Biology & Philosophy

, Volume 24, Issue 3, pp 359–374

Physical explanations and biological explanations, empirical laws and a priori laws


    • Philosophy DepartmentCalifornia University of Pennsylvania

DOI: 10.1007/s10539-007-9096-4

Cite this article as:
Press, J. Biol Philos (2009) 24: 359. doi:10.1007/s10539-007-9096-4


Philosophers intent upon characterizing the difference between physics and biology often seize upon the purported fact that physical explanations conform more closely to the covering law model than biological explanations. Central to this purported difference is the role of laws of nature in the explanations of these two sciences. However, I argue that, although certain important differences between physics and biology can be highlighted by differences between physical and biological explanations, these differences are not differences in the degree to which those explanations conform to the covering law model, which fits biology about as well as it does physics.


Biological lawsPhysical lawsCovering lawDeductive-nomologicalEmpirical lawsA priori lawsDispositionsRosenbergSoberKitcher

When philosophers of biology attempt to characterize the differences between biology and physics, they often focus on differences between biological and physical explanations. These differences are often presented as differences in the degree to which the two kinds of explanations adhere to the once generally accepted covering law model of scientific explanation. After all, the story goes, the progenitors and chief early adherents of the covering law model were all either physicists themselves, or at least much enamored of, and familiar with, physics. Consequently, it is no wonder that their model of explanation is a better fit for their science of choice than it is for biology. Identifying biological deviations from this model of explanation, it is said, will therefore also call attention to significant differences between biology and physics. However, although there is fairly widespread consensus that biological explanation is less suited to the covering law model than physical explanation, there is considerably less consensus about how exactly the two diverge. Much of this disagreement has focused on the status, or even the purported non-existence, of biological laws.

In the first part of this paper I intend to concentrate on the views of three influential philosophers of biology, Alex Rosenberg, Elliot Sober, and Philip Kitcher, along with their interlocking criticisms of each other. Although these three philosophers hold widely divergent views on the nature of biological explanations, they share an interest in the degree of applicability or inapplicability of the covering law model to biology. Rosenberg sees the least divergence from this model, arguing that biological explanations do appeal to empirical laws. However, he claims that such laws are comparatively scarce in biology. Sober claims that, unlike the empirical laws of physics, the laws of biology are a priori. Kitcher, on the other hand, rejects biological laws altogether.

For the most part, I intend to withhold my own analysis of these positions until the second part of the paper, in which I will argue that this focus on laws and divergence from the covering law model has led us astray. Though the covering law model has few true adherents today, the approach of these philosophers of biology tacitly grants that it applies reasonably well to physical explanations: well enough, at any rate, for it to make sense to assert that physical explanations fit the model better than biological explanations. In this paper, I intend to share this assumption. But though I will, as a result, effectively be defending the covering law model’s applicability to biological explanations in what follows, I do not mean to commit myself to that model as anything like a perfect account of scientific explanation. Rather, the point of defending the applicability of the covering law model to biological explanation is to argue that, although certain important differences between physics and biology can indeed be highlighted by differences between physical and biological explanations, these differences do not come down to differences in the degree to which those explanations conform to the covering law model. Biological explanation, I claim, conforms to the covering law model about as well as physical explanation.

Part I

1.1 A challenge for biological laws

On the face of it, biology does seem short on the sort of laws required by the covering law model. Alexander Rosenberg has suggested that this is because constant modification of species by natural selection undermines most attempts to form biological empirical generalizations. For example, present-day zebras have stripes, in part because they confuse the visual systems of lions. But ‘Zebras have stripes’ is not even successful as a ceteris paribus generalization because, given the constantly changing relationships between zebras, lions, and their shared environment, this probably has not always been, nor will it probably forevermore be, the case. A generalization that is currently perfectly serviceable, like ‘Zebras have stripes,’ might be confusing or misleading at other times, and is at any rate almost certainly false, even as a ceteris paribus generalization, when construed as applying to all zebras existing at all times (Rosenberg 2001a, pp. 739–740).

‘Zebras have stripes’ might not seem like a particularly good candidate for a biological law simply because it would be such a low-level law. But the same sort of reasoning will apply to apparently more robust generalizations. It once would have seemed (had there been anyone around to whom it might have seemed) a reasonable generalization to say that reproduction takes the form of mitosis. That would not be a plausible law of biology today, and before we go formulating some law that would be plausible today, we should consider what might be the case tomorrow. For, while any attempt to generate laws of biology will be an attempt to cut biological nature at the joints, the problem with biological nature is that the joints keep shifting places.

Rosenberg’s argument has been criticized by Kenneth Waters (1998), on the grounds that it conflates two sorts of biological empirical generalizations. Generalizations like ‘zebras have stripes’ describe the distribution of traits among populations and taxonomic groups, and are indeed constantly modified by natural selection. But the generalizations involved in, for example, predicting the relative fitnesses of horses and zebras in lion-infested environments needn’t quantify over the groups mentioned in these distributional generalizations. Instead, this second sort of generalization would identify causal regularities between, say, certain types of visual systems and certain types of visual targets, regardless of how those systems and targets might be taxonomically distributed.

