Natural kinds and dispositions: a causal analysis
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Abstract
Objects have dispositions. Dispositions are normally analyzed by providing a meaning to disposition ascriptions like ‘This piece of salt is soluble’. Philosophers like Carnap, Goodman, Quine, Lewis and many others have proposed analyses of such disposition ascriptions. In this paper we will argue with Quine (‘Natural Kinds’, 1970) that the proper analysis of ascriptions of the form ‘x is disposed to m (when C)’, where ‘x’ denotes an object, ‘m’ a manifestation, and ‘C’ a condition, goes like this: (i) ‘x is of natural kind k’, and (ii) the generic ‘ks are m (when C)’ is true. For the analysis of the generic, we propose an analysis in terms of causal powers: ‘ks (when C) have the causal power to m’. The latter, in turn, is analyzed in a very precise way, making use of Pearl’s probabilistic graphical causal models. We will show how this natural kindanalysis improves on standard conditional analyses of dispositions by avoiding the standard counterexamples, and that it gives rise to precise observable criteria under which the disposition ascription is true.
Keywords
Natural kinds Dispositions Generics Causality1 Introduction
According to Quine’s suggestion, x has a certain disposition if and only if x is of natural kind k and objects of kind k show manifestation m (under condition C). We will argue in Sect. 3 in favor of such an analysis, following authors like Fara (2005) and Manley and Wasserman (2007, 2008) making use of habituals, or generics, to account for disposition ascriptions. In Sect. 4 we want to go beyond the latter analyses by basing our proposal on a very precise comparative analysis of generic sentences. In Sect. 5 we will ground this analysis of disposition ascriptions on a causal relation between the natural kind and the manifestation. We will do this by making use of Pearl’s (2000) causal model theory, relating our analysis to the traditional idea that dispositions are, or involve, causal powers. In the end, our analysis proposes to analyze sentences like (1a) and (1b) as follows: We will argue that such an analysis will be less mysterious than it may look at first by showing how, and under which circumstances, dispositional ascriptions can be empirically tested after all.Intuitively, what qualifies a thing as soluble though it never gets into water is that it is of the same kind as the things that did or will dissolve. (Quine 1970, ‘Natural Kinds’, p. 16)
2 A conditional analysis
Carnap (1936–1937) proposed a conditional analysis of disposition ascriptions. According to it, ‘This cube is water soluble’ is true just in case this cube dissolves, if, or when, it is immersed in water. It was soon recognized that this conditional analysis cannot be worked out making use of the material conditional (because this conditional is true if the antecedent is false), or any other extensional connective (e.g. Burks 1955). Goodman (1954) suggested the natural alternative: making use of counterfactual conditionals: x is soluble iff xwould dissolve if it were immersed in water. Similarly, this sample of liquid is poisonous, means that someone would die, if she would drink it. Following Lewis’ (1973a) analysis of counterfactual conditionals, this would mean that x has the disposition of being soluble iff x dissolves in all most similar worlds to the actual one where x is immersed in water.
2.1 Standard worries involving mimicking and masking

x is disposed to m when C iff x has an intrinsic property B such that, if it were the case that C, and if x were to retain B for a sufficient time, then x would have property m.
When confronted with such counterexamples, there are always two ways to go. First, one can try to save the analysis by providing more specific conditions under which the to be observed effect, or manifestation, will show. As observed by Fara (2005) and others, however, this move is problematic for two closely related reasons: (i) it is doubtful whether it possible to spell out the content of this more specific ceteris paribus clause; (ii) if it is possible at all, the challenge is how to save the resulting analysis from being vacuous, or circular. An alternative strategy would be to weaken the conditional analysis.
One way to weaken the conditional analysis is to appeal to normality. Instead of demanding that x would be, or show, m in all most similar worlds where C would be true, one only demands that x would be m in all most normal worlds where C would be true (Mourreau 1997; Bonevac et al. 2011).^{4} In this way, all of the counterexamples could be explained away as being abnormal. Although formally appealing, and in line with much work on nonmonotonic logic, we find this proposal conceptually wanting. Without a more elaborate story of what it means to be a normal Cworld it is unclear what will be predicted. The danger of spelling out the appropriate normality conditions is obvious: how to avoid the analysis from becoming vacuously true?^{5} A, we feel, more attractive way to weaken the conditional analysis will be discussed in the next section.
2.2 Towards a generic conditional analysis, with natural kinds

x is disposed to m when C if and only if x has an intrinsic property in virtue of which it ms when C.

