## 1. Historical background



Conflicts between reductionist views and their dualist and pluralist
metaphysical rivals have loomed large in the history of philosophy.
The classical debate between materialism and mind-body dualism is
simply the most familiar of many such disputes. The Cartesian dualist
is committed to distinct types of substances--minds and bodies
each with its own essence. Different versions of
dualism--interactive, epiphenomenal, parallelist - disagree
about whether the two causally interact, but they all assert the
ontological difference between the mind and body. Neither reduces to
the other. They are distinct, and both are real.



Most contemporary reductionist as well as eliminativist positions
include some commitment to *materialism* or
*physicalism*--the view that the physical or material
provides the fundamental reductive base. However reductionism *per
se* is ontologically neutral. The concept of reduction entails no
specific ontological positions. It is logically independent of
physicalism or materialism. Indeed reductionism could be true of the
entities in a totally nonphysical world as long as it had some base
and everything in that world reduced to that base. Reduction is a
general relation between entities or theories that might hold in many
sorts of specific cases.



Some historical reductivists have been resolutely opposed to
materialism and physicalism. Bishop Berkeley's phenomenal
idealism reduced ordinary tables and chairs to collections of ideas
and denied the existence of matter. On Berkeley's reductive view
everything real reduces to minds and ideas. Idealism, as proposed by
Fichte can also be given a reductive interpretation. On this view, the
non-mental does not exist over and above the mental. The non-mental
can be fully explained in terms of the mental, and it can be
ultimately assimilated to the mental. According to this
interpretation, everything just *is* ultimately mental.
Idealists are reductionists whose reductive base is mental.



Idealism does not have many contemporary supporters, but there are
current reductive theories that are logically independent of
physicalism though consistent with it. Nominalism, for example, can be
regarded as a reductive theory about kinds. The nominalist argues that
the supposed reality of abstract objects--whether properties and
kinds or numbers and sets--can be reduced to facts about concrete
objects and our ways of talking about them. French structuralists can
be seen as reductionists about subjects, or subjectivism; they suggest
that the functioning of individual intentions and goals reduces to the
inner workings of larger discourses.



Carnap's mid-20th century unificationism presents an interesting
example of ontologically neutral, phenomenological reductionism.
Carnap's reductionism states that:



> 
> 
> ... science is a unity, [such] that all empirical statements can
> be expressed in a single language, all states of affairs are of one
> kind and are known by the same method. (Carnap 1934: 32)
> 
> 
> 



This interpretation covers several aspects--semantic ones
(concerning the expressive power of a language), metaphysical ones
(states of affairs being of *one kind*), and methodological
ones. Assume that any empirical statement, or proposition, can be
expressed in a single language. This language is, on Carnap's
view, a language that employs observational concepts only. This seems
to amount to an ontologically neutral form of reductionism. In
principle, any reductionist--the idealist as well as the
materialist--can subscribe to this thesis. Disagreement will
start only when it comes to specifying the ontological status of the
kinds designated by this language's observational expressions.
Are they external mind independent properties of objects or are they
phenomenal properties directly present in perceptual experience?



In sum, though most contemporary reductionists are also physicalists,
there are many actual reductionist positions that are independent of
materialism and physicalism. Such positions have played an important
role in the history of philosophy, and some are actively supported
today even as forms of *scientific* reduction.



The notion of reduction itself has become a target of philosophical
investigation. If science is to provide (at least partial) answers to
questions of reduction, such as whether the mental reduces to the
neural, then we need to understand what reduction requires. What are
the necessary conditions and how might scientific evidence show
whether they are fulfilled?



J.J.C. Smart provided an influential, tentative description of the
reduction relation according to which an entity *x* reduces to an
entity *y* only if *x* does not exist 'over and
above' *y* (Smart 1959: 143). Smart frames the question as
one of existence, an *ontological question*: What really
exists? How many basic kinds of things exist? Approaching the issue
from a different slant, the philosophical notion of
'reduction' might be cashed out in terms of an
*explanatory relation*. If *x* reduces to *y*, then
it can in a relevantly strong sense be explained in terms of *y*.
Take an example from Ernest Nagel. Describing the possible reduction
of headaches, he writes that when



> 
> 
> the detailed physical, chemical, and physiological conditions for the
> occurrence of headaches are ascertained ... an explanation will
> have been found for the occurrence of headaches (Nagel 1961: 366)
> 
> 
> 



Alternatively, reductions can be conceived of as consisting in a
specific form of *assimilation*--a description that can
also be found in Nagel (1961): 



> 
> 
> [In a reduction, a] set of distinctive traits of some subject matter
> is assimilated to what is patently a set of quite dissimilar traits.
> (Nagel 1961: 339f.)
> 
> 
> 



Reductivists are generally realists about the reduced phenomena and
their views are in that respect *conservative*. They are
committed to the reality of the reducing base and thus to the reality
of whatever reduces to that base. If thoughts reduce to brain states
and brain states are real, then so too are thoughts. Though
conservative realism is the norm, some reductionists take a more
anti-realist view. In such cases the reducing phenomena are taken to
replace the prior phenomena which are in turn *eliminated*. The
idea of mental illness as a type of psycho-neural disorder replaces
the idea of demon possession. Demons and their voices have no role or
reality in the new theory. The oxygen theory of combustion replaced
the phlogiston theory and phlogiston was eliminated.



Whether to count such *eliminativist* views as a variety of
reduction is a matter of theoretical choice. Some might argue
reduction entails realism about the reduced phenomena. If so,
elimination is not reduction. On the other hand, replacement has
sometimes been classed as a form of reduction (Kemeny & Oppenheim
1956), and it corresponds to an everyday sense in which theories are
described as 'reductive'. There is clearly a reductive
aspect to the eliminativist's claim that nothing is real other
than the replacing phenomena, e.g., that there is nothing real other
than brain states. Nonetheless most reductive views are realist, so
'reduction' hereafter should be understood in the
conservative sense that entails the realism of the reduced phenomena
unless otherwise noted.



Attempts to characterize the reduction-relation go back to the early
years of logical empiricism. Neurath (1931) and Carnap (1934)
described reduction as a relation of translation, or at least of
unification by the universal requirement that all statements have
testable consequences that can be stated in a shared observation
language.



In more recent decades two closely related strands of debate have
focused on the nature of reduction, one within the philosophy of
science and the other within the philosophy of mind. Philosophers of
science have proposed several explications of the notion of reduction,
often with an eye on broader conceptions of explanation, scientific
change, and unification. Philosophers of mind concerned with specific
questions about whether the mental just *is* physiological or
even physical in nature have asked what notions of reduction might be
relevant and how they might apply to particular cases in psychology
and neuroscience. The former philosophy of science strand is the topic
of
 section 2,
 and the philosophy of mind debate is discussed in
 section 3.


## 2. Models of scientific reduction in the philosophy of science



The philosophical discussion of reduction-relations in the sciences
was inspired by cases such as the alleged reduction of Newtonian
mechanics to relativity theory, of chemistry to atomic physics, and of
gas laws to statistical mechanics. However, in what respects these
actual episodes of theory change count as reductions is a complex
matter. Many factors might be relevant, not merely logical and
metaphysical relations but also epistemic, psychological, and
pragmatic ones. Questions also arose about the generalizability of
these cases and their role in scientific progress or change, which led
to the idea that an appropriate model of reduction should mainly focus
on cases of actual theory succession. The main target in the reduction
debate within the philosophy of science, however, was not a
reconstruction of the process of scientific change, but rather an
explication of the relation between successive theories that justified
the relevant change, i.e., relations alleged to hold between pairs of
theories such as Newtonian mechanics and relativity theory, or
thermodynamics and statistical mechanics. In this context, the relata
of reductions were conceived of as theories or models, rather than
properties, events, or kinds. Thus, they are often referred to as
'*theory* *reductions*'.


### 2.1 Reduction and theory-succession



The term 'reduction' oscillates between two different uses
(at least). Kemeny and Oppenheim (1956) regard the idea that
reductions somehow relate to scientific progress as being crucial for
understanding reduction:



> 
> 
> The label 'reduction' has been applied to a certain type
> of progress in science. As this process has been the subject of much
> philosophical controversy, it is the task of the philosophers of
> science to give a rational reconstruction of the essential features of
> reduction. [...] Scientific progress may broadly be divided into
> two types: (1) an increase in factual knowledge, by the addition to
> the total amount of scientific observations; (2) an improvement in the
> body of theories, which is designed to explain the known facts and to
> predict the outcome of future observations. An especially important
> case of the second type is the replacement of an accepted theory (or
> body of theories) by a new theory (or body of theories) which is in
> some sense superior to it. Reduction is an improvement in this sense.
> (Kemeny & Oppenheim 1956: 6f.)
> 
> 
> 



According to this use of the term, reduction is a relation of actual
theory-succession, such that reductions happen at a certain time,
namely, when scientists did bring it about. Often, philosophers employ
the term 'reduction' in this sense, thereby designating an
activity of scientists, or the outcome of such an activity. Whether or
not a theory actually becomes the successor of another theory will
depend not only on logical and metaphysical features of the respective
theories, but also on pragmatic and epistemic aspects. The most
relevant of these aspects are to be captured in a rational
reconstruction, which plays a major role in some models of
reduction.



When discussing the status of bridge-principles (intuitively,
statements that connect the vocabulary of the reducing theory to the
reduced theory), Ernest Nagel suggested that they might play different
roles in the *discovery* of reductions than in after the fact
analyses. They may function as hypotheses to be tested, or they may
rely on empirical evidence already gathered. Similarly, Kenneth
Schaffner suggests that an appropriate model of reduction should take
into account not only logical or metaphysical aspects regarding the
reduced and the reducing theory. Rather, it should also be sensitive
to pragmatic and epistemic aspects underlying the actions of
scientists who carry out reductions (Schaffner 1993: 515f.)--an
idea he labels 'logical pragmatism'. New Wave
reductionists criticized other models, especially Nagel's, for
not being adequately sensitive to how reductions are actually carried
out, a point also made by Ager, Aronson, and Weingard (1974).



The distinction between reduction as an activity and reduction as a
relation that is independent of what scientists actually do is
tentatively captured by the distinction between *diachronic*
and *synchronic reduction*.



The main idea of diachronic reduction is not merely that reduction is
a *temporal* affair but that they consist in the replacement of
one theory by another theory, such that one theory (the reducing one)
becomes the successor of the reduced theory (Dupre 1993;
Rosenberg 2006). It is also often described as a sort of
theory-improvement or scientific progress: particularly one in which
the laws of the prior theory apply to a proper subset of the cases
covered by the laws of the succeeding theory (Dizadji Bahmani et al.
2010).



The notion of reduction as a scientific activity, or as an event of
scientific change, is discussed in the entries
 scientific revolutions
 and
 scientific progress.
 The remainder of this entry will deal with "synchronic"
aspects of reduction.


