1. Introduction and Context

According to modal necessitarianism (MN), all possible worlds are nomologically identical. MN thus makes a stronger claim than dispositional essentialism (DE; e.g., Ellis Reference Ellis2001; Bird Reference Bird2007). According to DE, the laws are necessary conditional on the instantiation of the relevant properties or kinds, and worlds may differ with respect to which properties and kinds are instantiated. Bird (Reference Bird2004, Reference Bird2007) and Wilson (Reference Wilson2013), among others, have argued that MN is consistent (contra Schaffer Reference Schaffer2005) and that it has some theoretical advantages over DE.

A counterlegal is a counterfactual with a physically impossible antecedent. The idea that the laws are metaphysically necessary has been criticized on the grounds that it is incompatible with prima facie substantive counterlegal discourse as well as with many everyday counterfactuals, which turn out to be implicit counterlegals. The concern is that to realize a counterlegal antecedent a law must be violated. But if the laws are necessary, then there is no possible world in which an actual law is violated and all counterlegals come out vacuously true. A counterfactual will be an implicit counterlegal when, to avoid excessive backtracking, a law violation is required to realize the counterfactual antecedent.

Handfield (Reference Handfield2004) has offered a solution to the problem of counterlegals as it affects DE, which, I argue, does not help MN. Interest in MN is partly motivated by a desire to do without the mysterious notion of metaphysical possibility by assimilating it to physical possibility, thus providing a naturalized metaphysics and epistemology of modality. That Handfield’s proposal does not respect this motivation for MN is evident from his invocation of possible alien properties, that is to say, possibilities that are not grounded in the actual world.

The modal necessitarian is thus tasked with reconciling the prevalence of apparently substantive counterlegals with the nomological identity of all possible worlds. I offer an account of how this may be achieved, which takes inspiration from fictionalism about scientific models.Footnote ¹

My discussion will proceed as follows. In the rest of this section, I outline Handfield’s response to the problem of counterlegals on behalf of the dispositional essentialist. In section 2, I discuss MN and argue that Handfield’s response will not help the modal necessitarian. In section 3, I discuss scientific modeling and highlight the use of counterlegals in modeling. In section 4, I present Frigg’s fictionalist account of scientific models. In section 5, I argue that counterlegals can be given a similar fictionalist treatment, and I respond to some objections.

Handfield’s Suggestion.—Handfield argues that counterlegal utterances include an implicit presupposition of the falsity of necessitarianism about laws and that if we make this presupposition explicit the counterlegals of concern are nonvacuously interpretable via possible worlds semantics. Consider the following statement:

Parsing 1 in accordance with Handfield’s suggestion yields something like:

Handfield suggests that, even assuming the metaphysical necessity of the laws, in the sense implied by DE, 1* is nonvacuously true.

What, then, do we mean by “if it turns out that mass could have obeyed an inverse cube law” and “if mass had obeyed an inverse cube law” given the laws’ metaphysical necessity? The type of possibility invoked here is what Chalmers (Reference Chalmers, Lepore and Smith2006) has called deep epistemic possibility, although Handfield prefers the term conceptual possibility (I will stick with the latter). Conceptual possibility concerns how things could turn out (according to one gloss, P is conceptually possible if and only if [iff] not-P is not a priori). P is conceptually necessary, then, if for all worlds w, if it turns out that w is the actual world, then P will be true (see Handfield Reference Handfield2004, 408–9).

While it may be metaphysically necessary that mass obeys an inverse square law, it is at least conceptually possible that mass obeys an inverse cube law, by which we mean that it could have turned out that our world is such that the property that plays the mass role obeys an inverse cube law. Although we are granting for the sake of argument that the laws are metaphysically necessary (in the sense implied by DE), it could nonetheless turn out that the laws are different because we could discover that our world is in fact a world in which schmass, say, obeys an inverse cube law.Footnote ² To the utterer of the counterlegal, we attribute this presupposition that the laws turned out to be other than they actually are (because the schmass world turned out to be the actual world, say) and a fortiori that they could have been other than they actually are.

We thus have a proliferation of conceptual possibilities where there are no metaphysical possibilities. However, there is still a limit on conceptual possibility. It is not conceptually possible, for example, that $2+2=5$ because whatever world turned out to be actual—schmass world, Hume world, or whatever—two plus two would still equal four.