x is disposed to m when C iff x is of natural kind k such that xms when C in virtue of belonging to kind k.
Fara (2005) provides such a more principled answer by proposing that ‘ks are m when C’ should be analyzed as an habitual, or a generic. For instance, for the habitual ‘John smokes, if he is nervous’ to be true, it is not required that John always smokes when he is nervous. The same holds for habituals, or generics, involving natural kinds, like ‘Benzene burns when put into fire’. Fara argues that because habituals and generics tolerate exceptions, such an analysis can account for the counterexamples involving mimicking and masking. Unfortunately, Fara (2005) doesn’t provide any specific treatment of generics his analysis depends on. We agree with Fara that an analysis of dispositional statements in terms of generics is promising. But we also agree with YliVakkuri (2010) and others that without a specific treatment of generics, or without one that is sufficiently different from an analysis of counterfactuals, we can’t be satisfied with Fara’s proposal as it stands.
2.3 Structural worries about the conditional analysis
They note that this is a problem for conditional approaches, because the truthconditions of counterfactual conditionals don’t allow for the consequent to hold to more or less of a degree.
One might think that comparative and gradable disposition statements are easy to account for on a conditional analysis. After all, Lewis (1973b) makes use of a notion of comparative similarity to account for his counterfactuals. Although for counterfactuals he uses only the most similar worlds that make the antecedent true, one might think that less similar worlds could still be used to interpret comparative dispositional statements, or disposition statements involving degrees. Likewise for an analysis in terms of normality. Example (4a) can then be analyzed as being true, for instance, because when it would be put into fire, x would burn in more similar worlds to the actual one, or in more ‘normal’ worlds, than y would. Unfortunately, such an analysis would give rise to the consequence that because (4a) is considered to be true, ‘y is flammable’ is predicted to be false, because the counterfactual ‘y would burn, if put into fire’ would then be false. But this prediction seems much too strong, which falsifies the straightforward proposal under discussion.^{7}
But perhaps this seemingly false conclusion could be saved by an appeal to context. Perhaps all that comparative and gradable disposition ascriptions show is that the truth of disposition statements are context dependent. Indeed, as pointed out by Vetter (2013), in the context of aeronautics, many more things will be seen as ‘fragile’ than in more ordinary contexts. Some linguists have proposed (cf. Kamp 1975; Klein 1980) to reduce gradability to comparatives, and the analysis of comparatives, in turn, to context dependence. So perhaps the conditional analysis may be saved in this way from the threat of comparative and graded dispositions statements. Perhaps—but we feel that this kind of savior would be rather unnatural.
Another possible way to account for comparative and gradable disposition ascriptions on a conditional analysis would be to make use of probabilities. Just like many authors feel it is natural to account for indicative conditionals making use of standard conditional probabilities, one might make use of counterfactual conditional probabilities to account for the intuition that if an antecedent would hold, then one consequent would be more likely than another. On such a proposal, the acceptance of (4a) ‘x is more flammable than y’ need not give rise to the counterintuitive conclusion that y is not flammable. Indeed, we will make a related proposal ourselves later in this paper. But what, on such a proposal, would make it the case that ‘x is flammable’ is accepted? It cannot be that x would burn in all relevant worlds where it would be put in fire, for then the undesirable consequence that y is not flammable would follow after all. Perhaps one can get rid of the problem if one rejects the idea that counterfactual conditionals have truth conditions in the first place, or by proposing that a counterfactual is true if the probability that the consequent would be true if the antecedent would hold is greater than the probability that the consequent is not true. Perhaps—but we wonder how many proponents of the conditional analysis of dispositions would accept any of these consequences for the analysis of counterfactual conditionals.
A natural idea would be to (i) argue with Fara (2005) that a disposition statement like ‘x is flammable’ should be analyzed in terms of a generic, and (ii) that generics should be accounted for in terms of (counterfactual) conditional probabilities. This is a natural proposal, because generics allow for exceptions. It is thus only natural to propose that a generic can be true even though the relevant conditional probability is less than 1. We will see later that, indeed, a proposal along these lines was given by Manley and Wasserman (2007, 2008) and will be given by us as well. But before we come to that, let us first discuss an additional motivation for a treatment of disposition statements like this: the fact that dispositions don’t seem to come with a single stimulus condition. Manley and Wasserman (2007, 2008) suggest that this is another structural problem for the conditional analyses, and this point is strengthened by arguments due to Vetter (2013, 2014).
2.4 Multi and zerotrack dispositions
Multitrackdispositions pose a problem—independent of the problems of mimicking and masking—for the thought that the disposition of x can be characterized by the single conditional ‘If x were in condition C, then x would have, or show manifestation m’, because it falsely predicts that x has monly if C. Manley and Wasserman argue that to save the conditional analysis (that doesn’t make use of probabilities), we would need to think of a disposition as correlated with a nondenumerably infinite number of conditionals, each of which specifies in its antecedent a ‘fully specific scenario that settles everything causally relevant to the manifestation of the disposition’ (Manley and Wasserman 2007, p. 72). It is hard to imagine that such an infinite set of conditionals could state the meaning of disposition statements. But the problem is worse. As noted by Fara (2005) , some disposition statements do not mention conditions at all:In the simplest cases, dispositions are said to be singletrack. That is, they can manifest themselves in only one kind of circumstance, and then only in one kind of way. Such dispositions are characterized by a single subjunctive conditional. More often than not, however, dispositions are multitrack. That is, they can reveal their presence in a range of antecedent circumstances, yielding a range of different consequent events. Fragility is obviously a multitrack disposition, for a fragile object can manifest this disposition in a wide range of antecedent circumstances: for instance, after being dropped, knocked, struck, stretched, or compressed, resulting in such effects as shattering, cracking, splintering, rupturing, or cleaving. Consequently, to specify fully the meaning of a multitrack dispositional term like ’fragility’, it is necessary to enumerate the full list of subjunctive conditionals which operationally define the term.
I may be disposed to stutter, my drainpipe disposed to leak, and Mr. Magoo disposed to bump into things. Conditional accounts do not even apply to such cases, since they contain nothing to serve as antecedents to a conditional. (Fara 2005, p. 70)
there are plenty of dispositions that do not have any particular stimulus condition. Suppose someone is highly disposed to talk, but there is no particular kind of situation that elicits this response in him. He is disposed to talk when happy, when sad, with others or by himself—he is just generally loquacious [...]. We are not given an ascription of the form ‘x is disposed to give m in C’, we are only given ‘x is disposed to m’ [...]. So with nothing to put in place of ‘C’, how can we construct a conditional of the form ‘x would m if C’? (Manley and Wasserman 2008, pp. 72f)^{9}
Vetter (2013) argues that the problem is more general: all dispositions are multitrack dispositions. Even if there seems to be only one condition, the condition is, in fact, still multitrack. Consider the disposition of being fragile as analyzed by ‘to break if struck’. The problem is that one can strike the vase with more or less power. If we want to analyze fragility in terms of a single condition, we have to make a choice on the desired power with which it should be struck. But, intuitively, we don’t associate one strength of hitting with fragility, or the predicate ‘fragile’, in general: for different types of objects we think of different strengths.
Vetter (2013) convincingly argues that there is no way to reduce ‘apparent’ multitrack dispositions to singletrack dispositions, such that they could be analyzed by a single condition. In case one specific stimulus condition is singled out as the defining one for manifestation, it is wrongly predicted that x does not manifest the disposition in case x shows m in any of the other conditions. And in case all the conditions are combined in one large disjunctive condition, the condition might be too weak. Because fragility might have different stimulus conditions for different types of objects, it might be that this vase will not break under condition \(c_1\), but only under conditions \(c_2\) and \(c_3\), although \(c_1\) does make the disjunctive condition \(c_1 \vee c_2 \vee c_3\) true.
3 Manley and Wasserman’s and Vetter’s proposals
Having argued against the standard conditional analysis, Manley and Wasserman (2007, 2008) and Wasserman (2011) propose that the conditional analysis should be restated in a radically different way, while Vetter (2013, 2014) proposes that dispositions are not conditional at all. We will now discuss these proposals in turn.