### 2.2 Nagelian models of reduction



Recent reduction debates in the philosophy of science were initiated
by Ernest Nagel's model of theory reduction (Nagel 1949, 1961,
1970), which has also received considerable attention in the
philosophy of mind (see, e.g., Fodor 1981: 150; Kim 1993: 150, 248).
More recent approaches to reduction depart from or were developed in
opposition to the Nagel model (Hooker 1981; Churchland 1985; Schaffner
1993; Bickle 1998, 2003; Dizadji-Bahmani, Frigg, & Hartmann 2010;
van Riel 2014), though it has been argued that most of these
approaches merely echo the Nagel model instead of proposing
fundamentally new interpretations (Endicott 1998, 2001; Dizadji
Bahmani et al. 2010).


#### 2.2.1 Nagel's model



Nagel describes his model as follows:



> 
> 
> A reduction is effected when the experimental laws of the secondary
> science (and if it has an adequate theory, its theory as well) are
> shown to be the logical consequences of the theoretical assumptions
> (inclusive of the coordinating definitions) of the primary science.
> (Nagel 1961: 352)
> 
> 
> 



The basic idea is simple: a theory *TR* reduces to a theory
*TB* if and only if *TR* is derivable from *TB* with
the possible help of the relevant bridge laws (here labeled
'coordinating definitions'), often with an emphasis of the
derivation of the laws of the reduced theory. If we add the remarks
Nagel opened his discussion on reduction with--namely, that
reduction has to be understood as a certain kind of explanation (1961:
338)--the core idea of the Nagel model is fully characterized.
Adding Nagel's idea of reduction as a kind of explanation, the
so called "Nagel model of reduction" can be fully
specified as follows: Reduction is (i) a kind of explanation relation,
which (ii) holds between two theories if and only if (iii) one of
these theories is derivable from the other, (iv) with the help of
bridge laws under some conditions. The basic model covers two sorts of
reduction, one in which bridge laws are not required (homogeneous
cases) and one in which they are (nonhomogeneous cases; for a
presentation of homogeneous cases of reductions and the question of
whether or not alleged cases of reductions really should count as
reductions in the Nagelian sense, see the entry on
 intertheory relations in physics).
 Nagel conceives of sciences or theories as developing entities that
undergo changes, across which their vocabulary remains unchanged
(though it is, presumably, sometimes extended). These successive
states of theories are covered by the notion of homogeneous
reductions--deduction of an early stage from a later stage of a
theory is possible without bridge laws since they share a common
vocabulary. Nonhomogeneous cases of reduction hold between pairs of
different theories, employing different vocabularies. Whereas the
former variant of reduction did not attract much attention (by Nagel
and others), the latter has been a subject of intense discussion since
Nagel introduced it in 1949.


#### 2.2.2 Direct and indirect reduction



The Nagel model of reduction parallels the so called "deductive
nomological model" (DN-model) of explanation according to which
explaining some fact involves showing how its truth can be deduced
from a set of other truths containing at least some laws. For example
according to the DN model, one explains the acceleration of a falling
body by showing how it can be deduced from Newton's laws. The
Nagel model of reduction thus seems to suggest that a reduced theory
is *directly* explained by the corresponding reducing
theory.



In this sense, Nagel reduction is sometimes (cf. Schaffner 1967: 137,
1993: 423) interpreted as a kind of *direct* reduction, in
contrast with *indirect reductions*--those that hold in
virtue of the fact that the reducing theory explains the occurrence of
the phenomena of the reduced theory (rather than the reduced theory
itself)--a view associated with Kemeny and Oppenheim (1956) and
Friedman (1982), and captured as one variant of reduction by
Schaffner's model (Schaffner 1993).



However, some have argued that the distinction itself may not be
viable (van Riel 2011). First, it is not clear what it is to explain a
theory (except for explaining it to, say, a student). Second, the
examples Nagel discusses suggest that he believed his model to be
*indirect*--an "explanation" of a theory in
terms of derivation from a reducing theory amounts to using the
reducing theory to explain the phenomena dealt with by the reduced
theory.


#### 2.2.3 Bridge principles



Nagel's major (and perhaps most controversial) contribution was
the introduction of bridge principles. The notion of bridge
principles, (also: 'bridge laws' and 'coordinating
definitions') can be spelled out in several different ways. The
most common way in accord with Nagel (1961: chap. 11, Sec. II.3) is to
describe them syntactically as bi-conditionals linking terms in the
vocabularies of the two theories. In the same context, however, Nagel
describes them in terms of the 'nature of the linkages
postulated' (Nagel 1961: 354). He distinguishes three such
linkages:


* The links mimicked by the bridge laws are 'logical
connections' (1961: 354), which are understood as meaning
connections.
* The links postulated by bridge laws are conventions or
stipulations ('deliberate fiat'; 1961: 354).
* The links postulated by bridge laws are 'factual or
material' (1961: 354); that is, bridge laws state empirical
facts (these truths are then described as empirical hypotheses).



These alternatives characterize bridge laws semantically in ways that
seem more fundamental than their syntactic status and role in
derivation. There is a gap between characterizing a sentence as a
bi-conditional and as postulating the relevant sort of link.
Bi-conditionals in themselves can express nothing other than truth
functional relations, and real bridge laws cannot be defined in purely
syntactic terms. Bi-conditionals are rendered true by a great variety
of "links" (but 'being rendered true by' is
not to be conflated with 'postulating'): contingent
co-instantiation or co-occurrence is not enough for reduction, even
though it is enough for some bi-conditionals to hold (and, therefore,
for some pairs of--possible--theories to instantiate the
official Nagel model). Therefore, logical form (in first-order
predicate logic) is not what we should focus on when we consider the
syntactic nature of bridge laws. But what, then, is the relevant
characterization? This is a matter of dispute among Nagel interpreters
and those who refined the Nagel-model. Some tend to suggest weaker
notions, such as (nomologically necessary) co-instantiation or
co-relation (Dizadji Bahmani et al. 2010, also discussed in Klein 2009
and Kim 1998: 90, 2008), whereas others suggest stronger
interpretations in terms of nomic conections or identities (Sklar
1967; Schaffner 1993; Esfeld & Sachse 2007; van Riel 2011).



In Nagel 1970, he explicitly states that bridge-laws state identities
of the properties signified by the relevant predicates or nomic
relations among extensions. If bridge laws should thus be conceived of
as stating identities or relations among the relevant terms'
extensions, then clearly reduction on such a view incorporates
essential reference to the theories' ontologies and is more than
just a two-place relation holding among theories.



Consider the following passage:



> 
> 
> In ... cases [of reduction in which the vocabularies of the two
> relata of the reduction relation differ], the distinctive traits that
> are the subject matter of the [reduced] science fall into the province
> of a theory that may have been initially designed for handling
> qualitatively different materials.... The [reducing] science thus
> seems to wipe out familiar distinctions as spurious, and appears to
> maintain that what are prima facie indisputably different traits of
> things are really identical. (Nagel 1961: 340)
> 
> 
> 



Thus, Nagel-reduction is a relation that holds not just between
theories but also between their ontologies. Sometimes, however, the
Nagel model has been characterized as an *epistemological*
model of reduction, because it is a model of theory-reduction (Sarkar
1992; Hoyningen-Huene 1989; Silberstein 2002), because it is a model
of explanatory reduction (Sarkar 1992), or because bridge-laws are to
be interpreted epistemologically (Fazekas 2009, following Klein
2009).


#### 2.2.4 Arbitrary and non-arbitrary reductions



Finally, it is worth noticing that Nagel introduces a distinction
between arbitrary and non-arbitrary reductions: These criteria are
"non-formal" (1961: chap. 11, Sec. III). The first
criterion Nagel mentions is that the premises of a reduction--the
bridge laws and the reducing theory--should be well established
rather than arbitrarily chosen (1961: 358). Second, Nagel alludes to
the fact that the reducing theory should be better established than
the reduced one. The third criterion states that reduction is
concerned with unification. The fourth criterion states that the
reducing theory corrects and augments the reduced theory.



Ideas similar to those of Nagel have been pursued, amongst others, by
Robert Causey and Lawrence Sklar (Sklar 1967). Causey's model of
microreductions suggests that reductions are based in cross-theoretic
identities and are, in a sense, decompositional (Causey 1972a,b).
Kenneth Schaffner explicitly built upon Nagel's model, extending
it especially to cover cases of correction and replacement.


#### 2.2.5 Reduction and replacement



Kenneth Schaffner explicitly built upon Nagel's model, with a
special focus on extending it to cover cases of correction and
replacement. Appealing to the facts of actual scientific theory
succession, Schaffner proposed an alternative model of reduction
(Schaffner 1993: chapter 9; earlier versions can be found in his 1967,
1974, and 1976) based on the following main idea: If we want our
definition to cover the relevant sort of cases of actual scientific
developments, we must not focus solely on straightforward reductions;
mere replacements and cases of theory-correction should be
incorporated as well because (most, or at least a considerable number
of) actual theory-successions do not fit into the picture of
reductions in the more narrow sense (see, for example, Schaffner 1993:
426, 427f).



It thereby reflects the most powerful criticism that has been raised
against the Nagel-model (see
 Section 5.3).
 Schaffner proposes a complex definition comprising several disjuncts,
which are intended to encompass both genuine reductions and different
sorts of replacement ranging from complete replacements via a
continuum of relations to cases of correction fully within the reduced
theory. Schaffner notes in many cases what gets strictly reduced in
intermediate cases is not the original theory but a corrected version
of that theory constructed within the vocabulary of the reducing
theory.


### 2.3 Problems for Nagelian models of reduction



Many criticisms have been raised against both the original Nagel model
and its variants. The original Nagel model was faulted as too narrow
because it allows only for theory reduction (Wimsatt 1972; Hull 1976;
Darden & Maull 1977: 43; Sarkar 1992), whereas an appropriate
model would cover cases of reduction of mere models and the
like--sciences like biology and neuroscience should be regarded
as being possible candidates for reduction, although they do not
contain full-fledged theories (see also the entry on
 reductionism in biology;
 for a discussion of this and the following criticisms, see van Riel
2011).



In a more general sense, the Nagel model has been criticized as
exemplifying all the shortcomings of the orthodox view on science. For
example, it conceives of theories as syntactic entities, and it views
reduction as explanation cashed out in terms of the DN model (Hempel
& Oppenheim 1948), which has itself been challenged on many
grounds, especially those regarding the asymmetry of explanation (for
an overview that focuses on problems arising from reduction as
explanation, see Craver (2007: chap. 2), and for problems concerning
the DN model, see Salmon 1989).



The Nagel model describes reduction in terms of direct theory
explanation, whereas an appropriate model of reduction should shape
the notion in terms of indirect theory explanation, that is, in terms
of explanation of the phenomena of a theory (Kemeny & Oppenheim
1956; Schaffner 1967; Friedman 1982).



