Can Dispositions Replace Laws in the Description of the Physical World?

Abstract

In this paper, it is argued that, contrary to some suggestions in the philosophical literature, dispositions cannot replace laws in the description of the physical world. If for a certain type of physical situation a well-working law-based account is available, then it is not possible to describe that situation equally well in terms of dispositions. Using an example consisting of four laws (Coulomb’s law, Newton’s law of gravitation, the rule for the composition of forces and Newton’s second law), it is shown that when one attempts to replace a law-based account with a disposition-based account, the latter turns out to be either less informative or inferior with respect to theoretical virtues, such as simplicity, unification, non-triviality of predictions and explanatory power. Laws are also better suited to deal with interactions and their screening off. Using inference to the best explanation, one can conclude from this comparison that it is laws rather than dispositions that should be included in our ontological picture of nature.

1 Introduction

Throughout its history, humanity has gained quite a large amount of knowledge about nature. A significant part of this knowledge is systematised in the form of laws of various sciences (the most prominent of which are the laws of physics), some of which are widely believed to approximate the actual laws of nature. These successes suggest that the laws of nature should be one of the components of our metaphysical picture of the physical world. However, this transition from scientific successes to metaphysics is not an obvious one. In particular, one can worry that the usage of laws in successful scientific theories is only a matter of historical contingency and that some other ontological posits, such as dispositions, might work at least equally well. In this paper, I will investigate the question of whether dispositions can replace laws in the description of the physical world. In short, the proposed answer is “no”.¹ The general argumentative strategy is to show that dispositions are structurally too simplistic to effectively encode the knowledge we have about the physical world given by laws, and it is hard to refine them without making them eventually refer explicitly to the laws (in which case we cannot talk about replacing laws with dispositions). This difference between laws and dispositions can be made visible by comparing the theoretical virtues of law-based accounts and corresponding disposition-based accounts of certain physical phenomena. I will argue that if for a certain type of physical situation a well-working law-based account is available, then a corresponding disposition-based account will be either less informative or inferior with respect to theoretical virtues such as simplicity, unification, non-triviality of predictions and explanatory power. From this comparison, a metaphysical conclusion can be obtained by means of the inference to the best explanation: if laws provide better explanations of natural phenomena than dispositions, then it is reasonable to regard as real the former rather than the latter.

There are some terminological subtleties concerning the relation of dispositions to similar notions, such as potencies, potentialities, capacities, tendencies, powers, etc. However, these issues are tangential to problems that we are going to consider, as these differences do not concern the aspects of dispositions that are under investigation here. We will extract a minimal core of the concept of dispositions that also applies to potentialities, powers and other similar notions. Therefore, for reasons of uniformity and simplicity, even if some authors use one of these similar notions instead of the notion of dispositions, they will all be regarded as providing disposition-based accounts.

The paper starts with explaining the notions of dispositions (Section 2) and laws (Section 3). Section 4 mentions some projects in contemporary metaphysics that seem to declare that it is possible to replace law-based accounts of nature with disposition-based accounts. Section 5 formulates the criteria that such a replacement should satisfy to be regarded as successful and sketches the argument that it is impossible to find a replacement that satisfies these criteria. This argument is fleshed out in Sect. 6. Section 7 summarises my considerations and evaluates their metaphysical significance.

2 Dispositions

Dispositions are features of objects that prescribe their behaviour in certain types of situations, both actual and counterfactual. For example, a glass is fragile, which means that it has the disposition to shatter when struck. That a glass is fragile is true even if in its entire history it would never be struck (and, because of this, never shatter). Other common examples of dispositions are solubility, poisonousness and irascibility. Salt is soluble, which means that it has a disposition to dissolve when put into water. Some chemical substances are poisonous, which means that they have a disposition to kill a human when ingested. Some people are irascible, which means that they have a disposition to get angry when provoked.

There have been various attempts in the philosophical literature to provide a general analysis of dispositions (or, equivalently, truth conditions for disposition ascriptions). What should be the criteria of assessing such analyses? One reasonable proposal is that, first and foremost, they should adequately capture the paradigmatic examples of dispositions, such as the four mentioned above. This will be the constraint employed here.

Philosophers typically analyse dispositions as being specified by two components, a stimulus (denoted by S) and a manifestation (denoted by M). The stimulus is the condition under which the manifestation occurs; in the case of the fragility of the glass, the S is striking. The manifestation is what happens when the stimulus is present; in the case of the fragility of the glass, the M is shattering. A very popular analysis of dispositions called Simple Conditional Analysis is as follows: an object is disposed to M when S iff it would M if it were the case that S.

This simple analysis leads to various problems, and several attempts have been made to refine it (for an overview, see Choi and Fara 2018). However, because our considerations are intended to be very general, we should not rely on the details of any particular analysis of dispositions. Fortunately, it will suffice for our purposes to extract a few features that various such analyses have in common and treat this as a minimal core of the concept of dispositions. It seems uncontroversial that this core should contain the following two conditions:

• Dispositions are associated with individual objects.

• Dispositions are specified either by a manifestation M alone or by a stimulus S together with a manifestation M.

The first condition says that dispositions should always be associated with (or attributed to/rooted in) some individual object. Any true disposition ascription will then involve such an individual; that is, it will have the form: an object O has a disposition D. All our examples of dispositions satisfy this condition; we always ascribe fragility, solubility, poisonousness and irascibility to some particular objects, such as glasses, pieces of chemical substances or people.

The second condition says that dispositions convey relatively simple information, as they can be specified by two main elements, a stimulus S and a manifestation M (or even one, M alone). Most analyses of dispositions in the literature appeal to two elements, a stimulus and a manifestation (as in Simple Conditional Analysis and many of its refinements, such as Lewis 1997, 157; Malzkorn 2000, 464; Fara 2005, 70; Manley and Wasserman 2008, 75). However, some authors claim that even this is too much and it is only manifestation that matters for the identity of all or at least some dispositions (e.g., Molnar 2003, 60–61 and 83–94; Manley and Wasserman 2008, 72–73; Vetter 2015, 65; Rumberg 2016, 268). The second condition is also satisfied by our exemplary dispositions. The solubility of salt is its disposition to dissolve (M) when put into water (S). This explication seems to exhaust the content of the disposition ascription (“salt is soluble”); that is, one does not need to provide any further information to explain what solubility is (and analogously for other examples).

The above minimal core of the concept of dispositions is in agreement both with the literature and (even more importantly) with our paradigmatic examples of dispositions; the latter means that it satisfies the imposed criterion of adequacy. It will work as a constraint for our investigations. That is, anything that does not satisfy the above two conditions will not be regarded as a disposition; if one changes the core, one obtains a different concept. The core can be enriched to a certain extent, for example, by adding the reference to time or normal conditions, as long as our paradigmatic examples seem to require this for their analysis (e.g., salt dissolves not under any conditions but only in normal conditions, and the time it takes is not arbitrarily long). I assume that any entity whose description requires much more complex structures is too dissimilar to solubility, fragility, etc. to be called a disposition.

The discussion that will follow will be based on the understanding of dispositions given by the above minimal core. This will enable us to make it very general and applicable to various more specific conceptions of dispositions that are considered in the literature. It should be stressed that the arguments in Sects. 5–6 are intended to apply to any conception of dispositions that is in agreement with the above minimal core.

In addition to the above minimal core, one distinction will be needed, namely between “precise” and “imprecise” (specifications of) dispositions. It captures tendencies in theorising rather than particular complete theories. The “imprecise” strategy is when we give up an attempt to make specifications of dispositions more precise even though they could be made so, for example, by characterising dispositions only qualitatively when a quantitative description is available (e.g., “striking” instead of “striking with force of 8, 35 N in such-and-such direction”). In contrast, the “precise” strategy is when we try to refine our specifications of dispositions as much as possible, building into them as much information as we can (in particular, both the stimulus and the manifestation should be described quantitatively whenever possible). It may be disputable whether this distinction should be thought of as applying at the metaphysical level (i.e., dispositions existing in the world themselves can be precise or imprecise) or only at the level of our linguistic descriptions (in such a case, one and the same disposition can be specified in a precise or imprecise way). However, this controversy will not matter much for our purposes, as we will be considering accounts based on specifications of dispositions rather than directly considering dispositions as metaphysical entities; therefore, it is enough if this distinction makes sense at the linguistic level. The metaphysical conclusions will come out of this only at the very end (see the last two paragraphs of Sect. 5).

3 Laws

The philosophical literature concerning the laws of nature often assumes a very simple form of laws (it can be called the “Simple Analysis of Laws”, in short SAL):

$$\begin{aligned} \forall _x \left( F(x) \Rightarrow G(x) \right) , \end{aligned}$$

(1)

where F and G are some properties. This analysis can be found, for example, in Hempel and Oppenheim (1948, 153), Dretske (1977, 248), Armstrong (1983, 7), Goodman (1983, 17-27), Ellis (2001, 204; 2002, 89) and Bird (2007, 46). For some of these authors, the above form is a starting point rather than a full analysis of laws, as, according to them, it does not involve necessity of an appropriate type (so it should be supplemented by modal operators of some kind). The details of their analyses will not be reviewed here because they concern subtleties that are not important for our considerations.