x is disposed to m when C if and only if x would m in mostCcases.

x is disposed to m when C if and only if x would m in some suitable proportion of Ccases.

x is disposed to m when C if and only if x would m in manyCcases.
Whereas Manley & Wasserman’s (2008) and Wasserman’s (2011) proposals are still conditional in form, and rather vague, Vetter (2014) derives a drastic conclusion from the arguments given in the previous section, and comes up with a specific proposal. The drastic conclusion she proposes is to radically reject the conditional analysis: we should not analyze dispositions conditionally like ‘x is disposed to m, if C’, but we should analyze them nonconditionally, simply as ‘x is disposed to m’. Thus, she proposes that dispositions are individuated by their manifestations alone. But how many times, or under which circumstances, do they have to show their manifestations for the disposition statement to be true? Her specific proposal is to count ‘x is disposed to m’ as true exactly if it is possible for x to m.^{10} She proposes to account for comparative disposition ascriptions as follows: x would m in more of the relevant situations than y would. A similar analysis is proposed for dispositions involving gradability.
Vetter’s (2013, 2014) proposal is much less demanding than any analysis of dispositions discussed so far. She provides several arguments for this weak analysis. First, in this way the standard counterexamples to conditional analyses can easily be accounted for: mimics and masks are only counterexamples if it is demanded that in all relevant cases x is m, not if it is only required that x is m in at least some relevant case. Second, she notes that in lexicography, the most natural paraphrases of disposition suffixes like ‘able’ and ‘ible’ is not in terms of conditionals, but rather in terms of possibility statements like ‘can’. According to the Oxford English Dictionary, for instance, ‘soluble’ is described as ‘able to be dissolved’ and ‘fragile’ as ‘liable to break or be broken’.
For example, there will be contexts in which an atom does not count as disposed to remain stable even though it would remain stable in some muchhigherthannegligible proportion of nomologically possible situations. (Manley and Wasserman 2011, p. 1223)
To pose a real threat to the possibility conception, an opponent would need to adduce a disposition which (i) is not plausibly construed as just the negative of some disposition, as (I have argued) stability or robustness is, and (ii) comes with a modality that is clearly stronger than a (mere or graded) possibility, and closer to necessity, of its manifesting. (Vetter 2014, p. 152)
We think counterexamples are easy to find. Take ‘soluble’. Whatever the Oxford English Dictionary tells us, it certainly is not the case that substance x is soluble already if it can be dissolved. If we would say that, just about any substance is soluble. Just consider a metal heated to the (almost) extreme. It will be a liquid, just about any substance will dissolve into. Or take fragile: just about any (kind of) object will break, if you struck it hard enough, etc.. Of course, Vetter (2014) could argue that these examples could be explained away, making use of the context dependence of what counts as a relevant possibility. But that reaction would make Vetter’s proposal much less specific than it seemed at first sight, and thus much less attractive. Moreover, the Oxford English Dictionary states something more about the meaning of dispositions like ‘fragile’ and ‘irascible’ than the mere possibility of its manifestation. It also claims, for instance, that something is ‘fragile’ if it can be ‘easily destroyed’. Similarly for someone being ‘irascible’, or ‘hottempered’: he or she has to be ‘easily provoked to anger or resentment’. But the demand that something is easily destroyed, or provoked, is something quite different from the demand that it can be destroyed or provoked. For members of kind k to be easily destroyed it has to be the case that they will, or would, be destroyed in more situations than members of alternative kinds \(k', k^{''}, \cdots \) would be destroyed. What this suggests is that dispositions like being fragile and being hottempered demand a comparative analysis.
But we don’t have to rely on the ambiguity of what is claimed about the meaning of predicates like ‘fragile’ and ‘irascible’ in the Oxford English Dictionary to argue in favor of a comparative analysis of disposition ascription. First, as acknowledged by Vetter (2014) herself, there are dispositions that at least seem to pose a threat to her proposed analysis. Negative charge seems to be the disposition to attract positively charged particles and repel negatively charged ones. The opposite of being negatively charged is most naturally being positively charged. But to characterize any of those opposites in terms of what it can attract and repel seems much too weak. But there is a deeper worry. Disposition statements are not just statements that ascribe arbitrary properties