We can now discern two distinct operations on the same possibilia (worlds) corresponding to metaphysical and conceptual possibility, respectively: Counterfactual, where the denotation of our rigid designators is locked in based on how the actual world is; and Counteractual, where the denotation of our rigid designators is up for grabs, to be settled by how things are at the alternative world considered actual (Handfield Reference Handfield2004, 409). According to Handfield, the dispositional essentialist can interpret “if mass had obeyed an inverse cube law” as an invitation to consider some other world actual, one in which some other property, schmass, plays the mass role and which is such that schmasses are universally attracted in accordance with an inverse cube law. So long as the dispositional essentialist maintains that when we make statements such as 1 what we are doing is imagining that the schmass world is actual and that ‘mass’ denotes schmass, the counterlegal’s nonvacuity is consistent with the metaphysical necessity of the laws; its truth or falsity will depend on how things are at the closest schmass world considered actual. In essence, Handfield proposes deploying conceptual possibilities where there are no metaphysical possibilities to rescue DE from the problem of counterlegals.

2. Modal Necessitarianism

MN is the view that all possible worlds are nomologically identical—two-dimensional semantics cannot help MN to overcome the problem of counterlegals. For the counteractual operation to yield substantive truth conditions for counterlegals, we must admit possible worlds with different laws and properties to those present at the actual world. But this is exactly what MN denies.Footnote ³

A brief digression: possible alien kinds/properties conflict with the thesis of actualism and with the desire to give a naturalized account of the metaphysics and epistemology of modality. My contention is that a naturalistic modal metaphysics suggests grounding facts about modality in the concrete constituents of the actual world.Footnote ⁴ The resultant shrinking of the set of true modal propositions is thus motivated by a desire to resist appeal to mere possibilia and to resist the tendency to ground modality somehow outside of robust reality,Footnote ⁵ and thus, to assimilate the epistemology of modality to familiar scientific epistemology.Footnote ⁶ Similar considerations motivate MN—the idea that all possible worlds are nomologically identical and hence that physical possibility/necessity is the highest form of possibility/necessity. We are thus moved to resist possible alien properties in the interest of retaining the link between what is robustly actual and what is possible.

Something like the dispositional essentialist’s account of laws holding in virtue of the essentially dispositional properties of kinds is a plausible pass at an ontological underpinning for MN. But to yield the nomological identity of worlds, one must also deny the that there are any possible alien kinds and properties that would ground alien laws. However, with this supplement in place, Handfield’s attempt to reconcile necessitarianism about the laws of nature with apparently substantive counterlegal assertions cannot be invoked. Given MN, there is nowhere that the counteractual operation on worlds can take us that the counterfactual operation cannot. Thus, Handfield’s suggestion does not offer the modal necessitarian a way of reaping the benefits of a realistic semantics for counterlegals.

3. Models and Idealization

If MN yields the result that all counterlegal discourse is vacuous, then it is at odds with apparently substantive counterlegal claims common in scientific contexts. This would, in turn, spell trouble for MN’s desire to achieve continuity with science.

4. Models as Fictions

Frigg presents six questions that an account of scientific models ought to answer. Questions I4 and I6 concern the semantic question and the question of metaphysical commitment, respectively:

It is Frigg’s contention that the pretense theory of fiction (Walton Reference Walton1990) can provide the tools to account for truth in models, which, unlike possible worlds semantics, incurs no metaphysical commitment.

Two of the notions central to Walton’s (Reference Walton1990) pretense theory, of which Frigg makes use in his account of scientific models, are props and rules of generation. Props are objects, where the scope of ‘objects’ is very wide, which prompt us to imagine something “due to the imposition of a rule or ‘principle of generation’” (Walton Reference Walton1990, 38). The rule of generation prescribes “what is to be imagined as a function of the presence of the object” (Frigg Reference Frigg2010, 258). The idea, then, is that works of fiction and models can be considered props in this sense. From this we get an account of fictional truths in terms of what is prescribed for imagination, which can be extended to provide an account of truth in a model.

Props, together with the rules of generation, generate fictional truths in the form of propositions that are prescribed for imagination. Some fictional truths follow directly from the prop, while others result only indirectly given some rule of inference. For example, that Piggy in Lord of the Flies has asthma is given explicitly by the prop—the work of fiction—it is a direct truth. Whereas that Piggy lacks a rapport with the other boys is an inferred truth that the reader may deduce; the prop and the rules of generation prescribe the imagination of this proposition. The question of what the rules are that may be used to discern indirect truths is a source of controversy, but Frigg argues that for his purposes it suffices simply to note that there are such rules (Reference Frigg2010, 259).

Models, like works of fiction, can be considered props in games of make-believe. Truth in a model is, then, much the same as truth in fiction, where both types of truths follow from the prop and certain rules of generation. In Frigg’s words: “What is explicitly stated in a model description (that the model-planets are spherical, etc.Footnote ¹³) are the primary truths of the model, and what follows from them via laws or general principles are the implied truths” (Reference Frigg2010, 260).