Interestingly, Friend (2016) argues that all laws of nature we can think of have this structure, that is, “laws are conditionals of some quantified form with a single bound variable” (Friend 2016, 127), although in some cases the antecedent of the conditional is only implicit. Whether or not the universality claim holds is an interesting question, but it will not be important in what follows. What will be important is the fact that the above simple form of laws does not reveal some crucial properties that the known laws of physics have.

An important methodological remark is in place here. There is a distinction between the laws of nature (a metaphysical concept referring to something real in the world) and the laws of sciences (which are our linguistic constructions intended to capture those “real” laws but which themselves have different ontological status). However, any conception of the laws of nature must be based on what is accessible to us, that is, on the known laws of sciences. I will use the terms “laws”, “laws of physics”, etc. mainly to refer to our linguistic constructs but with the assumption that whatever we can tell about the laws of nature in the metaphysical sense, this must conform to what we know about the laws of sciences. In other words, it will be assumed that our only route to the concept of the laws of nature is via their known examples discovered by the sciences (even though presumably all of them are only approximations of the fundamental laws). A similar distinction for dispositions is between dispositions as something real in the world and specifications of dispositions. My later terminology of law-based accounts and disposition-based accounts refers to laws in the linguistic sense and specifications of dispositions instead of their metaphysical counterparts. However, our conclusions concerning such accounts clearly have metaphysical significance, as our main (if not the only) source of reliable information about the laws of nature and dispositions understood metaphysically is via the successes and failures of such accounts in the description of the world.

Even if laws known from physics can be written in the simple form (1), the predicates F and G would be very complex, and, as we will see, what makes laws so efficient in the description of the physical world are some features of the internal structure of F and G and not just the “global” conditional structure captured by SAL.² I am not going to give a more sophisticated general analysis of laws (although I believe there is a need for such an analysis in philosophy); instead, I will focus on one particular (but paradigmatic) example of a law-based account and six exemplary scenarios (i.e., types of physical situations) this account can deal with. Any general conception of laws can be regarded as correct only if it captures the crucial features of this example.

Consider a group of N bodies with masses $m_i$, electric charges $q_i$ and positions $\vec {R}_i (t)$ (in some fixed reference frame) with $i = 1, \ldots , \,N$. The distance between any two bodies is equal to $\vec {r}_{ij} (t) = \vec {R}_i (t) - \vec {R}_j (t)$. To describe situations of this type, four laws are needed. The first, called Coulomb’s law, establishes the electric force with which an object with index j acts on an object with index i:

$$\begin{aligned} \vec {F}^e_{ij} (t)= k q_i q_j \frac{\vec {r}_{ij}(t)}{r_{ij}^3(t)}. \end{aligned}$$

(2)

Newton’s law of gravitation determines the gravitational force between the bodies:

$$\begin{aligned} \vec {F}^g_{ij} (t) = G m_i m_j \frac{\vec {r}_{ij}(t)}{r_{ij}^3 (t)}. \end{aligned}$$

(3)

Then, the resultant force acting on a body with index i is a vector sum of all the forces acting on it:

$$\begin{aligned} \vec {F}^r_{i} (t) = \sum _{j \ne i} \vec {F}^e_{ij} (t) +\sum _{j \ne i} \vec {F}^g_{ij} (t) + \ldots , \end{aligned}$$

(4)

where “$\ldots$” indicates that if there are any other forces than the two mentioned above, they should also be added to this sum. Finally, the equation of motion (Newton’s second law) prescribes how these forces influence the motion of bodies:

$$\begin{aligned} \vec {F}^r_{i} (t) = m_i \frac{d^2}{dt^2} \vec {R}_i (t). \end{aligned}$$

(5)

This is a second-order ordinary differential equation,³ whose solution is a trajectory of the ith object, $\vec {R}_i (t)$, which specifies its positions at every moment of time. To solve it, we need additional factual information about the initial conditions, that is, the positions and velocities of all objects at some chosen time $t_0$ (the masses and charges are assumed to be fixed). From these conditions and equations (2)–(5), the evolution of the whole system (i.e., of all N objects) at later times follows.⁴

Let us consider the following six scenarios (to which we will refer later when trying to find a disposition-based account that can successfully replace the presented law-based account):

• Scenario 1: one massive and electrically neutral object.

• Scenario 2: two massive and electrically neutral objects.

• Scenario 3: $N \ge 2$ massive and electrically neutral objects.

• Scenario 4: $N \ge 2$ massive and electrically neutral objects, one of which has much larger mass than the others.

• Scenario 5: two massive objects of the same mass, one with a negative charge and one with a positive charge.

• Scenario 6: two massive objects of the same mass with a negative charge.

The laws (2)–(5) are the same for all the above scenarios, and for each of these scenarios they determine the trajectories of all objects, $\vec {R}_i (t)$. More often than not, it is difficult to calculate this solution even approximately, but it is a mathematical fact that the solution exists and is unique (unless some of the caveats of footnote 4 apply). In scenario 1, there is only one object, so no force is generated and the object’s state of motion does not change (i.e., it either stays at rest or moves with a uniform velocity all the time), which may be calculated by solving the equation

$$\begin{aligned} m_1 \frac{d^2}{dt^2} \vec {R}_1 (t) = 0, \end{aligned}$$

(6)

which arises from combining the laws (2)–(5).⁵

In scenarios 2 and 3, all objects are electrically neutral, so the electric force is zero and only gravitational force determines the motions. Scenario 2 is the so-called Kepler problem, and its well-known solution is that the trajectories of objects are ellipses, parabolas or hyperbolas, which may be calculated by solving the following set of equations:

$$\begin{aligned} \left\{ \begin{array}{l} m_1 \frac{d^2}{dt^2} \vec {R}_1 (t) = G m_1 m_2 \frac{\vec {r}_{12}(t)}{r_{12}^3 (t)}, \\ m_2 \frac{d^2}{dt^2} \vec {R}_2 (t) = G m_2 m_1 \frac{\vec {r}_{21}(t)}{r_{21}^3 (t)}, \end{array}\right. \end{aligned}$$

(7)

which, again, arise from combining the laws (2)–(5). For all other scenarios, the equations can be obtained in the same way; therefore let these two explicit calculations suffice.

Scenario 3 for $N>2$ is the so-called many-body problem, which is much more difficult to solve than the $N=2$ case. Scenario 4 is a special case of scenario 3, but the difference in mass makes it possible to use an approximation when solving the equations describing this scenario. In scenarios 5–6, both gravitational force and electric force are non-zero, and the details of trajectories depend on the details of a system under investigation (in particular, on whether the gravitational force is stronger or weaker than the electric force).

My reason for focusing on actual laws of physics rather than using SAL is that the former have several features that SAL overlooks. Below, I mention five of them.

First, the equations expressing laws can (and should) be combined. In order to calculate the actual motion of bodies, we need to perform the composition of forces, then put the result into the equation of motion (5) and finally solve the differential equation obtained in this way. Laws as represented by SAL can also be combined: if F(x) leads to G(x) and G(x) leads to H(x), then, using the logical rules, we can conclude that F(x) also leads to H(x). But in our exemplary law-based account, the combination of laws involves more than such a simple logical inference—some mathematical operations need to be performed. The difference, therefore, lies not in the possibility of combining laws but in the complexity of involved operations.

Second, laws are flexible schemata that can be easily adopted to various situations, differing, for example, by the number of objects or by the types of forces involved. This is illustrated by the above six scenarios. Some of them involve only non-zero gravitational force; the others also involve non-zero electric force, but the laws stay the same. This flexibility is clear when we analyse our exemplary law-based account, but nothing like this is suggested by formula (1).

Third, the crucial law, namely (5), is (a scheme of) a differential equation that (when supplemented by a concrete formula for the resultant force) leads to the description of the evolution in time. Our law-based account gives us the information of what exactly happens at each of the later moments, given that the initial situation is specified. This is a much more detailed description than just saying that if something happens, then something else must also happen (the latter is the kind of information that we learn from laws as represented by SAL⁶: if F(x), then G(x)).

Fourth, laws do not give us a straightforward description of what will happen (cf. Smith 2002, 243-251). Instead, they provide a recipe that allows us to calculate what will happen (in different scenarios) using the relevant information about the particular situation (the number of objects, their masses and charges and the initial conditions). It should be noted that it is often technically difficult to extract this description. This, again, is not something we can see when we just look at SAL, and, in fact, it is most natural to interpret (1) as giving a straightforward description of the actual behaviour of objects (namely, the possession of a property F by x leads to x’s possessing a property G) rather than a recipe for calculating such behaviour.

