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Can A Regularity Theory of Laws Provide an Effective Account of Causation?

The search for causes is natural to the human mind, which believes that nothing happens without reason. In the words of philosopher and metaphysicist Alyssa Ney (2014), an effective account of causation is incredibly important since it could amount to “a complete account of the nature of our universe and what it is like” (p.220). So, for example, it can inform us of the relation of a bullet fired (the cause, ‘c’) and John F. Kennedy’s assassination (the effect, ‘e’). Whilst theories of causation explore this in considerable detail, theories of laws can potentially enhance such theories to account for causation. A regularity theory of laws (hereafter ‘RVL’, since it is a regularity view of laws) seemingly develops nomic regularity theory, for instance, the regularity theory which aims to account for causation via the laws of nature (hence 'nomic'). RVL apparently develops the nomic-based theory by (i) maintaining that the laws of nature describe c and e since laws regularly result in e following from c, thus accounting for the relation between cause and effect to account for causation, and (ii) explaining what the laws of nature—which are meant to explain causation—consist of themselves. Or so the story goes.


This article will explore point (ii) to assess whether RVL really provides a plausible account of causation by developing the nomic regularity theory based on the laws of nature. Though the regularity view of laws certainly helps to understand causation, it does not provide a particularly satisfying account of causation.


Figure 1. Scottish Philosopher David Hume, famous for his radical empiricism and account of causation (Jessop, 2022).

Regularity Theories of Causation


Though RVL may improve general understanding of causality, something keeps going wrong when causation is accounted for via regularity whilst avoiding logical necessity. One must start from the beginning.


First, there is the so-called ‘simple’ regularity theory (‘SRT’), originally introduced by David Hume in 1740. Causes are regularly followed by the same kind of effect according to SRT, thus accounting for causation. Consider Stathis Psillos’ (2014) formalisation of SRT, whereby c causes e iff (if and only if):

  1. c is spatiotemporally contiguous to e;

  2. e succeeds c in time; and

  3. All events of type C (i.e., events that are like c) are regularly followed by—or are constantly conjoined with—events of type E (i.e., events like e).

Causation on SRT is ‘explained’ by the fact that things like c are regularly followed by things like e. At first glance, this makes a plausible enough case for saying that c causes e.


Figure 2. A Treatise of Human Nature: Hume here introduces regularity to explain causation (Hume, 2022).

The issue, however, is that causation needs to be a strong relation. Part of what makes a theory of causation an effective theory is that it genuinely accounts for causation by explaining causality (recall Alyssa Ney's (2014) words that an effective account of causation could result in "a complete account of the nature of our universe and what it is like") (p.220). Not sheer coincidences that appear like causation. Mere regular succession is of course not the same as causation, and either too many regular (non-causal) occurrences on this account may qualify as causation or the problem of singular causal relations arises (Psillos, 2014). The latter problem runs as follows. Assume, plausibly, that a giant meteor caused the dinosaurs to go extinct. For the meteor to be a cause on SRT, there must be a regularity saying that events ‘like’ giant meteors hitting Earth are followed by events rather like the extinction of dinosaurs. Yet it seems as if such regularity does not exist: the next giant meteor hitting Earth will not cause any dinosaurs to go extinct since dinosaurs already are extinct. The problem is that the meteor is only a cause of the extinction on one single occasion. According to SRT, however, this one meteor hitting Earth is only a cause if some other meteors hitting the Earth also satisfy regularity. But why would events far away in space and time determine whether this giant meteor caused the extinction of the dinosaurs on this occasion? Causation without regularity on a single occasion apparently satisfies the causation condition on SRT (Holger and Mario, 2021).


Consider another problem with Hume’s SRT, that there can be regularity without causation. The cry of the cock precedes the sunrise and the two events are regularly associated, for instance. Yet the cry of the cock is not a cause of the sun to rise. A special case of this problem arises from joint effects of a common cause (see Figure 3). Suppose an instantiation c of C is a common cause of the instantiations a of A and b of B, where the token event a precedes the token event b in time. Further suppose C is instantiated whenever A is. That is, whenever a token event of type of A occurs, a token event of type C occurs. Then there is a regular connection between A and B, which one does not normally count as a causal relation. This problem remains a central challenge for regularity theorists to date (Holger and Mario, 2021). Indeed, single-case regularities and non-causal regularities pose serious issues for any regularity theory. Clearly the simple view requires much development, hence being known as the ‘simple’ view. Cue nomic regularity theory (‘NRT’), which does exactly this.


Figure 3. Joint effects of a common cause (Holger and Mario, 2021).