Although truth in fiction/models is being accounted for in terms of what we are prescribed to imagine, it is argued that such “truths” are nonetheless independent of anyone actually imagining them. There will be a fact of the matter as to whether some proposition P is prescribed by a given prop and rule(s) of generation, regardless of whether anyone actually does any imagining. Thus, truth in fiction/models on this account suffers no mind dependence or context sensitivity.

The application of pretense theory to the practice of modeling can be made explicit as follows: descriptions of model systems in conjunction with the laws and principles taken to be at work in the model can be such that they prescribe the imagination of certain propositions about the model system, which can be thought of as the truths in/about that system. Again, in Frigg’s words: “For instance, ‘the solar system is stable’ is true in the Newtonian model of the solar system iff the description of the system together with the laws and principles assumed to hold in the system (the laws of classical mechanics, the law of gravity, and some general assumptions about physical objects) imply that this is the case” (Reference Frigg2010, 262).

Frigg maintains that this account of scientific models incurs no metaphysical commitments because it inherits all of the antirealist flavor of Walton’s pretense theory from which it derives (Frigg Reference Frigg2010, 264). Indeed, Frigg highlights that the notion of ‘truth in fiction’ is really distinct from truth and that the former is not a species of the latter (261). This fact is at least partly responsible for the metaphysically innocuous character of Frigg’s pretense theory of models since it is often felt that truth simpliciter requires truth-makers and that these things, whatever they may be, will constitute metaphysical baggage.

Fictionalism about Impossibilities.—One final point to note before I attempt to make the connection between the foregoing and the problem of counterlegals; if modelers really do trade in impossibilities in the strictest sense,Footnote ¹⁴ then this would constitute additional support for the view of models as fictions understood in accordance with the antirealist pretense theory. Importantly, fictions can involve impossibilities due to their metaphysical innocence.

Works of fiction can describe impossibilities such as the round square. And it can be true in these fictions that, for example, Archimedes squared the circle and that the square circle shocked the realm and that Archimedes is a hard worker, without thereby incurring any odd metaphysical commitments. Possibilities, however, seem to require some form of metaphysical grounding, whether it be possible worlds or the dispositions or laws of the actual world. Indeed, a lack of any such grounding tends to be what we mean when we deem something impossible. But that is not to say that we cannot, in a sense, represent impossibilities imaginatively. Fictionalism, with its essential appeal to the imaginative capacities, thus seems an appropriate scheme for fruitfully considering impossibilities.

5. Fictionalism about Counterlegals

If there are no counterlegal worlds, as MN implies, we can nonetheless understand counterlegals qua counterpossibles along fictionalist lines. My suggestion is that we think of a counterlegal antecedent as a prop, in the sense employed by Walton (Reference Walton1990) and Frigg (Reference Frigg2010). Given this prop and the rules of generation, certain consequences will be prescribed for imagination. A counterlegal will be “true” on this account iff the consequent is prescribed for imagination given the antecedent and the rules of generation. Reconsider the following counterlegal:

The counterlegal N can be understood as asserting that from the prop (i.e., the antecedent A: if gravity were Newtonian), we are prescribed to imagine the consequent C (a beam of light passing through the sun’s gravitational field would follow a curve of x arc seconds), provided the rules of generation. The counterlegal prop thus constitutes a fiction on this account, and the entire counterlegal will be true iff the consequent, C, is true according to that fiction, by which we mean it is prescribed for imagination.

Will (Reference Will1988) discusses calculations of Newtonian deflections of light by a gravitational mass. The value calculated for the angle of Newtonian deflection of a ray of light grazing the sun’s surface is 0.875 arc seconds (414). General relativity doubles this predicted value, the reason being that general relativity adds the effect of the curvature of space-time near the gravitational mass (414). So, the Newtonian fiction includes the absence of curved space-time, among other features such as Newtonian laws of motion, a corpuscular theory of light, and so on, all of which are relevant to the angle of deflection of a ray of light grazing the sun’s surface that is prescribed for imagination. I refer the reader to Will’s paper for details of the derivation of C (a beam of light passing through the sun’s gravitational field would follow a curve of 0.875 arc seconds) from the prop A (if gravity were Newtonian).