Fifth, laws are assumed to be universal, but in their standard form (that may be found in textbooks, etc.) quantifiers are only implicit. If they were to be made explicit, this would be a multiple quantification (i.e., quantification over many variables at once). In our case, the quantifiers range over all N’s (which are natural numbers), over all possible N-tuples of particles and over all times t. This is in contrast to (1), which contains only one quantifier over individuals. For the laws (2)–(5) to fit into the scheme (1), one would need to somehow hide the quantification over $N-1$ particles in predicates F and G or quantify over systems (taken as wholes) instead of individuals (and again hide the rest of the quantifiers in predicates); in both cases, some quantifiers will be invisible in the general scheme of a law, which makes these strategies rather artificial.

To repeat, I will not provide any general philosophical analysis of the notion of the laws of nature here. However, any such analysis should be guided by the best candidates for the laws of nature that we know, that is, by the laws of our best scientific theories, most prominently by the laws of physics. Equations (2)–(5) express laws that are such candidates, as they belong to relatively well-established physical theories. Therefore, whatever the right philosophical analysis of the laws of nature is, it needs to take into account the above five features that the exemplary laws of physics have.⁷ Although, strictly speaking, these features are not excluded by SAL, they are also not in any way indicated by it, which shows that SAL is at best an incomplete analysis of the concept of the laws of nature. I believe that a fully adequate account of the laws of nature should not only be consistent with their known examples but also reflect the features of these examples that are crucial for their scientific success. Therefore, in this sense SAL (and any theory based on it) cannot be regarded as adequate.

4 Dispositions Instead of Laws?

Dispositions are well known from everyday life. This intuitiveness makes them metaphysically attractive. Their other (supposed) virtue is that they are rooted in individual objects, which makes them an alternative to theories that postulate abstract principles detached from objects and somehow “externally” impose these principles on them, like law-based accounts do. There are also authors who prefer dispositions because they seem to be better suited than laws to ground natural necessity (e.g., Mumford 2004, 143–159).

Those who favour dispositions over laws have two strategies to employ. First, they can say that the description in terms of laws is a derivative of the description in terms of dispositions (e.g., Bird 2007; Demarest 2017; Kimpton-Nye 2017). Second, they can say that laws should be eliminated in favour of dispositions (e.g., Mumford 1998, 2004; Mumford and Anjum 2011). Let me briefly present each strategy in turn.

Bird’s (2007) main claim is that the fundamental natural properties are essentially dispositional; but laws involve precisely the properties that are natural, so the nature of laws is also dispositional. What is more, there is a sense in which we can derive laws from dispositions. This derivation proceeds, according to Bird, as follows (Bird 2007, 43-48). Assume an object has a feature P (which he calls potency), whose nature is dispositional, which means that $\Box \forall _x \left( Px \Rightarrow D_{(S,M)}x \right)$, where $D_{(S,M)}$ is a disposition with stimulus S and manifestation M. The disposition itself is analysed in terms of a simple conditional, that is, $\Box \forall _x \left( D_{(S,M)}x \Rightarrow Sx \Box \!{\rightarrow } Mx \right)$, with an implicit ceteris paribus clause. From these two formulas, the law of nature of the form $\Box \forall _x \left( (Px \wedge Sx) \Rightarrow Mx \right)$ follows.⁸ This derivation suggests that, in principle, we can describe the world with no reference to laws without losing information. However, Bird does not conclude that we should get rid of the laws; according to him, dispositions reveal the true nature of laws and both have their place in the metaphysical picture of nature.

Another version of the first strategy is to combine the dispositional ontology with the best system analysis of laws. This view, called “Potency-Best System Account of Laws”, was developed by Demarest (2017) and Kimpton-Nye (2017). In the latter paper, this account is summarised as follows: “The basic laws of nature at w are the axioms of the simplest, most informative, true systematization of all w-potency distributions, where a w-potency-distribution is a distribution of only potencies appearing in w at a world instantiating all and only those potencies instantiated at w” (Kimpton-Nye 2017, 12). The two main differences with the standard best system analysis is that potency distributions are considered instead of Humean property distributions, and they are summarised not only taking into account the actual world but also other possible worlds in which exactly the same potencies appear. What these worlds look like is supposed to be determined by the modal profiles of these potencies, so potencies of w determine the set of allowed worlds involving them, and this set in turn determines the laws of w. This means that all the modal information about w is already contained in dispositions, and laws are only a way of presenting (parts of) it, which is why this view is an instance of what I called the first strategy.

The second strategy is represented, among others, by Mumford (1998, 221), who says that “dispositions can do all that laws were required to do and more besides”. In his book, Mumford regards dispositions as real properties instantiated by individuals. He claims that dispositions understood in this way on the one hand ground natural necessity (which laws cannot do, according to him) and on the other hand do not render nature’s behaviour logically necessary because the fact that a given individual has given properties is logically contingent. Because of these metaphysical differences and because, according to Mumford, all the work that is often ascribed to laws can be done by dispositions, he concludes that we do not need the former and that everything in nature can be explained using dispositions alone.

5 Outline of the Argumentation and Its Methodology

In the next section, I will argue that in the cases where there is a well-working law-based account, it is impossible to replace it with an equally good disposition-based account. Before we start, we need to make explicit what exactly such a replacement would mean.

Consider a physical system consisting of N objects. By its account, I mean the construction consisting of:

• general assumptions (e.g., in the form of laws or specifications of dispositions, where by the latter I mean specifying the stimulus S and the manifestation M that are defining for a given disposition),

• factual assumptions about the system at some moment t and

• standard rules of inference (logical and mathematical).

The account of the physical system in this sense allows one to make predictions, that is, to make statements about the system at some moments $t'$ later than t. The general assumptions and the factual assumptions about the system (i.e., the first two points on the above list) will together be called “assumptions” of the derivation of the predictions, whereas the predictions themselves will be called the “conclusion” of this derivation.

If all the general assumptions used in a particular account are specifications of dispositions, then such an account will be called “a disposition-based account” of the system. The word “all” is important here: any account that uses some general assumptions other than dispositions (e.g. dispositions and laws) does not count as a (purely) disposition-based account. For example, as a system we may take a glass, and its disposition-based account could be:

• general assumptions: the specification of fragility (that it is a disposition to shatter when struck),

• factual assumptions: the glass is fragile and it is struck at t and

• a rule of inference that allows us to conclude from an object having a disposition to M when S and the fact that the object is S at t, the fact that the object is M at some $t'>t$.

The above disposition-based account allows us to make a prediction that at $t'>t$ the glass will shatter.

If all the general assumptions used are laws, then such an account will be called “a law-based account”. An example has already been discussed in Sect. 3:

• general assumptions: laws expressed by equations (2)–(5),

• factual assumptions: the number of objects, their masses and charges, the initial conditions of the system at t (i.e., the position and momentum of each object at t) and

• rules of inference: standard logical and mathematical rules that enable us to appropriately combine equations (2)–(5) into one equation and solve it (at least approximately).

The above law-based account allows us to predict what the positions of objects at any moment $t'>t$ are.⁹

Assume that we have a system for which a successful law-based account has been provided. By replacing it with a disposition-based account, I will mean to provide a disposition-based account such that:

• it gives at least equally accurate and precise predictions as the law-based account;

• it is at least equally theoretically advantageous as the law-based account (this can be made more precise by an appeal to the so-called theoretical virtues¹⁰);

• the dispositions invoked by it are not just dispositions to behave in accordance with the laws (i.e., they are not specified by just citing the laws), as only then are the two accounts genuinely different.

I do not claim that the authors mentioned in Sect. 4 had exactly such a project in mind, but I believe that these are the minimal conditions for the replacement to be regarded as successful. One can object here that only eliminativists with respect to laws are interested in such a replacement, whereas those who want to reduce laws to dispositions might regard laws as ineliminable from our explanatory practices and as such not replaceable with dispositions but only derivable from them (or reducible to them). If this was the case, then my argument that the replacement (as characterised above) is impossible would not undermine the idea that laws are derivable from (or reducible to) dispositions. However, I think that both eliminativists and reductionists should be interested in the replacement in my sense. For the derivation of laws to be non-circular, we must start from the basis that does not contain laws and construct the laws solely from the ingredients belonging to this basis. Therefore, a reductionist must be able to express in terms of dispositions everything that can be expressed in terms of laws; this leads to my first and third conditions for replacement. The second condition is more subtle. In principle, a reductionist might claim that laws are reducible to dispositions even though the description in terms of laws is more theoretically virtuous than the description in terms of dispositions. However, in my view, this would undermine the metaphysical significance of such a reduction, rendering it merely a linguistic exercise. This is because I assume that our best guidance in answering the question of whether entities of type X are more fundamental than entities of type Y (in this case, whether dispositions are more fundamental than laws or the other way around) is the comparison of theoretical virtues of accounts in terms of X’s and Y’s.¹¹

My next aim will be to argue (in Sect. 6) that replacement in the above sense is not possible. The first argument (Sect. 6.1) will show that imprecise (specifications of) dispositions¹² cannot satisfy the first condition because by using them we always lose some information that is present in the respective law-based account, which leads to worse predictions. Therefore, the only remaining candidates to replace law-based accounts are disposition-based accounts that use precise dispositions. However, they do not satisfy our second condition: they score worse with respect to such theoretical virtues as simplicity (Sect. 6.2), unification (Sect. 6.3), non-triviality of predictions, explanatory power and fruitfulness in applications (Sect. 6.4). Moreover, laws are more suitable than dispositions for describing interactions and their screening off in experimental set-ups (Sect. 6.5). Finally, it will be argued (Sect. 6.6) that the attempts to enrich the concept of dispositions by allowing dispositionalists to use the elements of the mathematical framework that make law-based accounts so effective eventually lead to the violation of our third condition, that is, to referring to dispositions to behave in accordance with the laws (i.e., dispositions whose specification amounts to just citing the laws).