However, in the context of the present question concerning the applicability of the covering law model to biology, these generalizations identifying causal regularities are no help. The problem is that the gradual and haphazard nature of natural selection still tends to produce poorly delineated kinds over which to generalize. Both within and beyond the lion species lies a near continuum of visual systems, each of which may behave slightly differently when confronted with zebra stripes. As a result, Waters himself admits that, although these causal generalizations possess some properties of laws of nature, they fall short of actually being laws (Waters 1998, p. 29). Consequently, if these generalizations are meant to take the place of laws in biological explanations, there would still be a difference, albeit perhaps a less significant one, in the applicability of the covering law model to physics and to biology.

1.2 Applying the covering law model to biological explanations

Rosenberg’s solution to his own problem is to claim that the only laws of biology are the empirical generalizations that make up Darwin’s theory of natural selection. It is these laws, plus the laws of molecular bio-chemistry, upon which biological explanations rely (Rosenberg 2001b). He axiomatizes natural selection (by his own admission, very roughly) in the following way:
  1. (1)

    Biological systems not on the verge of extinction or fixity reproduce with heritable variations.

  2. (2)

    If heritable variation obtains among biological systems, then there will be fitness differences among the biological systems.

  3. (3)

    In the long run, the more fit variants will leave a higher proportion of descendants than the less fit variants.

  4. (4)

    [It follows that] until fixity or extinction is attained, there will be descent with modification, i.e. evolution (Rosenberg 2001a, p. 752).

These principles do not fall prey to his criticism of other biological laws since

None of them is subject to qualifications or ceteris paribus clauses in virtue of the operation of selective forces on the earth. After all, these principles constitute the mechanism of natural selection itself; there is no scope for natural selection to qualify, limit, or shape its own operation (Rosenberg 2001a, p. 752).

Unfortunately, these laws are not necessarily as broadly applied as Rosenberg claims. Since (1)–(3) are empirical universal generalizations, the argument from (1)–(3) to (4) does conform to the covering law model. It explains a further empirical generalization: that evolution occurs in all species. However, many biological explanations are more particular than this. Biologists explain not only the occurrence of evolution simpliciter, but also the evolution of this or that trait in this or that population. But, when they use the sort of reasoning exemplified above for such particular purposes, what goes in the place of (1)–(3) need no longer be universal generalizations. For example,
  1. (1′)

    The monkeys in population P (which is not on the verge of extinction or fixity) reproduce with heritable variations of tail dexterity.

  2. (2′)

    If heritable variation of tail dexterity obtains among the monkeys in P, then the prehensile-tailed monkeys in P will be fitter than the less-dexterous-tailed monkeys in P.

  3. (3′)

    In the long run, the fitter, prehensile-tailed monkeys in P will leave a higher proportion of descendants than the less fit, less-dexterous-tailed monkeys in P.

  4. (4′)

    [It follows that] until fixity or extinction is attained, the proportion of monkeys in P with prehensile tails will increase.


In virtue of their reference to a particular population and a particular trait, (1′)–(3′) are no longer universal, and hence the explanation apparently fails to conform to the covering law model. And yet, as an explanation, it seems no worse off for this. Of course, it could be that contemporary evolutionary biologists always come to believe particular propositions like (1′)–(3′) by deducing them from Rosenberg’s laws (1)–(3). But, in principle, a pre-Darwinian (or just very poorly schooled) biologist, unaware of the universal truths (1)–(3), but aware of the particular truths (1′)–(3′), could nevertheless construct this second explanation. This is because both the inference from (1)–(3) to (4) and the inference from (1′)–(3′) to (4′) are underwritten by the same complex model of the process of natural selection. As it happens, natural selection applies to all living things, but this universal applicability appears to be irrelevant to explaining its effects in any given case. Consequently, it seems that biological explanations do not necessarily appeal to Rosenberg’s laws of natural selection. At the same time, biological explanations like (1′)–(3′) do not obviously appeal to the laws of bio-chemistry either, and so it looks as though it fails to appeal to any of the laws Rosenberg accepts.

Additionally, Philip Kitcher has objected to this sort of axiomatization of natural selection on the grounds that it fails to accurately describe the scientific advance effected by Darwin. The problem with taking the main premises of Darwin’s argument as comprising the revolutionary theory of natural selection is that these purported laws are now, and were in Darwin’s own time, stunningly obvious. Even biological novices can confirm that individual members of a species vary, that some of this variation leads to variations in fitness, and that offspring tend to resemble their parents.

But, says Kitcher,

We [legitimately] expect that the fundamental principles of a novel scientific theory should be those statements introduced by the theory that most stand in need of defense and confirmation (Kitcher 1985, p. 130).

Basically, Kitcher’s argument here is that, whereas the introduction of Darwin’s theory of natural selection was revolutionary, the laws of natural selection are not at all revolutionary, and hence the laws of natural selection cannot be the fundamental principles of biology.

1.3 Alternative models of biological explanation

Since the covering law model requires that all explanations involve derivation of the phenomena to be explained from a set of premises which include empirical laws of nature, if the above considerations, or others, decisively rule out the existence of most empirical biological laws, biological explanation will generally fail to conform to the covering law model. But if we grant that the explanatory power of physical explanations does come from something like the covering law model, those who reject that model for biological explanation need an alternative account of the explanatory power of biology.

1.3.1 A priori laws

One such account suggests that the difference between physical and biological explanation lies in the fact that, unlike the laws appealed to in the former, the laws appealed to in biological explanations are a priori. Elliot Sober, for example, has suggested that biological laws are actually mathematical models. He offers the following as an example of a mathematical model that might be considered a law of biology.

Hardy–Weinberg Law: If the frequency of the A gene is p and the frequency of the a gene is q at some locus in a population, then the frequencies of the three genotypes AA, Aa, and aa will be p2, 2pq, and q2, respectively (Sober 2000, p. 73).