The definition of ‘nomic’ is having the general force of natural law. NRT accounts for causation by maintaining that the laws of nature explain the regular relation between c and e. It is thus a development of SRT, with the laws of nature providing the much needed ‘back-up’. The laws “work to avoid counting cases of succession by coincidence as cases of causation”, as Ney (2014, p.224) puts it. NRT also importantly conforms with the belief that we cannot deductively or intuitively predict the effects of every single cause. Following in Hume’s footsteps, post-Hume regularity theorists avoid logical necessity through NRT (Psillos, 2014). Put differently, they avoid claiming that causation is a necessary relation. This proves particularly important since necessity entails that causation could not have been otherwise. If logical necessity is so, certain effects are inevitable after certain causes. This means that humans would be able to form ideas about causal relations without experiencing them, resulting in absolutely no understanding of causation at all. Indeed, many events appear to be the results brought about by identifiable causes, and the human mind is geared to look for these cause/effect relationships since humans use sensory experiences to guess what kind of cause can bring about a certain kind of effect. With the logical necessity view, however, it would prove possible to think of a particular instance of something without any experience of it whatsoever (hence no actual understanding of causation is the result). One may guess that a stone thrown at a glass window will cause the window to break, for example, without any experience of this kind of event prior. This is of course something to avoid, and is avoided on the NRT account. Here, the laws of nature instead don’t have to be logical truths therefore steering clear of this issue altogether whilst also preserving Hume’s notorious empiricist (i.e., knowledge via sensory experience) stance to eliminate the notion of necessary relations of causation (Holger and Mario, 2021). Alternatively, causation is contingent according to NRT, and the laws of nature could have been otherwise (which is logically coherent).


In short, NRT is a regularity theory incorporating the laws of nature. Here is Psillos’ (2014) updated formalisation, that c causes e iff:

  1. c is spatiotemporally contiguous to e;

  2. e succeeds c in time; and

  3. It is a law of nature that all events of type C are regularly followed by events of type E.

NRT is clearly more sophisticated than that of SRT. The effects regularly follow the cause, but it is a law of nature that they regularly follow. Hence the problems faced by SRT no longer pose such a threat. NRT supposedly provides a strong—or at least stronger—case for c causing e. Just one question remains, and proves unfortunately perplexing: what is a law of nature?


Figure 4. Billiard table with convex obstacle (Hoefer, 2022).

Regularity View of Laws


As discussed, NRT develops SRT by incorporating the laws of nature. NRT, however, now must explain what laws themselves consist of if such laws are to account for causation. RVL seemingly develops NRT by providing this explanation, yet rather unspectacularly, RVL (as the name suggests) simply claims that the laws of nature are regularities. As Psillos (2011) explains, RVL “is a metaphysical thesis about law-hood. The worldly stuff that laws consist of is regularities. RVL denies that laws, as they are in the world, are anything over and above stable patterns of events.” (p.80). RVL therefore holds that laws themselves are stable regularities – nothing more than the sorts of things which regularly account for the uniformity in the world and do not change object from object or subject to time.


Can laws be explained by regularities and account for causation which is itself a regularity on NRT? Setting aside the unconvincing ‘regularity to explain regularity’ NRT account of causation for now, one must consider some of the merits of NRT first, for NRT (with RVL) not only minimizes necessity conditions but also preserves Humean Supervenience (‘HS’). HS is a notable merit and must be considered before the theory’s merits and downsides are weighed up. Introduced by David Lewis, the HS doctrine takes a kind of ‘less is more’ approach. HS is described by Lewis (1973) as follows: “It is the doctrine that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another.” (p.556)


Broadly speaking, HS says that the fundamental properties in the world are local qualities: perfectly natural intrinsic properties of points, or of point-sized occupants of points (Weatherson, 2016). In other words, there is quite simply nothing to reality except the spatio-temporal distribution of local natural properties (Lewis, 1973). 'Supervenience' is notably a topic-neutral, dependency relation that typically holds between facts or sets of properties. Lewis' (1973) stance is then a Hume-inspired position towards the nomological. Call a property “Humean” if its instantiation requires no more than a spatiotemporal point and its instantiation at that point has no metaphysical implications concerning the instantiations of fundamental properties elsewhere and elsewhen. To give an example, Lewis (1973) suggests that some instances of Humean properties are the values of electromagnetic and gravitational fields and the presence or absence of a material particle at a point. Most importantly, it is then the HS view that every contingent property instantiated at our world holds in virtue of the instantiation of Humean properties (such as mass) (Loewer, 1996).


