## 1. Two Conceptions of Natural Selection



Natural selection is chiefly discussed in two different ways among
contemporary philosophers and biologists. One usage, the
"focused" one, aims to capture only a single element of
one iteration of Darwin's process under the rubric
"natural selection", while the other, the
"capacious" usage, aims to capture a full cycle under the
same rubric. These are clearly alternative, non-competing uses of the
term, and distinct philosophical controversies surround each one. This
section distinguishes these two uses and the following two sections
are dedicated to the debates that surround each one. Sections 4 and 5
consider how natural selection connects to explanation and causation.




In Darwin's wake, theorists have developed formal, quantitative
approaches to modeling Darwin's process. The
"focused" usage of natural selection finds its home as an
interpretation of a single term in some of these formalisms (and only
some of them). Below, we will consider two formal approaches, type
recursions and the Price Equation, elements of which have been
interpreted as quantifying selection. In the Price Equation, the
covariance of offspring number and phenotype is interpreted as
quantifying selection; in type recursions, fitness variables (or,
equivalently, selection coefficients) are interpreted as quantifying
selection. What makes these interpretations focused is that they
quantify only a single element of Darwin's process using the
notion natural selection; other facets of Darwin's process are
handled in other ways. So, in type recursions for instance, type
frequency variables quantify how the population varies, and
"spontaneous" variations are quantified by mutation
parameters. Similarly, in the Price Equation, inheritance is captured
by a different term than the term that is interpreted as quantifying
natural selection. The point is that each formal apparatus as a whole
is understood to capture Darwin's process, while only a single
element of that apparatus is said to refer to natural selection.



Some philosophers' definitions of natural selection are clearly
intended to capture this focused usage of the term. Millstein, for
instance, characterizes selection as a discriminate sampling process
(Millstein 2002: 35). Otsuka identifies natural selection with the
causal influence of traits on offspring number in causal-graphical
models (Otsuka 2016: 265). Okasha interprets the covariance of
offspring number and offspring phenotype as quantifying the causal
influence of selection (Okasha 2006: 28) Clearly, these uses of
"natural selection" are meant to capture only an element
of Darwin's process; they make no mention of inheritance or
replication. As discussed further below, controversies over the
focused notion of selection have to do with whether the focused notion
of selection can be distinguished from that of drift
 (section 3),
 and whether selection, in the focused sense, should be counted as a
cause
 (section 5).



The alternative, "capacious", usage of the notion of
natural selection is to capture Darwin's process in its
entirety, rather than a single contributor to it. Because
Darwin's process is cyclical, specifying what is sufficient for
a single cycle of it, a single instance of, say, replication of genes
for long legs caused by long-legged wolves making narrow captures, is
sufficient to specify a process that may explain adaptation and
speciation. This is true, anyway, when it is added that the process
gets repeated. The capacious notion, capturing a cycle of
Darwin's process, is used by Lewontin and later authors working
in the same vein, who put forward conditions for evolution by natural
selection: these include variation, inheritance, and reproduction.
While falling within the scope of "natural selection" in
the capacious sense used by Lewontin, these elements of Darwin's
process are treated as distinct from natural selection when that
notion is used in its focused sense.


## 2. Evolution and the Conditions for Natural Selection



Philosophers and biologists have been concerned to state the
conditions for evolution by natural selection, many because they take
it that there is a single theory, evolutionary theory, that governs
Darwin's process. In many ways, the attempt to state the
conditions for natural selection is a typical philosophical
undertaking. We know, for instance, that confirmatory evidence may be
used to raise our confidence in what it confirms and this recognition
spawns a debate over what, exactly, should count as confirmation (see
entry on
 confirmation).
 Similarly, Darwin's theory shows how some natural phenomena may
be explained (including at least adaptations and speciation), and thus it is a
similar philosophical concern to state exactly when the deployment of
the theory is licensed. Such a statement would then issue in a verdict
on what, beyond the phenomena targeted by Darwin, is equally explicable
using his theory. Such a verdict could be used to arbitrate whether
the spread of cultural variations, "memes", is genuinely
explicable using the theory, as Dawkins (1982) suggested. The
mammalian immune system may equally involve dynamics that are
explicable as selection processes (see section 4.1 of the entry on
 replication and reproduction).
 Zurek (2009) has even defended using the theory to explain phenomena
in quantum mechanics.



Here are the three principles that form the "logical
skeleton" of "Darwin's argument", according to
Lewontin (1970: 1):


1. Different individuals in a population have different morphologies,
physiologies, and behaviors (phenotypic variation).
2. Different phenotypes have different rates of survival and
reproduction in different environments (differential fitness).
3. There is a correlation between parents and offspring in the
contribution of each to future generations (fitness is heritable)



Lewontin's principles invoke, implicitly or explicitly, a number
of causal processes, including development, reproduction, survival,
and inheritance. Though it is controversial whether Lewontin succeeds,
clearly his three principles aim to capture at least what is
sufficient for a cycle of evolutionary change by natural selection,
something which, if repeated, could be used to explain adaptation and
speciation.



Lewontin's 1970 statement of the requirements for selection
should be contrasted with a similar statement in a later work, in
which that author once again states that "the theory of
evolution by natural selection" rests on three principles:



> 
> 
> Different individuals within a species differ from one another in
> physiology, morphology, and behavior (the principle of variation); the
> variation is in some way heritable, so that on the average offspring
> resemble their parents more than they resemble other individuals (the
> principle of heredity); different variants leave different numbers of
> offspring either immediately or in remote generations (the principle
> of natural selection). (Lewontin 1978: 220)
> 
> 
> 



This later statement is in many ways similar to the earlier one, but
there are some crucial differences in the formulations. According to
the later article, the different individuals must be within the same
species, and it is phenotypic variations, rather than fitness, that
must be inherited. Many authors have echoed Lewontin's
influential principles (see Godfrey-Smith (2007) for a thorough
review); for markedly different approach to formalizing the theory,
see Tennant (2014)).