What is the relevance of such an impossibility of replacement for ontology? To relate these two, I invoke a variant of inference to the best explanation, which in our case might be formulated as follows:

Premise 1: For phenomena of kind X, law-based accounts provide more theoretically virtuous explanations than disposition-based accounts (where for X one can substitute any of the numerous kinds of phenomena that have a well-working law-based account).

Premise 2: If for a certain kind of phenomena there exist various accounts, one should regard as real (or at least more fundamental) the ontological posits of the account that provides the most theoretically virtuous explanation of these phenomena rather than ontological posits of other accounts.

Conclusion: Therefore, for phenomena of kind X, one should regard as real (or at least more fundamental) laws rather than dispositions.

There are two premises in this reasoning. Premise 2 is just the affirmation of the inference to the best explanation as a rational type of reasoning. I do not have space here to discuss all the controversies around inference to the best explanation (see, e.g., van Fraassen 1989, 142–149 for a criticism and Psillos 1999 for an appraisal). However, it is worth noting that rejecting this type of reasoning would be unlikely to be of any help for dispositionalists—instead, it would rather lead to agnosticism about anything that is beyond the immediate reach of human perception (as is the case in van Fraassen’s constructive empiricism), which includes both laws and dispositions. The rest of this paper will be devoted to arguing for the impossibility of replacing law-based accounts with disposition-based accounts, which implies premise 1 but is a stronger claim. This is because one account might be more theoretically virtuous than another without being better in all relevant aspects (i.e., by being better on balance), whereas the impossibility of replacement is the claim that law-based accounts (wherever they exist) fare better in all relevant aspects.¹³ Importantly, premise 1 is about accounts, not directly about ontology, and the same is true about the impossibility of replacement thesis; therefore, the strategy here is to first compare accounts with respect to theoretical virtues and only after that make ontological conclusions from this comparison using inference to the best explanation.

Observe that the thesis defended here is relatively modest. I do not say that in any context laws are more basic than dispositions but only that in the contexts in which there is a law-based account that works well (and from the successes of sciences we know that this class is non-empty), any disposition-based account will be worse (and as such would be less ontologically plausible). This leaves open the possibility that some dispositions exist independently of laws and even that they are equally fundamental (although I doubt that this is the case). The only option excluded is that in the cases well described in terms of laws it might still be the case that in the underlying metaphysical picture it is the dispositions that play a primary role, and laws can be either eliminated or derived from them/reduced to them.

6 Argumentation Against the Possibility of Replacement

Let us consider our exemplary law-based account using laws (2)–(5) and see what the prospect of replacing it with some disposition-based account is.

6.1 The Loss of Information in Accounts Based on Imprecise Dispositions

To successfully replace the law-based account given by equations (2)–(5), a dispositionalist would need to provide dispositions that are rich enough in content to encode all the information that is contained in these laws or at least the part of the information that we effectively use in making predictions. In this section, I will argue that there are no prospects for doing so if dispositions are specified in an imprecise way (the distinction between precise and imprecise dispositions is explained in the last paragraph of Sect. 2).¹⁴

Our law-based account (given by (2)–(5)) allows us to calculate the values of electric and gravitational forces at each moment of time, the resultant force $\vec {F}^{r}_i (t)$ at each moment of time and the positions of bodies $\vec {R}_i (t)$ at each moment of time (together with their velocities and accelerations, which are simply derivatives of the position function). We do not effectively use all this information in testing our predictions (e.g., because of the limited precision of our measurement devices), but we can predict and check, for example, that a given body after $t \pm \delta _t$ units of time will be $x \pm \delta _x$ units of distance from its actual position, where $\delta _t$ and $\delta _x$ are relatively small error bounds. A non-trivial example taken from real science is the use of scenario 3 or scenario 4 by astronomers to model the positions of celestial bodies. In what follows, I will argue that imprecise dispositions are not able to lead to equally good predictions.

Consider scenario 5. Using imprecise dispositions, we can say, for example, that bodies with a negative charge have a disposition to attract bodies with a positive charge. However, the law-based account enables us to say more than this. It allows us to calculate what the exact values of the forces are, given the values of charges and distances; then, it allows us to predict how exactly the bodies will move. It is not possible to extract information of this kind from imprecise dispositions.

The expression “exact values” used in the previous paragraph is slightly exaggerated. The result is exact only in the sense that there is, mathematically speaking, one exact solution to the equations. However, in more complex cases (such as scenario 3 with $N>2$), the solution can often only be known approximately or numerically. But the fact is that there are cases where our law-based account can be used (and is actually used) to get empirically adequate predictions for scenarios 1–6 (or similar ones). And such a weak claim is enough as an argument against imprecise dispositions, as they are, in principle, not able to replicate this success. They intentionally resign from providing more detailed information than is already available.

A special case of the problem of the loss of information is the loss of temporal information when we replace laws with imprecise dispositions. The Simple Conditional Analysis of dispositions does not involve time. We do not know whether a stimulus should precede a manifestation temporally or should be simultaneous with it, and if it precedes a manifestation, what exactly the temporal delay should be. There are more sophisticated approaches that introduce time to the analysis of dispositions explicitly (e.g., Lewis 1997; Malzkorn 2000), but this is always done in an imprecise way: we know only that the moment at which a manifestation happens is later than the moment of stimulus. The appeal to our exemplary dispositions suggests that this is how it should be. In typical examples of dispositions, a manifestation occurs later than a stimulus and we can roughly tell what this time interval is, but it cannot be given precisely. The shattering of a glass happens just after it is struck, and for the dissolution of salt in water we need to wait a few seconds or a few minutes, whereas the death of a poisoned person will happen a few hours after the ingestion of a poisonous substance. This is the highest level of precision achievable in these cases.

In contrast, our laws explicitly include the temporal parameter (via equation (5)) and give us the evolution that prescribes the exact behaviour of massive bodies at each of the later moments. To restore this information in a disposition-based account, we should somehow add time to dispositions and do so not in an imprecise way but in the form that will allow us to tell what will happen after $t$ units of time from a given moment for any $t \in \mathbb {R}^+$. So, again, we should opt for a precise strategy in specifying dispositions, and from now on we will consider precise dispositions only.¹⁵

6.2 The Number of Independent Assumptions (Simplicity)

One of the features most often mentioned as desirable for a good theory is simplicity, which is a vague concept, but there are several ways to make it precise, and one of them refers to the number of assumptions of a theory. A good theory should be simple in the sense of having a small number of independent assumptions compared to its competitors. This is still ambiguous, as there can be many ways of counting assumptions. However, it is possible to say something about the number of assumptions in law-based accounts and related disposition-based accounts without choosing the exact way of counting. In the latter case, the number of independent assumptions grows much faster with the increasing complexity of a scenario (i.e., involving a larger number of objects or a larger number of types of forces).

Consider scenario 1. In the law-based account, we start from a large number of assumptions. We need to make some factual assumptions (one object with mass m, position $\vec {x}$ and velocity $\vec {v}$), but we also need to postulate the laws in the form of equations (2)–(5) and the whole mathematical framework to deal with these equations (i.e., mathematical analysis). In a disposition-based account, we have the same factual assumptions, and one disposition is needed that states that if there are no other objects (S), a massive body moves with a uniform velocity (M). Therefore, at this stage, the disposition-based account is much simpler (and both accounts give the same predictions, which is our first constraint on replacement; see Sect. 5).

