3.a The ordinary reading

For the following discussion, I will use ‘xx’ as a plural objectual variable and ‘ ’ as a binary connective that takes a plural objectual variable as its first argument (notated as its subscript) and a sentence as its second argument such that ‘ ’ is interpreted as ‘the xx have (had or will have) an iterated or noniterated potentiality that p.’

With this notation in place, I turn to the following question: What is the potentialist reading of a modal formula? The most straightforward way to apply Potentialism seems to suggest the following two-step procedure: in the first step, use the duality between possibility and necessity to transform the modal formula $ \Phi $ into an equivalent formula that does not involve any necessity-operators. In a second step, replace every possibility-operator with an instance of the schema ‘

’ such that the plural variable that replaces the schematic ‘ $ \chi \chi $ ’ does not occur in $ \Phi $ and is not used in more than one instances of the schema.Footnote ⁷ For example, this yields

as the potentialist reading of ‘ $ \square \diamond p $ .’ The steps can also be reversed to arrive at a modal formula by replacing instances of ‘

’ with ‘ $ \diamond $ .’

Let p stand for the true proposition that Vetter exists. Assume that Vetter possibly not exists. This gives us $ \diamond \neg p $ . According to Dependence, we get the result that possibly nothing has a potentiality for Vetter to exist. This yields . Using the procedure just introduced, we can substitute ‘ ’ with ‘ $ \diamond $ ’ to receive $ \diamond \neg \diamond p $ . We assumed p, so we get $ p\wedge \neg \square \diamond p $ , which is equivalent to $ \neg \left(p\to \square \diamond p\right) $ . The introduced procedure leads to the result that there is a counterexample to the B-axiom, as the conclusion of Strict Argument says.

3.b The hyperactualist reading

This subsection presents a further way to generate the potentialist reading of a modal formula. It has two significant differences to the ordinary reading presented in section 3.a. First and foremost, the quantifiers that are introduced during the translation of a modal formula $ \Phi $ into a potentialist formula $ {\Phi}^{\prime } $ will be assumed to range over the objects that exist at the context of evaluation of $ {\Phi}^{\prime } $ . Given that we wish to actually evaluate modal and potentialist formulas, this is tantamount to assuming that the quantifiers introduced during the translation quantify over the actual objects. Let ‘ $ {\exists}_0 $ ’ stand for a an embedding resistant quantifier that is such that if it occurs in a formula $ \Phi $ , then it quantifies over the objects that exist at the context of evaluation of $ \Phi $ , independent of whether the quantifier is embedded under any modal, potentialist, or other intensional operators. For example, if actually evaluated, ‘ $ \diamond {\exists}_0 xFx $ ’ says that possibly an actual object is F (and is hence equivalent to ‘ $ \exists x\diamond Fx $ ’). In contrast, ‘ $ \diamond \exists xFx $ ’ says that possibly there is an object that is F. To give a potentialist example, ‘ ’ says that some (actual) objects have a potentiality for some of the actual objects to have the potentiality that p. In contrast, ‘ ’ says that there are some objects that have the potentiality for there to be some objects that have the potentiality that p. What the embedding resistant quantifier does can be expressed using many Vlach store-and-retrieve operators $ {\uparrow}_n $ and $ {\downarrow}_n $ as they are described by Fabrice Correia (Reference Correia2007). Metaphorically speaking, $ {\uparrow}_n $ stores the world of evaluation at the nth position of a list and $ {\downarrow}_n $ retrieves the world stored at the nth position of the list. Differently put, $ {\downarrow}_n $ exempts what follows it from the scope of the modal operators that occur after the previous occurrence of $ {\uparrow}_n $ .Footnote ⁸ Now we can make the quantifiers in a potentialist formula $ \Phi $ embedding resistant by (i) putting an instance of ‘ $ {\uparrow}_1 $ ’ at the beginning of the formula (which stores the context at which $ \Phi $ is evaluated) and (ii) replacing every instance of ‘ $ \exists \chi \chi $ ’ with ‘ $ {\uparrow}_2{\downarrow}_1\exists \chi \chi {\downarrow}_2 $ .’ Whenever we come to an existential quantifier, we store the modal cotext at which we are at position 2, briefly go to the world at which $ \Phi $ is evaluated (which we stored at position 1), do the quantification, and retrieve the context we stored at position 2.