Lewontin's conditions have been criticized from many directions.
For one thing, they make no mention of the populations in which
selection occurs, and though the second set of conditions refers to
species, all the members of a single species may not form a single
population for the purposes of applying selection theory. Populations
must be appropriately circumscribed for some of the key vocabulary of
evolutionary theory (focused selection, drift) to be deployed in a
non-arbitrary fashion (Millstein 2009). Lewontin's formulations
also make no mention of the struggle for existence, a seminal notion
for Darwin (see discussion in Lennox & Wilson 1994). The
principles have also been subject to counterexamples. On the one hand,
authors put forward cases of evolving populations that fail to meet
Lewontin's conditions. Earnshaw-White (2012) puts forward a
population in which offspring do not inherit their parents'
traits, but whose members exhibit differential survival such that the
population will evolve to a stable equilibrium. Equally, differential
heritability may lead to evolutionary change without differential
fitness (Earnshaw-White 2012; but see Bourrat 2015). Despite failing
to meet Lewontin's conditions, populations will evolve that
feature variants that differ not in average offspring number, but only
in generation time, such that one variant reproduces twice as fast as
the other (Godfrey-Smith 2007: 495). On the other hand, authors put
forward cases of systems that fail to evolve despite meeting
Lewontin's conditions. A system exhibiting stabilizing
selection, such that it tends to evolve toward an equilibrium state
where multiple variants remain within the population, may meet either
set of Lewontin's conditions despite failing to evolve
(Godfrey-Smith 2007: 504). Counterexamples of these sorts depend upon
the assumption that we can tell whether a system is undergoing
selection without applying any set of criteria for doing so (see
Jantzen 2019 for a discussion of how a Lewontin-style approach to
whether a system undergoes selection is question-begging).



Okasha makes an important point about the intransitivity of covariance
that is relevant to Lewontin's principles. Okasha writes that Lewontin's last two conditions essentially require covariance
between offspring number and parental character (Principle 2), along
with covariance between parental character and offspring character
(Principle 3). (Here, Okasha is interpreting "fitness" in
Lewontin's conditions as offspring number rather than as a
variable in a formalism quantifying selection in its focused sense.)
Both those last two covariances might be positive without the system
exhibiting an evolutionary response (as would occur if some of the
parents with a given character have especially many offspring that do
not especially resemble them, while other parents with the same
character have offspring that especially resemble them but do not have
especially many of them). For a system to exhibit an evolutionary
response, Okasha requires that the covariance between parental
offspring number and average offspring character be positive (2006:
37).



Okasha's alternative condition is intended to replace a pair of
Lewontin's principles. Interestingly, Okasha further differs
from Lewontin in allowing that systems that do not evolve may meet his
requirements. A system in which selection is exactly offset by
transmission bias will not evolve but will undergo natural selection,
according to Okasha (2006: 39).


### 2.1 Replicator Selectionism



An alternative approach to stating conditions for natural selection
involves attention to replicators, of which genes are the paradigm
instance. This approach was motivated by the discovery of genetic
variations that spread despite *not* being conducive to the
reproduction of the organisms that bear them, for instance, genes that
exhibit meiotic drive. Replicator selectionism, which makes
genes/replicators the units of selection, was developed in contrast
with the Lewontin-style approaches, which make larger entities,
organisms and even species, units of selection (see entry on
 units and levels of selection).



Dawkins defines replicators as anything in the universe of which
copies are made (1982: 82). Hull has a similar definition: a
replicator is an entity that passes on its structure largely intact in
successive replications (Hull 1988: 408). Germ-line replicators have
the potential to have indefinitely many descendants; they contrast
with somatic replicators, the genes found in body cells, which produce
copies only as part of mitosis and whose lineages of descendants end
when the body dies. Active replicators have some causal influence on
their probability of being copied in contrast to inactive,
"neutral" ones, which do not have any effects on
development. Natural selection will occur wherever we find active
germ-line replicators (Dawkins 1982: 83). Dawkins distinguished
replicators from vehicles, his notion meant to replace and generalize
that of organism. Hull proposed the notion of interactor as a similar
complement to the notion of replicator. Neither notion, however, is
meant to further delineate the circumstances in which selection
occurs, or to narrow the scope of application of evolutionary theory
(for further discussion of these notions, see entry on
 units and levels of selection).
 This is evident, for Hull at least, insofar as genes may be both
replicators and interactors (1988: 409). 



The view that evolutionary theory is a theory that applies to active
germ-line replicators has come under fire from a multitude of
directions. Genes need not be germ-line to undergo selection, as it is
at least arguable that the immune system exhibits selection processes
(Okasha 2006: 11). Copying is beside the point, since only similarity
across generations, rather than identity, is necessary for
evolutionary change (Godfrey-Smith 2000, 2007 and entries on
 units and levels of selection
 and
 replication and reproduction).
 For his part, Hull seems to agree with this last point, as he allows
that organisms might well count as replicators, at least in cases in
which they reproduce asexually (Hull 2001: 28-29).



Despite the bevy of attacks on replicator selectionism, replicator
selectionists have not, to my knowledge at least, been criticized for
being too permissive and allowing that systems that do not evolve
count as undergoing selection. But germ-line replicators may exert a
causal influence on their probability of being copied without
spreading in a natural population as a result, as in some cases of
frequency-dependent selection of systems already at equilibrium. In
cases of frequency-dependent selection, variant genes cause their own
reproduction, but the extent of influence on reproduction is a
function of their frequency. Suppose each type spreads when it is
rarer. At 50% frequency, the influences of each gene on reproduction
cancel each other out, and a stable intermediate frequency is the
result (for a case of a natural system that behaves this way, see Hori
1993). Because causing replication may not lead to differential
replication in these and other cases, replicator selectionists do not
effectively take evolution to be necessary for selection while
Lewontin and those who follow his basic approach typically do do
so.


### 2.2 Is Evolution Necessary for Natural Selection?



One natural way to arbitrate the issue of whether systems that undergo
selection must evolve is to attend to the point of statements of
principles of natural selection, or statements of the requirements for
selection. Many theorists take it that the point of these principles
is to set out the scope of a theory in the special sciences that deals
with selection and evolution, evolutionary theory. Lewontin claims
that the theory of evolution by natural selection rests on his three
principles (1978: 220). Equally, Godfrey-Smith claims that statements
of conditions for evolution by natural selection exhibit the coherence
of evolutionary theory and capture some of its core principles (2007:
489). Finally, Maynard-Smith (1991) offered a statement of conditions
for selection that include evolution as a necessary component, calling
the theory so delineated, "Darwin's theory". For
these writers, the (or at least *a*) point of the principles
seems to be to capture the domain of application of the theory we have
inherited from Darwin.



