Speaking for Haecceitists

1. The Problem of Curious Indeterminism

1.1 Haecceitism

The first task is to isolate exactly which assumptions are needed to generate the problem of curious indeterminism, and to clarify how best to formulate them. I will begin with the thesis of haecceitism.

Haecceitism is usually characterized as the thesis that there are worlds which share the same qualitative facts but not the same non-qualitative (or ‘haecceitistic’) facts. To formulate this more carefully, we need to characterize the notions which are used to state it, such as those of a ‘world’ and ‘sharing the same qualitative facts’. This can be done in a simple framework which extends a standard quantified propositional modal language with another propositional operator—an expression which combines with a formula to produce a formula—whose intended interpretation is the property of propositions is qualitative (‘$Q$’). I will assume that this framework is governed by the principles of classical logic, a classical quantification theory for the propositional quantifiers, and the modal system $\mathrm{𝖲𝟦}$, the principles of which are as follows:³

K    $\mathrm{\square }(p\to q)\to (\mathrm{\square }p\to \mathrm{\square }q)$

T    $\mathrm{\square }p\to p$

4    $\mathrm{\square }p\to \mathrm{\square }\mathrm{\square }p$

Necessitation    If $⊢A$, then $⊢\mathrm{\square }A$

For the time being, I do not make any assumptions about qualitativeness. However, at various points I discuss how natural principles about qualitativeness interact with the discussion.

In this setting there is a natural conception of worlds which has its roots in Prior (1957) and Fine (1970). According to this conception, worlds are just possibly true, maximally strong propositions, and for a proposition to be true at a world is for that world to necessitate that proposition. This conception of worlds and the corresponding notion of truth-at-a-world are formalized as follows:

$\mathit{\text{Wp}}:=\mathrm{♢}p\wedge \forall q(\mathrm{\square }(p\to q)\vee \mathrm{\square }(p\to \neg q))$

$p\models q:=\mathrm{\square }(p\to q)$

To improve readability, I introduce a convention for simplifying claims about worlds. According to this convention, formulas with the form of the schemas on the left may be rewritten in the manner displayed on the right:

$\forall p(\mathit{\text{Wp}}\to \phi )$	$\forall w\phi [w/p]$
$\exists p(\mathit{\text{Wp}}\wedge \phi )$	$\exists w\phi [w/p]$

Here, $\phi [w/p]$ is obtained by replacing all free occurrences of $p$ in $\phi$ with $w$. I reserve the variables $w$, $v$ and $u$ for this convention, which do not belong to the official object-language. It is worth noting that there may be multiple, necessarily equivalent worlds which make true exactly the same propositions; nonetheless I will often informally quotient that multiplicity away and speak as if any such worlds are one.⁴

To supplement this understanding of worlds, I will assume a principle which guarantees that worlds play their presumed theoretical role of being the witnesses of possibility claims. This is captured by a familiar Leibnizian principle:

Leibniz Biconditional    $\mathrm{\square }\forall p(\mathrm{♢}p\leftrightarrow \exists w\hspace{0.17em}w\models p)$

This permits us to move freely between talk of existing propositions being possible and their being true at a world. As one would expect, this principle is equivalent in the current setting to the (necessitation of the) claim that a proposition is necessary iff it is true at all worlds.

To state the worlds-based haecceitist thesis, I introduce a relation of qualitative equivalence between worlds. Intuitively, this relation holds between a pair of worlds when they make true exactly the same qualitative propositions. In the framework I am using, one may formalize this notion as follows:

$p\approx q:=\forall r(Qr\to (p\models r\leftrightarrow q\models r))$

It is straightforward to see that this relation is an equivalence relation on worlds.

This allows us to state the worlds-based haecceitist thesis as the claim that there exists a pair of qualitatively equivalent worlds which disagree over the truth of some proposition (equivalently: they are not necessarily equivalent worlds).

Haecceitism   $\exists w\exists v(w\approx v\wedge \neg \mathrm{\square }(w\leftrightarrow v))$

Since qualitative equivalence is an equivalence relation on worlds, at each world we may think of the worlds as partitioned into equivalence classes, with membership of each class determined solely by which qualitative propositions are true at each world.

Fig. 1. Left: a representation of modal space (at a world) according to anti-haecceitism, in which no qualitative equivalence class contains more than one world. Right: a representation of modal space (at a world) according to haecceitism, with some qualitative equivalence classes containing more than one world.