to objects. Instead, they appear to ascribe characteristic properties of (kinds of) objects. A characterizing property of a (kind of) object is a property which sets it apart from other (kinds of) objects. At first one might think that a characteristic property of a kind of object is a property that all and only all objects of this natural kind possess. But that would be much too strong a requirement. Intuitively, one of the characterizing properties of fish, for instance, is that they (can) swim. But we all know that some other (kind of) animals can swim as well: humans and whales. Similarly, ‘flying’ is one of the characterizing properties of birds, even though some kinds of birds, i.e., penguins, don’t fly. In fact, for f to be a characteristic feature of natural kind k, not even the majority of ks have to have the property. Although ‘having manes’ is a characteristic property of lions, only adult male lions it them.
Characterizing sentences like ‘Fish swim’, ‘Birds fly’ and ‘Lions have manes’ that ascribe characteristic properties to groups of individuals are normally called ‘generic sentences’. If disposition statements ascribe characterizing properties to objects, it seems only natural to analyze such statements in terms of generics. We have seen above that Fara (2005) already proposed to do so, but that his analysis could not deal with the structural worries of other conditional analyses of dispositions. Manley and Wasserman (2008) and Wasserman (2011) proposed an analysis in terms of generics as well, that was constructed explicitly to account for the structural worries. But their accounts were much too unspecific.
In sum, we believe that Vetter’s (2013, 2014) possibilitybased analysis of dispositions is too weak. x has the disposition to m not just because it can show manifestation m. Rather, it has this disposition because it shows this manifestation easily, i.e., more easily than other types of objects. We concluded from our discussion of her proposal that we should go for a comparative analysis of disposition statements. The proposals of Manley and Wasserman (2008) and Wasserman (2011), on the other hand, are too vague, or unspecific. Although we think that their analyses are on the right track, we would like to have a more specific proposal. We think it is possible to come up with such a more specific proposal when we take the conclusion from our discussion of Vetter’s proposal into serious account: the analysis has to be comparative. In the ideal case, we would like to have an analysis that is specific enough even to allow disposition statements to become empirically testable, as originally demanded by Carnap (1936). In the next sections we will propose a specific analysis that, we argue, not only can account for the various problems other analyses of dispositions face, but also is empirically testable under many circumstances.
4 A generic analysis using natural kinds
We propose that (1a) ‘This drop of liquid is flammable’ is true, not just because this drop can burn when put in contact with fire, or that it would burn in suitably many occasions where one put it in contact with fire, but because this drop of liquid is of a kind such that it would burn under (significantly) more circumstances than a drop of (most) alternative types of liquid because of being of this kind. We will come to this analysis in three steps, which will be given below. The first step involves the semantics of characterizing, or generic, sentences.

\(\Delta P^m_k \quad = \quad P(m\mid k)  P(m\mid \lnot k)\).

\(\Delta P^m_{k,C} \quad = \quad P(m\mid k,C)  P(m\mid \lnot k,C)\).

x is disposed to m iff x is of (some) natural kind k and (ii) ks are m.

x is disposed to m if C iff x is of (some) natural kind k and ks are m, if C.

ks are m just in case \(\Delta P^m_k> \!\!> 0\), and

ks are m, when C just in case \(\Delta P^m_{k,C}> \!\!> 0\). (where \(x>\!\!> 0\) iff x is significantly above 0)

ks are m (when C) just in case \(\Delta P^m_{k,(C)}> \!\!> 0\).
Because our proposed analysis of generics is explicitly comparative in nature, it is able to account for disposition statements involving comparatives and degrees. According to the analysis of generics in terms of \(\Delta P^m_{k, (C)}\), the generic ‘ks are m (when C)’ is true if comparatively manyks have property m (when C). This, in turn, means that the conditional probability of individuals having property m given that they are of kind k, i.e. \(P(m\mid k, (C))\), is significantly higher than the conditional probability of individuals having property m given that they are of an alternative natural kind, \(P(m\mid \lnot k, (C))\). In terms of this it is easy to account for a comparative disposition statement as ‘x is more flammable than y’. The analysis would come down to the claim x is of a natural kind \(k_1\) and y of a kind \(k_2\) such that \(k_1\)s are more flammable than \(k_2\)s, i.e., \(P(f\mid k_1, (C)) > P(f\mid k_2, (C))\).^{11}

\(\Delta \!^* P^m_{k, (C)} \,= \,\frac{\Delta P^m_{k, (C)}}{1  P(m\mid \lnot k, (C))}\).