Darwin would have been surprised to hear that his theory of natural
selection was circumscribed so as to apply only to evolving
populations. He himself constructed an explanation of a persistent
polymorphism, heterostyly, using his own theory. Plants exhibiting
heterostyly develop two, or sometimes three, different forms of flower
whose reproductive organisms vary in a number of ways, principally
length. Some plants exhibit different forms of flower on the same
plant, while some are dimorphic and trimorphic, with only one sort of
flower per plant. Darwin interpreted the flower variations as
conducive to intercrossing, which he thought was beneficial, at least
for many organisms. Populations should not evolve directionally such
that a single form of flower spreads throughout the population;
instead, multiple variants should be retained, a polymorphism. Darwin
thinks it clear that heterostyly is an adaptation: 



> 
> 
> The benefit which heterostyled dimorphic plants derive from the
> existence of the two forms is sufficiently obvious [....] Nothing
> can be better adapted for this end than the relative positions of the
> anthers and stigmas in the two forms. (Darwin 1877: 30; thanks to Jim
> Lennox for this reference) 
> 
> 
> 



Even though the population is not evolving, but instead remaining the
same over time, it exhibits an adaptation that consists in this
persistent lack of change, an adaptation that Darwin thought
explicable using his theory. For a more recent and especially
compelling case of a selectionist explanation of a polymorphism, see
Bayliss, Field, and Moxon's selectionist explanation of a
fimbriae polymorphism produced by contingency genes (Bayliss, Field,
& Moxon 2001).



Darwin thought his theory could explain a lack of evolution, and
Darwinists in Darwin's wake have explained not only stable
polymorphisms, but unstable ones, cyclical behaviors, protected
polymorphisms, and a variety of other behaviors that differ from
simple directional evolution. These sorts of behaviors result from
specific assignments of values for theoretical parameters in many of
the very same models that are used to explain simple directional
selection (where a single variant spreads throughout a population, as
in the wolf case discussed in the introduction). The point is that
systems seemingly governed by evolutionary theory exhibit a variety of
different sorts of dynamics, and this variety includes both different
sorts of evolution, including at least cyclical and directional, as well
as a lack of evolution at all, as in cases of stabilizing
selection.



Consider in particular how the difference between stabilizing and
directional selection in the simplest deterministic models of diploid
evolution lies in the value of a single parameter in the genotypic
selection model, heterozygote fitness:

 

\[
\begin{align}
p' & = \frac{w\_{D}p^{2} + w\_{H}pq}{w\_{D}p^{2} + 2w\_{H}pq + w\_{R}q^{2}}\\
q' & = \frac{w\_{R}q^{2} + w\_{H}pq}{w\_{D}p^{2} + 2w\_{H}pq + w\_{R}q^{2}}\\
\end{align}
\]


where



* \(p'\) is the frequency of one allele (the \(A\) allele) in the
next generation
* \(q'\) is the frequency of the alternative allele (the \(a\)
allele) in the next generation
* \(p\) is the frequency of \(A\) alleles in this
generation
* \(q\) is the frequency of \(a\) alleles in this
generation
* \(w\_{D}\) is the fitness of the organisms bearing 2 \(A\)
alleles
* \(w\_{H}\) is the fitness of the organisms bearing 1 \(A\)
allele and 1 \(a\) allele
* \(w\_{R}\) is the fitness of the organisms bearing 2 \(a\)
alleles.




In the above simple, "deterministic" models, if we set the
fitness parameters such that \(w\_{D} > w\_{H} > w\_{R},\) the
\(A\) allele spreads throughout the population, together with
whatever traits it causes. Suppose we include within the purview of
Darwin's theory models of this sort, together with the systems
they (approximately) govern. If we change the value of the fitness
coefficients such that \(w\_{D} < w\_{H} > w\_{R},\) the system
will exhibit a different sort of dynamics. It will evolve to an
equilibrium point, where there exist some \(A\) alleles as well as
some \(a\) alleles in the population, and remain there, resulting
in a stable polymorphism. If we hold evolution as a condition for
selection, we will issue the curious ruling that a system governed by
the first sort of model falls within the scope of evolutionary theory
while a system governed by the second sort of model only does so up
until it reaches a stable intermediate state but then no longer.
Moreover, populations exhibiting stable polymorphisms resulting from
heterozygote superiority, or overdominance, are just one case among
many different sorts of systems that equally exhibit stable
polymorphisms.



Consider further that it is more realistic for systems' dynamics
to be a function of effective population size in the binomial sampling
equation, as well as fitness. The above models are deterministic,
while the dynamics of natural systems are to some extent random. A
more realistic systems of equations, one capturing the randomness
involved in evolutionary change, would feed the \(p'\) value issued by
the above equations into the binomial sampling equation by making \(p'
= j\):

 

\[
x\_{ij} = \frac{(2N)!}{(2N - i)! i!} \left(1-\frac{j}{2N}\right)^{2N-i} \left(\frac{j}{2N}\right)^{i}
\]


where \(x\_{ij}\) is the probability of \(i\) \(A\) alleles in
the next generation given \(j\) \(A\) alleles in this
generation, and \(N\) is effective population size. A system
governed by both the deterministic equations and the binomial sampling
equation is said to undergo drift; all natural systems do so. (For
more on drift, effective population size, and randomness in
evolutionary theory, see entry on
 genetic drift).
 A system exhibiting heterozygote superiority whose dynamics are a
function of the binomial sampling equation will not simply rest at its
stable intermediate frequency but will hover around it, in some
generations evolving toward it, more rarely evolving away, and in some
generations exhibiting no evolution at all. Which of these cases are
cases in which the system undergoes natural selection in the capacious
sense? That is, which cases are cases in which the system falls within
the purview of evolutionary theory? A natural answer is all of them.
To answer in this way, however, we must not make evolution necessary
for natural selection. If we do make evolution necessary, we issue the
verdict that systems governed by the above systems of equations
sometimes fall within this purview of evolutionary theory and
sometimes fall outside it, on a generation to generation basis,
despite being governed the whole time by a single same system of
equations, ones undeniably developed as an outgrowth of Darwin's
initial theorizing.