In this setting, haecceitism can be understood as the thesis that at least some of these equivalence classes contain more than one world.

It is worth noting that this picture becomes even simpler in the presence of further assumptions. In particular, the underlying modal system could be strengthened to $\mathrm{𝖲𝟧}$ by extending it with the following principle:

B    $\mathrm{\square }\forall p(p\to \mathrm{\square }\mathrm{♢}p)$

Under this assumption, it becomes non-contingent which worlds exist and which propositions are true at a given world. Alongside this extension, one could make the further, popular assumption that qualitativeness is a necessary status of the propositions which have it:

Qualitative Persistence    $\mathrm{\square }\forall p(\mathit{\text{Qp}}\to \mathrm{\square }Qp)$

In $\mathrm{𝖲𝟧}$ this is equivalent to qualitativeness being a non-contingent status of propositions. Thus on this strengthened picture qualitative equivalence becomes a non-contingent equivalence relation on worlds. This would mean that the partition of worlds into qualitative equivalence classes is modally invariant: it does not change from world to world. There is a certain elegance to this picture, but since neither $\mathrm{𝖲𝟧}$ nor Qualitative Persistence are needed for my arguments, I will not take them as assumptions.⁵

1.2 $\phi$-Haecceitism

The general thesis of haecceitism has now been stated. But recall that a specific form of this thesis featured in the problem of curious indeterminism. In particular, the problem concerned a haecceitist who countenanced a world that shares the actual world’s qualitative facts and its history up to a certain time.

Such particular haecceitist theses can also be captured neatly in the current framework. The basic idea is twofold. In addition to the relation of qualitative equivalence, one can introduce a partial equivalence relation on worlds that holds between a pair of worlds when they are qualitatively equivalent and both make true a certain proposition $\phi$. For example, that proposition could be one describing the history of the actual world up to a certain time.

$w{\approx }_{\phi }v:=w\approx v\wedge w\models \phi \wedge v\models \phi$

Moreover one can say what it is for a world to bear this relation to the actual world:

$w{\approx }_{\phi }\mathrm{@}:=\exists v(v\wedge w{\approx }_{\phi }v)$

I pronounce the former claim as ‘$w$ and $v$ are qualitatively equivalent $\phi$-worlds’ and the latter as ‘$w$ and the actual world are qualitatively equivalent $\phi$-worlds’. Note that $w$ and $v$ are qualitatively equivalent $\phi$-worlds only when they both make true $\phi$.

Fig. 2. A representation of modal space (at a world) according to haecceitism, with some qualitative equivalence classes containing more than one world; the dotted rectangles represent equivalence classes under the partial equivalence relation ${\approx }_{\phi }$.

With these definitions in view, we can state the type of haecceitist thesis that figures in the problem of curious indeterminism:

$\phi$-Haecceitism    The actual world and some non-actual world are qualitatively equivalent $\phi$-worlds.

This may be formalized as: $\exists w(\neg w\wedge w{\approx }_{\phi }\mathrm{@})$. To obtain the instance of $\phi$-Haecceitism that was used in the opening example, consider any time just before the particle collision, and let $p$ be a proposition describing the intrinsic history of the world up to that time. Substituting $p$ for $\phi$ in $\phi$-Haecceitism delivers the relevant haecceitist thesis: there is a non-actual world qualitatively equivalent to the actual world that shares the actual world’s intrinsic profile up to just before the particles $a$ and $b$ were created.

2. Haecceitistic Realizations

Upon being confronted with the indeterminism argument, haecceitists might just acquiesce in its conclusion. Indeed many have been tempted just to accept the relevant instance of $\phi$-Indeterminism and retreat to a weaker qualitative determinist thesis that is not undermined purely by the presence of mere haecceitistic differences between worlds:¹⁷

Qualitative $\phi$-Determinism    Any physically possible $\phi$-worlds are qualitatively equivalent.

What is tempting about this reaction is that haecceitists would appear to have few alternatives. If one accepts a suitable instance of $\phi$-Haecceitism, the only alternative to embracing $\phi$-Indeterminism is to reject either Nomic Inclusion or Nomic Entailment. Yet Nomic Inclusion is just the thought that the physical laws are true qualitative propositions—a widely accepted, partly empirical claim whose rejection looks difficult to shoulder. Moreover, as we have seen, Nomic Entailment stems from a natural conception of restricted necessities.