Three formal worries of the Nagel model merit mention: if reduction is
derivation plus (sometimes) bridge laws, then any theory would reduce
to itself (because any theory is derivable from itself); moreover, any
theory would reduce to any inconsistent theory; and contrary to what
one might expect, reduction would not turn out to be an asymmetric
relation--derivability does not entail asymmetry.



Feyerabend (1962, 1966), Churchland (1986), Schaffner (1993), and
Bickle (1998), amongst others, have argued that Nagel's model
allows neither for correction in the reducing theory nor for
replacement, and that, therefore, it is fundamentally misguided.
Recall, however, that Nagel suggests that at least some interesting
reductions involve correction. He writes that:



> 
> 
> [t]he [reducing] theory must also be fertile in usable suggestions for
> developing the secondary science, and must yield theorems referring to
> the latter's subject matter which augment or correct its
> currently accepted body of laws. (Nagel 1961: 360)
> 
> 
> 



At first sight, it is not clear how correction is possible if
deduction is the basis for reduction. The truth of the reducing
theories entails the truth of the reduced theories. This is what
deduction is about. So, it is worth noting that Nagel introduced this
criterion, even though it seems hardly compatible with his official
model. However, as suggested by Putnam (1965: 206ff., esp. n. 3),
there are a number of ways to incorporate correction within
Nagel's model: by speaking of approximate truth, by
contextualizing the reduced theory, or by introducing probabilistic
notions. Moreover, in discussing Feyerabend's criticisms based
on the alleged incommensurability of expressions in different
sciences, Nagel explicitly stated that reductions may rely on
approximations (1970: 120-21, 133). A similar idea seems tied to
Nagel's suggestion that some reductions may require boundary
conditions (Nagel 1961: 434) in order to connect the relevant kind
terms in an appropriate bridge law, thereby altering and adjusting one
of the terms' extensions. As has been pointed out by Endicott
(1998), any account that allows for boundary conditions, correction,
approximation and the like (such as the account proposed by Schaffner)
will be able to cover such cases. Kemeny and Oppenheim suggest that
idealization may be justified in developing models of reduction,
noting that without idealization things would become "hopelessly
complex" (Kemeny & Oppenheim 1956: 13).



Finally, Nagel's model has been criticized for the role it
attributes to bridge-laws. New Wave reductionists in particular have
argued that bridge-laws do not play the role they are supposed to play
in discovering reduction-relations. Contrary to Nagel, we do not use
bridge-laws when discovering for example that parts of chemistry
reduce to atomic physics. Rather, we infer bridge-laws from the
discovery of relevant similarities between the two theories (for the
explanatory role of bridge-laws and identity statements, see also
Marras (2005: 351) and Block & Stalnaker (1999: 28)).



In response to those critiques, some, like Schaffner and
Dizadji-Bahmani et al. (2010) and Butterfield (2011a&b) have
developed variants of the Nagel model that retain its spirit but avoid
many of the problems faced by the original. Prominent suggestions
concern refinements of the logical framework in which Nagelian
reduction is described (Dizadji-Bahmani, Frigg, and Hartmann (2010)
and Butterfield (2011a)) as well as a relaxation of the condition that
reductions must be complete. The idea is that even in physics,
reductions are "most likely partial, and thus not fully
systematic" (Schaffner 2012: 559). As Schaffner observed, even
"almost sweeping reductions" (Schaffner 2006: 379) in
physics, such as the reduction of physical optics by Maxwell's
electromagnetic theory, fail at some point. Partial reductions are
conceived of as "largely completed reductions containing
exceptions or failures of attempted reductions" (Schaffner 2012:
563). An interpretation of partial reductions for sciences such as
biology has been developed in Schaffner (2006) and further developed
by Winther (2009). His suggestion heavily draws on explanatory notions
and mereological relations: In a partial reduction, the objects
referred to in the explanans are parts, or a microstructure of the
phenomenon to be explained. The phenomenon to be explained is a
macro-object. The microstructure needs to be linked via
"connectability assumptions" to the macro-object. These
assumptions may suggest causal links or, in more straightforward
cases, identity. Such explanations may turn out to be partial for
various reasons: For example, they often do not explain all cases of
the target phenomenon, the reduction base may span levels of
organization, and they may fall short of straightforward derivations
of the target phenomenon. Such partial reductions share important
features with mechanistic explanations; for example, they are local,
postulate a causal mechanism that is responsible for the occurrence of
the target phenomenon, refer to constituents in the explanation, and
may span different levels.



Partial reductions thus allow for emergence; we get a partial
reduction only, some aspects of the target phenomenon are emergent. On
Schaffner's interpretation, this sort of emergence is purely
pragmatic (Schaffner 2006: 382).



Others have rejected the Nagel model and used different frameworks to
account for reduction. The most relevant alternatives will be
discussed in the next sections.


### 2.4 Structuralist models of reduction



One alleged problem of Nagelian models is that they start from a
mistaken background view of science in that they conceive of theories
as syntactic objects and rely on dubious notions about the nature of
laws and theoretical terms. Structuralists tried to overcome these
alleged problems (as well as others, such as the problem of
incommensurability) by revising the notion of a theory.



Part of the classical structuralist program consists in the idea that
philosophy of science, properly construed, pursues an ideal of
rational reconstruction of science (cf. Stegmuller 1979; Balzer
1984: 331), which is to be understood as a reconstruction of actual
scientific developments. Since it is common knowledge that cases of
straightforward reductions are rare, if they exist at all,
structuralists focused primarily on cases of mere replacement or on
similarities between theories, which are cashed out in terms of
approximation or (semantic counterparts of) derivation under ideal
conditions (Moulines 1984; Scheibe 1999; Stegmuller 1979 &
1986: ch. 4).



According to the structuralist picture (Sneed 1971; further developed
in: Stegmuller 1979), to give an account of a theory *T* is
to define the predicate '\_is a (model of) *T*' in
set-theoretical terms. A theory basically consists of a set *M*,
a set of potential models *MP*, a set of potential partial models
*MPP* and a set of intended applications *I*, which is
supposed to be "open", or non-extensionally individuated.
The relation between the different variants or stages of *I* is,
as Stegmuller suggests, best thought of as being pragmatically
fixed (Stegmuller 1979). Building on such a notion of a theory,
Patrick Suppes proposes the following characterization of inter-theory
reduction:



> 
> 
> Many of the problems formulated in connection with the question of
> reducing one science to another may be formulated as a series of
> problems using the notion of a representation theorem for the models
> of a theory. For instance, the thesis that psychology may be reduced
> to physiology would be for many people appropriately established if
> one could show that, for any model of a psychological theory, it was
> possible to construct an isomorphic model within physiological theory.
> (Suppes 1967: 59)
> 
> 
> 



These set-theoretical structures are then used to define a reduction
relation in terms of mapping functions from one structure to another.
It is worth noting, however, that structuralists believed their models
of reduction to be advantageous *because* they believed their
concept of a theory to be more appropriate. The general status of
structuralism is an important factor in assessing the problems these
models of reduction seem to face.


### 2.5 Problems for structuralist models of reduction



Schaffner has argued that the structuralist notion of reduction is too
weak a notion to be useful, and that (official) Nagelian reduction is
a special case of Suppes' model. That is, whenever there is a
Nagel-reduction, then there is a Suppes-reduction. Moreover, according
to Schaffner, Suppes' model covers cases that should not be
covered, namely, arbitrary reductions (Schaffner 1967). Similar
problems arise for successors of Suppes' model that try to cash
out reduction in terms of an isomorphism or an analogy-relation, which
have been developed and refined by Sneed (1971) and Stegmuller
(1979). The basic problem of these accounts has been described by
Moulines (1984: 55) and a similar idea can be found in Hoering (1984).
Moulines puts it as follows:



> 
> 
> There is at least one further aspect of reduction that is overlooked
> by [the structuralist conception]. This is what I would like to call
> "the ontological aspect". I wish to argue that, for a
> complete picture of a reductive relationship between two theories, one
> has to take into account some sort of relation between the respective
> domains. Otherwise, when confronted with a particular example of a
> reductive pair, we would feel that all we have is an ad hoc
> mathematical relationship between two sets of structures, perhaps by
> chance having the mathematical properties we require of reduction but
> not really telling something about "the world". We could
> have a reductive relationship between two theories that are completely
> alien to each other. (Moulines 1984: 55)
> 
> 
> 



Thus, the problem is that mere structural similarity does not
guarantee that both theories are actually concerned with the same
phenomena, or the same properties. This is a problem Nagelian models
do not face if they describe bridge-laws in semantic terms, i.e., in
terms of extensional or intensional relations among the relevant
predicates or kind terms. Relatively independent of structuralist
models, philosophers who work at the interface of the philosophy of
mind and the philosophy of science have developed accounts that later
have been labeled 'New Wave' reductionism.


### 2.6 "New Wave"-models of reduction



Relatively independent of structuralist models, philosophers who work
at the interface of the philosophy of mind and the philosophy of
science have developed accounts that have come to be known as
'New Wave' reductionism.



New Wave reductionists take it that their models give an appropriate
reconstruction of a specific kind of scientific development
(Churchland 1986: 279ff.; Bickle 1998: chapter 1). At the same time,
however, this movement is associated with a reductionist metaphysical
position. Arguing that neuroscience reduces or eliminates psychology,
some protagonists try to show that we should embrace an eliminative
form of reductionism about the mind (e.g., Paul Churchland 1981, 1985;
Patricia Churchland 1986; Bickle 1998). Thus, there is a tendency to
continue debates originating from the philosophy of mind tradition.
The so called New Wave model (or better: a family of similar models),
which was developed in a series of articles and books by Clifford
Hooker (1981), Paul Churchland (1979, 1985) and more recently John
Bickle (esp. 1998), builds on some aspects of Schaffner's
current model (1993) and its earlier versions (Schaffner 1967, 1974,
1976). Here is a formulation of Hooker's definition:



> 
> 
> Within *TB* construct an analog, *T\*R*, of *TR* under
> certain conditions CR such that *TB* and *CR* entail
> *T\*R* and argue that the analog relation, *AR*, between
> *T\*R* and *TR* warrants claiming (some kind of) reduction
> relation, *R*, between *TR* and *TB.* Thus (*TB*
> & *CR* - *T\*R*) and (*T\*R* *AR*
> *TR*) warrants (*TB* *R* *TR*). (Hooker 1981:
> 49)
> 
> 
> 



The conditions, *CR*, will consist of limiting assumptions and
boundary conditions that guarantee that if *TB* is more
comprehensive than TR, the application of elements of
*TB*'s vocabulary is restricted to the domain relevant for
*TR.* The idea is to effect reductions that include corrections
by working on the reducing theory (here '*TB*') and
not by working with corrections formulated in the vocabulary of the
reduced theory. Ronald Endicott (1998: 56) identifies four core
aspects of New Wave reduction: (i) "New Wave
construction": the corrected version of the reduced theory is
constructed within the vocabulary of the reducing theory; (ii)
"New Wave deduction": this corrected version is deduced
from the reducing theory; (iii) "New Wave relation":
analogical relations play the role bridge-laws play within Nagelian
models; (iv) "New Wave continuum": these analogical
relations may differ in degree of strength--reductions based on
strong analogical relations justify a conservative stand towards the
(ontology of) the reduced science, whereas reductions based on weak
analogical relations justify elimination. Identity (a limiting case
for the continuum), for example, ensures that the ontology of the
reduced theory is preserved by the reducing theory.