Figure 5. Philosopher David Lewis, responsible for developing “Humean supervenience”(Ng, 2017).

HS is the conjunction of two distinct theses; (i) truth supervenes on being, meaning that all the truths about the world supervene on the distribution of perfectly natural properties and relations in that world, and (ii) the perfectly natural properties and relations in this world are intrinsic properties of point-sized objects and spatiotemporal relations (Bigelow and Pargetter, 1989). This is to say that there are only local qualities in the sense of intrinsic properties instantiated by space-time points or point-sized particles at space-time points. Of course, however, the world requires more than intrinsic properties of point-sized objects and spatiotemporal relations. As such, in his distinctive modal metaphysics (modal metaphysics concerns the metaphysical underpinning of our modal statements. These are statements about what is possible or what is necessarily so), Lewis (1973) locates the nomic in the 'humean mosaic'. To do this as precisely as possible, Lewis (1973) distinguishes between ‘simple’ and ‘strong’ truths. A simple truth might be that person x’s nails are painted red. Such a truth differs to any strong truth which instead can inform us about the world. Every truth in Einstein’s Theory of Gravity, for example, that what goes up must come back down (Nordtvedt et al., 2022). Yet, Lewis notes, there is an exception: for the laws of nature can be simple and strong. Lewis (1973) argues that (1) the truth that any two objects are attracted to one another with a force proportional to the product of their masses and inversely proportional to the distance between them is relatively simple, but (2) the fact that this truth tells us a lot about the forces between many distinct objects also makes it strong. Hence the laws are simple but strong truths (Weatherson, 2016). HS is therefore an “elegantly simple picture of the world as a whole”, Lewis (1973, p.567) writes. NRT can indeed preserve this picture of simple but strong truths about causation and the world:


We have geometry: a system of external relations of spatiotemporal distance between points (…) and at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. (Lewis, 1973, p.556.)


Figure 6. David Lewis' Counterfactuals includes his theory of laws of nature (Lewis, 1973)

HS is not something one wants to give up easily and is preserved by HS, providing regularity theorists with an arrangement of qualities. There is simply nothing more complex that one must grasp. Lewis (1973), for instance, notes that there is no difference without difference in the arrangement of qualities. So, all supervenes upon that. Everything that humans see in the world around them ‘supervenes on’ (i.e., is composed by) qualities. This then has an arrangement, for those qualities are instantiated at points. Such is reality in the HS picture. It is hard to deny that HS is worth defending since it is a doctrine that explains all there is to the world in a particularly elegant and simplistic manner. HS, most importantly, is preserved by NRT (and the RVL) since it is an approach to scientific laws which denies that laws imply necessary connections between distinct existences.


This view, that the laws of nature are just patterns or ways of describing patterns in the mosaic of events, is an undeniably appealing part of NRT. The idea that the laws of nature reduce to the patterns of occurrent and non-modal events occurring in the world in NRT indeed further eliminates the notion of necessary relations of causation. NRT is thus permitted by HS, which is particularly important since Hume’s theorising. Hume’s argument is persuasive, suggesting that there cannot be any necessary connections between existences, for if this was the case, it is possible to infer a priori—without experience—from the idea of A existing, that B exists. Yet forming ideas without experiencing them first is impossible. One could not comprehend the existence of a Jaguar E-type if one has never experienced a car, for example. As Hume argues, and as is preserved by the NRT picture, this is simply not possible. Experiences of cars throughout one’s life means that one can form ideas about particular cars and their shapes, colours, engine details, and such like. In short, ideas may only be gained via sensory impressions. The same applies with causation: unless experienced events like triggers being pulled on guns that are regularly followed by events like John F. Kennedy’s death, humans would not be able to see the pulling of a trigger and guess that John F. Kennedy will die.


Figure 7. ‘Hume's Fork’: genuine knowledge may only be derived from experiences and the senses. Copper drypoint by Amy Hiley (2021).

Avoiding logical necessity is essential and makes NRT a contingent account of causation. Such inclusion of contingent laws of nature also means that NRT may preserve HS. NRT indeed has the potential to form an effective and interesting account of causation as such. This potential, however, proves unreached. The so-called ‘merits’ of NRT unfortunately does not amount to an effective account of causation. This is discussed next.