This is a law about the frequency of genotypes in a population of randomly sexually reproducing organisms. However, if we strip the law of its biological vocabulary, we are left with:

Mathematical truth: If the frequency of property A is p and the frequency of property a is q for some set of objects, then (if the possession of A is independent of the possession of a) the frequencies of the property sets {A, A}, {A, a}, and {a, a} possessed by those objects will be p2, 2pq, and q2, respectively.

When the properties are interpreted as genes and the objects are interpreted as organisms, we get the Hardy–Weinberg Law. But we could just as well interpret the properties as Heads and Tails, and the ‘objects’ as (weighted or unweighted) coin-toss events and then we would get a law about the frequency of coin-tosses. Either way, what makes the law true is entirely a priori.

Sober thinks that this just demonstrates an interesting way in which biological laws differ from physical laws.

In physics, general laws such as Newton’s Law of Gravitation and the special theory of relativity are empirical. In contrast, many of the general laws in evolutionary biology…seem to be nonempirical (Sober 2000, p. 72).

However, though mathematical models obviously play a significant role in the biological sciences, it is not clear that they can assume the explanatory duties of empirical laws. As Rosenberg argues:

Necessarily true mathematical models will have a role in explanation only to the degree that they are reflected in generalizations that describe actual causal processes (Rosenberg 2001a, p. 746).

A mathematical model cannot explain or predict the behavior of some biological system unless we first know that the model can appropriately model the system. That means we have to know there is a coherent way of interpreting the various mathematical objects, properties, and relations that appear in the model as biological objects (species, organisms, organs etc.), properties (traits, etc.), and relations (descendant/ancestor relations, mating relations, predator/prey relations, etc).

Knowing this involves knowing certain empirical facts. If the model is to have general applicability, then it involves knowing certain general empirical facts. For example, if we know that, say, all mammal populations satisfy the background conditions of the Hardy–Weinberg Law (i.e. sexual reproduction with random mating), then the Hardy–Weinberg Law will allow us to conclude that the genes in mammal populations are always distributed as it suggests. But if so, it seems that it is this empirical generalization (i.e. that all mammals satisfy the background conditions) which is the real biological law doing the explanatory work.

But if it must be the empirical generalizations that serve as laws, we are stuck with Rosenberg’s problem again. There is no guarantee that the assumptions of the Hardy–Weinberg Law will always be satisfied by mammals, since sexual reproduction may itself be altered by natural selection or even supplanted by some other form of reproduction. It is a matter of empirical law whether the model has any general application, and Rosenberg has claimed that such laws are not in the cards.

On the other hand, just as with Rosenberg’s laws of natural selection, one need not seek general applicability of mathematical models. The Hardy–Weinberg Law could be applied just as well to some particular population satisfying the conditions of the model, even if this satisfaction were a peculiarity or an accident. Something like this seems to be what Sober has in mind. In a reply to Rosenberg’s earlier claim that he had denied the existence of biological laws altogether, Sober writes

Of course, if one stipulates that laws must be empirical, the [mathematical models] will not be counted as [laws]…I have no strong objection to this usage of the term “law,” but the resulting claim—that “evolutionary models do not provide laws”—gives quite the wrong impression. The point is not that evolutionary biology is exclusively given over to narratives about historical particulars (Sober 1996, p. 467).

However, if we apply the Hardy–Weinberg law in this way, nothing explanatory appears to happen at all. If a biologist knows that, as a matter of, possibly particular, fact, this population of koala bears reproduces sexually and randomly, it is merely a matter of calculation, enshrined as the Hardy–Weinberg Law, that their genes will be distributed as the law says. It is no more explanatory of the resulting distribution than the fact that I have two apples in my left hand and three oranges in my right hand, along with the ‘law’ that 2 + 3 = 5, is explanatory of the fact that I am holding five pieces of fruit. In either case, the mathematical law allows us to re-describe a narrative of historical particulars in order to make explicit certain facts implicit in that narrative, but it does not add to our understanding of those particulars, or to the narrative.

1.3.2 Historical biological explanations without laws

Kitcher’s view of biological explanation arises from his general view of scientific explanation, which conveniently sidesteps the issue of natural laws. On this view (Kitcher 1999) an argument will form the basis of an explanation when it instantiates an argument pattern that unifies the explanation of many phenomena. In identifying such patterns, we are supposed to look for patterns of argument that are simultaneously relatively specific and yet widely applicable to the phenomena to be explained. A theory is a collection of these patterns that unifies the explanations within its domain. Whether or not a science has any empirical laws becomes a matter of whether universal empirical generalizations make an appearance in these unifying patterns.

In evolutionary biology, Kitcher thinks the relevant argument patterns are what he calls ‘Darwinian histories.’

A Darwinian history for a group G of organisms between t1 and t2 with respect to a family of properties F consists of a specification of the frequencies of the properties belonging to F in each generation between t1 and t2 (Kitcher 1985, p. 139).

Unlike the unifying patterns of physics, which frequently incorporate empirical laws, Darwinian histories make no reference to empirical laws. The general idea is that any explanation in terms of natural selection will critically involve a recitation of the traits possessed by the ancestors of the organism, population of organisms, or species possessing the trait to be explained. Certain kinds of questions, Kitcher says, can be answered by an appeal to this sort of minimal pattern alone. Other sorts of evolutionary questions might require that additional detail be added to the pattern. For example, if we want to explain why gibbons have prehensile tails, we will describe the history of the changing frequency of the prehensile-tail trait among gibbon ancestors and add a gloss about the fitness advantages conferred upon those ancestors by their prehensile tails.