However, when turning to richer scenarios, the initial simplicity of the disposition-based account becomes illusory. Let us restrict ourselves to situations with gravitational force only and add further bodies successively (this corresponds to scenario 3 with growing N, whose simplest special case is scenario 2). In the law-based account, the only things that change are the factual assumptions: we need to specify the masses, positions and velocities of each newly added body. However, the laws (and the mathematical rules of using them) remain the same. In contrast, to account for the behaviour of the bodies, a dispositionalist needs not only to change the factual assumptions but also to add further (specifications of) dispositions, as from the disposition to behave in a certain set-up of N bodies nothing follows about the disposition to behave in a similar set-up with $N+1$ bodies.¹⁶

Although this argument does not rely on any particular way of specifying precise dispositions, let me give an example to illustrate the general idea. Let us say that for the 2-body set-up the relevant disposition has a stimulus “being at $t_0$ in the vicinity of a body with mass 2, position (1, 1.5, 3) and velocity (1, 0, 0)” (all quantities are given in some fixed units and position and velocity are 3-dimensional vectors in a fixed reference frame) and a manifestation “moving with trajectory $\vec {R}_1 (t)$ at times $t > t_0$”, where $\vec {R}_1 (t)$ is a particular function with times as arguments and position vectors as values (it can be calculated by solving the equations (3), (4) and (5), but here it is just assumed as given, as otherwise this disposition would rely in its specification on laws). Observe that to calculate $\vec {R}_1 (t)$, we need not only the information provided in the stimulus but also the mass, position at $t_0$ and velocity at $t_0$ of the object under consideration. Therefore, under this method of specifying dispositions, these specifications will be different for bodies with different masses and initial conditions. From the above specification of a disposition without further assumptions nothing follows about the situation in which to such a set-up the third body is added (in which case the specification of the stimulus should mention masses, positions and velocities of both the old body and the new body, as the trajectory depends on all these factors, and the specification of manifestation needs to be changed accordingly).

It is disputable whether the additions described above amount to taking into account new dispositions (in the metaphysical sense) or adding new specifications of the same disposition. The latter case would involve the so-called multi-track dispositions. However, as our considerations at this stage concern accounts, this difference does not matter. The number of newly added assumptions is the same irrespective of whether we interpret this as adding new dispositions or as adding new specifications of the same multi-track disposition, and it is this number that matters for the assessment of the simplicity of the account.

One can object that in this way I do not take into consideration the relevant kind of simplicity, which is ontological simplicity. Is it possible that A is ontologically simpler than B even though its full description is more complex? Maybe; for example, in some traditions of philosophical theology, God is claimed to be simple, and at the same time these traditions give a long and complex characterisation of God (although the consistency of such a combination might be contested both in the case of God and of more mundane metaphysical posits). This line of reasoning suggests that multi-track dispositions might be ontologically simple even though their descriptions are very complex. However, if ontological simplicity is a feature that is invisible at the level of accounts, it is irrelevant for my approach (cf. the last two paragraphs of Sect. 5), as my aim is to compare the accounts with respect to theoretical virtues, including simplicity (and infer ontological conclusions from this comparison). Therefore, it is only simplicity at the level of accounts that matters for me. For the same reason, it is not the number of dispositions invoked in the account that is important but the simplicity of the account as a whole. One can decrease the number of dispositions by “merging” them into a single multi-track disposition, but this would not make the account based on such dispositions simpler.

Another possible counter-argument of a dispositionalist could be that even if in the law-based account one can formulate appropriate equations, they very soon become too complicated for us to be solved, so, in practice, they cannot be used to make predictions. The case of scenario 3 is symptomatic. For $N=2$ (in which case it reduces to scenario 2), the equations can easily be solved for an arbitrary choice of masses and a wide range of initial conditions, but moving only one step higher to $N = 3$ leads to significant practical problems (i.e., the equations are very difficult to solve), not mentioning higher values of N. However, the fact is that the law-based account has not been given up by scientists despite these problems. First, in some special cases, the equations can be solved approximately. An example is our scenario 4: one of the bodies is much larger, so we can assume that the gravitational force coming from it is much stronger than the ones coming from the other bodies, and disregarding the latter would not change our predictions significantly. Second, the equations can be solved numerically for quite large numbers of bodies. The numerical solution means that we do not get a function (in the form of a mathematical formula) that prescribes the positions of bodies for each moment of time, but at least, knowing the initial conditions, we are able to give numerical values of the positions at each moment of time with high accuracy.

To perform numerical calculations, we need to use some methods that are not in the basic theoretical framework of mathematical analysis, which may suggest that this amounts to the growth of the number of independent assumptions. But this is not the case: mathematically there are no assumptions over and above mathematical analysis and the equations (2)–(5). The techniques of obtaining numerical solutions are not arbitrary—if they are valid, this is only because they give the results in accordance with the results prescribed by our equations. Similarly, all the approximations used are not additional assumptions about the physical situation, but their validity relies on the fact that the neglected factors are indeed not significant for the final result. All these seemingly additional assumptions only uncover what is already present in the description given by equations (2)–(5). The fact of what approximations and numerical methods are good and to what extent they are good is already encoded in the initial framework.

Dispositions deal worse than laws not only with the growing number of objects but also with the growing number of types of forces that should be modelled. Our exemplary law-based account contains two types of forces, electric and gravitational. The scenarios considered so far in this subsection (i.e., scenarios 1–4) involved electrically neutral objects, so the electric force was zero, and the resultant force was simply equal to the gravitational force. However, we can easily deal with scenarios 5 and 6 without adding any general assumptions. We just need to calculate the electric force (which now will be non-zero) using equation (2) and take it into account while calculating the resultant force using equation (4). We can also imagine adding forces of yet different kinds to our law-based account. This would not require a radical change of the framework we already have. All our equations (2)–(5) remain valid; the only thing that is needed is one additional equation expressing the new force, and we need to remember to take it into account when calculating the resultant force using (4).

In the case of dispositions, the situation is different. To each of the existing dispositions specifying how the motion of a given object would look if it were presented with gravitational force of a given magnitude, direction and orientation, there corresponds infinitely many dispositions that determine what would happen in a similar situation that differs only by the fact that a new force (e.g., electric) has such and such magnitude, direction and orientation.¹⁷’¹⁸ The infinity comes from the fact that there are infinitely many possible specifications of the values of magnitude, direction and orientation of this new force. Therefore, in the case of a law-based account, capturing a new type of force requires adding only one new assumption (irrespective of how many objects the system contains), whereas in the case of a disposition-based account, this requires infinitely many new (and independent) assumptions.

Before ending this section, let me come back to the views developed by Demarest (2017) and Kimpton-Nye (2017), which might seem to be a counter-example to my argument from the lack of simplicity of (precise) dispositions, given that they regard laws as derivative with respect to dispositions and build into them the feature of simplicity by defining them in a best system-like way. However, it is laws that are simple in their account (or at least as simple as they can be without sacrificing too much informativeness) and not specifications of dispositions. We do not learn much from these authors about how the description of the world in terms of dispositions would look (that is, what would be their stimuluses and manifestations), and they do not attempt to show that such a disposition-based account would be simpler or in any other way more theoretically virtuous than the description in terms of laws.

Concerning the difficulty with finding the specifications of dispositions (i.e., the explicit formulation of what their stimuluses and manifestations are), it seems to be anticipated by Demarest (2017, 52), who writes:

The difficulties that beset attempts to state dispositional essences do not beset similar attempts to state the laws of nature. For instance, even if the potencies are restricted to include only mass, there is a great deal of controversy as to what the dispositions are. Are masses simply disposed to attract one another, with perhaps a meta-disposition for how those massive attractions add to or subtract from one another? Or are there separate dispositions for each particular possible mass-instantiation distribution? If, instead, the laws systematize the possible distributions of potencies throughout spacetime, there is the possibility of ‘backsolving’ to arrive at the potencies. And, it is likely that this is exactly what metaphysicians do when they discuss dispositions such as mass and charge. But, of course, the metaphysical direction of explanation need not point the same way as the epistemic one.

Demarest considers various ways of specifying dispositions, all of which are discussed in my paper. Her “masses simply disposed to attract one another” correspond to my imprecise dispositions (Sect. 6.1), her “separate dispositions for each particular possible mass-instantiation distribution” would lead to the proliferation of (specification of) precise dispositions discussed in the current subsection and her “meta-disposition” seems to be a disposition specified by citing the law given by equation (4), so it cannot be used in a genuine disposition-based account (cf. Sect. 6.6). Eventually, she claims that assuming that laws summarise possible distributions of potencies (as she does), one can derive what the potencies are from laws by “backsolving” them. However, it is difficult to see what the result of such a backsolving could be other than the options investigated in this paper.

Demarest also makes an intriguing remark about the independence of metaphysics from epistemology. Perhaps one can believe that dispositions are more basic than laws without being able to say what exactly these dispositions are (i.e., what are their stimuluses and manifestations)? This is of course not a self-contradictory view, but it is difficult to see how it might be justified based on our practices of predicting and explaining by means of laws. Why believe that something known and well-working is derivative from something else, where this “something else” and the alleged derivation are almost entirely unspecified?

6.3 Unification

The issue of the number of independent assumptions is related not only to simplicity (as explained in the previous section) but also to unification, which is another desirable feature of theories. To unify means to explain different phenomena by the same (general) assumptions instead of using different (general) assumptions to explain each of these phenomena separately. Many of the greatest achievements in the history of science amount to finding one theory that is able to deal in a uniform way with phenomena that were previously thought to be unrelated. For example, Newton’s theory of gravitation explains both the behaviour of falling bodies near Earth and of planets moving around the Sun, which in Aristotelian physics were regarded as phenomena of fundamentally different types. Another example is Maxwell’s theory of electrodynamics, which unified the phenomena of magnetism and electricity.