It was suggested in the above discussion of pretense theory that the notion of rules of generation, because of its vagueness, may be a weakness for that theory. However, when understanding the types of counterlegals that are relevant to scientific practice along fictionalist lines, it should be more obvious what the rules of generation are. If we ask what sort of rules could be such that from the fiction of Newtonian gravity we are prescribed to imagine light following a curve of 0.875 arc seconds through the sun’s gravitational field, the rules of mathematics immediately present themselves. From the Newtonian “prop” (which includes no space-time curvature, Newtonian laws of motion, a corpuscular theory of light, etc.), the consequent, C, of N follows mathematically. So, the rules of generation, which will allow us to arbitrate on whether a given consequent is true according to a given counterlegal fiction and, hence, on the truth of the overall counterlegal conditional in question, will be the familiar rules of mathematics (and perhaps logic) central to scientific inquiry.Footnote ¹⁵

One might worry that the example of Newtonian gravitation is an especially simple one and that problems may arise in more complex cases that do not arise here.Footnote ¹⁶ In response, I would again refer the reader to Will’s (Reference Will1988) discussion of Newtonian gravitational lensing, which includes a fair deal more complexity than I have space to reproduce and which may thus assuage the concern that the present example is especially simple. For example, Will is concerned with differing assumptions about the speed of light in the Newtonian “fiction” leading to (negligibly) different predictions for Newtonian deflection of a ray of light (414). Perhaps in more complex cases there will be additional scope for different assumptions to lead to divergent “prescribed imaginings.” But this is not a particular problem for the current account of counterlegals. Rather, it just follows from the fact that in order to draw out the implications of scientific hypotheses, those hypotheses need to be sufficiently unambiguous, and disambiguating hypotheses may be harder, the more complex those hypotheses are.

The approach outlined so far does constitute a concession: if we want to be modal necessitarians, our account of truth, as it pertains to counterlegals, will be different from our account of truth elsewhere, since elsewhere we may want a more realist theory of truth. If one found this dualism objectionable, there is of course the option of embracing a thoroughgoing fictionalist account of truth, but such a drastic measure would be ill motivated.

The dualist result should come as no surprise to the committed modal necessitarian, because really it constitutes a restatement of her position. If there are no possible worlds in which the laws are different from those at the actual world, then there are no things that may provide a realist semantic underpinning for statements about what would be the case if the laws were different. If we were to embrace thoroughgoing fictionalism in the interest of avoiding semantic dualism, MN would lose its substance because it would be unclear in virtue of what counterlegal suppositions are different from counterfactual suppositions consistent with the laws, in which case, the claim that the laws are strongly necessary loses its bite.

One might object that the concern is not really with possible worlds but with restricting oneself to just the use of possible worlds—if we admit impossible worlds, the problem of counterlegals goes away (see, e.g., Nolan Reference Nolan1997; Brogaard and Salerno Reference Brogaard and Salerno2013; Priest Reference Priest2016; Berto and Jago Reference Berto and Jago2019, for discussions of the metaphysics and theoretical virtues of impossible worlds). In response, I would first emphasize that, for the modal necessitarian, possible worlds do not ground or make modal propositions true. One motivation for maintaining that the laws are metaphysically necessary is the desire to avoid the problems engendered by taking possible worlds to be the truth-makers for modal propositions (e.g., what do the goings-on at some other possible world have to do with what is possible at this world?). The modal necessitarian will likely understand metaphysical possibility as (at least roughly) consistency with the laws, and, hence, modal propositions will be made true by the actual world in a way that seems intuitive. This does not preclude all appeal to possible worlds (their use in model theory, e.g., may remain intact). The modal necessitarian may understand possible worlds as something like maximal consistent sets of propositions, where consistency will require consistency with the laws. Maximal sets of propositions that are inconsistent with the laws might then be among the impossible worlds. Possible worlds, so understood, are a useful way of representing modal space, and perhaps impossible worlds can be invoked to do some theoretical work too (such as modeling impossible beliefs; see Nolan [Reference Nolan2013] for discussion of this and other applications of impossible worlds). But, ultimately, it will be the actual world that is of the utmost importance to the metaphysics and semantics of modality because it’s laws will determine which “worlds” are possible.

Now the impossible worlds—sets of propositions detached and free-floating from actuality—look awfully like fictions. Hence, pretense theory may present a useful way of understanding these things. Perhaps, then, the suggestion that we admit impossible worlds is not at odds with my suggestion in this article. I have just sought to offer more in the way of an understanding of these impossible worlds and the contrast with possible worlds. A concern with admitting impossible worlds is that it makes a mystery of the distinction between possibility and impossibility; such an admission demands an explanation of why some worlds are possible and others impossible. In a sense, I agree with the above “objection”; in a sense, what I have established here is that the modal necessitarian might appeal to impossible worlds in her semantics for counterlegals. But I have also attempted to meet an explanatory demand that the admission of impossible worlds raises. It has been argued that, for the modal necessitarian, the distinction between possible and impossible worlds corresponds to the distinction between those worlds that really do represent ways the world could be, given its laws, and those that are mere fictions.Footnote ¹⁷