It should be noted that for my aim, it is only relevant that the actual objects are quantified over in embedded contexts, not that no further objects are quantified over. Hence, a salient alternative to quantifying over the actual objects in every context (as I suggest) is to cumulatively quantify over the actual objects together with further objects that one encounters as one moves through modal space. This idea has been made precise by Teitel in the context of the reduction of modality to essence (see Teitel Reference Teitel2019, sec. 4.4), who calls it a “cumulative reduction.” The resulting cumulative picture would be less attractive, however. In addition to potential qualms concerning quantification over nonactual objects, the cumulative picture is unnecessarily contextual, for it does not allow to uniformly resolve the meaning of the $ \diamond $ -operator. What a $ \diamond $ -operator means would be dependent on the number of further modal operators it is embedded in.Footnote ⁹

The second difference to the ordinary reading is that quantification over objects that (fully) constitute a potentiality will take place—not only over those who possess the potentiality. In the first section, I introduced the distinction between the objects that possess a potentiality and those that constitute it. I assumed that the objects that possess a potentiality also help to constitute it, but not vice versa. To reiterate the motivating example: Vetter helps to (and is needed to) constitute the potentiality for her to not exists, but she does not possess it. Let ‘ ’ say that the xx constitute the potentiality that p. This means that (a) some yy among the xx are such that they possess the potentiality that p and that (b) the other objects among the xx (if there are any) are further objects needed to constitute the potentiality. Plausibly it is necessary that ‘ ’ holds just in case ‘ ’ holds (necessarily, every possessed potentiality is constituted). Still, the subtle difference between the objects that constitute and those that possess a disposition will be relevant in what follows.

With this notation in place, I provide the following method for giving the hyperactualist reading of a modal formula $ \Phi $ : in the first step, use the duality between possibility and necessity to transform the modal formula $ \Phi $ into an equivalent formula that does not involve any necessity-operators. In a second step, replace every possibility-operator with an instance of the schema ‘

’ such that no plural variable that instantiates the schematic plural variable ‘ $ \chi \chi $ ’ occurs in $ \Phi $ and is used in more than one instances of the schema. For example, this yields

as the potentialist reading of ‘ $ \square \diamond p $ .’ In words, according to the hyperactualist reading, for a truth to actually be necessarily possible is for there to not be any (actual) objects that constitute a potentiality for no actual objects to constitute a potentiality that p.

Given the hyperactualist reading, the claim that potentialism validates the 5-axiom can hence be put as follows: If some xx constitute a potentiality that p, then no yy constitute a potentiality that none of the actual objects constitute a potentiality that p. It is this feature of the hyperactualist reading that allows to deny premise (IV) of Strict Argument. Only what the actual objects are like and which potentialities there are for them to be different is relevant to what is necessarily possible. This leaves open that a proposition is necessarily possible although possibly there are no objects that have a potentiality for it to be true.

However, this only shows that premise (IV) of Strict Argument can be denied by proponents of the hyperactualist reading. A positive argument to the conclusion that holds for all possible p is still lacking. Such an argument is needed to justify the claim that potentialism yields an S5 modal logic when the hyperactualist reading is assumed.

Whether potentialism yields an S5 modal logic depends on metaphysical assumptions that go beyond the question how potentialism is spelled out (a point that is also acknowledged in Vetter (Reference Vetter2015, 213). The hyperactualist reading guarantees an S5 modal logic when combined with two reasonable auxiliary assumptions, or so I will argue.

Here are two assumptions that are sufficient for the present aim:

Talk about the state of the world at a time (as it occurs in the Triviality Thesis) is often used in philosophy (see, e.g., van Inwagen [Reference van Inwagen1975] for a prominent example). As usual, I understand it such that the state of the world at t does not include truths about the future of t (e.g., being such that there will be someone walking on the moon in two hundred years is not part of the state of the world in 1768). I also understand the state of the world at t as only concerning objects that are present at t. Here is why this might be important: Vetter suggests accepting strong eternalism—the thesis that “it is always true that everything is always something” (Reference Vetter2015, 293). According to the strong eternalist, in 1800 Vetter already existed. Still, potentialists should like to allow that later than 1800 something had a potentiality for Vetter to not exist.

The Triviality Thesis is argued for by Vetter, who argues that “we have no potentialities for the past to have been different, though we have potentialities for the past to have been just as it was” (Reference Vetter2015, 189) and it is defended by Kimpton-Nye (Reference Kimpton-Nye2021, 355).