This last pattern of argument can be extended. Indeed, given that
every natural system undergoing selection also undergoes drift,
evolutionary theory is arguably applicable also to systems that
undergo drift even in the absence of selection (in the focused sense).
Gradually reduce the importance of focused selection in the above
system of equations, that is, gradually reduce the differences between
\(w\_{D}\), \(w\_{H}\), and \(w\_{R}\). Is the point at which the values
equalize so momentous that it marks the point at which systems
governed by the equations cease to fall within the purview of one
theory and instead fall within the purview of another? Does
Kimura's neutral theory of genetic polymorphism belong in a
different theory than does Ohta's (1973) competing nearly
neutral theory? Brandon (2006) argues that the principle of drift is
biology's first law, writing that neutral evolution is to
evolutionary theory what inertia is to Newtonian mechanics: both being
the natural or default states of the systems to which they apply. If
Brandon is right, then conditions for the application of evolutionary
theory must not even include conditions for selection in the focused
sense, much less conditions for evolutionary change.



The point of stating conditions for evolution by natural selection
need not be to state the conditions of deployment of a particular
theory in the special sciences. Godfrey-Smith mentions that the
principles may be important to discussions of extensions of
evolutionary principles to new domains. Statements of the conditions
for evolution by natural selection might have value for other reasons.
But evolutionary theory is, despite the name, at least arguably a
theory that is applicable to more systems than just those that evolve,
as the replicator selectionists would have it.


## 3. Natural Selection as the Interpretation of a Component of a Formalism



One of the two chief uses of the notion of natural selection is as an
interpretation of one or another quantity in formal models of
evolutionary processes; this is the focused sense distinguished above.
Two different quantities are called selection in different formal
models widely discussed by philosophers. On the one hand, fitness
coefficients are said to quantify selection in type recursion models
of selection; the \(w\)'s in the above genotypic selection
model are fitness coefficients. This is standard textbook usage (Rice
2004; Hedrick 2011). The recursive structure of these models is
important. They can be used to infer how a system will behave into the
future (though of course only if causes of the variables in the system
do not change their values in dynamically-relevant ways that are not
explicitly modeled in the recursive equations). At their simplest,
type recursions make system dynamics a function of fitness
coefficients weighting type frequencies/numbers together with
effective population size (quantifying drift), as in the genotypic
selection model discussed above. More complex variants of type
recursions include frequency-dependent selection models, for cases in
which population members' fitness is a function of the type
frequency variables; density-dependent selection models, for cases in
which fitness is a function of population size; spatially and
temporally variable selection models, for cases in which fitness
varies as a function of a varying environmental variable that
interacts with type differences, and many more. Writers working with
type recursion models have developed explicit interpretations of their
theoretical terms, including the fitness variables quantifying
selection. So, for instance, Beatty and Millstein defend the view that
the fitness coefficients representing selection in type recursions
should be understood as modeling a discriminate sampling process,
while drift, controlled by effective population size, should be
understood as indiscriminate sampling (Beatty 1992; Millstein
2002).



Philosophers have also contended that particular terms in models of
systems featuring the formation of groups (or collectives) should be
understood as quantifying the influence of selection at different
levels. So, some type recursions of systems involving groups feature
"group fitness" or "collective fitness"
parameters, ones analogous to individual or particle fitness
parameters. Kerr and Godfrey-Smith (2002) discuss one such system of
recursions; Jantzen (2019) defends an alternative parameterization of
group selection as part of different system of equations. (See also
Krupp 2016 for causal-graphical conceptualization of the notion of
group selection.) For much more on multi-level selection, see entry on
 units and levels of selection.



The other formal model of particular interest to philosophers is the
Price Equation. The Price Equation represents the extent of evolution
in a system with respect to a given trait across a single generation
using statistical functions:

 

\[
W\Delta Z = \textrm{Cov}(w\_i,z\_i)+\mathrm{E}(w\_i \Delta z\_i)
\]


where



* \(z\_i\) denotes the character value of the
\(i\)thpopulation member
* \(\Delta Z\) is the change in average character value in the
population
* \(w\_i\) denotes the number of offspring produced by (the fitness
of) the \(i\)thpopulation member
* \(W\) is the average number of offspring produced by population
members
* E is expectation
* Cov is covariance




In the Price Equation, selection is associated with the first
right-hand side quantity, while the second represents transmission
bias.



Identities among algebraic functions of statistical functions make
possible the mathematical manipulation of the Price Equation such that
it may feature a variety of different quantities. As with type
recursions, quantities in various transformations of the Price
Equation are equated with selection at different levels for different
systems; Okasha, following Price, treats the covariance of the fitness
of collectives with the phenotype of collectives as collective-level
selection, while the average of the within-collective covariances
between particle character and particular fitness is identified with
particle-level selection. The persistence of altruistic variants in
natural populations has been explained as the result of the
stabilizing conflict between selection at these different levels
(Sober & Wilson 1998; and again see entry on
 units and levels of selection.
 The Price Equation can equally be manipulated to yield distinct
notions of inheritance; Bourrat distinguishes temporal, persistence,
and generational heritabilities and argues for the temporal notion as
appropriate for the purposes of stating conditions for evolution by
natural selection (Bourrat 2015). 