x has the disposition to m (when C) iff x is of kind k, and \(\Delta \!^* P^m_{k,(C)}\) is high.
Intuitively, the difference between the generic sentence (6a) and the explicit dispositional sentence (6b) is that the former is already true if there exists a regular pattern, while for the latter it needs to be the case that the pattern is caused by something within John internally. In fact, Fara (2005) and Wasserman (2011) argue that for the disposition statement there does not even have to exist a pattern.^{13} What this suggests is that an analysis of dispositions cannot be grounded solely on a frequencybased analysis of generics. The second problem of our second provisional proposal is that it does not predict that generics, or disposition ascriptions, support counterfactuals. But this seems to be required for a proper analysis. Indeed, in section 2.2 we argued that x is disposed to m (when C) not just when (i) x of kind k, and (ii) the generic ‘ks show m, when C’ is true, but rather when x of kind k such that xms when Cin virtue of belonging to kind k. What is missing of our analysis so far is an explication of what it means to show m (when C) in virtue of belonging to kind k.

x has the disposition to m iff x is of natural kind k, and the generic ‘ks are m (when C)’ is true because of a principle connection between (being of) kind k and manifestation m.
5 A causal analysis
An analysis of dispositions in terms of the measures \(\Delta P^m_{k, (C)}\) or \(\Delta \!^* P^m_{k, (C)}\) is very Humean, built on frequency data and probabilistic dependencies and the way we learn from those. Many linguists and philosophers feel that there must be something more: something hidden underlying these actual dependencies that explains them. A most natural explanation is a causal one: the probabilistic dependencies exist in virtue of objective kinds which have causal powers, capacities or dispositions.^{14} Indeed, traditionally philosophers have assumed that the natural world is objectively divided into natural kinds, which have essences, a view that has gained popularity in the 20th century again due to the work of Kripke and Putnam. This is much in accordance with the neoAristotelean position—defended by psychologists like Keil (1989) and Gelman (2003)—that people (and children especially) have an essentialist view of the world. A closely associated modern view that has gained popularity recently has it that causal powers (Harré and Madden 1975) or capacities (Cartwright 1989) are that in virtue of which many generalities exist. Although any view that makes use of causal powers or capacities has been for a long time unpopular in modern analytic philosophy,^{15} the above authors paved the way to current proponents of such views in philosophy (Ellis, Molnar, Mumford & Anjum, and Bird) and in psychology (Cheng). In this paper we want to provide a down to earth analysis of these concepts making use of causal Bayesian models (cf. Pearl 2000). Our analyses will be closely related to that of Cheng (1997), who derived the measure \(\Delta \!^* P^m_k\) from the assumption that generalities are (taken to be) due to unobservable causal powers.
There is a closely related way why we should be unsatisfied with our (still provisional) analysis of disposition ascriptions in terms of the measures \(\Delta P^m_{k, (C)}\) or \(\Delta \!^* P^m_{k, (C)}\). These measures are essentially population data, data about the members of kind k seen as a whole. But many disposition ascriptions are about individual objects, and we want to be able to account for these as well. A causal counterfactual analysis can make a connection between the two.
5.1 Causal models and Carnap’s testability criterion
where \(pa_i\) denotes the values of the endogenous variables that are the parents of \(V_i\), and where the \(u_i\) are the set of exogenous variables on which \(f_i\) depends.
What the parents of \(V_i\) are can be directly seen in the directed acyclic graph (DAG) associated with the causal model \({{\mathcal {M}}}\). Such a directed graph represents the casual dependencies between the members of V graphically. In such a graph, the members of V are represented by nodes, and if in \({{\mathcal {M}}}\) there exists a mapping f from \(pa_i\) (the members of V that are the parents of i) to \(V_i\), this will correspond to directed edges from the nodes corresponding to the members of \(pa_i\) to \(V_i\).
Notice that the value of the exogenous variable U determines the value of all other variables immediately: if \(U = 1\), all endogenous variables will receive value 1 as well, and similarly when \(U = 0\). But at many cases we are interested in what would be the case if something went different from how they actually went. Given that the prisoner is dead and that rifleman A shot him, we would like to know, for instance, whether the prisoner would still be dead had rifleman A not shot him. To answer such a counterfactual question, Pearl makes use of interventions.
The counterfactual can now be handled as follows: First we use abduction: given that the prisoner is dead and that rifleman A actually shot him, we can conclude that the captain ordered a shooting, which means, in turn, that the court ordered this as well. Then, we make use of intervention (break the law \(A := C\), and replace this by \(A = 0\)). Finally, we make use of prediction: what follows in the new imagined situation? Obviously, rifleman B is still ordered to shoot, so the prisoner would still be dead.
Notice that if \(P(y_x) > \frac{1}{2}\), this won’t be due to the fact that most situations where x were true, y would hold, but rather due to the fact that in most situations, y would hold after an intervention with x. Thus, like in the proposal made by Manley and Wasserman (2007), the ‘quantifier’ most has wide scope over the counterfactual conditional. Alternatively, if we think of u as an object and x and y as properties, \(Y_x(u) = y\) represents the ‘fact’ that after an imagined (minimal) change of u such that it will become an x (or will have property x), it will have property y. \(P(y_x)\), then, measures the relative amount of objects that would be y after ‘becoming’ x due to an intervention.