Be that as it may, I want to explore a response to the indeterminism argument that challenges Nomic Entailment. This response accepts the claim that the physical laws are true qualitative propositions, but it rejects the thought that physical necessity is a restriction of metaphysical necessity by only qualitative propositions. My line of argument is general: my aim is to argue that haecceitists have reason to reject such thoughts about many such necessities as a matter of principle.

Bringing out why this is the case will involve looking at haecceitism in a slightly different light. Haecceitism is typically viewed as an exclusively metaphysical doctrine, something within the purview of metaphysics and only of metaphysics. But its scope is actually much broader: haecceitism has ramifications for how we must view ordinary modal talk and the general practice of modal theorizing. The aim of this section is to highlight these ramifications; in the next section, I return to the indeterminism argument with them in view.

Turning to the details, suppose that in Particle Collision a scientist arrives hours before the collision occurs and closely observes the entire process unfold, including the trajectories of the different particles emitted from the collision. In describing this scenario, many would find it difficult—if not absurd—to reject the following counterfactuals (where $t$ is the trajectory along which $a$ is in fact emitted):

• (1) Had the scientist arrived one nanosecond later, he would still have observed $a$ being emitted along $t$.

• (2) Had the scientist cleared his throat one more time before arriving, he would still have observed $a$ being emitted along $t$.

• (3) Had a particle on Mars been moving slightly faster, the scientist would still have observed $a$ being emitted along $t$.

Let us say that the emission of $a$ along $t$ is counterfactually stable with respect to the different antecedents in question. This counterfactual stability arises because the emission of $a$ along $t$ is not contingent on minutiae such as the scientist’s exact arrival time or events on Mars.

This counterfactual stability is not an isolated phenomenon. In describing the scenario, many would also find it difficult to reject any of the following claims:

• (4) Given that the scientist was observing the collision so closely, he couldn’t easily have missed $a$ being emitted along $t$.

• (5) Given that the scientist was observing the collision so closely, it was practically impossible for him to miss $a$ being emitted along $t$.

• (6) Given that the scientist was observing the collision so closely, it was humanly impossible for him to miss $a$ being emitted along $t$.

Indeed it would be utterly natural for one to assert just the following:

• (7) Given that the scientist was observing the collision so closely, he couldn’t have missed $a$ being emitted along $t$.

The initial counterfactual stability judgments are accompanied by other stability judgments. The scientist’s witnessing the emission of $a$ along $t$ is modally stable according to many modalities—easy possibility, practical possibility, human possibility, and modalities regularly invoked in ordinary discourse. Let us call this family of modalities the local modalities.

Now, although such stability is present in the realm of local possibility, haecceitists must recognize that there is a broader realm of possibility from which it is absent. Given any antecedent-satisfying world $w$, haecceitists will maintain that there is a world $v$ which differs from $w$ by a mere permutation of $a$ and $b$. For example, suppose $w$ is a world where the scientist arrives one nanosecond later and $a$ is emitted along $t$. Typically, haecceitists will countenance a world with the same qualitative facts as $w$, at which the scientist also arrives one nanosecond later, but where the careers of $a$ and $b$ have been permuted.

Despite this, I take it that no one would seriously claim that haecceitists must reject (1)-(7). The reason for this is straightforward: just because a certain haecceitistic swap is metaphysically possible, it does not follow that it is locally possible. This allows counterfactuals to discriminate between qualitatively equivalent worlds, indeed discriminate between certain qualitatively equivalent $\phi$-worlds, for various substitution instances of $\phi$ (e.g., where $\phi$ is a detailed haecceitistic proposition which necessitates that the scientist arrives one nanosecond later).

These considerations generalize to the other examples. If modal judgments like (1)-(7) are correct, and if they are representative of how the local modalities generally behave, the local modalities must discriminate between certain qualitatively equivalent $\phi$-worlds.

Fig. 3. Two representations of the modal space (at a world) from Fig. 2. Left: the arrows represent what is possible from some world according to a necessity that does not discriminate between qualitatively equivalent $\phi$-worlds. Right: the arrows represent what is possible from that world according to a necessity that does discriminate between qualitatively equivalent $\phi$-worlds.