I consider one final objection before concluding. Jenkins and Nolan (Reference Jenkins and Nolan2012) have argued that there are such things as impossible dispositions—dispositions with impossible stimulus or manifestation conditions. They appeal to two kinds of examples: those in which agents are disposed to react in particular ways to metaphysical impossibilities, for example, Jane’s disposition to be surprised at a perceptible round square object in front of her (738), and those involving nomologically impossible reactions to nomologically impossible circumstances, for example, photons’ disposition to “have a speed of $3\times {10}^{10}\text{m/s}$ in circumstances where they are in a vacuum and the speed of light is $3\times {10}^{10}\text{m/s}$ ” (743). Since the current concern is with MN, metaphysical and nomological necessity need not be distinguished, and the above two examples can both be understood as metaphysically impossible dispositions. Now if there are indeed impossible dispositions, then perhaps they could provide the modal necessitarian with truth-makers for counterlegals. Impossible dispositions are consistent with MN because it can be accepted that there is no possible world in which they are manifested, so there is no possible world in which an actual law is violated. And, contra my fictionalist proposal, they would allow for a unified account of modal truth.Footnote ¹⁸

I have two things to say in response to this concern. First, the impossible dispositions approach is inconsistent with a dispositionalist/hardcore actualist account of modality. Dispositionalism is characterized by adherence to something along the following lines:

If there are impossible dispositions, then there are counterexamples to POSS because something could have a disposition whose manifestation consisted in p without ‘possibly p’ being true. But dispositionalism is naturally allied with MN. Dispositionalism presents an attractive modal metaphysics that shares similar motivations with MN, and, what is more, dispositionalism is plausibly understood such as to imply the nomological identity of all possible worlds (see also n. 4 above). So, admitting impossible dispositions causes trouble for a natural ally of MN.

Second, and perhaps more importantly, Vetter (Reference Vetter2016) has provided strong reasons to doubt Jenkins and Nolan’s conclusion that there are impossible dispositions. On the one hand, Vetter takes issue with Jenkins and Nolan’s heavy and explicit reliance on linguistic intuitions to draw their metaphysical conclusions (2688). Vetter has argued (Reference Vetter2014, 146–48) that linguistic intuitions about the expressions ‘disposed’ and ‘disposition’ are an unreliable guide to the metaphysics of dispositions. But, more generally, Vetter argues that there cannot be any impossible dispositions, because impossible dispositions give rise to referential opacity and dispositions never give rise to referential opacity (see Vetter Reference Vetter2016, sec. 3, for details of the argument). The examples that Vetter offers of impossible dispositions giving rise to referential opacity are ones in which the disposition ascription involves a name. Thus, one might respond that the kinds of impossible dispositions that would serve as truth-makers for counterlegals (e.g., a photon’s disposition to have a speed of $3\times {10}^{10}\text{m/s}$ ) need not involve any name and so are unproblematic. But this would be too quick. The intuitive evidence adduced for impossible dispositions seems just as strong in the case of disposition ascriptions involving names as in the case of disposition ascriptions that do not involve names. If one accepts Vetter’s argument, it “suggests that something was wrong with the intuitive considerations [in favor of impossible dispositions] in the first place, so why rely on them elsewhere?” (2692–93). In a nutshell, Vetter has given us good reason to be very wary of impossible dispositions in general by casting doubt on the veracity of the evidence in their favor and by showing that they have an unpalatable consequence.Footnote ²⁰

I thus suggest that the fictionalist treatment of counterlegals is favorable to the impossible dispositions approach because it leaves open the option of twinning MN with a dispositionalist account of modality, and there are compelling reasons to be skeptical about impossible dispositions in general.

6. Conclusion

Anyone who claims that the laws of nature are metaphysically necessary is tasked with responding to the concern that their view implies the vacuity of apparently substantive counterlegal discourse. Dispositional essentialists claim that the laws are metaphysically necessary in the sense that they admit of no counterexamples because the laws hold in virtue of essentially dispositional properties. Toby Handfield thus proposes invoking two-dimensional semantics to overcome the problem of counterlegals as faced by this type of necessitarianism about laws. However, this strategy cannot help the modal necessitarian, who maintains the stronger view that all worlds are nomologically identical. I thus propose supplementing MN with a fictionalist semantics for counterlegals. This yields a kind of semantic dualism whereby counterlegals (qua counterpossibles), but not ordinary counterfactuals, are understood along antirealist lines, but this, I have suggested, is quite a natural position for the modal necessitarian to adopt.