Observe that in the Fire Squad example, \(P(d_a\mid \lnot a, \lnot d)\) will be 1. Indeed, the shooting of A suffices to cause the death of the prisoner. On the other hand, \(P(d_a, \lnot d_{\lnot a})\) depends on the initial probability of \(U = 1\). Suppose that \(P(U = 1) = \frac{3}{4}\), then \(P(d_a, \lnot d_{\lnot a}) = P(d_a, \lnot d_{\lnot a} U = 1) \times P(U = 1) + P(d_a, \lnot d_{\lnot a} U = 0) \times P(U = 0) = 0 \times \frac{3}{4} + 1 \times \frac{1}{4} = \frac{1}{4}\), where \(P(d_a, \lnot d_{\lnot a} U = 1) = 0\) because \(P(\lnot d_{\lnot a} U = 1) = 0\).
The effects of interventions are observable by making explicit experiments. One of the most striking consequences of Pearl’s causal analysis is, however, that the effect of interventions can in many circumstances be predicted from nonexperimental data alone, i.e., in case only observational data are available! We take this to be very relevant for the analysis of unobservable dispositions. Pearl (2000) shows that if a condition that he calls ‘monotonicity’ is assumed (the assumption that from a change from \(\lnot x\) to x, y cannot change into \(\lnot y\), or formally, \(\forall u \in U: Y_x(u) \ge Y_{\lnot x}(u)\)), \(P(y_x\mid \lnot x, \lnot y)\) reduces to \(\frac{P(y_x)  P(y)}{P(\lnot y, \lnot x)}\). It is easy to show that \(\frac{P(y_x)  P(y)}{P(\lnot y, \lnot x)} \) is equal to \(\frac{P(y_x)  P(y \mid \lnot x)}{1 P(y\mid \lnot x)} \), and thus that \(P(y_x\mid \lnot x, \lnot y) = \frac{P(y_x)  P(y \mid \lnot x)}{1  P(y \mid \lnot x)}\). Likewise, \(P(y_x, \lnot y_{\lnot x})\) comes, under these circumstances, down to \(P(y_x, \lnot y_{\lnot x}) = P(y_x)  P(y_{\lnot x})\). From this it follows that the measures \(P(y_x\mid \lnot x, \lnot y)\) and \(P(y_x, \lnot y_{\lnot x})\) can be tested by observational data only when the causal effects \(P(y_x)\) and \(P(y_{\lnot x})\) can be empirically tested in this way. Pearl (2000) famously shows that this can be done by taking the confounding factors (taken to be Z) into account and condition on their various values and averaging over the results: \(P(y_z) = \sum _{z \in Z}P(z) \times P(y x,z)\).
5.2 A causal analysis of dispositions
Before we will discuss our final proposal, let’s first see how to think of the variables in the causal model to make sense of it. We have seen above that it depends on the application of a causal model how we should think of the exogenous and endogenous variables. In some applications, the set of exogenous variables U should be thought of as a set of situations, while in others it could be, for instance, a set of objects. We will think of the set of endogenous variables, V, as functions from U to truth values. If U would be a set of situations, any binary endogenous variable X would be thought of as a propositional variable. If U would be a set of objects, an endogenous variable X would be thought of as a property object u does or does not have. We propose that for our purposes it is best to think of the exogenous variables as situations, or better, of being an arbitrary object in one of these situations. If we think of x as this arbitrary object,^{18} an intervention of u with condition c or with natural kind k, in turn, should be thought of as the imagined (minimal) change such that c would be true in the new situation, or that object x would be of kind k. If u is a value of an exogenous variable thought of as a particular situation, and M an endogenous variable in the graphical model, M(u) states whether object x has property m in situation u. Similarly, \(M_k(u)\) states whether x would have property m or not, after an intervention of x in situation u with kind k, i.e., whether x in u would have property m or \(\lnot m\) when x were of natural kind k. A related interpretation should be given to \(M_c(u)\), when c is a condition. Because we assume in this paper that the variables can normally take only two values, we will write \(M_{(k)}(u) = m\), instead of \(M_{(k)}(u) = 1\), and similarly for \(M_{(k)}(u) = \lnot m\). As a result, \(P(m_k)\) would measure the proportion of circumstances in which x would have property m if it were of kind k.
Natural and familiar as the use of \(P(m_c\mid \lnot m, \lnot c, k)\) might seem (because the condition c can be thought of as the antecedent of the counterfactual), analysis (15) is not what we will propose. There are several reasons for this. First of all, we have argued above that condition (ii), stated in terms of a counterfactual, should correspond with a generic. Although there is one—‘Things in situation c, when of kind k, show manifestation m’—it is not the one we thought of earlier: ‘Things of kind k, when in situation c, show manifestation m’. Second, and related, the use of \(P(m_c\mid \lnot m, \lnot c, k)\) doesn’t really appeal to the desired principle connection. Intuitively, the principle connection should be between natural kind k and manifestation m. But on the above formulation, a principle connection is assumed between condition c and m. Third, on this analysis it is completely unclear how to account for nonconditional dispositions, things that we have argued in Sect. 2.4 exist as well. Finally, on this analysis it is unclear how to account for Vetter’s (2013) observation that all dispositions are multitrack: the manifestation of the condition can vary with the strength of the condition (one type of stuff can be bent if heated under less high temperatures than other stuff). The proposal under discussion doesn’t allow for this.