The need for unification is particularly pressing when there are some visible similarities between the behaviour of different systems. This is the case in our disposition-based account described above. In scenario 4, when we change the number of bodies from N to $N+1$ (i.e., we add a new small object to the system with one large object and $N-1$ small objects), the behaviour of the system does not change much: the previous N bodies behave very similar to the way they behaved in the absence of the new object (i.e., they orbit around the large body). From the point of view of the disposition-based account, this is an unexplained similarity: the (precise) disposition to behave in a certain way in the presence of N objects is something different from the (precise) disposition to behave in a certain way in the presence of $N+1$ objects, and in a disposition-based account they need to be two independent assumptions. The problem seemingly might be solved by appealing to dispositions specified in a less precise way (e.g., an object has a disposition to orbit around another object when the latter is much larger than the former and than all other objects nearby), but, to recall, we have abandoned the imprecise strategy of specifying dispositions because it leads to the loss of information (see Sect. 6.1).

To get rid of the independence of assumptions (and thus make the whole account simpler and more unified), a dispositionalist could try to impose some additional constraints that would allow us to infer some dispositions from others (e.g., from the disposition to behave in a certain way in the presence of N massive uncharged bodies to the disposition of behaving in the presence of $N+1$ such bodies). But how could such connections be drawn? On the one hand, we know that the law-based account given by (2)–(5) works well. Therefore, any way of formulating these connections must be consistent with equations (2)–(5). On the other hand, to avoid the loss of information, all the content of these equations should be recovered. What we can do is to postulate that dispositions should be “generated” in accordance with the equations (2)–(5). The problem is that then the real predictive and explanatory work would be done by the rule of generating dispositions rather than by dispositions themselves—and such an account may not be regarded as genuinely disposition-based (in fact, this would rather be a law-based account in disguise).

6.4 The Distance Between the Assumptions and the Conclusions

Another crucial difference between law-based accounts and disposition-based accounts is what I want to call “the distance between the assumptions and the conclusions”. Recall from Sect. 5 that the main purpose of our accounts is to give predictions based on some assumptions (general and factual) and inference rules. Therefore, our assumptions are some general truths and some factual truths (holding at some time) about the system under investigation, and our conclusion is a prediction concerning the behaviour of the system (at later times). The conclusion is reached from the assumptions by means of the inference rules. Once we fix the set of assumptions and the set of inference rules, it is an objective fact how many inferential steps are needed to reach a particular conclusion from the assumptions (i.e., what is the smallest number of such steps). This number will be called “the distance between the assumptions and the conclusion”.

For disposition-based accounts, the distance between the assumptions and the conclusion is very small. If one knows that a glass is fragile (and that fragility is a disposition to shatter when struck) and that the glass has been struck at t, very few inferential steps are needed to conclude that at some later time $t'>t$ the glass will shatter. One might be tempted to say that it requires almost no inferential work; the conclusion (i.e., the prediction that at some $t'>t$ the glass will shatter) is already visible in the assumptions and we immediately grasp the conclusion when looking at the assumptions. This makes performing inferences based on dispositions very easy and fast: from the occurrence of S and an object having a disposition to M when S, the occurrence of M follows directly. Fragility understood as a disposition to shatter when struck is an instance of an imprecise disposition (and imprecise dispositions have been excluded as a possible replacement for laws on other grounds; see Sect. 6.1), but the point holds equally well for precise dispositions, as this argument relies only on the structural features of dispositions (that is, their being specified by a stimulus S and a manifestation M). Whatever S and M are, the reasoning would look exactly the same (to have a concrete illustration, the reader might consider what the reasoning using an exemplary specification of a precise disposition given in the fourth paragraph of Sect. 6.2 would look like).

For our exemplary law-based account, however, quite the opposite holds: in order to get the predictions from the factual assumptions and the laws, some complicated calculations need to be performed. Therefore, the distance between the assumptions and the conclusion is large, sometimes even too large for humans to overcome in practice.

At first glance, it might not be obvious what is better—to have a smaller or a larger distance between the assumptions and the conclusion. The answer is that this depends on the context. When the time for performing inferences is limited, only accounts with a small distance can be useful. The knowledge that tigers have a disposition to eat humans is useful because when faced with a tiger we can conclude in a very short time that we should run away. If a longer inference were required, it would be too late to escape. However, when it comes to science, the larger distance turns out to be connected with some of the features of science we value most—non-trivial predictions, fruitfulness in applications and explanatory power.

Both law-based accounts and disposition-based accounts enable making predictions. However, the type of these predictions differs significantly. Laws allow us to make predictions that are less obvious. We cannot tell just by looking at the equations and the initial conditions what the positions and velocities of the celestial bodies will be five years later; this needs to be calculated. In simpler scenarios, we can perhaps guess using only our intuitions but with precision much lower than that given by the calculations based on laws. In contrast, we can tell just by looking at the specification of a disposition of solubility that a soluble object will dissolve when put into water (i.e., the operations needed to draw this conclusion are so simple that it is almost unnoticeable that we perform any operations at all). More generally, if we know that an object has a disposition to M when S and that S happened, the conclusion of M is obvious, so making such an inference does not teach us anything different from what we have already known.

The non-triviality of predictions that can be made on the basis of laws makes them more fruitful in the context of practical applications. One can invent new machines without discovering new laws just by applying the known laws to the arrangements of matter that have not yet been considered. Knowledge about the dispositions of objects and materials is also very important in practice, but its constructive potential seems to be much more modest.

The feeling of “non-obviousness” is also important for explanation. The exact account of explanation is disputable in philosophy. But the key point is that in practice nobody would call an explanation the claim that just states the fact that is to be explained or the claim that states something so close to this fact that, by looking at it, we already see what this fact is. In short, some minimal distance between the assumptions and the conclusion is needed to call something an explanation of a fact rather than a statement of this fact.

The contestability of the explanatory power of dispositions is not a new idea. Molière famously criticised explaining why opium causes people to sleep with reference to virtus dormitiva, which is basically the power to make people sleep. According to him, dispositions do not have any explanatory power at all but they just re-state what is to be explained. This view has been challenged by some philosophers who argue that dispositions have explanatory power, against Molière’s suggestion (see, e.g., Rosenberg 1985, 160; Radcliffe 2007, 118). However, to make my point, I do not need the radical thesis that dispositions do not have any explanatory power at all. It is sufficient to say a much less controversial thing, namely that accounts with a moderate or large distance between the assumptions and the conclusion have greater explanatory power than those with an extremely small distance. Therefore, in particular, law-based accounts have greater explanatory power than disposition-based accounts.

This argument may look somewhat “soft” because it refers to the subjective feeling of non-obviousness or non-triviality. In some sense, the objection may go, even in law-based accounts everything is put in by hand, as mathematical proof only reveals what was already hidden in the assumptions, and it never adds anything genuinely new. Therefore, the perceived difference between disposition-based accounts and law-based accounts relies on our limited capacity to perform deductions. A hypothetical subject with infinite capacity would see the trajectory by looking at the laws and initial conditions in the same way as we see M when looking at S and “a disposition to M when S”. However, there is an objective difference on which this subjective difference is based: the minimal number of operations that need to be performed in order to get the conclusion from the assumptions is objectively determined once we fix the set of inference rules. Therefore, the difference between trivial and non-trivial predictions as well as between smaller and larger explanatory power is grounded in this objective fact about our accounts (predictions are non-trivial and explanatory power is reasonably large if the minimal number of operations needed to deduce the conclusion from the assumptions is not extremely small).

6.5 Interactions and the Screening Off of Interactions

Dispositions are attached to single individual objects. This means that typically S and M are attributed to a single individual object. We can make them more complex by allowing relational properties; then, S and M could involve both an object’s intrinsic properties and an object’s staying in the relation with some other objects (see, e.g., Heil 2005). But still a disposition is always centred on a single object. This is the first ingredient of the minimal core of the concept of dispositions presented in Sect. 2. Because of this centring on a single object, a disposition-based account of any actual situation has two “sides”—the object (with its properties, including dispositions) and everything else that influences it, glued together into S. Dispositionalists do not have the means to partition the stimulus S into separate factors and see how they affect each other or, alternatively, to analyse how the object behaves in the presence of two or more different stimuli (any such “calculus of stimuli” would be a proper addition to specifications of dispositions in accordance with the S–M scheme). In particular, they do not have the means to analyse the situation where different stimuli cancel each other out (fully or only partially). Doing so would require moving beyond the simple S–M scheme, which amounts to moving beyond dispositions.