So, what is the basic refinement of New Wave reduction? According to
Bickle (1998: 29 & 1992: 224) and Paul Churchland (1985: 11),
avoiding reference to bridge-principles has a great epistemological
advantage: The epistemological virtue of reduction does not depend
upon knowledge of bridge-principles. Rather, comparison of reduced and
reducing theories justifies identity claims.



New Wave reductionists typically allow for a great variety of
relations backing up the analogical relation established between
*TR* and *TR\**. In New Wave reduction, bridge-laws are
replaced by ordered pairs of elements of the descriptive parts of the
vocabularies of the two theories, which enable us to judge the degree
of similarity between the derived image (which is formulated in the
language of the reducing theory) and the reduced theory (Bickle 1992:
223). Reduction is associated with a space of theory-relations ranging
from "perfectly smooth" or "retentive"
reductions to "bumpy" reductions, which are best
understood as mere replacements (Bickle 1992: 223; Hooker 1981: 45).
Bickle describes smooth reductions as follows:
('*IN*' designates the derived image and
'*TO*' designates the reduced or the
"old" theory):



> 
> 
> In cases lying at or near the retentive endpoint, the *IN* is the
> exactly equipotent isomorphic image of the *TO*, and no
> counterfactual limiting assumptions or boundary conditions are
> required for the derivation of *IN.* (Bickle 1992: 223)
> 
> 
> 



In this case, the pairing of the terms corresponds to identity of
referents (Bickle 1992: 224) and *IN* can be directly obtained
from *TR* (the reducing theory), whereas in cases on the opposite
point of the spectrum, pairing is achieved only by reference to
counterfactual limiting assumptions and boundary conditions. In this
case, ontology is eliminated, as in the reduction of phlogiston-theory
case.


### 2.7 Problems for "New Wave"-models of reduction



The main criticism raised against New Wave reductionism has been that
it either misses the target of reduction, or that it is not new at
all. Ronald Endicott argues that New Wave reductionists fail to
recognize the important distinction between *mere* replacements
on the one hand (that correspond to eliminativism, rather than
reductionism), and proper reductions on the other (Endicott 1998,
2001). Insofar as New Wave reductionists conceive of reduction in
terms of replacement, they just miss the target. If they conceive of
reduction in terms of smooth reductions, then, basically, they repeat
the Nagelian model in new terminology. A similar point has been made
by Dizadji Bahmani et al. (2010). The authors point out that,
especially when Schaffner's model is conceived of as being
basically Nagelian in spirit, New Wave reductionism just is Nagelian
reduction.



Endicott as well as Dizadji Bahmani et al. (2010) argue that the sole
difference between the classical Nagel-model and New Wave reduction is
that New Wave reductionists highlight the procedural aspect of how
reductions are carried out--or of how scientists may argue that
reduction-relations are instantiated, whereas Nagel was mostly
concerned with questions regarding the relation that is so discovered.
How scientists may arrive the truth of bridge-principles is a question
different from the question as to the nature of bridge-principles.


### 2.8 Reduction and mechanisms



The recent debate on mechanistic explanation took its point of
departure in reflections on sciences other than physics. Most
proponents of a mechanistic turn in the philosophy of explanation
oppose a number of mainstream interpretations of core notions in the
philosophy of science, especially that of a scientific explanation, a
law, the layer-cake model of a hierarchy of sciences (Wimsatt 1976),
and the notion of a theory. Most often, mechanistic explanation is
characterized as (i) being related to the use of 'by'
rather than 'because' (Craver 2007), (ii) being such that
the functioning of a complex entity is explained in terms of the
coordinated activation of its constituents, (iii) by being tied to a
hierarchical order of local mechanisms, whose constituents can
themselves become the target of a mechanistic explanation, and (iv) by
the relevance of the notion of organization--mechanisms are
organized to do something. This is how Craver puts it:



> 
> 
> Mechanistic explanations are constitutive or componential
> explanations: they explain the behavior of the mechanism as a whole in
> terms of the organized activities and interactions of its components.
> (Craver 2007: 128)
> 
> 
> 



A number of aspects concerning the relation between reduction and
mechanistic explanation are dealt with in the entry on
 reductionism in biology.
 Thus, suffice it to say that some philosophers, like Craver, reject a
reductionist interpretation of mechanistic explanation (Craver 2007),
whereas others seem to suggest that a conservative form of
reductionism may be compatible with mechanistic explanation (Bechtel
2007 and, especially, Schaffner 2006). Interestingly, the example of
the reduction of headaches as described by Nagel (see above,
 Section 1)
 seems to suggest that he conceives of reductive explanation as
mechanistic explanation. This is consonant with his first publication
on reduction: In his early paper on reduction (Nagel 1935), he
describes reduction as a constitution relation. Van Riel (2014)
suggests that reductive explanation is a cognate of mechanistic
explanation.


### 2.9 Reduction and causation



Mechanistic explanation is non-causal, and so is reductive
explanation. However, sometimes, reductionist models are phrased in
causal terminology. Enc (1976) is a case in point. He describes
reductive explanation and reductive dependence as causal. The
description he gives of this relation, however, does not fit the usual
use of the term 'causation'. Enc describes it as a
'generative' relation that may hold between an object
*a* and an object *b* even if *a*=*b.* Hence,
despite the fact that he uses the term 'causation', he
should not be interpreted as stating that literally, reductive
explanation is causal explanation. This has been overlooked by
Schaffner (1993) who reflects upon Enc's description,
criticizing him for assuming that reduction is causation.


## 3. Models of scientific reduction in the philosophy of mind



Reductionist positions have gained attention within the philosophy of
mind in four main respects. Firstly, type-identity theory was proposed
as a reductionist position about the mind. Secondly, token-identity
theories are sometimes conceived of as being reductionist in spirit as
well. Thirdly, models of functional reduction have been intensively
discussed. And, finally and corresponding to the previous
distinctions, there are different conceptions of reduction that are
discussed and rejected by non-reductive physicalism.


### 3.1 Type-identity theories



Type-identity theory claims that for types of some domain
*D*1, (e.g., psychology) there are types of a domain
*D*2 (e.g., neuroscience) such that for any member
*x* of *D*1, there is a member *y* of
*D*2 such that *x*=*y*. This aspect is
revealed in the theory's name--it is *type-identity*
theory. It is not just the items falling under the types that are said
to be identical but also the types themselves; e.g., being a pain is
said to be the very same type as being a certain pattern of neural
firing. However, just to claim that for any type *x* of some
domain *D*1, there is a type *y* of a domain
*D*2 such that *x*=*y* does not make you a
type-identity theorist. Rather, it is a specific reductionist as well
as a naturalist stance that accompany these identity-claims that make
them proper "type-identity" claims.



The main concern was to defend this position against various
counterarguments. The question of what exactly makes this position
*reductive* did not receive considerable attention. Primarily,
this theory is a monistic theory; but monism is not identical to
reductionism. Herbert Feigl gives the following characterization of
type-identity theory when answering the question of "whether the
mental and the physical can in some sense be identified":



> 
> 
> [I]t is proper to speak of "identification [...]".
> Concepts of molar behavior theory like habit strength, expectancy,
> drive, instinct, memory trace, repression, superego, etc., may yet be
> identified in a future psychophysiology with specific types of
> neural-structure-and-process patterns. (Feigl 1967: 77)
> 
> 
> 



This quote suggests a connection between type-identity theories and
models of theory-reduction: It suggests that reduction, understood as
an inter-theory relation, is closely related to type-identity theory.
The idea seems to be this: Within a future psychophysiology, the fact
that mental types are identical to physiological types might be
revealed, and the nature of mental states will ultimately be given in
terms of psychophysiology, rather then in terms of psychology. Similar
positions have been held by Place (1954, 1956, 1960), who primarily
speaks of constitution rather than identity, whereas Smart explicitly
stated this thesis in terms of identity (Smart 1959).


### 3.2 Token-identity theories



Type-identity theory was largely, though not completely, abandoned
when arguments from multiple realizability were put forward to show
that due to the fact that at least some mental kinds can be multiply
realized--that is: instantiated in different physiological
kinds--an identification of mental with physiological kinds is
impossible. Still, this left space for token-identity theories,
according to which any mental kind is physically instantiated. The
identity is that of the objects or items that are of the two types or
that possess the two relevant properties, even though the types or
properties themselves are not identical; e.g., one and the same item
might have both mental and neural properties. The bearers of those
properties, i.e., the items falling under the types, are identical,
but the properties or types themselves are distinct. Token-identity
theory is still a reductionist position in that it opposes any form of
interactionist substance-dualism, substance-parallelism, and
substance-epiphenomenalism. Even though it is not a very common term,
in the literature, we sometimes find variants of the expression
'token reduction' (Cartwright 1999: 32ff.; Hooker 1981:
part III; Bickle 1998: 223ff.).



Token-identity theories perfectly match weak interpretations of theory
reduction. If for reduction, bridge-laws that state correlations or
merely extensional relations are sufficient (Richardson 1979), or if
bridge-laws are not laws in any robust sense, (Klein 2009; Dizadji
Bahmani et al. 2010) then token-identity may be sufficient for
Nagelian theory reduction.



Token-identity theories also match theory-reduction according to
Schaffner's and the New Wave reductionist's models. Within
these models, token-identity theories do not back up strong or
perfectly smooth cases of reduction, but they nevertheless support
reductions that are stronger than those that are merely extreme cases
of replacement, not involving correction in any robust sense.


### 3.3 Local reductions and disjunctive kinds



Anti-reductionist arguments based on the assumption that mental kinds
are multiply realizable have been attacked on at least two different
grounds.



Clapp (2001) and Walter (2006) argued that high-level kinds can be
legitimately identified with disjunctive kinds. The high-level type is
identical to a kind we can refer to by a term consisting of a
disjunction of expressions each of which refers to one of the
different low-level kinds that realize the high-level kind:
*H**x* = (*L*1*x* or
*L*2*x*
or......*L**n**x*). We may be able
to specify the relevant term only in principle in cases where the
disjunction is infinitely long. This strategy is also pertinent in
Dizadji Bahmani et al. 2010.