Much more could be said about the details of Kitcher’s account. However, the preceding is enough to show that, elegant as it is, it is insufficient. It is certainly true that scientific explanations tend to follow patterns, and that explanations that unify a wide range of phenomena are often highly prized. But the fact that explanations often have the properties Kitcher identifies as relevant does not entail that it is those properties that make them explanatory. In order to produce an explanation, it is necessary to show that the phenomenon to be explained was, at least to some degree, to be expected. Nothing in Kitcher’s Darwinian histories by themselves demonstrates this. As Rosenberg puts it:

If evolutionary explanations really do not appeal implicitly or otherwise to an ampliative ‘principle of natural selection’, the explanatory force of Kitcher’s selectionist explanatory pattern remains ungrounded. There is only a narrative (Rosenberg 2001a, p. 751).

Consider the Darwinian-history explanation of the prehensile tails of gibbons. Many generations ago, there were more or less gibbon-like organisms that lived in trees. A very few of these organisms had prehensile tails. Many of these organisms died or were seriously injured as a result of falling from trees, but the organisms with prehensile tails suffered considerably fewer such incidents than their cousins with less-dexterous tails. In subsequent generations, the frequency of the prehensile-tail trait increased, and eventually became more or less universal among the descendants of the first organisms. Modern gibbons are descendants of these organisms.

In some sense, this is just the sort of thing one expects from an evolutionary explanation. However, none of these statements is even relevant to the current possession of prehensile tails by gibbons unless some empirical generalization or other is also true. For example, in order to understand the relevance of prehensile tails to falling, one must know that, ceteris paribus, animals with prehensile tails are less likely to fall out of a tree than those without such tails. In order to understand the relevance of the death and injury reports, one must know that, ceteris paribus, animals who die or are severely injured tend to leave fewer offspring than those who do not. And in order to understand the relevance of all these historical events to modern gibbons, one needs to know that offspring tend to resemble their parents. The point is that, even if explanations are arguments that fit a pattern, they must surely also be good arguments. Without laws, the explanantia of Kitcher’s Darwinian histories fail to entail their explananda.

Part II

2.1 Re-evaluating differences

The tangle of objections and counter-objections involved in this debate seems to me to provide sufficient motivation to look more closely at the contention that the best way to evaluate the differences between biology and physics involves comparing the degree to which they conform to the covering law model. The alternative accounts of explanation offered by Sober and Kitcher identify two main possible departures from this model. First, biological explanations may appeal to a priori laws whereas physical explanations typically fail to do so, appealing instead to empirical laws (Sober). Second, biological explanations may fail to appeal to empirical laws, either because they appeal to a priori laws instead (Sober), or because they do not appeal to laws at all (Kitcher). What I aim to do now is argue that neither of these purported differences between biology and physics is as deep as is often supposed.

2.1.1 A priori physical laws

Professor Malcolm Wells, a recently retired physicist at the end of a distinguished career, still teaches now and then. Truth be told, he has never been entirely satisfied with his performance in the classroom, and now that he has the time, he wants to get it right.

The problem has always seemed to be that, no matter how thoroughly he explains something, he fails to call attention to some crucial bit of background information. Ruminating on this conundrum, his thoughts drift back to the sixties…and his undergraduate philosophy of science course. Philosophical methods, he remembers, have a remarkable tendency to force one to be explicit. Perhaps if he were to present his lecture on Snell’s Law tomorrow as an attempt to fill out a complete covering law explanation, thorough student comprehension would be ensured.

So, the next day Wells begins by conducting a quick experiment (see Fig. 1). First, he attaches his laser pointer to the side of a fish aquarium at a fixed angle. He turns the pointer on and marks the point on the bottom of the aquarium where the resultant red dot appears. He next fills the aquarium with 30 cm of water. As the water rises, the dot moves. After the tank is filled, he marks the new position of the dot, which has moved 27 cm. Snell’s Law, he says, explains this phenomenon. He then writes the following on the board:
  1. (1)

    Snell’s Law: If a light ray passes from one medium to another, the light will be refracted such that the angle of incidence α, the angle of refraction β, and the index of refraction of the new medium i, are related as sinα = isinβ.

  2. (2)

    The laser beam passed from air to water.

  3. (3)

    The index of refraction for water is 1.333.

  4. (4)

    The laser pointer is oriented such that α = 60°.

  5. (5)

    The water is 30 cm deep.

  6. (6)

    Therefore, the red dot moved 27 cm when the water was added.
Fig. 1

 Professor Wells’ experiment

Confident of having included every essential premise, Wells asks “Any questions?”

Sadly his confidence almost immediately evaporates in a flurry of raised hands. Some of the questions are simply demands for definitions of unfamiliar terms. Wells dutifully directs the students asking these questions to their textbooks, gently but firmly reminding them of their ability to read. Most of the questions, however, are devoted to calculation. Some of the students are not sure how to solve the Snell’s Law equation for β. Others fail to see how determining the angle of refraction is relevant to the distance the dot traveled.

Though inwardly cursing the public schools’ lack of emphasis on the practical application of mathematics, Wells admits that the students have a point. Covering law explanations are not generally presented as explicitly including mathematical truths, since such truths are not supposed to carry any of the explanation’s content. But that doesn’t mean that one can follow the explanation without knowing the mathematical truths necessary to carry out the derivation. And this calculation isn’t completely trivial. So Wells runs through the calculations, drawing Fig. 1 as he proceeds.