This demonstrates a crucial point about how haecceitists should view ordinary modal talk. As they must see it, the local modalities may ‘hold fixed’ a great deal of haecceitistic information that is neither made explicit nor salient on the relevant occasion of use. For example, consider again the modality of easy possibility. A familiar thought is that, as used in (4), this modality holds fixed lots of information. It holds fixed propositions about the scientist observing the collision, precise features of the immediate environment, what preceded the observation, certain laws, certain aspects of the qualitative state of the world, and so on. But additionally the haecceitist must recognize that, on the same occasion, it also holds fixed information about which particular haecceitistic realization a qualitative state may achieve. For were such information not held fixed, our ordinary modal talk—as exemplified by (1)-(7)—would be pervaded by error.

This phenomenon can be described much more precisely. Let us say that a proposition is a world selector when it is compatible with exactly one member (modulo necessary equivalence) of any qualitative equivalence class of worlds:

$\text{Selector}:=\lambda p.\forall w\exists v\forall u(\mathrm{\square }(v\leftrightarrow u)\leftrightarrow (w\approx u\wedge \mathrm{♢}(p\wedge u)))$

For each total possible qualitative state, world selectors pick out a unique haecceitistic realization that witnesses that qualitative state. Intuitively, one may think of them as large, haecceitistic conjunctions of material conditionals, each linking a total possible qualitative state to a particular haecceitistic realization of it, with no two material conditionals in the conjunction sharing the same antecedent.

Fig. 4. A representation of modal space (at a world) according to haecceitism; the dotted rectangles represent a world selector.

Next, say that a necessity is selecting when it applies to a world selector:

$\text{Selecting}:=\lambda X.\text{Nec}X\wedge \exists p(\text{Selector}\hspace{0.17em}p\wedge \mathit{\text{Xp}})$

For any metaphysically possible total qualitative state, such necessities admit as possible at most one world that instantiates that state. (There may, however, be some metaphysically possible qualitative states which those necessities deem impossible.) As a consequence, they will behave like ‘anti-haecceitist necessities’. By this, I mean they will license the thesis of anti-haecceitism (i.e. the negation of haecceitism), according to which any qualitatively equivalent worlds they deem as possible are the same:

$⟨X⟩:=(\lambda p.\neg X\neg p)$

$\text{AH}:=\lambda X.\forall w\forall v(w\approx v\wedge ⟨X⟩w\wedge ⟨X⟩v\to \mathrm{\square }(w\leftrightarrow v))$

Theorem 3. $⊢\text{Selecting }X\to \text{AH}X$

The selecting necessities, then, are those in which haecceitists may see a grain of truth in anti-haecceitism.

Before putting selecting necessities to use, it is important to head off one potential confusion about them. As Theorem 3 captures, the thesis of anti-haecceitism is true with respect to anti-haecceitist necessities. However, these necessities license anti-haecceitism because they hold fixed a great deal of haecceitistic information. This haecceitistic information is codified in the world selector to which they apply, which, recall, one can think of as linking each metaphysically possible total qualitative state with a unique haecceitsitic realization of it. This is what makes selecting necessities anti-haecceitist. For any total qualitative state they deem possible, there is only one haecceitistic realization of it they also deem possible: the one ‘given’ by the world selector to which they apply. So a necessity is anti-haecceitist by virtue of holding fixed so much haecceitistic information.

To now put selecting necessities to use, what I take the above examples of ordinary modal talk to indicate is that haecceitists should think that the local necessities are selecting.¹⁸ On this picture, when we engage in modal theorizing we often invoke modalities that ‘hold fixed’ propositions about which haecceitistic description will be realized if the world is in a certain total qualitative state. In other words, it is a picture on which ordinary modal discourse is permeated with haecceitistic content in unexpected ways. This would explain why seemingly obvious truths like (1)-(7) are still true in the presence of haecceitism.

To conclude this section, I want to highlight two important points about the claim that the local necessities are selecting. According to the definition above, for a necessity to be selecting is just for it to apply to some world selector—for it to select a unique haecceitistic realization of each total qualitative state it deems possible. The first point I want to make is that, in one respect, this is quite a weak condition. It is natural to assume that there are many different world selectors, some of which will select haecceitistic realizations of qualitative states that, intuitively, are not possible according to any of the local necessities. For example, consider a haecceitistic realization of the actual qualitative state in which you occupy the qualitative role of an inanimate object. In saying that the local necessities are selecting—that they apply to some world selector—I am not trying to specify precisely which world selectors they apply to. That is a difficult question best deferred to the metasemantics of those expressions. And there is nothing unusual about doing so: all questions about which facts get held fixed in modal discourse are best deferred to metasemantics, and this is simply one of them.