The Downfall of RVL – Regularity Cannot Explain Regularity


First, it is important to reconsider why the laws of nature in NRT are so important. There are three principal reasons for this, namely that (a) they fix SRT flaws, (b) they avoid necessary relations, and (c) they preserve HS. In developing SRT and ‘adding in’ the laws of nature, NRT firstly overcomes some of the serious issues most regularity theories face. For instance, regular succession cannot qualify as causation in NRT since c only causes e if it is a law of nature that all events of type C are regularly followed by events of type E (Psillos, 2014). The NRT is a vast improvement from the SRT since the laws of nature provide the kind of stable relation that is required between c and e. Further, perhaps most importantly, NRT also avoids logical necessity and is thus permitted by HS. Undoubtedly, the laws of nature are thus a significant part of understanding causation. The appealing aspects of NRT nevertheless end here, and the 'stable relation' that was required to explain causality goes unexplained itself.


Regularity and nomic-based theories certainly help to understand causation. This article argues that they simply cannot, however, explain everything there is to know about causation. Most particularly, this paper suggests, the regularity view of laws is unable to form a strong account of causation. As reflected above, the laws of nature are admittedly important, yet the actual description of the laws of nature do not explain anything at all about causation. RVL is supposed to offer the tools to develop NRT. It must therefore refine the very explanation of the laws of nature if it is to do so properly. Instead, however, the laws of nature are mere regularities. The laws of nature, which apparently explain causation as regularity, surely cannot be regularities themselves. This only shifts the goalpost, for regularity cannot explain regularity. All that remains is mere regularity. This will not do.


Figure 8. Diagram representing the simplicity of the Humean mosaic view (Sartenaer, 2019).

To show just how serious an issue this is, Norman Swartz (1997) lists out the sorts of features the laws of nature could have in the RVL account. Swartz sets out five conditions for the laws of nature, describing what kinds of features separate laws from factual claims. To show the severity of the issue at hand, Swartz’s list notably still results in mere regularity – amongst all the conditions of the laws of nature listed. According to Swartz, a law of nature (1) expresses a factual truth (not a logical one), (2) is true for every time and every place in the universe, (3) contains no proper names, (4) makes universal or statistical claims, and finally (5) makes a conditional claim, not a categorical one (Swartz and Carroll, 1997).


These five conditions describe the laws of nature as regular things in the universe. Condition (1) refers to the fact that a law of nature must be contingent (not necessary, as discussed). Condition (2) then states that the laws of nature must apply everywhere (i.e., there obviously cannot be laws that hold only in London, for instance). Condition (3) states that laws must not contain any proper names – laws, in other words, shouldn’t ever refer to specific instantiations. The fourth condition says that laws must be quantified. Finally, condition (5) asserts that a law of nature must make a conditional claim. So, a law of nature cannot make a categorical claim such as ‘there are clouds’. The point to take home from Swartz’s list is essentially that a law of nature can’t simply be a claim about the world. Even still, this doesn’t seem to matter. The five conditions above result in nothing more than mere regularity. It might well be true that laws are regular and with the above characteristics (‘conditions’). It might also be true that we consistently see laws of nature in action and with cases of causation, but this shouldn’t have to allude to the idea that the laws of nature are mere regularities. This simply does not account for causation if causation is explained by regular laws of nature which themselves are mere regularities. Regularity cannot provide the answer to everything.


Figure 9. In the quantum realm, cause doesn’t necessarily come before effect (see Ojala, 2020).

Conclusion


It is no surprise that theories of causation have been refined time and time again. The idea that a cause is regularly followed by its effect is at the very core of regularity theories of causation, the most influential of which has been explored in this article – namely Humean regularity theory. Since Hume’s time, regularity theories indeed have more and more fallen into disuse. This article illustrates (part of the reason) why.


A regularity view of laws, broadly speaking, fails to provide an effective account of causation. If RVL postulates that regularity (via laws of nature) can explain causation, then it must also thoroughly explain what the laws of nature consist of. RVL is supposed to develop NRT, which in turn develops SRT to account for the relation between c and e. Not everything can be explained by regularity, however. RVL must describe what law-like regularity is at the very least to become an effective account of causation. All in all, the NRT merits like avoiding logical necessity and HS are outweighed by the major—perhaps foundational and engrained—disadvantages. A regularity view of laws doesn’t truly provide an effective account of causation if laws are merely regularities themselves. Causation remains unexplained. On the one hand, a more developed and plausible definition of a law of nature is first required if such phenomena are going to explain causation. On the other hand, perhaps the ‘consequences’ of logical necessity are worth addressing so (nomic) regularity theory may avoid mere regularity. The latter might well defeat the purpose, but NRT as it stands does not account for causation effectively, if at all.


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Weatherson, B., (2016). David Lewis. In: The {Stanford} Encyclopaedia of Philosophy. Metaphysics Research Lab, Stanford University.

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