The potentialist has to accept the Stability Thesis if they wish to uphold that whatever is possible at some point in time has always been possible (a claim that is forcefully defended in Dorr and Goodman [Reference Dorr and Goodman2020]). More specific to potentialism, Kimpton-Nye argues that if objects in the past did not have an iterated potentiality for the further potentiality P to come into existence, then it would be “a mystery why P should just pop into existence” (2021, 355).Footnote ¹⁰

Now, if there is a first moment of the universe, then the Triviality Thesis yields that nothing ever has a potentiality for the state of the world at the beginning of time to be different. Accordingly, given Potentialism, the state of the world at the beginning of time is necessary.Footnote ¹¹ If there is no first moment of the universe, then we still get the result that necessarily, the state of the world is identical to the actual state of the world up to some branching point in the past. Here is why: the Triviality Thesis gives us that at no time t does anything have a potentiality for the state of the world to be different at any time $ {t}^{\prime }<t $ . Consequently, there never is a potentiality for the entire history of the world to be different from actuality, there are only potentialities for the world to be different from a point of departure onwards. This is compatible with no time being such that the state of the world at this time is necessary, there might be no earliest possible point of departure. Still, there necessarily is a point of departure.

How does this give us $ \left(@\square \diamond \right) $ ? Here the Stability Thesis comes into play. It guarantees that if some things have a potentiality that p at some point in time t, then at every $ {t}^{\prime }<t $ , something has a potentiality that p. Footnote ¹² Together with the claim that nothing has a potentiality to change the state of the world at all times in the past, we get the result that nothing has a potentiality to remove all potentialities that p. This is exactly what we need to get $ \left(@\square \diamond \right) $ .Footnote ¹³ It is an interesting disanalogy to the dialectics with respect to essentialism that no essentialist principles akin to the Triviality Thesis and the Stability Thesis have been discussed. The main reason for this disanalogy seems to be that it is generally accepted that essences are immutable and necessary, whereas it is widely held that things can change their potentialities.

At this point, some clarification is in order. Someone might object that clearly some objects xx have a potentiality for any objects yy that actually constitute the potentiality that Vetter exists to not constitute this potentiality. They might hold that the xx have a potentiality for some among the yy to not exist and that if the yy did not all exist, then they would not constitute the potentiality that Vetter exists. This objection can be rebutted by clarifying that for some objects to be such that they do not constitute a given potentiality, they have to be a certain way, namely such that they do not constitute the given potentiality (and that for to be a certain way they have to exist). ‘ ’ is to be read as saying that nothing has a potentiality for every plurality of actual objects to be a certain way—namely such that the objects among it fail to constitute the given potentiality.Footnote ¹⁴ This can be motivated by arguing (as I will in the next subsection) that the objects needed to constitute a potentiality that p are also needed for the proposition that p to exist. The intended reading of ‘ ’ yields that no possible proposition can be turned into an impossible proposition, which gives us a legitimate reading of the claim that the possible cannot be impossible.

It should be clear by now why it is a central ingredient of the hyperactualist reading that quantification over all objects that constitute a potentiality takes place (i.e, why I used the -operator rather than the -operator). To elucidate this by means of an example, assume once more that Vetter necessarily helps to constitute, but never possesses the potentiality for her to never have existed. Plausibly there are some objects that have a potentiality for this potentiality to never be possessed by ensuring that Vetter never comes into existence. However, there is no potentiality for changing the objects in every plurality that constitutes the potentiality that Vetter does not exist (and that hence includes Vetter) in a way that it does not constitute this potentiality any more.

This shows that there are independently plausible assumptions endorsed by Vetter and other defenders of potentialism that, when added to the hyperactualist reading, guarantee that the logic of metaphysical modality is S5.Footnote ¹⁵

What is the dialectical relevance of the hyperactualist reading and the result that it enables the potentialist to defend an S5-modal logic? One might think that just presenting some way to transform modal formulas into potentialist formulas that can lead to a picture on which S5 is validated does not show much. To make the hyperactualist reading dialectically relevant, it has to be shown that it is compatible with both the spirit of potentialism and with the core claim of potentialism. If this is shown, then the hyperactualist reading becomes an attractive option for those potentialists who wish to uphold that the modal logic is S5.