The distinction between type recursions and the Price Equation is
important, because selection is interpreted differently in each. The
two formalisms will issue in different verdicts about whether, and the
extent to which, focused selection operates within a single system. If
we fix upon a single natural system, and ask how selection operates
within it, we will get different answers if we associate selection
with fitness variables in type recursions rather than
\(\textrm{Cov}(w\_{i},z\_{i})\) in the Price Equation. To see this,
consider how type recursions are structured such that inferences about
dynamics over multiple generations may be made by means of them. If
fitness coefficients in these models quantify selection, and these
take fixed values (as they do in the genotypic selection model
considered above and a great many others), then the extent of
selection will remain the same over the time period governed by the
model: the fitness variables remain at fixed values so selection
remains an unchanging influence. But, in just the same cases, the
value of \(\textrm{Cov}(w\_{i},z\_{i})\) in the Price Equation will
change across generations as the system evolves: the covariance
function in the Price Equation will be highest at intermediate
frequencies, when evolution proceeds quickly, and lowest at liminal
ones, when evolution goes more slowly. Consider, for instance, the
extent to which the population evolves, according to the genotypic
selection model above, when the following values are plugged into the
model:

 

\[
p = 0.9, 
q = 0.1, 
w\_{D} = 1, 
w\_{H} = 0.8, 
w\_{R} = 0.6
\]


In that case, the frequency of the \(A\) alleles moves from 0.9 to
0.92 across a single generation; there is some covariance between
\(A\) types and reproduction. Inputting \(p = 0.5,\) and
\(q = 0.5,\) the frequency change is greater, \(A\) types go
from being half the population to frequency 0.56, and the covariance
between offspring production and being an \(A\) type is
correspondingly greater as well.



To see even more starkly how what is called selection in the Price
Equation differs from what is called selection in type recursions,
consider a system exhibiting heterozygote superiority like the one
from earlier, where \(w\_{D} = w\_{R} < w\_{H}.\) Recall that a system
of this sort will evolve to a stable equilibrium (provided drift is
idealized away). Using the recursive equations above, the reader can
input values that meet the constraint that heterozygotes have the
highest fitness, set \(p' = p,\) and compute the stable equilibrium
value for \(p\), the frequency of the \(A\) alleles. If we
understand selection as quantified by the fitness coefficients in this
sort of set-up, then the whole time, selection operates in a constant
fashion, since the fitness coefficients remain fixed. In particular,
the operation of selection is the same when the system is evolving
toward its stable equilibrium as when it remains at that stable
equilibrium. By contrast, the covariance term in Price Equation model
of the system will diminish in value until it reaches zero as the
system evolves to its equilibrium state. When selection is identified
with the covariance between type and reproduction, the frequency of
the different types matters to the extent of selection. When selection
is identified with fitness variables in type recursions, the frequency
of different types has no influence on the extent of selection in the
system. Thus, the different interpretations of selection that
correspond to different quantities in different formal models are
actually incompatible. We should expect, then, at least one of these
interpretations of selection to fail, since focused selection cannot
be two different things at once, at least if what counts as natural
selection is non-arbitrary.



One way to reconcile these competing interpretations of selection is
to make first right-hand side term in the Price Equation quantify
*the extent of the influence of selection* in a system. If we
assume that focused selection accounts for whatever covariance exists
between parental offspring number and phenotype, then we may treat the
first right-hand side term of the Price Equation as a measure of the
extent of the influence of focused selection, at least at a given type
frequency (see Okasha 2006: 26). This approach puts the logical house
in order, allowing for a univocal concept of selection, but it does so
at the expense of other commitments. To note just one, the Price
Equation will no longer be causally interpretable, since its
quantities may no longer be said to represent causes (but instead
measure the extents of their influences given further limiting
assumptions). There exists a sizable literature on which of multiple
alternative manipulations of the Price Equation represents the actual
causal structure of different sorts of system (see Okasha 2016 and
 section 5
 below for more on this issue).



It was noted earlier that if what counts as selection is
non-arbitrary, then it cannot be the case that what is quantified by
the \(w\)'s in type recursions and the covariance of
offspring number and phenotype both count as selection. A substantial
debate has arisen over the question of whether what counts as
selection is indeed non-arbitrary. In particular, some philosophers
dubbing themselves "statisticalists" have challenged the
non-arbitrary character of the distinction, claiming that it is
model-relative (Walsh, Lewens, & Ariew 2002; Walsh 2004, 2007;
Walsh, Ariew, & Matthen 2017). (For more on the arbitrariness of
the selection/drift distinction, see entry on
 genetic drift).
 A related issue, discussed in the subsequent section, concerns the
causal interpretability of the theory: Advocates of the non-arbitrary
character of selection also typically treat selection and drift not
only as non-arbitrary quantities, but also as causes, while those who
allege that the distinction is arbitrary typically equally challenge
the treatment of selection and drift as causes. 



There exists very little discussion of formal models in statisticalist
writings; various toy scenarios are put forward instead, and these are
not analyzed using the techniques of formal population genetics
(Matthen & Ariew 2002; Walsh, Lewens, & Ariew 2002; Walsh
2007; Walsh, Ariew, & Matthen 2017). When biologically realistic
scenarios are discussed, systems of equations for inferring how such
systems behave are not made part of the discussion (for more on
population genetics, see entry on
 population genetics).
 We consider next a case they discuss because it provides a way of
contrasting how the contrast between selection and drift is made in
type recursions and how it is made in the Price Equation. There is a
sort of arbitrariness here, but it emerges only from analysis of a
hypothetical system using population genetics modeling techniques.



In a recent paper, Walsh, Ariew, and Matthen put forward a case of
temporally variable selection and claim that it could be treated as a
case either of selection or of drift (2017). The case is of a
discrete-generation system with yearly reproduction in which each of
two types of organisms produce different numbers of offspring
depending on whether the year is warm or cold, with each type of year
being equally probable: the H types produce 6 offspring in warm years
while the T types produce 4, and the reverse holds for cold years. The
authors take the position that the system "looks like"
both a case of selection and a case of drift, but without
consideration of any formal model of it. Instead, we are invited to
think it is a case of selection because the fitnesses of the different
types vary year-to-year; we are invited to think it is a case of drift
because, allegedly, over long stretches of time the two types have
equal fitnesses and we should, allegedly, predict no net increase in
frequency for either type (Walsh, Ariew, & Matthen 2017:
12-13).



The scenario is illuminating because it involves randomness that
cannot be quantified by effective population size in a type recursion
but *can* be quantified as such by the drift parameter in Price
Equation. When deploying type recursions, we must treat cases of
temporally variable selection as cases of selection, but we are under
no similar constraint when it comes to the Price Equation.



