening of one of those possibilities.
How could Disjunction fail on the possibilities picture? We must fail to have the nearest \(A \vee B\)-possibility be a \(C\)-possibility. But that’s impossible if the nearest \(A\)-possibility is a \(C\)-possibility, and the nearest \(B\)-possibility is a \(C\)-possibility, and all \(A \vee B\)-possibilities are \(A\)-possibilities or \(B\)-possibilities. On the possibilities picture, that last clause fails. It might be that the nearest possibility that makes \(A \vee B\) true does not make either \(A\) true or \(B\) true; it just guarantees that a sequence of refinements will eventually make one or the other true. Also note that we don’t require that the nearest \(A \vee B\) possibility makes \(\neg C\) true; it could be that Disjunction fails because \(C\) is true at some refinements of the nearest \(A \vee B\) possibility and \(\neg C\) is true at others.
That’s what happens with our man who wisely doesn’t enter the room. What’s the nearest possibility in which he does enter the room? It’s the incomplete possibility where he is killed by one of the gods. For each i, j with i > j, the possibility where he is killed by Godi is closer than the one where he is killed by Godj. But the indeterminate possibility where he is killed, without specifying which god kills him, is closer to actuality than any complete possibility that specifies the homicidal divinity.
So rejecting Disjunction is compatible with the similarity approach to counterfactuals, as long as we use possibilities. The other worry about rejecting Disjunction, one Fine (2023) stresses in his response to Bacon, is that we use Disjunction a lot in ordinary counterfactual reasoning. We should be cautious about giving it up. Of course, there are plenty of rules that we use in ordinary reasoning that work in all but a few edge cases. If we could show that Disjunction was truth-preserving in all but some rare exceptional cases, we could justify using it as a rule of inference. After all, we don’t think that the failure of Axiom V in full generality means that it’s always a mistake to infer from the existence of some things to the existence of a set containing all and only them.
As Fine stresses, it’s hard to see how on Bacon’s view Disjunction would even count as typically fine. The reason Bacon says that it fails in the paradoxical case from Bernadete generalises to more humdrum uses.
But that’s not true for the possibilities model. You need to have some very unusual relationships between the family of similarity relations \(S_w\) and \(\leqslant\) in order for Disjunction to fail. For one thing, you need the nearest possibility \(x\) where \(A \vee B\) holds to be an incomplete possibility where neither \(A\) nor \(B\) holds. Clearly \(x\) has to have refinements where \(A\) is true, and refinements where \(B\) is true. To get a Disjunction failure, you need one of the refinements of \(x\) where one of the disjuncts is true to not be (a refinement of) the closest possibility to actuality where that disjunct obtains. I can’t construct an intuitive case where that happens that doesn’t involve infinite sequences like in Bernadete’s case, though I also don’t have a proof that no such case can be constructed. This is all far from conclusive, but it seems plausible that on the possibilities model, failures of Disjunction will be rare. And that would be enough to explain the fact Fine appeals to, that we are usually happy using it in everyday inferences.
5 Conclusion
I’ve gone over two uses of the possibilities framework, but there are many more things we could imagine using it for. I’ll end by briefly mentioning two of them.
As I alluded to in Section 4, possibilities can do a lot of work that philosophers have tried to make supervaluations do. As well as using possibilities instead of supervaluations in preserving Conditional Excluded Middle, it’s worth exploring whether they are useful in thinking about vagueness, or about the open future.
Humberstone (1981) mentions that possibilities do a better job than possible worlds at making sense of talk about ‘belief worlds’. We could say the same thing about the worlds of fiction. David Lewis (1978) ends up treating the operator In this fiction as box-like, because he thinks otherwise we’d have to make arbitrary choices about some details of how the story is to be filled out. Using possibilities here seems smoother. Any coherent fiction, I conjecture, picks out a particular possibility. That will always be a less than fully refined possibility, but a possibility nonetheless. On this approach we don’t get left with the unfortunate triple, to which Lewis is committed, of its being true in the story that \(A \vee B\), but false that it’s true in the story that \(A\), and false that it’s true in the story that \(B\). I also think there might be some uses of possibilities in characterising the distinctive relationship between a story and its sequel. That a sequel is a refinement of the original story might do a better job of capturing some features of this relationship than anything we can do in a worlds framework.
But I’ll leave those tasks for another day. The main point of this paper is to remind the reader how many uses Humberstone’s notion of a possibility has, and to explore what happens to the logic of possibilities when we add quantifiers or infinitary connectives.