“Since we know that α = 60° and that i = 1.333, we can use Snell’s Law to calculate that β ≈ 40° by taking the sine of 60°, dividing that by 1.333, and taking the cosecant of the result. Now, suppose we draw a line perpendicular to the surface of the water, through the point where the laser beam meets the water. Drawing this line constructs two right triangles. Both triangles have a side lying along the perpendicular and both have a side lying along the bottom of the aquarium. One triangle’s hypotenuse is coincident with the beam’s path when the aquarium is empty; the other’s is coincident with the refracted beam’s path after the water is added. The difference in the lengths of the bottom sides of these two triangles will be the distance traversed by the dot. The bottom sides of the smaller and larger triangles are (30 cm)(tan 40°) ≈ 25 cm and (30 cm)(tan 60°) ≈ 52 cm, respectively. Consequently, the dot moved 52 cm − 25 cm = 27 cm.”

“But Professor Wells,” interjects a student, “shouldn’t all these facts about calculation be added to your explanation?”

Wells tries for a moment to remember the logical positivists’ views about empirical vs. mathematical content, but quickly realizes that, given his present purposes, it doesn’t really matter whether mathematical truths count as a part of the explanation. He is trying to fill in as many gaps in his students’ knowledge as possible, so he agrees to put it all up. He adds several ‘premises’ to his explanation, including “sin 60° ≈ (1.333)(sin 40°),” “(30)(tan 40°) ≈ 25,” and “52 − 25 = 27.” Confidence regained, he again asks, “Any questions?”
  • “Professor, I understand all the calculations now, but there’s still something troubling me. How do we know that Snell’s Law is the right equation?”

  • “Well, Snell’s Law tells you how angles of incidence and refraction are related when light passes from one medium to another, and that’s what happened in this case.”

  • “How do we know that’s what happened?”

  • “You watched me do it, unless you were asleep. It’s recorded as premise (2).”

  • “No sir, I wasn’t sleeping, and I see premise (2). It’s just that I’m not seeing the relevance of premise (2) to premise (1).”

  • “Ah…well…(1) makes a hypothetical general claim about what happens any time a certain sort of event occurs, and (2) says that an event which satisfies that hypothetical condition has, in fact, occurred. We can therefore conclude that what the law says happens in such circumstances will, in fact, happen now. This is just a matter of logic. If I remember right, the philosophers have a special name for it. Modusponens…or something like that anyway.

  • “Oh…well if it has a special name, I guess it must be OK.”

  • “Indeed…let’s call it a day, shall we?”

I take it that the moral of this vignette is relatively obvious. Sober has claimed that at least one fundamental difference between biological and physical explanations is that only the former significantly involve a priori laws. But the story makes clear that, merely in virtue of being logical derivations, all explanations rely on a priori generalizations. When explanations are quantitative, many of these generalizations will be mathematical (like the generalization that whenever one has 52 of something, and then gets rid of 25 of them, one is left with 27). But even purely qualitative explanations must at least rely on the logical truths that link their premises (like modus ponens). So, far from identifying a unique feature of biological explanation, Sober has identified a feature that is omnipresent in explanation, especially if explanation is to be conceived on the covering law model, or on any model that represents explanations as arguments.

At the same time, Sober is onto something. While both physical and biological explanations appeal to universal a priori generalizations, such generalizations often seem to assume greater importance in biology. Consider another of Sober’s examples. R. A. Fisher explained why populations of sexually reproducing organisms tend to have male/female ratios close to 1:1. Suppose that there is a significant deviation from a 1:1 sex ratio in a population, and that there is heritable variation in the sex ratios of the offspring individual organisms produce. In such a situation, individuals who produce more offspring of the minority sex will have more grandoffspring than individuals who produce more offspring of the majority sex (Each individual in the grandoffspring generation requires a pairing of male with female, so offspring of the minority sex will necessarily participate in more pairings than offspring of the majority sex). Consequently, whenever the sex ratio of the population deviates from 1:1, there will be a selection bias in favor of production of the minority sex until such time as the balance is redressed (see Sober 2000, p. 17).

This explanation appeals to various empirical considerations. I would even contend that it appeals to empirical laws. I intend to say more about what these laws might be in the next section, but for now it will be sufficient to note that identification of these empirical considerations is not Fisher’s main contribution to biology here. It is relatively easy to confirm that many organisms reproduce sexually, that their pairings are often more or less random, that the sex ratio of their offspring is heritable and varies, and so on. Instead, the challenging part is realizing that organisms which fit this description are overwhelmingly likely, as a matter of mathematical necessity, to give birth to an even balance of males and females. If we try to fit Fisher’s explanation into the covering law model (using as yet unspecified empirical laws), Fisher’s main contribution seems to be recognizing the inference pattern, rather than recognizing some new empirical fact or law.