x has the disposition to m (when C) iff (i) x is of natural kind k and (ii) ks are m (when C), in virtue of a principle connection between k and m.
We have seen in the previous section that the measures \(\Delta P^m_{K, (C)}\) and \(\Delta \!^* P^m_{K, (C)}\) can be derived from \(P(m_k, \lnot m_{\lnot k}\mid (C)) \) and \(P(m_k\mid \lnot m, \lnot k, (C)) \), respectively. Measure \(P(m_k\mid \lnot m, \lnot k, C)\) can be thought of as the proportions of Csituations in which x (which is neither of kind k nor has property m) would have property m if it were of natural kind k. Something similar holds for \(P(m_k, \lnot m_{\lnot k}\mid C) \). Observe that \(P(m_k\mid \lnot m, \lnot k, C) \) is high just in case being an object of kind k one has a high causal power of being able to show manifestation m in a circumstance in which C holds.
Analysis (16)–(17) can account for comparative disposition statements like ‘Things of kind \(k_1\) are more disposed to m than things of kind \(k_2\)’. Indeed, \(P(m_{k_1}\mid \lnot m, \lnot k_2, (C)) > P(m_{k_2}\mid \lnot m, \lnot k_2, (C))\) will be the case if there are more situations u such that \(M_{k_1}(u) = m\) than situations where \(M_{k_2}(u) = m\). Take, for instance, the dispositions of being malleable or of being ductile. The manifestation condition for both dispositions is (can be) ‘bent’. Rubber is more ductile than iron, even if iron can be bent as well, at least under high temperature. Think of \(u_1\) and \(u_2\) as situations of x being of a particular temperature. Say that \(T(u_2) > T(u_2)\), meaning that the temperature in \(u_2\) is higher than that of \(u_1\). Suppose for simplicity that \(P(u_1) = P(u_2)\). Still, it will be the case that \(P(Bent_{rubber}\mid \lnot rubber, \lnot bent) > P(Bent_{iron}\mid \lnot iron, \lnot bent)\) although also iron can be bent when heated (enough). The reason is that rubber can be bent easier when heated. It could be that \(Bent_{iron}(u_2) = 1 \) and \(Bent_{iron}(u_1) = 0\), but that rubber could be bent in both situations: \(Bent_{rubber}(u_1) = 1 = Bent_{rubber}(u_2)\). It is obvious that gradable disposition statements can be accounted for in a similar way. The same example also shows that this analysis can account for nonconditional dispositions: Rubber can be bent in much more situations than other type of stuff. We take this all to be in support of our analysis of disposition statements in terms of \(P(m_k\mid \lnot m, \lnot k, (C))\).
We think that our causal power analysis (16)–(17) of dispositions and/or disposition statements is appropriate. We have also shown that this analysis is far less obscure than it might seem at first, because the clause ‘members of natural kind k have a relatively high causal power to show manifestation m’ can be tested by observational data alone, because it can be reduced to an observable criterium under various circumstances. Thus, it satisfies Carnap’s desire for a testable analysis of dispositions, at least under many circumstances.
Fortunately, because in most situations where something is in contact with fire, it either is not kerosene, or it burns, it will hold that \(P(b_k\mid \lnot b, \lnot k, f) \) is high because \(P(b_k{\mid } f)\) is high. In this way we have explained why the disposition statement supports the counterfactual.
6 Final thoughts
This paper was about dispositions, and disposition statements, that involve natural kinds. The examples we concentrated on were natural kinds like kerosene, sugar, and arsenic. That was no accident, because these are natural kinds the members of which are very similar to one another. Of course, we make disposition statements as well about objects that are of a natural kind where the members of that kind are not all that similar. And for such disposition statements, our analysis seems false: ‘This dog is dangerous’ can be true, even though this dog is a St. Bernard dog, and most of such dogs are not dangerous.^{19} Similarly, ‘This dog is not dangerous’ can be true, even though this dog is a Pit bull, a kind of dog we generally take to be dangerous. Would this mean that our analysis works only for an extremely limited type of disposition statements?