An alternative strategy suggested by David Lewis (1969, 1972) and
developed by Jaegwon Kim argues in favor of local reductions (Kim
1992). High-level kinds are multiply realized in virtue of the fact
that different physical kinds play the required role in different
organisms, systems or contexts. Thus, if we individuate the high-level
kinds in a more fine-grained way, we can save a weaker form of
type-identity theory. Even though kind *K* might be multiply
realized by kinds *P*1, *P*2, ...
*P**n*, there are more specific contexts
*C*1, *C*2
*C**n*, such that in each context
*C**i*, only one physical kind
*P**i* plays the role of *K* in that
context. Thus we make more local or contextual identifications. For
example, pain in general may not be identical with any one neural
kind, but we may be able to reductively identify human-pain with one
neural kind, lizard-pain with a second neural kind and octopus-pain
with a third. The relevant context could be species-specific,
individual-specific or even specific for conditions of occurrence
within an individual.



This idea was further developed by Brian Loar (1981: ch. 4), Jackson,
Pargetter and Prior (1982), and Michael Esfeld and Christian Sachse
(Sachse 2007; Esfeld and Sachse 2007). Esfeld and Sachse (2007) and
Sachse (2007) describe how this insight can be accommodated within the
high-level science by constructing fine-grained versions of the
high-level types--so called 'functional
sub-types'--which can then be linked to the relevant
low-level kinds, using the resources of high-level science.


### 3.4 Functional reduction



Functionalism about properties of a given sort is the view that such
properties are functional in nature. For example: Functionalism about
the *mental* is the view that *mental properties* are
functional. Intuitively, a functional property is a property whose
nature, essence or significance is fully captured by a description of
the role it plays within a system. Versions of functionalism can be
distinguished with respect to the sort of role they attribute to
functional properties and the sorts of resources required to specify
those roles. The relevant roles might be specified solely in terms of
causal input-output relations. Alternatively, they can be conceived of
in terms of relationally defined computational states (computational
functionalism), or in terms of goal-directed interactions
(teleological functionalism) (cf. Block forthcoming; Lycan 1987; Van
Gulick 1980; Sober 1990).



'Functional reduction' signifies a family of models which
are concerned with the reduction of such causal, computational, or
goal-related properties. These models describe reduction as consisting
of a relation between a functional property *P* and a
structural or substrate property *P\** that plays the role
determined by the functional property *P,* or at least does in
a restricted class of systems or contexts. This general description
leaves the question of the exact relation between *P* and
*P\** open, nor does it say anything about the specific
epistemic status of functional reduction. On this issue different
models of functional reduction disagree.



Some functionalists assume that the functional property *P* and
the property that plays the *P*-role, *P\**, are
identical. On this view (also called '*filler*-',
'*realizer' or
'occupant-functionalism*') a specification of the
relevant role--typically obtained by a logical construction (the
so called "Ramsey sentence") from the theory containing a
predicate that signifies this property--merely serves as a
description to pick out the property that plays the functional role in
the actual world. That is, '*P*' refers to the
underlying property that actually plays the role, namely *P*\*.
Others (so called 'role-functionalists') assume that the
functional property *P* is a "higher order"
property; it is simply the property of playing the relevant role and
is not to be identified with any given underlying property *P*\*
that does so. On this view, *P\** merely *realizes P*; it
is not identical with *P.* Given its denial of property identity,
this latter view seems less apt to a reductionist interpretation.
However, there may be means by which the higher-order properties might
be logically constructed from lower-order ones that count as reductive
(Putnam 1970).



Moreover, there is disagreement about how we arrive at a functional
definition of a property. Some tend to assume that this is a more or
less a priori affair. Others tend to think that it is an empirical
matter.



Thus functional reduction is primarily concerned with the connection
between reduction and the metaphysics of specific properties, namely,
functional ones. That is: *prima facie*, it is compatible (if
compatible at all, cf. Block *forthcoming*) with various
definitions of reduction. It can be incorporated within a Nagelian
framework (Esfeld and Sachse 2007), and it matches New-Wave
reductionist ideas. Occupant-functionalism can be regarded as being
compatible even with classical type-identity theory. Thus, models of
functional reduction do not give sui-generis models of reduction;
rather, they can be regarded as yielding detailed theories about how
functional properties behave with respect to reduction. In short, the
question addressed under the heading of 'functional
reduction' is this: How do functional properties fit into a
reductionist perspective?



If the following three are true: mental properties are functional in
nature, occupant-functionalism is correct, and the role of mental
properties is played by physiological properties, then mental
properties reduce in a straightforward sense to physiological
properties. Alternatively, it has been suggested that even if
identification of the functional property with its realizer is not
always possible, a reduction of such properties can still be achieved,
either by way of local reductions (Kim 1993), by way of the reduction
to disjunctive kinds (Clapp 2001; Walter 2006), or by way of the
reduction of functional sub-types (Esfeld and Sachse 2007).



Others have argued that functionalism best suits non-reductive
physicalism in the following sense: It maintains the
naturalist's intuition that there is unity among the underlying
property-realizers which are all ultimately physical; hence, token
identity theory is true. Nevertheless, it opens the possibility to
accommodate the threat of multiple realizability in a particular way:
Functional kinds can be multiply realized, and, hence, they can be
regarded as being distinct from the lower-level physical properties.
Moreover, if a pragmatic dimension is incorporated in the conception
of functionalism, functionalism seems to contradict reductionist
interpretations; it may be impossible as a matter of practice to
specify the relevant roles using only the resources of the lower-level
theories (cf. Van Gulick 2010).



Thus, differing models of functionalism and functional reduction might
be used to either justify or *criticize* various forms of
reduction*ism*.


## 4. Definitions of '\_reduces to\_'



As discussed above, the term 'reduction' can be
interpreted in many ways. These rival explications can be
distinguished (i) with respect to how they conceive of the
*relata* of the reduction relation, and (ii) with respect to
how they conceive of the *conditions* the relevant relata have
to meet in order to instantiate the relation. Often, an epistemic
aspect complements these characterizations. This section groups the
rival accounts with respect to the objects they offer as candidate
relata, and with respect to the links they offer as candidates for the
conditions the relata have to meet in order to instantiate the
reduction relation. Three remarks are in place: 



In what follows, only candidates for the *primary relata* of
the reduction relation are listed; it is the *primary* relata
with respect to which philosophers disagree. The distinction between
*primary* and *secondary relata* can be explained by way
of example: Patricia Churchland holds that



> 
> 
> [s]tatements that a phenomenon *P**R* reduces to
> another phenomenon *P**B* are derivative upon the
> more basic claim that the theory that characterizes the first reduces
> to the theory that characterizes the second. (Churchland 1986:
> 278)
> 
> 
> 



On this view, the relation signified by the reduction-predicate in its
basic, or proper use is instantiated by pairs of theories only. A
*derivative* notion of reduction (and, hence, a deviant use of
the predicate) allows for statements of the form '*a*
reduces to *b*' to be true, even if the respective
instances of '*a*' and '*b*' do
not designate theories. The truth of this statement then depends upon
a corresponding truth about theory-reduction. For example, if water
reduces to H2O, then on this view it does so *only if,
and if so: in virtue of the fact that* folk-chemistry reduces to
chemistry.



On the other side of the spectrum, those who view reduction in terms
of real world entities rather than theories would not claim that
statements of the form '*a* reduces to *b*'
are always false when the instances of '*a*' and
'*b*' designate theories, and true only when they
designate properties, or parts and wholes. Rather, they would hold
that claims about theory-reductions are derivative upon claims about
property-reductions or that the notion of theory reduction depends
upon a notion of property reduction in a more ontological sense. Thus
if folk-chemistry reduces to chemistry, it does so *because*
water reduces to H2O, and *because* Iron reduces to
Fe, and so on.



Secondly, this way of conceiving of the primary relata of reduction
suggests that the reduction-relation is a binary relation. Recall,
however, the observation that on Nagel's account, in heterogenic
cases of reduction, no theory reduces to another theory
*simpliciter*; it does so only with respect to certain
bridge-laws and/or boundary conditions. Depending on how one counts
the arity of a relation, i.e., the number of items it relates, this
might suggest that even on the classical interpretation, reduction is
not a binary relation. More recently, alternative explications have
been suggested according to which reduction is of an arity higher than
2. Thus, the intuitive notion of primary relata should be tied to the
designata of the expressions flanking the reduction predicate in true
reduction statements.



Finally, talking about candidate relata, we talk about *types*
of objects that possibly function as the relata of an instance of the
reduction-relation, rather than these objects themselves.


### 4.1 Representational and ontological reduction



Many reductionist views seem to involve ontological claims. If one
denies classical substance-dualism and advocates a version of
reductionism rather than eliminativism for minds, persons and mental
state, then one seems to assert that *objects* or
*substances* reduce to *objects*. If one rejects any
dualistic parallelism of mental and physical events and advocates a
version of reductionism rather than eliminativism for events such as
perceivings, willings or pain events, then one seems to assert that
*events* may reduce to *events*. If one denies
property-dualism, and advocates a version of reductionism regarding
properties such as that of being a person or of being a pain, then one
seems to be committed to the assumption that *properties*
reduce to *properties*. These different candidate-relata share
one important feature: They are non-representational in nature
(ignoring cases of objects that are used as vehicles for
representations); they are real world items, not representations about
items in the world. Especially in the philosophy of mind, these
candidate relata have received special attention.



In contrast, most classical explications of reduction developed in the
philosophy of science treat the reduction relation to be primarily
instantiated by representational entities, such as theories or models.
Building on these two lines of thought, one can, following Van Gulick
(2001), draw a distinction between representational and
non-representational relata of the reduction relation, and accordingly
between *representational* and *ontological* reduction.
Intuitively, the conditions on an appropriate model of ontological
reduction require the identification of a specific sort of
*intrinsic* similarity between non-representational objects,
such as properties or events. In contrast, the conditions on an
appropriate model of representational reduction require the
identification of a specific sort of *intentional* similarity;
if one representation reduces to another, it does so because of a
similarity in *what*, and maybe *how*, these
representations represent.



It is noteworthy that different candidate relata come with different
candidate relations: Whereas those who regard properties as the
primary relata of reductions might coherently suggest that identity is
the relevant link, those who assume that theories are the primary
relata of reduction should refrain from this move: No theory reduces
to itself.



A brief survey of the candidate-relata should thus pair them with the
candidate relations that best fit each in turn (Van Gulick 2001).


### 4.2 Theories



The most prominent candidates are *theories*. On the classical
interpretation, theories are sets of sentences. This suggests a
specific view on the reductive link: The reduction-relation is
instantiated when the reduced theory is syntactically derivable from
the reducing theory, with the help of bridge-laws. We thus get the
following definition where *x* is the theory being reduced,
*y* is the reducing theory and *z* is a set of bridge
laws.