This is significantly different from the Snell’s Law explanation. Professor Wells expects that listing the empirical laws and empirical particular facts necessary to deduce the movement of the dot will provide a sufficient explanation, and he seems within his rights to be a bit vexed when his students fail to make the necessary mathematical and logical connections. Surely a good physics student, given the empirical laws and facts, could at least manage to work through the trigonometry and algebra on his or her own time (The student who fails to grasp modus ponens may seem positively surreal unless one has attempted to teach introductory logic). It is virtually impossible to imagine an alternate scientific history in which the community of physicists comes to realize the truth of Snell’s Law without yet realizing its applicability to something like Professor Wells’ experiment. But something analogous appears to happen in biology all the time. I conclude that Sober is more or less right to focus on the role of a priori generalizations in characterizing the difference between physics and biology, but wrong to think that the difference is a matter of substituting a priori laws for empirical ones. Both physical and biological explanations depend upon the truth of both empirical and a priori generalizations, but, unlike physical explanations, the ‘Eureka!’ premise of biological explanations tends to be a priori.

I think this difference also accounts for Kitcher’s reluctance to accept that Darwin’s great achievement was just the introduction of the laws of natural selection. These empirical generalizations were old news, hence Kitcher claims they could not have revolutionized biology. However, as Kitcher himself says, “What was in dispute in the Darwinian revolution was not so much the truth of [the laws of natural selection], as their significance,” (Kitcher 1985, pp. 130–131). Darwin’s achievement was in demonstrating that the familiar process of natural selection was sufficient, by itself, to produce the entire biological world. Darwin was filling in an explanatory gap analogous to the one Professor Wells attempts to fill by explicitly listing mathematical truths. Just as most of Wells’ students are capable of doing their own trigonometry, most of Darwin’s contemporaries understood the logic of natural selection. But, just as many of Wells’ students failed to see how to apply their knowledge of trigonometry, Darwin’s contemporaries often failed to see natural selection when and where it was at work. In physics, filling in the inferential details may tend to be grunt work (although this is certainly not always true). But what we have seen is that filling in the inferential details in biology often leads to significant discoveries. Ironically, it is Kitcher’s insistence that “the fundamental principles of a novel scientific theory should be those statements introduced by the theory that most stand in need of defense and confirmation,” (Kitcher 1985, p. 130), which seems to be an artifact of the early covering law theorists’ preoccupation with physics. The inferences of natural selection, being a priori, may not have been in need of “defense and confirmation,” but it did take the genius of Darwin to demonstrate their importance.

Thus, I conclude that Sober’s recognition of the importance of a priori mathematical models in biology does not necessarily undermine the applicability of the covering law model to biology. Physical explanations, just as much as biological explanations, rely upon the truth of a priori generalizations, but in neither case do these generalizations fill the role of empirical laws.

2.1.2 Empirical biological laws

However, this similarity between physical and biological explanations won’t save the equal applicability of the covering law model to both sorts of explanation unless they both also rely upon empirical laws. So far, the only empirical biological laws that seem plausible are Rosenberg’s laws of natural selection. Though the considerations above appear to deflect Kitcher’s historical objection to those laws, we have already seen that explanations of particular evolutionary events, as opposed to the general fact of evolution, do not appear to essentially rely upon those laws. Combined with Rosenberg’s concerns about the shifting joints of biological kinds, this might lead us to doubt whether there are enough biological laws to satisfy the demands of the covering law model.

Consider again the explanation of prehensile tails. There do not appear to be any obvious laws hiding amongst its premises. The explanation contains assertions that in this particular population the prehensile-tail trait is heritable, that there is variation with respect to possession of this trait, and that those with the trait are fitter than those without it. Since these claims are all claims about a particular population and a particular trait, they lack the generality of laws. If this explanation contains empirical biological laws, they are hiding.

One possibility is that the explanation relies on implicitly assumed, well known laws. Uncovering such implicit laws is a common method of deflecting apparent counterexamples to the covering law model (Hempel 1966, pp. 51–52). However, in the current case, no such lurking premises appear to be necessary. If these particular facts about particular monkeys and their tails are true, it is a matter of mathematical necessity that the frequency of the prehensile-tail trait will probably increase in the population.

So, if there are empirical generalizations at work in this explanation, they must be hiding in the explicitly stated premises. Consider the claim that the prehensile-tail trait is heritable. This is a claim that the members of this population have a certain disposition (or propensity): the disposition of tending to pass the prehensile-tail trait to their offspring. Explicitly defining the concept of a disposition is a notoriously vexing task (for example, see Malzkorn 2001) that I don’t intend to undertake here. However, for present purposes I think we can make do with the relatively safe assertion that claims about complex, higher-level dispositions (i.e. dispositions like heritability or fitness, perhaps unlike fundamental dispositions such as electrical repulsion or proton decay) imply the claim that the disposition occurs because of some set of underlying natural regularities. We would not call some object’s tendency to behave in a certain way a disposition if we believed that this tendency were a mere coincidence or miracle. These implied natural regularities are empirical laws if anything is (Of course, some philosophers might want to reject talk of laws altogether, but since what is at issue in the present instance is whether or not biological explanations conform less well to the covering law model than physical explanations, and since any outright rejection of natural laws would equally undermine the covering law model for both sorts of explanation, I will assume that talk of laws is acceptable for present purposes).

One interesting property of these disposition claims is that, although they appear to have this implication, one can intelligibly make a disposition claim without necessarily being able to specify which laws underwrite the disposition. For example, Kepler, Newton, and Einstein all equally shared the belief that planets are disposed to travel in elliptical paths. But whereas Newton thought this disposition a result of a natural law that universally attributes an attractive force to massive objects, Einstein thought it underwritten by a law governing the effects of massive objects on the curvature of space-time. Kepler lacked any understanding of the underlying regularities at all, but certainly believed that they were there. His belief counts as a belief in the disposition of planets to orbit in ellipses merely in virtue of the fact that he believed there to be some set of underlying laws responsible for it.