The dynamics of the warm/cold system cannot be inferred from a type
recursion in which fitness coefficients are set equal and
next-generation frequencies are determined entirely by the
"drift" term, effective population size in the binomial
sampling equation. When fitnesses are equal, the frequency of each
type is determined solely by the binomial sampling equation above
(since post-selection frequency, the input to the sampling equation,
is just pre-selection frequency). Such a determination makes
next-generation frequency a normal, bell-shaped distribution whose
mean is the initial frequency of the types in the system. The
distribution of next-generation frequencies in the warm/cold system
does not look like this. The type recursions for the warm/cold system
must be calculated using a temporally variable selection model:



\[
p' = \frac{w^{t}\_{w} p}{w^{t}\_{w}p + w^{t}\_{c}q}
\]


where



* \(p\) is the frequency of warm types
* \(q\) is the frequency of cold types
* \(w^{t}\_{w}\) is the fitness of warm types in year \(t\)
* \(w^{t}\_{c}\) is the fitness of cold types in year \(t\)




What is inputted into the binomial sampling equation, \(p'\), when the
dynamics of the system are calculated using the temporally variable
selection model is never the same as what is inputted into the
binomial sampling equation when the dynamics of the system are treated
as solely a function of drift. In the latter case, the input into the
equation is just \(p\). But since the temperature always favors one
or another type, post-selection frequency, \(p'\) is never equal to
pre-selection frequency, \(p\). Treating a system of warm/cold type
that undergoes temporally variable selection as though it undergoes
drift would lead to making false inferences about its dynamics. Cases
of temporally variable selection of the warm/cold sort simply must be
modeled using type recursions in which fitness parameters are a
function of time; the pioneering analysis of such systems was done by
Dempster (1955). The above point generalizes: what is quantifiable as
selection and drift in type recursions is made definite by how fitness
variables and effective population size function in those models.



The story is different, however, with the Price Equation, owing to how
randomness is handled in that formalism. A version of the Price
Equation in which both selection and drift are represented is this
(Okasha 2006: 32):

 

\[
W\Delta Z = \textrm{Cov}(w\_{i}^{\*},z\_{i}') + \textrm{Cov}(\mu\_{i}, z\_{i}')
\]


where



* \(\Delta Z\) is the change in average character value in the
population
* \(W\) is the average number of offspring produced by population
members
* \(w\_{i}^{\*}\) is the expected fitness of the \(i\)th
population member
* \(z\_{i}'\) is the average character value of the
\(i\)th population member's offspring
* \(\mu\_{i}\) is deviation from expectation of the
\(i\)th population member's offspring
production




Here, the second term quantifies change due to drift (Okasha 2006:
33). Note how the \(\mu\_{i}\) parameter quantifies deviation from
expectation, and hence drift is a function of how the expectation is
determined. Nothing about the Price Equation formalism constrains such
determinations. In the warm/cold population, whether a given
year's evolution is quantified by the first right-hand side
term, and hence constitutes change due to selection, or the second
one, and hence constitutes change due to drift, will be determined by
whether the theorist treats the weather as contributing to expected
fitness. If, for instance, she is ignorant of how the weather changes
year to year, she may treat the weather as drift; if she can predict
it, she may not do so. Deployment of the Price Equation is compatible
with both treating the weather as contributing to expected fitness and
treating it as causing deviation from expectation.



The result is that a theorist deploying the Price Equation may treat
as drift (that is, quantify as deviation from expectation) what a
theorist deploying type recursions must treat as selection (quantify
by fitness variables). It is possible to make assumptions using the
Price Equation such that the drift term quantifies what is quantified
by the drift term in type recursions, but nothing about the Price
Equation proper forces one to do this, and indeed proponents of the
Price Equation, such as Grafen (2000), tout how the drift term in the
Price Equation may quantify all sorts of randomness, explicitly
including randomness that is not quantifiable as drift in type
recursions.



As noted earlier, selection and drift are construed in logically
distinct fashions in type recursions and the Price Equation. What is
called is selection in type recursions is one sort of definite thing:
it's whatever it is that must be quantified by fitness
variables. What is called selection in the Price Equation is another
thing, and this will be determined by researchers' understanding
of the system, in particular how they generate expectations about
fitness. Ultimately, the conflict between the two modeling approaches
with respect to what counts as selection may be resolvable in at least
a couple of ways. Perhaps one modeling approach is simply wrong about
what selection is. Alternatively, something having to do with
selection is arbitrary here. In particular, whether some or another
form of randomness should be treated as selection or drift is a
function of one's choice of which modeling approach to use,
where distinct approaches provide logically distinct understandings of
"selection" and "drift". Once one chooses to
model the system using type recursions, one's hand is forced:
the temperature must be modeled as selection. Once one chooses to
model the system using the Price Equation, one's hand remains
free: one may treat the changing temperature as relevant to expected
fitness, and hence treat it as selection, or instead treat it as
producing deviation from expectation, and thereby treat it as drift.
According to this second way of resolving the conflict, the choice
between the two modeling approaches is not dictated by nature, and is
thus at least metaphysically (if not pragmatically) arbitrary.


## 4. Natural Selection and Explanation



As noted above, evolutionary theory cannot do its job unless it has an
explanatory structure. Philosophers have contended selection explains
a variety of things in a variety of ways.


### 4.1 Explanatory Scope



There exists a longstanding debate among both scientists and
philosophers over exactly what natural selection can explain, one
begun by Sober and Neander who were concerned with what natural
selection can explain (Sober 1984, 1995; Neander 1988, 1995). In a
recent expansive treatment, Razeto-Barry & Frick 2011 distinguish
between the creative and non-creative views of natural selection. On
the non-creative view, natural selection merely eliminates traits
while doing nothing to create new ones; the latter phenomenon is the
result of mutation. Proponents of the creative view see natural
selection as a creative force that makes probable combinations of
mutations that are necessary for the development of at least some
traits. While Razeto-Barry and Frick grant that natural selection
cannot explain the origin of traits that arise by a single mutation,
they argue that it can explain the occurrence of sequences of
phenotypic changes that would otherwise be wildly unlikely to occur
without selection operating to cause the spread of the changes prior
to the final one in the sequence. Additionally, it is disputed whether
natural selection can explain why an arbitrary individual has the
traits it has (Walsh 1998; Pust 2004; Birch 2012). On the positive
view, selection affects the identity of individual organisms, and it
is part of the identity of an individual to have been produced by the
parents that produced it, so natural selection explains why
individuals have the traits they do. On the negative view, the
explanatory scope of natural selection is limited to population level
properties. Razeto-Barry and Frick further consider the question of
whether natural selection can explain the existence of individuals,
ultimately arguing against it.