The hyperactualist reading seems to be in line with the actualist spirit of potentialism. It allows to translate every modal formula into a formula about actual objects and the potentialities they have. Figuratively speaking, no looking past actuality is needed to account for modality.

However, it seems prima facie questionable whether the hyperactualist reading is consistent with the modal robustness of the core claim of potentialism. In the next subsection, I will argue that it is, given the assumption of higher-order contingentism. I will furthermore show that assuming higher-order contingentism is plausible in the given dialectical context.

3.c Hyperactualism and higher-order contingentism

I take potentialism to be a modally robust thesis about potentiality and possibility. Plausibly, if it holds, then it necessarily holds.Footnote ¹⁶ One might take this to give rise to the following objection. The modal robustness of potentialism forces us to accept: for every proposition $ \left\langle p\right\rangle $ (i.e, the proposition that p), necessarily, it is possible that p just in case some things have a potentiality that p. If we allow for quantification into sentence-position, this can be put more formally as follows:

The hyperactualist reading of potentialism does not validate $ \left(\square Pot\right) $ . According to the hyperactualist reading, there possibly not being any objects that have a potentiality for Vetter to not exist is consistent with Vetter necessarily possibly not existing. The reason is that according to the hyperactualist reading, the necessary possibility of a proposition has to do with the question which potentialities there are for actual objects to have different potentialities, not with the question which objects might fail to exist. Consequently, so the objection goes, the hyperactualist reading of potentialism fails to yield the modal robustness of potentialism.

In the remainder of this section, I will address this objection. I will argue that if higher-order contingentism is assumed, then the modal robustness of potentialism can be combined with giving up $ \left(\square Pot\right) $ . Higher-order contingentism is the thesis that some properties and propositions contingently exist (see, e.g., Fritz and Goodman Reference Fritz and Goodman2016; Stalnaker Reference Stalnaker2011; Adams Reference Adams1981).Footnote ¹⁷ A consideration in support of higher-order contingentism is analogous to the consideration that supports Dependence: propositions about (or properties concerning) objects depend for their existence on these objects; the objects are, to speak with Leech, needed to contribute to or constitute the proposition/property (see Leech Reference Leech2017, 461f; and section 1 of this paper). This consideration is surely defeasible (and it is famously opposed by Plantinga [1983]), but those who like the spirit of potentialism and wish to locate modality in actual objects will likely also feel drawn to the idea that there cannot be a proposition about, say, Vetter if Vetter does not exist.

Higher-order contingentists should distinguish between a de re and a de dicto reading of the modally robust core claim of potentialism. According to the de re reading, every proposition $ \left\langle p\right\rangle $ is necessarily such that it is possible iff something has a potentiality that p. According to the de dicto reading, necessarily, every proposition $ \left\langle p\right\rangle $ is possible iff there is a potentiality that p. $ \left(\square Pot\right) $ expresses the biconditional that results from the de re reading of the robustness of the core claim of potentialism. The de dicto version, in contrast, can be put as follows:

Higher-order contingentists hold that it is a contingent matter which semantic values for sentences there are. Accordingly, there is room for higher-order contingentists to hold that $ \left(\square Pot\right) $ fails whereas $ \left(\square \forall Pot\right) $ holds. An analogous structure arises with respect to the reduction of modality to essence and the distinction between a de re and a de dicto reading of essentialism plays a crucial role in my response to Teitel (Reference Teitel2019) in Werner (Reference Werner2021).

In the remainder of this section, I will argue for two claims. First, I will argue that the combination of higher-order contingentism and the hyperactualist reading of potentialism validates $ \left(\square \forall Pot\right) $ . Second, I will argue that, given higher-order contingentism, $ \left(\square \forall Pot\right) $ is the more plausible reading of the robustness of the core claim of potentialism than $ \left(\square Pot\right) $ . This will establish that the combination of higher-order contingentism and the hyperactualist reading of potentialism can do justice to the robustness of the core claim of potentialism.

In what follows, I will argue that the hyperactualist reading validates $ \left(\square \forall Pot\right) $ . To argue that the hyperactualist reading validates $ \left(\square \forall Pot\right) $ , it has to be shown that necessarily, the possible propositions that exist are exactly those for which some things have a potentiality to be true.