This makes salient the second point, which is that one must both permit and expect vagueness in which world selector a local necessity is based on in a given context. In fact, a natural model of this is already available: it is inspired by the Stalnakerian treatment of counterfactuals. This treatment validates ‘conditional excluded middle’, which, intuitively, amounts to there being a unique counterfactually closest world. To assuage concerns about this world being selected arbitrarily, however, the Stalnakerian permits the counterfactual conditional to be a source of vagueness. And so although determinately there is a unique counterfactually closest world, there is no unique world that is determinately the counterfactually closest one.¹⁹ Plausibly, this model will predict that the counterfactual often makes selections between qualitatively equivalent worlds. In such contexts, its different precisifications will hold different bodies of haecceitistic information fixed—in particular, information about the haecceitistic realizations of qualitative states. Different haecceitistic realizations of one and the same qualitative state will then be possible on different sharpenings of the counterfactual.²⁰ It is precisely this idea that I recommend the haecceitist use to allow for intra-contextual vagueness in which world selector is operative. The suggestion is that a given local necessity X, say practical necessity, determinately applies to some world selector, but there is no particular world selector such that determinately X applies to it. In other words, each local necessity determinately necessitates that a given possible qualitative state has a unique haecceitistic realization, but there is no particular haecceitistic realization of it which the necessity determinately necessitates that qualitative state to realize.

3. Nomic Selection

With these observations in view, we can return to the indeterminism argument. In that argument, Nomic Entailment was motivated by the thought that physical necessity is a restriction of metaphysical necessity by a body of only qualitative propositions, the physical laws. However, once the haecceitist has come to appreciate that ordinary modal discourse is permeated with unexpected haecceitistic content, this inference should strike them as much less obvious. If the necessities invoked in ordinary discourse are selecting, it would not be surprising if some of our utterances of ‘physical necessity’ expressed similar necessities. But then in those contexts Nomic Entailment would be false, for physical necessity would be a restriction of a selecting necessity—as opposed to metaphysical necessity—by only the physical laws.

How might utterances of ‘physical necessity’ express such a restriction? The question becomes particularly salient when one recalls that physical necessity is often equated with entailment by being a physical law, a property of only qualitative propositions. Nevertheless haecceitists have a natural answer: the world selector is given by the operative notion of entailment. For whereas initially only one entailment relation was identified, the reality is that there are many such relations, each generated by a given species of necessity. That is to say, given a necessity X we can characterize a notion of X-entailment:

${\le }_X:=\lambda Fp.\exists C(C\equiv F\wedge X({\bigwedge }_C\to p))$

Indeed for any property of propositions F and any necessity X, the operation being X-entailed by F is a necessity:²¹

$X^F:=(\lambda p.F{\le }_{X}p)$

Theorem 4. $⊢\text{Nec}X\to \text{Nec}{X}^F$

Thus if physical necessity is a restriction of a selecting necessity, it is still guaranteed to be a genuine species of necessity.

This makes salient an alternative to Nomic Entailment according to which physical necessity is a restriction of some particular selecting necessity by being a physical law:

Nomic Selection    For some selecting necessity X: physical necessity is X-entailment by being a physical law.

In formal terms: $\exists X(\text{Selecting}X\wedge \mathrm{■}=X^L)$. This implies that physical necessity itself is selecting, and so it will not deem possible distinct qualitatively equivalent worlds.

Crucially, even in the presence of $\phi$-Haecceitism and Nomic Inclusion, Nomic Selection does not imply $\phi$-Indeterminism, for, as just mentioned, physical necessity will no longer regard any distinct qualitatively equivalent worlds as both possible—at most one of them will be selected by it. To make this point more concrete, we can connect it with the opening example of Particle Collision. Thinking back to that example, we can now see that Nomic Selection implies that at most one of the qualitatively equivalent worlds that the haecceitist posited on the basis of Particle Collision will be physically possible. Accordingly, the mere existence of such a pair of worlds will not constitute a counterexample to determinism. Nomic Selection thus delivers to the haecceitist a conception of determinism that is compatible with the modal judgments they wish to make about Particle Collision.