> 
> 
> *Theory-Reduction I*:
> *x* reduces to *y* if and only if *x* is a theory
> & *y* is a theory & [?]*z* (*y*, *z*
> +
> *x*)
> 
> 
> 



The quantification is restricted to bridge-laws of the appropriate
kind. A variant of this definition interprets the relata as sets of
laws of two theories. Nagel's *homogeneous* reductions
are trivially covered (bridge-laws are irrelevant for the derivation;
however, they do exist and state trivial connections not connecting
distinct predicates, but the very same predicate in the reduced and
the reducing theory). Variants of this definition include
Schaffner's model that, in addition, mentions possible
replacements and catches aspects of the epistemological dimension of
reductions, concerning the way scientists actually arrive at
reductions (Schaffner 1993). These additional requirements might be
regarded as imposing further constraints on the reductive link;
however, especially in Schaffner's model as well as in New Wave
models of reduction, these further constraints are always
*disjunctive* and seem to play the role of characterizations
that accompany the definition. They do not form necessary conditions
for reduction.



Depending on how bridge-laws are characterized (epistemically,
semantically, metaphysically, or syntactically), a full-blown
characterization of the link might or might not be a purely syntactic
affair: If, for example, appropriate bridge-laws state
property-identifications, or relations among the predicates'
extensions, then the characterization of the reductive link contains a
metaphysical aspect.



On a structuralist interpretation, theories are complex
model-structures. Here, mapping-relations play the role derivation is
supposed to play on the syntactic interpretation of theories. We thus
get (schematically, since the relevant function *f* is left
unspecified):



> 
> 
> *Theory-Reduction II*:
> *x* reduces to *y* if and only if *x* is a theory & *y*
> is a theory & (*f*(*y*)=*x*)
> 
> 
> 



The conditions on this function differ according to different
accounts; however, the function is characterized solely in structural
terms. It is supposed to capture a relevant similarity relation
between the respective theories, such as a structural isomorphism
(Suppes 1967; Sneed 1971). Moulines' criticism that this picture
of reduction lacks an ontological link to rule out arbitrary
reductions (Moulines 1984) reflects the idea that bridge-laws
shouldn't be given a purely structural interpretation.


### 4.3 Models



The notion of a model is also used more widely outside structuralist
interpretations of scientific theories. Talk of models often refers to
fragments of theories, to mere approximations and to objects (or
descriptions of objects) that are used to represent a given target.
Given that the common interpretation of classical conceptions of
reduction requires the relata to be fully fledged theories, it is not
clear how sciences such as biology, psychology, and neuroscience fit
into the picture. In these areas, such imperfect, highly idealized or
merely approximately correct models dominate theorizing, and often
model-descriptions are accompanied by visualizations, which might
raise further questions as to the applicability of purely or mainly
syntactic notions of reduction (Bechtel & Abrahamsen 2005). There
are no definitions of reduction that accommodate that fact directly;
that is, there are no definitions of how one model reduces to another
that might serve as a counterpart to those we have for
theory-to-theory reduction. Rather, definitions of reduction directed
at model-based theorizing are often described in ontological terms: It
is the mechanisms, or phenomena captured by the models that allegedly
reduce. The models themselves reduce only in a derivative sense (a
discussion of these and related issues is covered in the entry on
 reductionism in biology.)



If, on the other hand, structuralist reconstructions of
"imperfect" theories are successful, then model-reductions
might be covered by *Theory-Reduction II*. Moreover, it has
been argued that Nagel's model should not be interpreted too
strictly; at least, it maybe was intended to cover model-reductions as
well (van Riel 2011).


### 4.4 Concepts



Reduction may also apply to concepts, where concepts are conceived of
loosely as whatever is expressed by a meaningful term, or as the
cognitively significance of a linguistic representation, or in Fregean
terms as the mode of presentation under which an object is given by a
representation. Concepts can be grouped into conceptual frameworks and
networks of interconnected concepts. Most notably, there seems to be a
fundamental difference between the framework that applies to our
psychological states from the first-person perspective, and those that
apply to our psychological states from the third-person perspective,
such as the framework of neurophysiology.



Whereas models and theories as candidate-relata for reduction received
considerable attention in the philosophy of science, within the
philosophy of mind reduction as a conceptual issue also played a major
role.


#### 4.4.1 Conceptual issues and analytic functionalism



One such view is tied to functionalism, although this connection is
mainly implicit: If a functional description of a property can be
obtained from an *analysis* of a functional *concept*,
then conceptual issues play an important role in the
*epistemology* of functional reduction. On one standard version
of functionalism, it is conceptual analysis which provides the
functionalist with the specification of the functional roles defining
mental properties that are needed to carry out the reduction. The
condition on analytic functional reduction is thus (schematically)
this:



> 
> 
> *Analytic Functional Reduction*:
> *P*1 reduces to *P*2 only if
> instances of *P*2 play role *R* and it is a
> conceptual truth (or analytic) that whatever instantiates
> *P*1 plays role *R*.
> 
> 
> 


#### 4.4.2 Conceptual issues and the explanatory gap



Another motivation for believing that conceptual issues are crucial
for reduction stems from the idea that the identity of properties or
kinds expressed by two predicates *P1* and
*P2* manifests in the fact that there is an a priori
route from *P1* to *P2*. This
interpretation is implicit in the received view attacked early on by
Smart (1959). It also plays a role in Levine 1993, 1998 and Chalmers
1996; the idea is that for reductions to be explanatory, they must not
leave out any aspect of the phenomenon to be explained. If
qualia-concepts or concepts of conscious states pick out mental states
in terms of features that cannot be fully explained in terms of the
vocabulary of the reducing science, the explanation and hence the
reduction is not successful. Even if our mental terms and neural terms
refer to the same states, if the former pick them out using mental
concepts that cannot be neurally explained, we will not have reduced
the mental to the neural. There will be a residual explanatory gap
(Levine 1993, 1998).



Thus, even if two theories instantiate a relation of the sort required
by Nagel's model, we should not speak of proper reduction, if as
a matter of epistemic possibility the connection between the domains
of the two theories remains too loose to support the materialist
claims. Levine (1993) gives this an epistemological interpretation;
Chalmers (1996) seems to suggest that the epistemic possibility of two
concepts coming apart implies that there is a metaphysical possibility
of the two properties coming apart. Many (e.g., Block & Stalnaker
1999; Van Gulick 1999; Yablo 1999) have criticized this view. Even a
hard-core physicalist will deny that the postulation of a mind-brain
identity as a brute fact would suffice for reduction, but at the same
time accept that it would suffice for the relevant sort of derivation.
However, this does not support the further claim that there must be a
conceptual path from concepts or predicates of the reducing science to
concepts or predicates of the reduced science. Explanations are not an
all-or-nothing affair; there might be explanations that are more than
mere brute-fact postulates, but still less than the tight conceptual
connections required by those who demand an a priori link between the
reduced and the reducing science or conceptual framework (Van Gulick
1992; Kim 1998).



This a priori link can, following Beckermann (2009: 162), be described
as an a priori implication of the truth of the reduced science,
theory, or framework by the truth of the reducing science, theory, or
framework:



> 
> 
> *A Priori Reduction-1:*
> *P*1 reduces to *P*2 only if the set
> of all truths about the relevant domain employing concepts of the
> *P*2-framework conceptually implies the set of truth
> formulated using concepts of the *P*1-framework.
> 
> 
> 



This way of putting it relies on an intuitive understanding of
different conceptual frameworks. Stronger versions require there to be
a conceptual connection between pairs of properties (Jackson 2005). At
least under some interpretations, this apriorist view has clear
predecents in early considerations about the unity of science.


#### 4.4.3 Conceptual issues and the unity of science



According to Carnap, scientific unification should be effected by way
of a theory that has the same expressive power as the reduced
theories, and, hence, by *translation*.



> 
> 
> ... science is a unity, [such] that all empirical statements can
> be expressed in a single language, all states of affairs are of one
> kind and are known by the same method. (Carnap 1934: 32)
> 
> 
> 



The notion of translation relevant here is not particularly precise;
however, as long as the notion of *expressing* the same thing
is not fully captured by that of *designating* the same thing,
then reference to conceptual aspects of language obviously plays a
role for this view on reduction. A particularly strong version, though
apparently not Carnap's, is this (this schematic conditional is
restricted to properties signified by predicates, which makes the
condition easier to formulate):



> 
> 
> *A Priori Reduction-2*:
>  The property signified by the predicate '\_*F*'
> reduces to the property signified by predicate
> '\_*G*' only if it is conceptually necessary that
> whatever is *F* is *G*, and vice versa.
> 
> 
> 



The demand for conceptually necessary links echoes the position of
Smart's (1959) anonymous opponent who claims that if a type
*T* is identical to a type *T\**, then a corresponding
conceptual relation holds between expressions signifying these
types.


#### 4.4.4 Conceptual issues and the pragmatic dimension of reductions



Van Gulick (1992, 2010) highlights another aspect of the relevance of
conceptual considerations for models of reduction. He suggests that
conceptual issues might play a role for reduction insofar as their
pragmatic function is concerned. People use representational devices,
such as concepts, but also theories, models, and expressions, in order
to successfully deal with objects in specific contexts. The
directedness of a representation is not exhausted by the object it
represents; a representation is typically embedded in a context in
which it is used for a certain purpose. Here, the epistemic difference
between different conceptual frameworks becomes apparent: The way the
first-person perspective enables us to evaluate and deal with our
mental states seems to be radically different from ways in which a
third person perspective could enable us to deal with these states. If
similarity in this respect is required by an appropriate conception of
reduction, then the prospects for reduction seem rather meager.


#### 4.4.5 Conceptual issues and the a posteriority requirement



The idea that reference to concepts is required by any appropriate
explication of the reduction-predicate is also implicit in
Nagel's work (Nagel 1961): Nagel describes bridge-laws as being
*a posteriori*, thereby opposing any sort of analytic
reductionism. It has been argued that this might be crucial for fully
understanding the concept of reduction. If so, then reduction turns
out to be a four-place relation, involving not only entities but also
the concepts by which they are picked out or referred to. Truth
conditions for sentences of the form '*a* reduces to
*b*', where the expressions '*a*' and
'*b*' designate properties, events, phenomena or
the like, are sensitive not only to what '*a*' and
'*b*' designate but also to how they do so, i.e.,
to the concepts or modes of presentation '*a*' and
'*b*' express (van Riel 2010, 2013 & 2014;
Jenkins 2011).


### 4.5 Properties



Consider the example of pain reducing to C-fiber stimulation. Under
one interpretation, the expressions 'pain' and
'C-fiber stimulation' pick out abstract objects, namely
natural kinds, properties or types. On this reading of 'pain
reduces to C-fiber stimulation', the primary relata of reduction
should be conceived of as properties or types. This view has played an
important role in the philosophy of mind, underlying classical
versions of type-identity theory and being the main target of
anti-reductionist's counter-attacks (Putnam 1967; Fodor 1974,
1997).