If you believe that it is a contingent truth that $ \left\langle p\right\rangle $ exists, then you will plausibly take $ \left\langle p\right\rangle $ to exist only if the objects $ \left\langle p\right\rangle $ is about exist. At this point, the following coordination-assumption is needed: Necessarily, the objects that a possible proposition $ \left\langle p\right\rangle $ is about exist if and only if some objects constitute a potentiality that p. From the considerations in the last section, we get that (given the necessity of the Triviality Thesis and the Stability Thesis) necessarily, if some objects constitute a potentiality that p, then necessarily, if these objects exist then they constitute a potentiality that p. We additionally need the claim that necessarily, the objects that constitute a potentiality that p exist if and only if the objects $ \left\langle p\right\rangle $ is about exist. This is supported by the observation that the motivations for higher-order contingentism and for Leech’s claim that, e.g., Vetter is needed for there to be potentialities for Vetter are analogous. As already mentioned above, both are based on considerations akin to Dependence. The objects $ \left\langle p\right\rangle $ is about plausibly just are the objects that have to be added to objects that possess the potentiality that p to arrive at objects that constitute the potentiality that p. Footnote ¹⁸

The next step is to argue that $ \left(\square \forall Pot\right) $ is the more plausible reading of the robustness of the core claim of potentialism than $ \left(\square Pot\right) $ . We can formalise Potentialism as

This should be read as the thesis that the possible propositions are those that are such that something has a potentiality for them to be true. If one takes Potentialism to be necessarily true, then one gets the claim that necessarily, the possible propositions are those that are such that something has a potentiality for them to be true. This just is the de dicto reading of the robustness of Potentialism. Taking Potentialism to be modally robust gives one $ \left(\square \forall Pot\right) $ , but it does not give one $ \left(\square Pot\right) $ .

Higher-order contingentism allows for upholding $ \left(\square \forall Pot\right) $ and denying $ \left(\square Pot\right) $ . An opponent could try to argue that a higher-order contingentist who subscribes to $ \left(\square \forall Pot\right) $ should also accept $ \left(\square Pot\right) $ . This would amount to demand that worlds in which propositions do not exist should still provide material concerning the modal status of these propositions. I find it hard to see why this should be a reasonable demand. To ultimately evaluate whether it is, a picturesque view of the structure of modal space that higher-order contingentism gives rise to might be helpful.

Higher-order contingentism yields the result that every world spans its own modal space, which worlds (possible or impossible) there are at the space spanned at w depends on which propositions there are at w. This becomes particularly clear if we take the (possible or impossible) worlds in the modal space spanned by w to be identical to classes of propositions that exist in w. $ \left(\square \forall Pot\right) $ demands that which of these worlds are possible from the perspective of w depends on the potentialities of the objects that exist in w. Footnote ¹⁹ Let v be a world that is in the modal space spanned by w, but that itself spans a modal space that does not include w (e.g., because in w there exists a proposition about Vetter and in v this proposition does not exist). (□Pot) would demand that material from v allows to classify w as possible or impossible, although w is not in the modal space spanned by v. Metaphorically speaking, it requires v to make judgements about worlds of which it cannot know. $ \left(\square \forall Pot\right) $ only demands that the material we find at each world allows to classify the worlds in its own space.

This sketch hopefully helps clarify why the higher-order contingentist has a point in denying (□Pot). To still insist on (□Pot) would surely require further argument. I therefore conclude that the combination of higher-order contingentism and the hyperactualist reading of potentialism can be reasonably claimed to do justice to the modal robustness of the core claim of potentialism.

Some might see the resulting situation as follows: both the ordinary reading and the hyperactualist reading are faithful to potentialism. One might argue that unless I provide independent compelling reasons for adopting an S5 modal logic, the simpler ordinary reading should be preferred. A general response goes as follows: given the initial plausibility of S5 (mentioned and discussed above), it is worthwhile to investigate whether there is a way for potentialism to yield an S5 modal logic. This paper shows there is such a way. A more example-driven response goes as follows: if the world developed such that Vetter’s parents never met and Vetter never came into existence, it would have still been possible in a metaphysically relevant sense that they did meet and that Vetter came into existence. The ordinary reading does not allow for this possibility, because without Vetter there were no Vetter-concerning possibilities. The hyperactualist reading gives us the desired possibility of Vetter’s existence in a world without Vetter. This is enough to warrant interest in it.