Several comments about this general point are now in order. First, notice that many of the local modalities are characterized in terms of entailment by certain propositions. For instance, it is common to see practical necessity characterized by what is entailed by our practical constraints. However, if practical necessity licenses claims like (5), as it should do, the underlying relation of entailment in terms of which it is characterized should be X-entailment, where X is not metaphysical necessity but rather some selecting necessity. Thus the key thought behind Nomic Selection is a generalization of a mechanism which underwrites much of ordinary modal discourse.

Second, it is important to return to the remarks from the end of §2, where I emphasized that saying the local necessities are selecting is, in one respect, quite a weak claim. Saying that a local necessity applies to a world selector leaves open the question of to which world selector it applies. Instead of trying to settle this, however, I deferred it to the metasemantics of the modal expressions in question. Here I do the same. Some world selectors better fit our discourse about physical necessity than others; the metasemantics decides which one it expresses. In fact, like before, I doubt that any particular one of these world selectors is uniquely best: there is vagueness in to which world selector physical necessity applies. But all this does is bring the treatment of physical necessity even further in line with that of the local necessities, and the Stalnakerian treatment of the counterfactual that guides it.

Third, I am not suggesting that speakers think of what is expressed by ‘physical necessity’ under the guise of ‘X-entailment by being a physical law’ (where X is a selecting necessity). For just as one can think of the different individuals under the same indiscriminate guise, one can think of different necessities under an indiscriminate guise too. Indeed in different contexts speakers may think of what is expressed by ‘physical necessity’ under the same indiscriminate guise, for instance ‘what follows from the laws’ or ‘a modal status of the laws but not of our practical constraints’.

Fourth, it is important to be clear about the status of Nomic Selection. I am not suggesting that the term ‘physical necessity’ could never be used to express (metaphysically necessary) entailment by being a physical law. In some cases, it is simply stipulated to do so. Rather the key point is that in contexts where ‘physically necessary’ licenses determinist theses, it does so by expressing a selecting necessity. The claim is that the permeation of ordinary modal discourse by unexpected haecceitistic content colors our judgments about physical necessity and determinism. It is not that the haecceitistic restrictions are inescapable. Thus I am offering Nomic Selection to haecceitists as a hypothesis that explains and vindicates our determinist-judgments in contexts where they seem so compelling.

For the fifth point, recall the response to the puzzle of indeterminism that I discussed at the beginning of §2. This response was simply to reject full determinism and accept only qualitative determinism with respect to physical necessity, which it equated with (metaphysically necessary) entailment by being a physical law. In other words, the response embraced Qualitative $\phi$-Determinism and Nomic Entailment. Together, these theses imply a principle closely related to Qualitative $\phi$-Determinism

Qualitative $\phi$-Determinism${\mathrm{◇}}^L$ Any ${♢}^L$-possible $\phi$-worlds are qualitatively equivalent.

In the presence of Nomic Entailment, Qualitative $\phi$-Determinism${\mathrm{◇}}^L$ is equivalent to Qualitative $\phi$-Determinism; but without Nomic Entailment, they are not equivalent. Interestingly, though, given Nomic Selection there is a neat argument from Qualitative $\phi$-Determinism${\mathrm{◇}}^L$ to $\phi$-Determinism:²²

$\phi$-Determinism Any physically possible $\phi$-worlds are identical.

For take any physically possible $\phi$-worlds $w$ and $v$. Since physical possibility is no more inclusive than ${\mathrm{♢}}^L$-possibility, $w$ and $v$ would also be ${\mathrm{♢}}^L$-possible. Hence by Qualitative $\phi$-Determinism${\mathrm{◇}}^L$, they would be qualitatively equivalent. Yet if physical necessity is selecting, any qualitatively equivalent worlds which are both physically possible are identical, which means that $w$ and $v$ must be the same. The philosophical upshot to this is that, on my view, full determinism is downstream from a qualitative determinist thesis to which other responses to the puzzle retreat.

To summarize the main contention, I have drawn attention to the fact that haecceitists should recognize that much of our ordinary modal discourse is permeated with haecceitistic content in subtle ways. In light of this, I have proposed that it is natural for haecceitsts to see this haecceitistic permeation as afflicting expressions like ‘physically necessary’ too. As a consequence they have a new determinist-friendly solution to the indeterminism problem that draws on a philosophy of language which sits naturally with their view.