The same was importantly true of heritability. Darwin famously lacked any theoretical understanding of why organisms tend to produce offspring that resemble them. But he realized that due to some sort of underlying natural process, organisms do have this tendency. Luckily, this was all he needed to know about heritability to construct natural selection explanations. In the explanation of prehensile tails, so long as the prehensile-tail trait is heritable, it doesn’t much seem to matter why it is heritable. All that is important is that heritability is a natural disposition, rather than a coincidence.

Assertions of fitness are also disposition claims. A claim that one type of organism is fitter than another implies a claim that, certain of the laws of nature being what they are, individuals of the former type are likely to leave more offspring than individuals of the latter type. Once again, one might believe that one type of organism is fitter than another without necessarily having an understanding of precisely which laws of nature are responsible.

I think this is where our missing empirical laws have been hiding. Though it is true enough that we can explain the evolution of prehensile tails entirely in terms of the properties possessed by particular organisms, some of these properties are dispositions. These disposition claims imply that there are certain unspecified universal natural laws which underwrite the dispositions. Clearly, if known, these unspecified laws would allow us to construct covering law explanations of the dispositions themselves, and these explanations could then be substituted into whatever higher-level explanations refer to those dispositions. In the case of heritability, for example, we now know enough about the underlying genetic processes to at least sketch a (bio-chemical) covering law explanation.

But the great value of dispositional language is that it allows us to refer to these laws without specifying them, or even knowing precisely what they are. The attribution of dispositions thus acts as a sort of placeholder for the unspecified laws in a covering law explanation by placing outer bounds on which natural laws might be true. The explanation of prehensile tails depends on the laws of nature being the sort of laws that, in the conditions that obtain, give rise to heritability of that trait in gibbon ancestors. Since heritability is easily verifiable in this case, whatever the laws of nature in fact are, they must be of the right sort. Similar limitations are placed on the laws by other disposition claims in the explanation, such as attributions of fitness. Since the explanation relies on these laws without specifying them, we can explain why it looks as though the biologist’s explanation makes no reference to laws at all, even though it in fact conforms to the covering law model.

This defense of the covering law model involves a greater degree, as it were, of implicitness than Hempel’s. Whereas his defense claims that covering law explanations can leave certain known premises about laws implicit, I am suggesting that the attribution of dispositions allows us to implicitly include in our explanations laws about which we are ignorant or which we understand only incompletely. After all, the point of including laws in covering law explanations is that this “fits the phenomenon to be explained into a pattern of uniformities and shows that its occurrence was to be expected,” (Hempel 1966, p. 50). So long as our knowledge of dispositions constrains the ways that the underlying laws might be sufficiently that, however they actually are, the phenomenon to be explained was to be expected, this purpose will have been fulfilled.

The same defense can be made, I suspect, of any true biological explanation. Fisher’s sex ratio explanation, for example, contains references to dispositions (both heritability and fitness), so it implicitly makes reference to unspecified laws. Furthermore, biology is rife with attributions of other dispositions. Hearts are disposed to pump blood, wolves are disposed to kill deer, and so on. Wherever explanations rely upon claims about dispositions, they rely upon laws, and this satisfies the covering law model.

This also allows us to avoid Rosenberg’s worry about the historical contingency of biological kinds. Rosenberg argues that empirical generalizations cannot be laws because they appeal to historically defined categories that allow for significant temporal variation in traits. But as we have seen, evolutionary biologists don’t really need to appeal to generalities concerning all members of a species. For example, if we want to explain why zebras are harder for lions to catch than horses, we will be able to do so by pointing out that present-day zebras have stripes, present-day horses don’t, and present-day lions have trouble seeing stripes. Over time, zebras, horses, and lions may change in ways that undermine these generalizations, but that doesn’t matter since these are not the laws to which the explanation appeals. Instead, the explanation appeals to the laws, whatever they may be, that govern the assorted complicated processes which underlie the disposition of the visual systems of lions to detect horses more effectively than zebras. These laws do not have to be explicitly mentioned in the explanation, or even known by the biologist who develops it, so long as it is clear that the visual systems of lions do, in fact, possess this disposition. The biologist’s assertion that this disposition exists places bounds on the laws that might exist, and this is enough to satisfy the covering law model.

This is consistent with Rosenberg’s claim that whenever biological explanations fail to refer to the laws of natural selection, they refer to the laws of molecular bio-chemistry, and that biology is thus reducible to bio-chemistry (Rosenberg 2001b). However, my thesis is only tangential to the issue of reductionism. First, while it may be that disposition claims in biological explanations always act as placeholders for bio-chemical laws, if it turned out otherwise, reductionism might be in trouble. But my thesis about the equal applicability of the covering law model to both biology and physics would not be similarly imperiled, since it makes no assertions about the sorts of laws underlying the disposition claims in biological explanations. Second, if, as Rosenberg maintains, the issue of reducibility comes down whether or not the explanations of biology can be complete without including reference to bio-chemical laws, my own view suggests that this issue is subject to an ambiguity. For, while I claim that biological explanations, via their disposition claims, refer to laws of nature, I do not require that these laws be made explicit, or even that they be known, so long as the disposition claim sufficiently constrains the possible ways the laws might be. So, if by ‘complete,’ we mean that an explanation is fully explicit, then I would agree that biological explanations cannot be complete without a full statement of the laws upon which they rely. But we might simply mean by ‘complete’ that an explanans contains all that is needed to deduce its explanandum. In this sense, I think that biological explanations that leave all reference to the laws of nature implicit in disposition claims can still be complete, since their disposition claims imply that the laws, whatever they may actually be, must be such that the explanandum was to be expected.