In contrast, a law-based account divides what surrounds an object into separate pieces and considers the interactions between them; all the pieces are on a par, including the object we started with. This difference makes it easier for law-based accounts to analyse interactions between objects. Consider scenario 6: because the objects are massive as well as charged, both gravitational and electric forces are non-zero, so in predicting the trajectory of the objects we need to take into account these two types of forces. Gravitational force is attractive, whereas electric force between objects of the same charge is repulsive, so it is non-obvious what the trajectories will be: will the objects move towards each other or away from each other (or follow some more complicated trajectories)? Equations (4) and (5) allow us to answer this question. In our particular case, the direction of both types of forces is the same (their vectors lie on the line joining the two objects), so what matters for the motion of bodies is only the differences between the values of the forces. Assume, for example, that the electric force is stronger. In this case, the objects will move away from each other, despite the fact that there is a gravitational force attracting them to each other. However, the fact that the gravitational force is present influences their motion: they will move slower than they would in the absence of this force. If we add further objects, we can treat them in exactly the same way: we just compose the forces generated by different influencing bodies (i.e., add them as vectors, in accordance with the law (4)) to find the resultant force acting on the body we are interested in, and from this we calculate its trajectory using equation (5).

The law-based account not only allows one to deal with interactions when they are present and significant but also to analyse cases in which they are (to some extent) screened off, and as such they can be ignored for the purposes of practical calculations. This is an important point because Demarest (2017, 47–48) regards such cases as indicating the “science-friendliness” of dispositions (cf. also Cartwright 1999):

Since potencies often do not exist in isolation, and are typically exposed to many stimulus conditions, it can be quite difficult to determine just how the particles that instantiate them will behave. Luckily, the effects of stimulus conditions get weaker with distance, and often can be shielded. (...) To illustrate this phenomenon, consider a representative physics experiment, which measures gravitational attraction at very small distances. In such an experiment, the physicists place two masses very close to each other and measure the force between them by how much they deflect the springs to which they are attached. Because the force of gravity is so weak, especially when compared to the electromagnetic forces, the biggest challenge of the experiment is to screen off as many outside effects as possible. Thus, the experiments are performed in a deep basement, in the dark, at night, and only during breaks in the traffic outside. This careful screening-off procedure is evidence that the physicists aim to discover the characteristic behavior (manifestation) of a specific kind of property (disposition).

However, it is a law-based account that has straightforward means to predict that the gravitational force will get weaker with distance, which is easily visible from equation (3). In contrast, if we specify dispositions as in the example in the fourth paragraph of Sect. 6.2 or in a similar way, reaching this conclusion requires comparing infinitely many different specifications of dispositions (and if we specify them imprecisely, we will likely not get the result of weakening at all—e.g., from the fact that particles with opposite charges attract each other nothing follows about the changes of this “attracting” with the distance between them).

What is more, in a law-based account we can understand why we should screen off as many interactions as we can in the experiments performed in order to test our hypotheses. As mentioned in Sect. 5, to derive any prediction, we need not only general assumptions (in our case, laws) but also factual assumptions (in our case, the number of bodies, their masses and charges, as well as their initial positions and velocities). However, knowing all the details of a given physical situation is very difficult, so it would be better for us if most of them were not significant for the final trajectories of bodies under investigation, as then we could make accurate predictions without knowing those facts exactly. Therefore, the entire business of screening off can be interpreted as an attempt to simplify the significant part of our factual assumptions (“significant” in the sense of their contributions to equation (4) having large values), which are used together with the general assumptions (i.e., laws) to derive the predictions about the system. If we grant this explanation, there is no need to seek an alternative for laws in order to understand the need for screening off certain interactions in experiments. It suffices to use our law-based account, paying sufficient attention to the distinction between general and factual assumptions.

Another related issue, which is regarded by some authors as providing a reason for invoking dispositions in the analysis of the laws of nature, is the so-called ceteris paribus clause. For example, Kimpton-Nye (2017, 15) suggests that his Revised Potency Best-System Account has the advantage of being able to explain the ceteris paribus nature of Coulomb’s law:

It is implicit in the formulation of Coulomb’s law (and other laws) that intervening factors are absent. All Coulomb’s law says explicitly is that separated charges exert a force on each other proportional to the magnitude of their charge and inversely proportional to the square of the distance between them. What is left implicit is that this is only the case in the absence of, say, a nearby black hole or indeed of anything else that may negate the tendency of charged individuals to interact in accordance with Coulomb’s law. What Coulomb’s law tells us, on the current conception, is that in the absence of intervening factors, that is, ceteris paribus, charged bodies will interact in this and that way.

However, I believe that there is no such fact to be explained¹⁹ and that the confusion stems from the ambiguity of the term “interaction”: it might be understood as referring to equation (2) alone or to all equations (2)–(5) taken together. In the latter case, there is no such thing as “interacting in accordance with Coulomb’s law”, as the effect of the interaction is determined by more than one law. In the former case, there is such a thing, but it does not amount to any particular motion because Coulomb’s law alone does not determine the trajectories of bodies. (For this purpose, the formulas for other forces, the law of the composition of forces (4) and the equation of motion (5) are also needed.) Therefore, if two identically charged bodies move towards each other, this does not mean that Coulomb’s law does not hold in this case but only that there is some other force present that contributes to the resultant force in accordance with (4), and the resultant force makes them move towards each other via (5). If we attempted to make predictions in this case under the assumption that Coulomb’s law does not hold and the Coulomb’s force is zero despite the presence of charged bodies, we would get wrong predictions because we would ignore one of the contributions to the resultant force (4). We do not need the ceteris paribus clause here because we have the law of the composition of forces, (4). Of course we can say that, ceteris paribus, two identically charged bodies move away from each other, but this is not the entire content of laws (2)–(5); the content of the latter is much richer. This is also not the content of (2), which on its own does not tell anything about the motion of bodies (such as moving towards or away from each other or following some more complicated trajectories). Sometimes, we do not know all the forces, and as a consequence we are unable to make accurate predictions (cf. the previous paragraph about screening off), but this reflects our lack of knowledge about factual assumptions and not the limitation of laws (2)–(5) that can be captured by adding to them the ceteris paribus clause.

6.6 Refinement Strategies

One can object that so far we have not been fair to disposition-based accounts. Laws as described by SAL suffer from the same problems as dispositions do, but this is not the case for more mathematically sophisticated laws. Therefore, one can ask whether there could be a refined form of dispositions as well. Perhaps they should be allowed to use variables, or even functions, and all the mathematical machinery that laws have at their disposal (in addition to mere numbers, which need to be used in the specifications of precise dispositions, such as the example in the fourth paragraph of Sect. 6.2). Where would this lead us?

Let us try to find a dispositional counterpart of (2). One can start like this: instead of considering separate dispositions for different values of charges and distances, we can use one disposition involving variables. For example, any object with charge $q_1$ has a disposition to exert a force $\vec {F}^e = k q_1 q_2 {\vec {r}_{12}}/{r_{12}^3}$ on any object with charge $q_2$. Importantly, here $q_1$, $q_2$ and $\vec {r}_{12}$ are now not any particular numbers or vectors but variables ranging over all of them, which is why we avoid the objection of Sect. 6.2. This move significantly reduces the number of independent assumptions, but we still need to postulate separate dispositions for situations involving different numbers of bodies (in the case of a law-based account, this is not needed because it uses (4)). What is more, introducing variables on its own does not give us enough resources to formulate a dispositional analogue of the law (5), as it requires explicit reference to time and a differential operator.

Therefore, perhaps we should supplement the above with a disposition specified by citing the rule for calculating the resultant force and an additional disposition that involves the law of motion: any physical object has a disposition to move as prescribed by equations (4) and (5). Now, our disposition-based account no longer suffers from the problems raised previously; all the information from the laws is restored, the number of independent assumptions is the same as in the respective law-based account, inferences become non-trivial (as the manifestation is not a ready description of what will happen but a rule that allows us to infer such a description), etc. But this is because the manifestations of the dispositions literally cite the respective laws. Therefore, we ended up with dispositions to behave in accordance with the laws (2)–(5), so this is not a genuine alternative to the original law-based account but rather a law-based account in disguise.

Observe that in the above proposal of a disposition-based account that relies on citing the laws, dispositions have been characterised only in terms of their manifestation. Because there are no conditions under which laws (2)–(5) do not hold (cf. the discussion of ceteris paribus laws in Sect. 6.5), there is no good candidate for a stimulus. (The presence of massive or charged objects does not need to be added as a stimulus because in their absence these laws still work: we just need to substitute 0 for all $m_j$’s and $q_j$’s in the equations and we will get the right results.) For this reason, the most natural way to specify the dispositions to behave in accordance with these laws is to use only the manifestation M instead of a full S–M scheme. This means that we can easily translate between such a disposition-based account and the original law-based account by just adding or removing the preamble “a disposition to” in front of the formulas expressing laws. However, the addition of such a preamble does not change the way in which this account is used to deal with the phenomena—to derive any conclusions about some scenario from this disposition-based account, we just need to remove the preamble “a disposition to” and use the laws as described in Sect. 3.