4. Conclusion: Cheap Anti-Haecceitism

To conclude, I want to highlight that there are close parallels between the position I have developed and Lewis’s (1986, chap. 4) famous doctrine of ‘cheap haecceitism’.²³

According to Lewis’s doctrine, worlds are disjoint, causally disconnected spacetimes. He also required that no two worlds differ ‘merely haecceitistically’, by which he meant that no two of them have the same qualitative character. However, in his theory there is an additional layer of complexity, which is that de re modal discourse is to be interpreted via the apparatus of counterpart theory. And according to the cheap haecceitist, different individuals may bear different counterpart relations to one and the same individual in a given world. So, as Lewis (1986, p. 230) puts it, possibilities are not possible worlds.

This allows the cheap haecceitist to recover both haecceitist and anti-haecceitist modal judgments in a way that many philosophers have found attractive. To take the example of Particle Collision, the cheap haecceitist will recognize only one spacetime with the relevant qualitative character, in which two duplicate particles of the same type are emitted from a particle collision along different respective trajectories. However, although there is only one such spacetime, there are many counterpart relations borne to the individuals there. There is one such relation which only the particle $a$ bears to the particle that is emitted along the trajectory $a$ in fact travels, and there is another relation which the particle $b$ (and perhaps still $a$ too) bears to the particle that is emitted along that trajectory. According to the former counterpart relation, the relevant thesis of determinism may be true and haecceitism false. This is because the interpretation of the modal operator furnished by that relation must deem it impossible for the qualitative state of the world to tolerate a haecceitistic swap between $a$ and $b$. But according to the latter counterpart relation, determinism is false and haecceitism true. This is because the interpretation of the modal operator furnished by that relation must deem it possible for that qualitative state to tolerate such a swap.

Seen from this perspective, the structure of the cheap haecceitist response closely parallels that of my own. Both approaches draw on their underlying metaphysics to identify different interpretations of modal operators which diverge over the truth of determinism and haecceitism respectively.²⁴ The metasemantics of the views are similar too. Lewis let context determine which counterpart relation is invoked, embracing vagueness in precisely which one is, just as my approach does with world selectors. The main difference between the approaches stems from their different underlying metaphysics. Whereas Lewis’s cheap haecceitist draws on a contentious metaphysics of worlds and counterpart relations, the version of haecceitism I have developed simply draws on a general theory of relative necessities. As I have tried to bring out, the underlying framework of relative necessities is grounded in assumptions that many will already accept. Thus, to reconcile the seemingly opposed haecceitist and anti-haecceitist judgments that generate the problem of indeterminism, one can avoid the detour through counterpart theory that might initially have seemed necessary.

This is important because, as Lewis emphasized, cheap haecceitism achieves its flexibility by divorcing possibilities from possible worlds. In effect, this involves rejecting his analogue of the Leibniz Biconditional. However, as others have emphasized, the consequences of doing this are not to be understated: indeed they raise the question of whether cheap haecceitism creates more trouble than it is worth—of whether cheap haecceitism genuinely is so cheap.²⁵ But since the version of haecceitism I have developed avoids these issues, the flexibility of cheap haecceitism can be had a better price.

Nevertheless, these close parallels notwithstanding, there is one respect in which my approach inverts the Lewisian one. Lewis described his approach as cheap haecceitism because he took no two worlds to differ merely haecceitistically. But on the approach I have proposed, it is anti-haecceitism which comes cheap: there is an unrestricted sense of ‘necessity’ according to which haecceitism is true, whereas anti-haecceitist judgments can be vindicated only when the sense of ‘necessity’ is restricted.²⁶ There is consequently a cheap substitute for anti-haecceitism available to haecceitists. And, as I have argued, they may use it to dispel one of the main concerns about their view.²⁷

Appendix

Theorem 2 (Indeterminism Argument).