‘x is disposed to m’ (when C) iff (i) x is extremely similar to all members of group G and (ii) the generic ‘Gs are m, (when C)’ is true because of a principle connection between being a G and m (when C).
This paper was concerned with the semantic analysis of disposition ascriptions. We have not dealt with the metaphysical question whether dispositions are basic, or whether they should be reduced to categorical properties. In principle, we take it that our analysis is independent of this issue. But that by itself is already noteworthy: our analysis is compatible with the view that dispositions are basic! Let us explain.
In the previous section we argued that on our analysis, disposition ascriptions support counterfactuals. Thus, we assumed there to be a link between dispositional statements and counterfactuals, just like traditional conditional analyses did. However, we don’t necessarily share the original motivation for a counterfactual or conditional analysis of dispositions. The standard conditional analyses define the meaning of disposition statements, and perhaps of dispositions themselves, in terms of these conditionals. Dispositions should be explained away, or reduced to something else that fits a Humean metaphysics. Proponents of such a metaphysics normally allow for categorial properties, but not for dispositional ones.
It is wellknown that Lewis (1973b) attempted to reduce causality to counterfactual dependencies for similar metaphysical reasons. Making use of causal models, Pearl (2000), instead, suggests to take causality as a primitive in terms of which counterfactuals can be defined. Similarly, the previous section suggests that we have assumed the existence of causality, or of causal powers, as basic, in terms of which we analyze the meaning of (certain type of) generic and dispositional sentences. On such a reading of our previous section, one can (though need not) think of dispositions as being these primitive causal powers, and thus as basic properties themselves. Our semantic analysis of disposition ascriptions is then not an analysis of the relevant dispositions, in the sense that the analyses do not define the dispositions. Instead, on this reading, our analysis ascribes basic properties to (kinds of) objects, and provides criteria under which the existence of these properties can be tested.
Although we make use of causal models, we don’t feel committed to the above metaphysical stance. In fact, we are quite attracted to Pearl’s earlier, more modest, position concerning causality. In Pearl (1988), he takes the view that humans, when facing complex phenomena, interpret these phenomena almost automatically into structures of causeandeffect relationships for computational reasons. A causal view of the world, including the assumption of hidden causes, allows us to represent and make use of empirical knowledge in an effective way. Whether this means that these assumed hidden causes actually exist is beyond our grasp. Whether a similar computationally motivated position should be taken with respect to the ageold question whether natural kinds are real, or are concepts that exist only due to our urge to view the world in an organized way, we must leave to the reader.
Footnotes
 1.
Another reason for their attention is the ontological question whether dispositional properties can and should be reduced to categorical properties. We will mostly ignore this issue in this paper.
 2.
Philosophers tend to use ‘finking’.
 3.
Notice that these type of counterexamples are familiar to students of philosophy of mind: where Ryle (1949) and others gave analyses of ‘pain’ and other psychological properties in terms of observable behavior, it was soon realized that this won’t do: pain could be mimicked (by a good actor), or masked (by a superman).
 4.
Formally, such an analysis would give up the (strong and weak) centering condition used in Lewis’s analysis of counterfactuals, and thereby giving up on Modus Ponens for the conditional involved.
 5.
A similar problem threatens another way to weaken the conditional analysis: instead of demanding that x would be m in all most similar or normal or worlds where C would be true, just add a ceteris paribus condition to it.
 6.
As explained in the final section of this paper, in the end this won’t be the case. Our final analysis will be more finegrained than that, and as a result closer to Fara’s intuition, or analysis.
 7.
On this straightforward proposal, dispositional predicates would be treated just like absolute adjectives as ‘straight’ and ‘flat’, and not like what linguists would call a relative adjectives as ‘tall’ (cf. Kennedy and McNally 2005). But now think of the disposition predicate ‘fragile’. This champagne glass is very fragile, and more fragile than that more ordinary glass. But this doesn’t mean that this ordinary glass is not fragile. Thus, ‘fragility’ doesn’t seem to behave like an absolute adjective at all. Similarly, we think, for most other dispositional predicates.
 8.
See also Ryle (1949).
 9.
With their ‘N’ replaced by ‘x’, and their ‘M’ by ‘m’.
 10.
Though she also argues that what is a relevant possibility is contextdependent.
 11.
This comes out on our analysis of generics when we adopt, for instance, Klein’s (1980) analysis of comparatives: we say that \(k_1\)s are more flammable than \(k_2\)s exactly if with respect to a contextual given set of objects c, \(k_1\)s are f, and \(k_2\)s are not f. ‘\(k_1\)s are f’ is true with respect to context c iff \(P(f\mid k_1 \cap c) > P(f\mid \lnot k_1 \cap c)\). Similarly, ‘\(k_2\)s are not f’ is true with respect to context c iff \(P(\lnot f\mid k_2 \cap c) > P(\lnot f\mid \lnot k_2 \cap c)\), which holds exactly if \(P(f\mid \lnot k_2 \cap c) > P(f\mid k_2 \cap c)\). Now assume that context c consists exactly of those objects that either are of kind \(k_1\) or of kind \(k_2\). In that case, (assuming that no object is of both kinds) the truth conditions of the generics ‘\(k_1\)s are f’ and ‘\(k_2\)s are not f’ both come down to \(P(f\mid k_1) > P(f\mid k_2)\), i.e., to that what we wanted to show. Because our analysis of generics is explicitly comparative, it can account for gradable disposition statements straightforwardly as well.
 12.Wasserman (2011) also argues that the difference in acceptability between (ia) and (ib) is another indication that disposition statements are not exactly on a par with habitual, or generic, ones.
 13.
Although something similar holds for generics as well. Even if there has been no post from Alaska thus far, the generic ‘Mary handles the post from Alaska’ can still be true.
 14.
It seems no accident that (general) causal statements typically are of generic form (‘Sparks cause fires’ and ‘Asbestos causes cancer’).
 15.
 16.
For those who are not familiar with causal models, it might help to think of \(P(y_x)\) as the probability of y after imagingP by x, as proposed by Lewis (1976), if X and Y correspond to variables associated with propositions.
 17.
For convenience, we will use a comma, instead of logical ‘\(\wedge \)’ below.
 18.
An arbitrary object should be thought of like a peg, or a placeholder: an object without any identifiable properties. This means that all arbitrary objects are indistinguishable, and thus the same.
 19.
Wasserman (2011) criticizes Lowe’s account of dispositions on much the same ground.
Notes
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