Some authors who seem to classify themselves as explaining laws in terms of dispositions or replacing laws with dispositions in fact specify these dispositions by just citing the laws (e.g., Cartwright 1999; Fisher 2018, 124; Hüttemann 2014; Vetter 2015, 285). However, this way of explaining or replacing laws seems to me only apparent, simply because the laws of nature remain in the same form they have in a law-based account. In such a non-genuine disposition-based account, laws do all the conceptual work that they do in the corresponding law-based account. The addition of “the disposition to...” before the equations (2)–(5) does not in any way improve the account with respect to theoretical virtues; its effectiveness in describing nature comes from the features of laws listed in Sect. 3 rather than from their being “embedded” in dispositions. It is the laws that are invoked for the purposes of predicting or explaining and not something else that replaces them or from which they can be derived.

Furthermore, there is another reason to doubt that expressions such as “the disposition to behave in accordance with laws (2)–(5)” indeed refer to some dispositions. In all our paradigmatic cases of dispositions, the manifestation concerns the actual behaviour of an object—dissolving in the case of solubility, getting angry in the case of irascibility, etc. As we have mentioned, law-based accounts provide a recipe to find the description of the actual behaviour rather than providing such a description directly, and this feature is inherited from laws by the manifestations of the above formulated specifications of dispositions. As our criterion for dispositionality is based mainly on conformity to paradigmatic cases, this change in the character of the manifestation is arguably enough to say that here we no longer have an instance of dispositions.

It can be objected that if one of our conditions for a successful replacement is that dispositions should recover all the information that is involved in laws, it should not be surprising that we ended up with dispositions that behave like laws. In response, let me make two points. First, what we ended up with here are not just dispositions that behave like laws in the sense of recovering all the information that is present in the laws; rather, these are dispositions which are specified by literally citing the laws. For this reason, all the theoretical virtues of this pseudo-disposition-based account are entirely derivative from the features of the laws that are cited in the specifications of these dispositions. The fact that we managed to overcome the drawbacks of disposition-based accounts discussed in Sects. 6.1–6.5 by using dispositions whose specification involves laws has nothing to do with the fact that we are using dispositions (i.e., structures specified by a stimulus and a manifestation or a manifestation alone); rather, it is only thanks to our invoking laws. The same result (i.e., the same theoretical virtues) would be gained (in a cleaner way) by simply using the law-based account we started with.²⁰

Second, as I mentioned earlier, if we want to talk about replacing laws with dispositions (or reducing laws to dispositions), there should be a genuine difference between them, and the specification of the latter should not rely on the former (irrespective of whether our ambition is the elimination of laws or only their derivation from/reduction to dispositions). Is this an excessively high demand? I do not think so; this seems to be a usual demand for a replacement or reduction. Think about the idea that the mental is reducible to the physical. An often-invoked example is the reduction of pain to C-fiber stimulation. Would it count as an instance of (genuine) reduction if “C-fiber” was defined as a fiber that is stimulated whenever one feels pain? Certainly not; “C-fiber” should be characterised in purely physical terms for this to be a successful example of reduction of a mental state to a physical state. Analogously, the (genuine) reduction of laws to dispositions requires that the latter are not specified in terms of the former. Interestingly, this requirement is satisfied, at least sometimes, in the other direction. For example, scenarios 5 and 6 can be described in terms of dispositions to attract objects with the opposite charge and repel objects with the same charge, respectively. This information is recovered (and refined) by a law-based account using (2)–(5) without literally citing the mentioned dispositions. Therefore, in this case, a law-based account can successfully replace the mentioned disposition-based account in my sense of replacement.

7 Summary

I have argued that whenever some physical situation can be given a well-working law-based account, it cannot be given an equally well-working disposition-based account—the latter will always be either less informative or inferior with respect to theoretical virtues such as simplicity, unification, non-triviality of predictions, explanatory power and fruitfulness in applications. The reason is that dispositions are structurally too simple: they are specified by just two elements (a stimulus and a manifestation) or even only one (a manifestation alone), they are about one individual only (its environment is treated as a non-analysable whole) and provide a straightforward description of the behaviour (expressed by the manifestation part). In contrast, laws can be arbitrarily complex, can involve many individuals and, most importantly, do not straightforwardly describe the behaviour of objects but are principles that allow us to derive such descriptions. I suppose that the reason why this difference has been largely overlooked is the prevalence of the simplistic analysis of laws based on formula (1), which obscures many important features possessed by laws and suggests that they are closer to dispositions than they actually are.

An analogy that might be useful in explaining the difference between dispositions and laws is that of a black box. A disposition-based account provides pairs of inputs and outputs (i.e., stimuluses and manifestations); everything that is “between” is a black box. In contrast, law-based accounts may be seen as an opening of the black box: they tell us the rules of generating outputs from inputs. I think that it is this difference that makes law-based accounts more theoretically virtuous than the corresponding disposition-based accounts.

One may worry that our conclusions can be undermined by observing that an exemplary account discussed here belongs to a physical theory that is currently not regarded as fundamental, namely to classical mechanics, which needs to be replaced with quantum mechanics (which in turn needs to be replaced with quantum field theory and perhaps finally with some quantum gravity theory). Are these more fundamental theories more disposition-friendly? It is clear that both quantum mechanics and quantum field theory are law-based accounts (using differential equations) and not disposition-based accounts in our sense. But are there some prospects for the replacement (in the sense of Sect. 5) in these cases? Dorato (2007) and Bigaj (2011) argue that there is a place for dispositions in quantum mechanics, although they do not consider these dispositions as a potential replacement for laws. They both give an example of spin as a candidate for a quantum disposition. At this point, one should ask: If it is a disposition, what are its manifestations? The starting idea might be that “the spin of an electron in a given direction is identical with the electron’s disposition to be deflected either up or down in a non-uniform magnetic field aligned in the selected direction” (Bigaj 2011, 212). However, this is definitely not the full specification of this disposition, as the value of spin influences an electron’s behaviour in many more contexts and not just in this particular experimental set-up. Therefore, one may try to characterise this disposition by providing a list of pairs consisting of an experimental set-up (playing the role of S) and the behaviour of a particle with a given value of spin in such a set-up (playing the role of M) and then another list for cases involving two particles with given values of spin, and so on. This reminds us of the issue of the number of independent assumptions discussed in Sect. 6.2 and indicates that when trying to replace the quantum mechanical law-based account with a disposition-based account, we will face the same problems as in the case of an attempt to replace the classical mechanical law-based account with a disposition-based account.

Another objection that has been posed to me is that the relevant contrast that my considerations point out is not between laws and dispositions but between a quantitative and a merely qualitative conception of properties. The differences between laws and dispositions analysed in Sect. 6, the objection goes, are due to the fact that dispositions are conceived in a qualitative way and laws are conceived in a quantitative way, and not due to the differences between dispositions and laws as such. However, this objection does not fully do justice to my argumentation. An exclusively qualitative conception of dispositions has been considered only in Sect. 6.1 (which dealt with what I called “imprecise dispositions”), but in later sections we investigated dispositions specified in a precise way, including the exact numerical values of properties that appear in the stimulus and the manifestation. The problem with dispositions is not that they cannot be made numerically precise but that they have so simple a structure that this precision requires us to add many new independent general assumptions to our disposition-based account, whereas laws, thanks to their more complex mathematical structure (which is not just the stimulus–manifestation pair), can encode the same information in a much more efficient way.

What is the metaphysical significance of these considerations? I agree that without additional assumptions they do not have any immediate metaphysical consequences, as they concern accounts rather than beings. Based on the fact that law-based accounts are more efficient in the description of the physical world than disposition-based accounts, it does not straightforwardly follow that laws are more real or more fundamental than dispositions. Whether one should draw this conclusion depends on one’s more general approach to building metaphysical theories. One reasonable approach is to build metaphysics in close relation to what our current best-working accounts of the world are (where “best-working” is understood in terms of theoretical virtues, such as those considered in this paper).²¹ For someone with this attitude, our considerations are highly metaphysically relevant (and I think that this attitude should be preferred). However, if for some philosopher other constraints on metaphysics are more important than what are the current best-working accounts of the world and these constraints are in conflict with the thesis that laws are more real or more fundamental than dispositions, then I do not have any further arguments to convince such a philosopher. For example, for some philosophers the fact that dispositions are rooted in concrete objects and the laws are not might be the feature of dispositions that outweighs the balance in favour of them, irrespective of all their drawbacks.

This is related to the final objection I want to consider, namely that the aim of dispositionalists is not to provide an alternative to the appeal to laws in making predictions but rather to provide a metaphysical story of what underlies these laws and the predictive power that they enjoy (or what are the truthmakers of the law statements). In other words, dispositions are not supposed to compete with laws in the territory of science but in the territory of metaphysics. My answer is that to do this work of “underlying” or making true, dispositions need to encapsulate all the information that is contained in the law statements. But we have seen that imprecise dispositions are not able to do this, whereas precise dispositions can do this at the cost of simplicity, unification, etc., which are theoretical virtues that both science and metaphysics aim for. The only remaining option seems to be dispositions specified by citing the laws, but I have argued that this is not a genuine alternative to laws.