### 4.2 Challenges to Explanatoriness



The capacity for natural selection to explain has come under fire from
several directions. An important challenge to the explanatory power of
selection was made by Gould and Lewontin who impugn the veracity of
selectionist accounts of the origin of traits whose correctness is
pinned on nothing more than their plausibility and capacity to explain
(Gould & Lewontin 1979: 588). Gould and Lewontin offer several
examples of traits for which such "just so" stories are
easy to construct but nonetheless taken to be true on the basis of
little or no evidence beyond the plausibility of the accounts and
their consistency with the theory. Lennox argues that, when not taken
as even purportedly true, "just so stories" can play a
genuine, if modest, role in the evaluation of scientific theories as
thought experiments used to assess the explanatory potential of the
theory (see also Mayr 1983; Lennox 1991; entry on
 adaptationism).
 In the debate over Darwin's theory, thought experiments of this
sort were deployed by both Darwin and his critics in ways both
legitimate and rhetorically effective (Lennox 1991).



Another attack, or set of attacks, on the ability of natural selection
to explain have to do with the threat that selectionist explanations
are circular. This sort of problem with a would-be explanation arises
when one must already know what one means to explain in order to
construct one's (thereby failed) explanation of it. This problem
surfaces, in different ways, in the contexts of Lewontin's
requirements, type recursions, and the Price Equation.



Lewontin's requirements for evolution by natural selection
contain an ambiguity. "Fitness" could refer either to
offspring number or to fitness variables of the sort found in the
genotypic selection equation above (the \(w\)'s).
Godfrey-Smith, in noting that some formalisms do not feature fitness
variables at all, is using the term in the latter sense (2007: 496),
while Okasha's treatment of Lewontin's requirements uses
fitness in the former sense. Suppose fitness means offspring number
and suppose further that the requirements play the role of determining
under what circumstances evolutionary theory may be deployed. If these
things hold, then the circularity problem arises: we must know that
the variants in the system have different numbers of offspring in
order to apply Lewontin's requirements to deploy the theory that
would explain the very same offspring production despite such
production serving to license the theory's deployment in the
first place.



Consider type recursions next. Suppose we must know actual
reproduction rates to assign relative fitnesses values in type
recursions. Were this so, an alleged explanation of the extent of
evolutionary change in the system that makes crucial use of type
recursions would be circular. The circularity problem does not come up
for practicing biologists deploying type recursions, as those workers
rely upon fitness estimates that are inferred from statistical facts
about a target system during the estimation phase in order to assign
values to variables in type recursions that are then deployed over the
system during the projection phase. What such biologists mean to
explain, the behavior during the projection phase, is different from
what they use to construct the explanation, the behavior during the
estimation phase (Mills & Beatty 1979; Bouchard & Rosenberg
2004; the estimation/projection phase contrast is due to Glymour
2006).



That the scientists largely agree about the practice of statistical
estimation shows that they largely share some tacit concepts of
selection and fitness, ones it would be an advance for philosophers to
define. Philosophers have developed definitions of fitness. Brandon
was the first to defend the propensity interpretation of fitness and
it has earned much discussion since (Brandon 1978; Mills & Beatty
1979; Rosenberg 1982; Brandon & Beatty 1984; Brandon 1990; Sober
2000). Rosenberg and Bouchard have developed a recent definition of
fitness based on the relationships between an individual and its
environment that contribute to the individual's success
(Bouchard & Rosenberg 2004; for more on "fitness" see
entry on
 fitness).



Turning finally to how the tautology problem surfaces in the context
of the Price Equation, consider how that equation formally represents
the extent of evolution across some time period in which reproduction
occurs. Assigning values to its statistical functions requires
information that spans that time period, such as information about the
offspring number of the parents and the phenotypes of the offspring.
The deployment of the Price Equation does not involve any
estimation/projection period contrast. It cannot be used a source of
new information about some time period that remains otherwise
unexamined. The Price Equation could not, for instance, be used to
make a prediction about the dynamics of some system into the future in
the same way that type recursions can do. This is because "an
application of the Price equation for the purpose of prediction would
presuppose the very information you want to predict with it"
(Otsuka 2016: 466). For this reason, Otsuka claims that the equation
is not explanatory (2016: 466).



Another difficulty faced by selectionist explanations has to do with
their reliability. Glymour has argued against the reliability of
explanations constructed using type recursions: "population
genetics models evolving populations with the wrong variables related
by the wrong equations employing the wrong kinds of parameters"
(Glymour 2006: 371). Glymour demonstrates that exogenous fitness
variables cannot be used to quantify environmental causes of
reproduction that change in their influence during the projection
period. Fitness cannot be a kind of summary variable that abstracts
away from the causal details of the natural population while also
functioning to reliably predict and explain the dynamics of the
systems to which type recursions apply. Gildenhuys (Gildenhuys 2011)
provides a response to Glymour, acceding that his arguments are valid
but contending that there are a variety of ways to complexify type
recursions to formally handle the influence of environmental causes;
Glymour (2013) provides a reply.


### 4.3 Natural Selection as a Mechanism



Philosophers interested in explanation have recently turned their
attention to mechanisms: "mechanisms are sought to explain how a
phenomenon comes about or how some significant process works"
(Machamer, Darden, & Craver 2000: 2). One way to secure
explanatory status for selection would be to show that it functions as
a mechanism. Whether natural selection qualifies as a mechanism is
controversial. Skipper and Millstein gainsay the position: natural
selection not organized in the right sort of way to count as a
mechanism, exhibits a lack of productive continuity between stages,
and further lacks the requisite sort regularity of operation (Skipper
& Millstein 2005: 335). Barros (2008) has argued that natural
selection may be characterized as a two-level mechanism, with a
population-level mechanism and an individual-level mechanism working
together. Havstad (2011) responds that the account Barros offers is
too general and so includes any selective process, not just natural
selection. Moreover, the parts/entities and activities/interactions
that supposedly make-up the mechanism of selection can only be
specified by the roles in the selective process, whereas mechanistic
explanation works by associating these roles with specific phenomena
(Havstad 2011: 522-523). Matthews (2016) offers a case study of
the debate over selection as a mechanism, while DesAutels (2016)
provides a defense of a mechanistic view.