#### 4.5.1 Property-identity



What sort of reductive link might suit those who assume that the
primary relata of reduction are properties? Doing justice to the idea
that if some *a* reduces to some *b*, then *a*
does not exist over and above *b*, one might tend to assume
that property-reduction requires identity:



> 
> 
> *Identity reduction:*
> *P*1 reduces to *P*2 only if
> *P*1 = *P*2.
> 
> 
> 



This condition is, under a certain interpretation, what underlies
Nagelian bridge-principles (van Riel 2011, see however Sarkar 2015 for
a different interpretation). It is also explicitly endorsed by Robert
Causey (1972a, b, 1977), who connects it to theory reduction via
micro-reductions. Just as in the writings of Smart (1959, 1963), the
idea is that the relevant identities are not a priori. In addition, he
suggests that by identifying the relevant kinds, attributes and
properties of the reduced theory with kinds, attributes and properties
of the reducing theory, we arrive at explanatory reduction because the
laws of the respective theories will thereby be shown to be
nomologically equivalent (Causey 1974: 5).



*Identity Reduction*
 is, at the same time, obviously compatible with various sorts of
*a-priorism*. Moreover, it has been suggested that it offers a
possible way to escape the explanatory gap: If some mental state just
is some physical state, then there is nothing to be explained about
how "these states" connect; it is a sheer misunderstanding
to ask for more than just the identity-statement. Identity claims are
basic in the sense that they cannot be successfully explained in the
relevant sense (Block & Stalnaker 1999). As is well known, it is
often hard to decide whether or not we are faced with identity or mere
correlation; Wimsatt (1976: 697ff) argues that when we conceive of
reduction as a scientific process, we are well advised to interpret
apparent correlations as identities.



Recently, van Riel (2014) has argued that the idea that reduction is
relevantly tied to identity generates a puzzle: Intuitively, identity
seemingly contradicts the directionality, or what is sometimes called
'asymmetry' of reduction. More precisely: if *a*
reduces to *b*, then *b* does not reduce to *a*.
But how can this be if for *a* to reduce to *b*, it is
required that *a*=*b*? The suggested solution is that
'\_reduces to\_' generates hyper-intensional contexts, and
that this is best explained by the idea mentioned above: For
*a* to reduce to *b*, two conditions must be satisfied:
(i) *a*=*b* and (ii) '*a*' expresses
a conceptual content that relevantly differs from the conceptual
content expressed by '*b*'. Another option would
consist in giving this a pragmatist reading as in (Van Gulick 1992,
2010), or giving up on the idea that identity is required for
(property-)reduction.


#### 4.5.2 Alternatives to property-identity



There are three rival suggestions on the market, all of which argue
that weaker notions of reduction still serve the purpose of the
reductivist: The reductive link could be spelled out in terms of
supervenience (Kim 1982, 1990), causation (Enc 1976) and
realization (Kim 1998; Melnyk 2003; Shoemaker 2007. Whereas usual
stipulations of supervenience do not rule out the possibility that
*a* supervenes on *b* and, at the same time, *b*
supervenes on *a*, at least causation seems to be relevantly
directional and so do justice to the directionality of reduction.
Nevertheless, causation is probably not what is required: the
occurrence of water is not *caused* by the occurrence of
H2O (in the usual meaning of the term
'causation'); there is no temporal component involved,
what is, on most accounts, required by causation.


#### 4.5.3 Supervenience



Supervenience does not fare better, though for different reasons: It
seems to be too weak a notion to capture the idea that guided
reduction-talk, although it surely is a necessary condition for
reduction. In a sense, if mental states merely supervene on
physiological states, they exist in a straightforward sense over and
above the latter. Such supervenient states are dependent; but being
dependent is not enough for reduction (Kim 1998). Note that this is,
up to a certain point, a matter of stipulation. However, the
stipulation is driven by an intuition: An appropriate definition of
the reduction-predicate should adequately mimic the intuitions that
guided the use of the term. If, for example, one hope was to formulate
a strictly monist position, then relying on supervenience does not
serve this goal: Supervenience makes room for a weak form of dualism.
For example, it is compatible with emergent property dualism. The
emergent properties on such a view are something over and above the
physical, but in so far they emerge from the physical base according
to natural laws of emergence, they would supervene on that base. Any
two cases in which the physical bases were the same would also agree
in their emergent properties. Thus supervenience is generally not
judged sufficient for reduction. However, many regard it as consistent
with reduction or even required. With regard to the mind body case,
for example, Kim argues that supervenience is needed for what he calls
"minimal physicalism", whether reductive or not. Under a
somewhat liberal though well established interpretation of Nagelian
reduction, there is a sense in which reduction is compatible even with
emergence, that is: a theory about "novel behavior" (and,
thus, emergent behavior) can be deduced from another theory
(Butterfield 2011a&b).


#### 4.5.4 Realization



Realization, which has figured prominently in recent versions of
non-reductive physicalism may seem too weak a relation to support a
reduction claim, but nonetheless strong enough to guarantee
ontological unity on the token level that underlies physicalism.
Realization can be described in terms of dependence; e.g., for a
particular physiological event-type to realize pain is for it to be
such that in virtue of its physiological organization, its instances
play the pain-role. This obviously suits the (role-) functionalist
interpretation of higher-level properties; these higher-level
properties are realized by lower level ones in the sense that
instances of the latter play the role of the former in virtue of their
physical organization.



Most functionalists, being opposed to type-identity theory, regard the
realization-relation as too weak a relation for reduction. Given a
strong interpretation of what reductionism consists in--e.g., the
type-identity of properties - the realization-interpretation
might not suffice to guarantee reduction. It might ensure reductionism
at the token-level, but it might, at the same time, be committed to
dualism at the property level--though perhaps only a relatively
innocent form of dualism. Options to reconcile role-functionalism with
reductionism have been introduced in
 Section 3.4.



### 4.6 Wholes and their parts



The idea that reduction is tied to part-whole relations is frequently
alluded to in the literature (esp. Schaffner 1993; Place 1956, 1960).
The most elaborate version of this assumption forms part of
Causey's more general picture of scientific unification.
Intuitively, and abstracting away from numerous details of his
elaborate model (described in terms of theory-reduction), a
micro-reduction in Causey's sense relies on an identification of
a property *P*1 with a property
*P*2, where the syntactical arrangement of the terms
forming the complex term '*P*2' reflect
the mereological composition of *P*1, whereas
'*P*1' does not.
'*P*1' simply refers to the whole
without explicitly referring to its parts as
'*P*2' does, but they in fact refer to
the same complex. Whereas mereological relations alone are compatible
with emergentist or dualist interpretations combining mereological
composition and identity seems to ensure a link strong enough to count
as a reductive link.



Another way of connecting reduction to composition is this: one could
try to describe reduction as a one-many relation. In one use of the
term 'composes', a given object *a* may be composed
out of several objects *b*1, *b*2,
*b*3 ... *b**n*. At
least at the face of it, there is nothing in the use of
'reduction' that should prevent us from, say, stating that
a whole reduces to its parts.



At the same time, it is not clear how tying reduction to mereology
could deal with cases like this: *Iron* reduces *Fe*
(the chemical structure kind). Both expressions, 'iron'
and 'Fe', are syntactically simple. More importantly,
'*Fe*' does not present us with Iron-atoms as
composed wholes. Similarly, the often-used example of pain and C-fiber
firing does not match this idea.



Moreover, this move would come at the price of building a sort of
atomistic materialism into the very notion of reduction; if we were to
define reduction in mereological terms, it would be conceptually
impossible that parts reduce to the whole they form. However, it seems
possible (though, in the light of science, maybe highly unlikely) that
in some cases parts do not exist over and above the whole they form.
Such was the view of certain historical varieties of holistic idealism
(see for example entry on
 F.H. Bradley),
 and some have argued it is true of quantum systems including
entangled particles (Silberstein 1999). Thus if we were to tie the
notion of reduction to mereology, it would not be general enough to
capture the full range of possible reductionist positions.



Finally, mereological relations are usually regarded as being
instantiated by concrete objects and the wholes they form. A
derivative notion can be defined for properties via reference to
mereological relations between their instances. However, it is not
clear how this could extend to events or event-types. Thus
interpreting reduction as a matter of mereology might prevent us from
applying it to events and deprive us of a fully general notion of
reduction.


### 4.7 Mechanisms



One solution to this problem would be to connect reductive explanation
to mechanistic explanation, even though mechanistic explanation is
often described by its proponents as non-reductivist in spirit
(Kauffmann 1970; Cummins 1975, 1983, 2000; Bechtel & Richardson
1993; Bechtel 1994; Glennan 1996; Machamer et al. 2000; Craver 2005,
2007).



The proposal would be to define a phenomenon *a* as reductively
explained by a phenomenon *b* *iff* *a* is
mechanistically explained by *b*. Mechanistic dependence might
be distinguished from a mereological relation, if mechanisms are not
the sorts of things that have parts in the ordinary sense of the term
as some have claimed (Craver 2005). On one reading, the entity
explained in a mechanistic explanation is an event. A mechanistic
explanation explains an event in terms of events that together form
the event to be explained. On this view, mechanistic dependence is
analogous to mereological dependence but not the same, and might thus
complement the mereological approach to yield a fully-fledged
characterization of reduction.


## 5. Unresolved issues



The mainstream in the philosophy of mind is, apparently, one version
or another of non-reductive physicalism. The majority within the
philosophy of science has nowadays abandoned the unificationist
program, to which reduction was intimately connected right from the
start. However, as became apparent only in recent years, some
questions regarding the concept of reduction have not successfully
been accounted for yet, and recent developments in metaphysics in
connection with metaphysical grounding may shed new light on the
concept of reduction.


### 5.1 Reduction and explanation



Philosophers do not agree upon the relation between reduction and
explanation. Some argue that the concept of reduction is on a par with
concepts such as causation in that it is intimately tied to certain
'because', 'in virtue of' or 'by'
statements, or that there is no fundamental split between explanatory
and ontological models of reduction (van Riel 2011; McCauley 1981).
Others have held that not only is there a difference between
explanatory reduction, which is an epistemological matter, and
ontological reductionism, but that there might even be a difference
between explanatory reduction and derivation-based models of theory
reduction (see the entry on
 reductionism in biology;
 see also Klein 2009; Dizadji-Bahmani et al. 2010).



For notions of reduction that are connected to explanation, the
different options are, apparently, these:


1. The notion of reduction is itself an explanatory *concept*,
in the sense that it is a concept which is to be cashed out in terms
of explanatory concepts: stating that *a* reduces to
*b*, one *states* that *b* is explanatorily tied
to *a*.
2. The reduction relation is an explanatory *relation*,
despite the fact that the concept is not to be cashed out using
explanatory notions: that *a* reduces to *b* merely
implies that *b* is explanatorily tied to *a*.