2.1.3 The covering law model and approximation

Some may doubt, however, that these references to unknown laws hidden in disposition claims really satisfy the covering law model’s demand for covering laws. Leaving known laws implicit (ala Hempel) is one thing, but if disposition claims merely imply that there are some underwriting laws or other, at best one has provided only the promise of an explanation.

But the covering law model has always been intended as an idealization. Besides explanations with implicit laws, Hempel also allows deviations from the ideal that he calls ‘partial explanations’ (Hempel 1962, pp. 16–18). For example, he considers a case in which Freud explained a slip of the pen as the result of wishful thinking. In this case, although the strongest law endorsed by Freud would be a law to the effect that strong unconscious desires will cause a slip that symbolically fulfills those desires in some way or other, Freud’s explanation is of a particular slip (a misdated letter). The explanation is partial in the sense that, while it succeeds in predicting a slip, it fails to predict the particular sort of slip that actually occurred.

Something similar happens when explanations involve quantitative approximations. If Professor Wells’ initial measurements involve some margin of error, that margin will be transmitted to the explanandum. Just as Freud’s explanation entailed only that some sort of slip would occur, Wells’ explanation from approximated initial conditions entails only that the laser dot will move some distance falling in a small range around 27 cm. So this also seems to count as a partial explanation in Hempel’s sense, and, while we might on that account want to say that it fails to be an ideal explanation, it would certainly seem overly fastidious, even within the covering law tradition, to deny that Wells had explained the motion of the dot.

The biological explanations that I claim invoke unknown laws via disposition claims do not merely involve this sort of approximation of initial conditions. However, I believe we can say that they involve an analogous ‘approximation of laws.’ If biological disposition claims implied nothing but the existence of some natural law or other, then biologists really would be out of the explanation business. But as I have said above, the attribution of a disposition also places a constraint on what the underwriting laws must be like: the laws must be laws that will actually bring the disposition about. For example, the disposition of monkeys to be more successful in reproduction when they manage to avoid falling out of trees places certain constraints on the law of gravity. That monkeys have this disposition does not tell us that objects in freefall at the Earth’s surface have an acceleration of −9.8 m/s2, but it does tell us that the acceleration of gravity at the Earth’s surface is sufficient to accelerate a monkey who falls from the height of a tree to a velocity high enough to kill or seriously damage it when it nearly instantaneously decelerates upon reaching the ground. The existence of the same fitness disposition also places limits on the laws governing the forces that account for the strength and flexibility of monkey bones, the effectiveness of various tissue repair mechanisms, and much more.

These constraints may often be fairly weak, but the point is that they appear to be strong enough to produce (partial) biological covering law explanations. Just as the explanation of the motion of the laser dot can survive some imprecision in its prediction resulting from approximated initial conditions, so the explanation of prehensile tails can survive approximated laws. When made fully explicit, the disposition in the prehensile tail explanation would be cashed in, not for claims about the precise set of laws from which prehensile tails may be deduced, but instead a range of sets of laws, any one of which predicts the evolution of prehensile tails. Since any one of the sets of laws leads to the same prediction, we can show that, even given what little knowledge of laws is contained in our knowledge of the dispositions, we should expect prehensile tails. Of course, just as with other sorts of partial covering law explanations, we could, and sometimes do, produce a better explanation by more precisely pinning down what has previously only been approximated. As we improve our knowledge of physics, chemistry, and molecular biology we can narrow down the range of covering laws to which our explanation appeals. But we needn’t necessarily do so to have a partial covering law explanation just as good in its own way as Professor Wells’ explanation of the laser dot or Freud’s explanation of his slip of the pen.

So once again we find that the differences between biological and physical explanations are merely a matter of degree. All sciences employ approximations of one sort or another in the construction of covering law explanations. In virtue of these approximations, the explanations in which they occur fall short of perfect realization of the covering law model, but still seem worthy to be called (partial) explanations. As one might expect, biologists, who deal with extremely complex systems, will need to rely relatively heavily on various sorts of approximation if they are to explain anything at all. This will include approximation of the covering laws through attributions of dispositions. But in so doing, they are not falling significantly shorter of the covering law ideal than scientists in other disciplines, and they need not be construed as engaged in some wholly different sort of explanatory activity.


Sober and Kitcher have indeed uncovered important differences between physical and biological explanations. Scientific progress in biology appears to rely more heavily on recognition and application of a priori principles or models governing explanatory inferences, and less heavily on discovery of new empirical laws or particular facts, than does scientific progress in physics. However, these differences should not be characterized as differences in the degree to which biological and physical explanations conform to the covering law model of scientific explanation. For, while a priori principles do appear to play a larger role in biological explanations than in physical explanations, such principles are essential components of any covering law explanation, including physical covering law explanations. And, while biological explanations do appear to place less emphasis on empirical laws, and often omit explicit mention of them, reference to empirical laws still plays an essential role in such explanations, just as it does in physical explanations. The appearance that biological explanations make no reference to empirical laws is the result of the fact that the laws involved in biological explanations are typically only implied by premises attributing dispositions, which place bounds upon the laws of nature without actually asserting the truth of particular laws.

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© Springer Science+Business Media B.V. 2007