## 5. Causation



Godfrey-Smith makes the point that some character could fit
Lewontin's conditions for evolution by natural selection by mere
chance. Parents with a particular trait could have more offspring than
parents without it, even though the trait does not causally influence
offspring number (Godfrey-Smith 2007: 511). In such cases, Lewontin
would rule there is evolution by natural selection, provided that the
trait is heritable, since individuals with it have a *de facto*
higher rate of survival and reproduction, if only by chance. Equally,
a trait could spread by hitchhiking and meet Lewontin's
requirements for selection (or Okasha's for that matter). A
hitchhiking trait is correlated with one that causes reproduction, and
as a result, individuals with the trait exhibit increased
reproduction, despite the trait not causing reproduction at all. But,
for Godfrey-Smith, there is no natural selection, at least in the
focused sense, on the trait in these sorts of cases because
differences in the character had no causal role in producing variation
in offspring number (Godfrey-Smith 2007: 513).



As noted earlier, theorists working on type recursions who defend the
view that selection is a distinct quantity from drift typically defend
the view that it is also a distinct cause. Forber and Reisman
explicitly invoke the interventionist account of causation and argue
that natural selection (and drift) qualify as causes (Reisman &
Forber 2005). They do so on the basis of an examination of
experimental work in which selection (and drift) are manipulated to
produce changes in population-level behavior. "Causalists"
working with type recursions must confront the natural philosophical
challenge of stating, precisely, what cause or causes count as
selection, that is, what exactly fitness coefficients quantify. The
Beatty/Millstein interpretation of selection as discriminate sampling
is explicitly causal. Equally, Otsuka's position that fitness
coefficients should be interpreted as linear path coefficients from
phenotype to offspring number in causal graphs is clearly causal as
well.



The Price Equation may be deployed for some system without regard to
what causal relationships hold within it. Okasha argues that variants
of the equation may constitute causally adequate representations under
special circumstances. It is important to distinguish this contention
from the contention that the Price Equation is causally interpretable.
The right-hand side terms of the Price Equation are (algebraic
functions of) statistical functions, while, in causally interpretable
equations, causes are variables (Woodward 2003: 93). A causally
interpretable system of equations is one that one features causal
variables on its right-hand side and effects on the left: \(a = b\) is
causally interpretable if \(b\) causes \(a\) but not if the
reverse. Okasha does not call any formulation of the Price Equation
causally interpretable. However, the notion of causal adequacy he does
deploy seems to have been developed solely in the context
Okasha's discussion of the Price Equation and thus risks being
about causation in name only.



Okasha's basic approach is to assess whether some formulation of
the Price Equation as a causally adequate representation in terms of
whether the statistical functions in the equation take positive values
while their arguments are causally related (2016: 447). In a
multi-level, causally adequate version of the Price Equation,



\[
W\Delta P =\textrm{Cov}(W\_{k},P\_{k}) + \mathrm{E}\_{k}[\textrm{Cov}(w\_{jk},p\_{jk})]
\]


where



* \(W\) is average offspring number
* \(\Delta P\) is change in average character value
* \(W\_{k}\) is the average fitness of the individuals in the
\(k\)th group
* \(P\_{k}\) is the average character value of the individuals in the
\(k\)th group
* \(\mathrm{E}\_{k}\) is the average across groups
* \(w\_{jk}\) is the offspring number of the \(j\)th
individual in the \(k\)th group
* \(p\_{jk}\) is the character value of the \(j\)th
individual in the \(k\)th group




Okasha proposes a test for causal adequacy that works this way: if the
first term takes a positive value, then \(W\_{k}\) and \(P\_{k}\) must
be causally related, or else the equation is not causally adequate.
Equally, a version of the Price Equation is not causally adequate if
its statistical functions take a positive value despite their
components failing to be causally related, or if its statistical
functions take value zero when their components are causally related.
Whether a version of the Price Equation is causally adequate is thus a
function of the causal relationships among the variables in the
system. On this approach, the concept of causal adequacy means
something different than causal interpretability does for equations
that exhibit dependencies between variables that represent causes.


## 6. Conclusion



This entry initially distinguished two usages of "natural
selection", the focused and the capacious one, the focused one
picking out a single formal element of the process captured by the
capacious sense. There are some intermediate cases. Brandon offers
this definition of selection: selection is differential reproduction
that is due to differential adaptedness to a common selective
environment (Brandon 2005: 160). This definition does not attempt to
capture the entirety of Darwin's process: while it identifies
natural selection with a type of reproduction, there is no mention of
inheritance or replication. Repeat a process specified to fit
Brandon's definition as often as you like and we still might not
have something that makes adaptation or speciation explicable. But
neither is Brandon's definition of selection appropriate as an
interpretation of a theoretical term in any formalism. Instead, that
definition connects with an effort to develop a *Principle of
Natural Selection* which would serve a seminal role in a
qualitative version of Darwin's theory. The Principle is
formulated using probability talk along with the notion of fitness.
The historical development of the Principle of Natural Selection,
beginning with Brandon together with Beatty and Mills (Brandon 1978;
Mills & Beatty 1979), has been guided by debates over the
interpretation of probability and fitness addressed elsewhere (Brandon
1990: ch. 1; Sober 2000; Campbell & Robert 2005; entry on
 fitness).



"Natural selection" has been used to pick out one, or
multiple, or all the elements of a single cycle of the recursive
process that we learned from Darwin. Seemingly, there is arbitrariness
in how one decides to deploy the term natural selection, such that any
part of Darwin's recursive process could be treated as *the
natural selection part*. It is difficult to say that either
Brandon, or Okasha, or Otsuka, or Millstein is wrong in their
characterizations of selection, even though the characterizations are
superficially logically incompatible. The several definitions pick out
genuine elements of genuine processes, each with their own significant
theoretical importance.