$⊢\phi \text{-Haecceitism}\wedge \text{Nomic Inclusion}\wedge \text{Nomic Entailment}\to \phi \text{-Indeterminism}$

Proof. Given Nomic Entailment, we assume throughout that physical necessity is simply entailment by the laws, i.e. $\mathrm{■}={\mathrm{\square }}^L$. We first show that given Nomic Inclusion, i.e. $\forall p(\mathit{\text{Lp}}\to p\wedge \mathit{\text{Qp}})$, physical necessity does not discriminate between qualitatively equivalent worlds:

For this, observe that to establish the consequent we must just show that the collection $C$ coextensive with $L$ is such that:

Given that $v$ is a world, this is equivalent to:

Moreover, since $C$ is a collection, this is equivalent to:

To derive (2), observe that, for similar reasons to the equivalence of (2) and (4), the assumption ${\mathrm{♢}}^{L}w$ gives us:

Moreover the assumption $w\approx v$ implies:

Thus by Nomic Inclusion, (1), (2), and the assumption that $C$ is coextensive with $L$ deliver (4). This suffices to establish (1).

Next, recall that $\phi$-Haecceitism implies the existence of an actual world $\mathrm{@}$, i.e. a world at which every truth is true. By Nomic Inclusion, we have:

For otherwise, we would have:

But, by Nomic Inclusion, every instance of $C$ is a truth.

Recall further that $\phi$-Haecceitism tells us that there is a non-actual world $w$ which is qualitatively equivalent to $\mathrm{@}$, each of which makes $\phi$ true. With (1) and (7), this implies that $w$ is physically possible:

It follows that there are two different physically possible $\phi$-worlds, which is what $\phi$-Indeterminism states. $\mathrm{\square }$

Theorem 3. $⊢\text{Selecting}\hspace{0.17em}X\to \text{AH}X$

Proof. It first helps to establish a lemma about world selectors, which states that they are compossible with exactly one world (modulo necessary equivalence) from any given qualitative equivalence class.

To see why this holds, assume $\text{Selector}\hspace{0.17em}p$ and observe that the definition of a world selector gives us:

Thus for any world $w$ we have the existence of a world $v$ such that for any world $u$:

Hence from the reflexivity of qualitative equivalence and the assumptions of $\mathrm{♢}(p\wedge w)$, $\mathrm{♢}(p\wedge u)$ and $w\approx v$, standard modal reasoning gives us $\mathrm{\square }(w\leftrightarrow u)$.

With (1) to hand, assume $X$ is a selecting necessity such that for worlds $w$ and $v$:

Since X is selecting, there is a world selector $p$ such that $\mathit{\text{Xp}}$. Thus, by the fact that $X$ is a necessity, we have:

Again, since $X$ is a necessity (which implies $\text{N}X$), this gives us:

And hence by (1), $\mathrm{\square }(w\leftrightarrow v)$. This suffices to establish $\text{AH}X$.☐

The final result concerns one of the last claims made in §3, that there is an argument from Qualitative $\phi$-Determinism${\mathrm{◇}}^L$ and Nomic Selection to $\phi$-Determinism. This is underwritten by the following formal result:

Theorem 5.

$\begin{array}{c}⊢\text{Selecting}X\wedge \forall w\forall v({\mathrm{♢}}^{L}w\wedge {\mathrm{♢}}^{L}v\wedge (w\models \phi )\wedge (v\models \phi )\to w\approx v)\to \hfill \\ \forall w\forall v(⟨X^L⟩w\wedge ⟨X^L⟩v\wedge (w\models \phi )\wedge (v\models \phi )\to \mathrm{\square }(w\leftrightarrow v))\hfill \end{array}$

Proof. To establish this, it helps to record the following lemma:

(1) holds because $X^{L}p$ unpacks to the following:

Thus, given $\forall p(\mathit{\text{Xp}}\to \mathit{\text{Yp}})$, we have:

This is just $Y^{L}p$, and so (1) holds.

Given (1), it straightforward to verify the following:

Moreover, given $\text{Selecting}X$ (and hence $\text{Nec}X$), we also have $\forall p(\mathrm{\square }p\to \mathit{\text{Xp}})$. And so, since it is straightforward to verify $\text{Nec}\mathrm{\square }$, (4) and Theorem 4 give us:

Turning now to the main claim, assuming $⟨X^L⟩w\wedge ⟨X^L⟩v$, (5) gives us ${\mathrm{♢}}^{L}w\wedge {\mathrm{♢}}^{L}v$. And so, assuming $w\models \phi$ and $v\models \phi$, we have $w\approx v$. Now, it is easy to verify that the assumption of $\text{Selecting}X$ implies $\text{Selecting}{X}^L$. Thus by Theorem 3 it follows that $\mathrm{\square }(w\leftrightarrow v)$.☐