If the concept of reduction is an explanatory concept, then a
definition of it is either of the form [*... because
...*], or it relevantly involves an analysis of explanatory
notions. If it is not, but still is such that the relation is
explanatory, then a definition would at least imply the truth of a
corresponding explanation.



Given this broad outline, it seems reasonable to assume that most
models of reduction are concerned with explanation in one of these
senses. Possible exceptions would include models that highlight
replacement issues, where an apparent phenomenon is not explained but
rather "explained away", and perhaps some more technical
notions of reduction that are guided neither by intuitions of the
"nothing over and above" sort, nor by any metaphysical
positions such as functionalism (e.g., Dizadji-Bahmani et al. 2010;
Bickle 2008; Butterfield 2011a&b).


### 5.2 Epistemological and ontological issues



In a similar spirit, notions of reduction are often characterized as
being either *ontological* or *epistemological*. There
is, again, no uniform use of these qualifications. For Oppenheim and
Putnam (1958), epistemological reduction is a reduction that is
"true in virtue of the meanings of the terms involved"
(1958: 3), i.e., reduction that is tied to analysis in the spirit of
 *A Priori Reduction-2*;
 this use, however, is relatively rare. First of all, it will prove
useful to draw a distinction between epistemological and ontological
forms of reduction*ism*.


#### 5.2.1 Epistemological and ontological reduction*ism* (and non-reductive physicalism)



An epistemological reductionist concerning a science or theory
*S* holds that in fact, we are (or at some point will be) able
to reduce *S* to a more fundamental science. It is a thesis
about what we can epistemically achieve. The ontological reductionist
is, in this respect, more modest: She just holds that in fact, there
is just one sort of objects and properties out there in the world;
however, due to our cognitive limitations, we might never be able to
actually carry out all the reductions that would be appropriate given
the actual ontological structure of our world (Scerri & McIntyre
1997; Esfeld & Sachse 2010).



This distinction motivates different versions of non-reductive
physicalism. On one interpretation, the non-reductive physicalist
opposes the idea that we can and should in fact reduce high-level
sciences; we need them for epistemic or pragmatic purposes (Fodor
1974, 1997; Putnam 1978; Van Gulick 1992). We need a plurality of
autonomous theories and frameworks. However, the non-reductive
physicalist accepts that what is actually out there might very well be
all of one kind, at least in some ultimate sense such that ontological
reductionism is true.



This version of non-reductive physicalism has to be distinguished from
an ontological version, which is best described as
*token-physicalism*, or *token-token identity theory*.
Token-physicalism is the view that what occurs in our world is just
physical; however, mental types are not identical to physical types.
Hence, in this form, non-reductive physicalism is compatible with a
weak form of ontological pluralism at least of properties or
kinds.



It is noteworthy that it is often rather difficult to draw the line
between these two versions of non-reductive physicalism. This is
mainly due to the fact that the notion of a *type* or a
*natural kind* is far from clear. Hempel, for example,
tentatively suggests that what makes a type/kind a mental type/kind is
that it is referred to within psychology or psychological theories
(Hempel 1969); thus the nature of a kind or type becomes dependent
upon the context in which we refer to it, which in turn might be
epistemically or pragmatically constrained. Fodor suggests that being
a kind is to be mentioned in a law; and that the locution 'it is
a law that...' generates intensional contexts, i.e.,
contexts that are sensitive to how things are referred (Fodor 1974:
109). Hence, being a kind again might not be an all-or-nothing affair.
If to be a kind is to play a certain epistemological role (this is, to
be sure, not Fodor's official point), then the distinction
between ontological and epistemic versions of non-reductive
physicalism (and, analogously, between ontological and epistemological
reductionism) might be hard to spell out.


#### 5.2.2 Epistemological and ontological reduction



Independent of issues regarding epistemic and ontological
reduction*ism,* philosophers sometimes allude to a distinction
between epistemic and ontological forms of *reduction*. Sarkar
(1992) suggests that a notion of reduction is epistemic if and only if
it cashes out reduction in terms of *explanation*, presumably
based on the assumption that the concept of explanation is an
epistemic concept. Thus insofar it enters the concept of reduction,
the latter is epistemic as well. On this view, an ontological concept
of reduction would be one that is defined solely in non-epistemic
terms.



Silberstein (2002), Sarkar (1992), Brigandt and Love (2012), and
Hoyningen-Huene (1989) suggest another use of 'epistemic
reduction': Reduction is epistemic if and only if its primary
relata are representational in nature (and, hence, this use is
equivalent to Van Gulick's use of 'representational
reduction' in 2001). Again this use seems motivated by the idea
that epistemic concepts enter this characterization of reduction; for
example, one might regard the very notion of a theory as an epistemic
notion insofar as a theory is a body of knowledge. Accordingly,
ontological reduction can be distinguished again in terms of the
relevant
*relata*: they must be non-representational entities.



It is noteworthy that this way of putting things is contentious and
possibly misleading; after all, on a technical understanding theories
are sets of sentences. The notion of a theory is thus a
"linguistic", rather than an epistemic notion. Labeling
theory-reduction as 'representational' rather than
'epistemic' leaves open the possibility that the concept
of theory-reduction is independent of concepts such as knowledge,
belief, justification or understanding. At least Nagel's
official definition does not make use of epistemic concepts.



Moreover, this use of 'epistemic reduction' suggests a
misleading distinction. Any description of bridge-laws that involves
reference to the predicates' extensions or intensions involves
an "ontological" aspect. In this sense, the classical
accounts of "epistemological" reduction are ontological,
or at least include an ontological element.


### 5.3 Reduction and Ground



Recently, explanatory connectives, such as 'because' and
'in virtue of' have attracted considerable attention in
the study of *metaphysical ground*. Very briefly, the term
'ground' is used to articulate dependence claims, without
commitment to specific modal or counterfactual interpretations of
various sorts of metaphysical dependence. Let us first provide some
background and then hint at possible connections between reduction and
ground.


#### 5.3.1 A very short introduction to ground



Following Schaffer (2009), one can draw a distinction between two
types of metaphysics in terms of their alleged subject matter.
*Quinean* metaphysics deals with existence questions. It asks:
What exists? What is part of the ontology of a given theory, such as
physics? In contrast, *(Neo-)Aristotelian* metaphysics is
permissive regarding existence. It is mainly concerned to identify
what is fundamental and what depends on the fundamental. In
Schaffer's terms, it is concerned with "what grounds
what" (Schaffer 2009). Examples of dependence claims the
Neo-Aristotelian is concerned with include:


1. The particularized property depends in its bearer.
2. The mental depends on the physical.
3. Aesthetic properties depend on physical properties and
attitudes.
4. Mechanisms depend on their constituents and these
constituents' interactions.
5. The whole depends on its parts.
6. Sets depend on their elements.
7. The hole in the Swiss cheese depends on the cheese.



At least some of these claims, or cognates thereof, have formed part
of metaphysics carried out in the Quinean spirit. Until recently,
however, metaphysics did not provide the means to address these claims
properly, says the defender of ground. Consider the claim that the
mental *depends* on the physical. Until recently,
metaphysicians would have suggested that this claim is to be cashed
out in modal or counterfactual terms. The claim could be given an
interpretation in terms of supervenience: Any shift in mental
properties necessitates a shift at the level of physical properties.




Any analysis along these lines will leave the grounding theorist
dissatisfied. One reson to be skeptical about an analysis of
dependence in terms of supervenience is that supervenience is neither
*irreflexive* nor *asymmetric* (unless by way of
stipulation). In contrast, the sort of dependence grounding theorists
are after when claiming that the mental depends on the physical
requires that the physical does not depend on the mental, neither in
whole nor in part. Unlike supervenience, dependence, at least of the
sort assumed in the grounding literature, is asymmetric and
irreflexive. And dependence claims may concern necessarily
(co-)existing entities; dependence, unlike supervenience, appears to
generate hyper-intensional contexts, i.e. contexts in which even
substitution of necessary equivalents is not guarantee to preserve
truth value (for further discussion, see Schaffer 2009, 364). 



The concept of *ground* is supposed to do the trick. Ground is
regarded as a primitive (Fine 2001). As a consequence, instead of
giving an analysis of the concept of ground, philosophers have studied
the structural features of ground - it is transitive,
irreflexive, asymmetric, and it generates hyper-intensional contexts
(Fine 2012; see also the contributions in Correia and Schnieder 2012).
But if this notion is primitive, how do we get access to it? After
all, 'ground' is, like 'reduction', a
philosophical term; reflection on ordinary usage will not give access
to its meaning in philosophical contexts.



One influential idea is that explanatory connectives such as
'because' or 'in virtue of' introduce a basic
notion of ground (Fine 2012). 'Because' and 'in
virtue of' are arguably asymmetric and irreflexive, and they
generate hyper-intensional contexts (Schnieder 2011). They are part of
ordinary discourse, and we are competent in using these expressions.
When taken in a non-causal sense, or in a context where these
connectives do not track causal dependencies, they may give access to
the sort of metaphysical dependence philosophers have sought to
capture for decades, if not centuries. 



Alternatively, one could give an ostensive definition and introduce
the notion of ground by way of examples; reflection on lists of
dependence claims, or pairs of objects that are supposed to
instantiate the grounding relation, may give access to the notion of
ground (Schaffer 2009). 



There are different interpretations of ground - in particular
with respect to what can bel labeled 'relational'
grounding and 'operational' grounding (Tahko 2013).
Moreover, philosophers who accept a relational interpretation disagree
about the primary *relata* of ground (candidates include:
facts, propositions, properties etc). 



Summing up: The primitive notion of ground is supposed to do justice
to dependence claims, without any aspiration to further reduce this
concept to modal or counterfactual concepts; and it has been suggested
that explanatory connectives express this notion of ground. These
preliminary remarks in mind, let us now turn to possible connections
between redutcion and ground. First, there are two very general
reasons to regard reduction as a particular variant of ground (5.3.2).
At the same time, there appears to be a tension between reduction and
ground. Either, reduction turns out to be distinct from ground, or the
structural features of reduction may shed light on the nature of
ground (5.2.3). Third, one may want to resist an assimilation of
reduction to ground, based on a general skepticism towards the
philosophical significance of grounding talk (5.2.4). 


#### 5.3.2 Reduction and ground linked



There are two paths leading from reduction to ground. First, reductive
interpretations of paradigmatic dependence claims may inspire a
reconstruction of reduction in terms of ground. Reductionists about
the mental argue that mental states, like pain, reduce to certain
physiological states and are, hence, identical to the latter.
Eliminativists argue that mental states or properties just do not
exist. Anti-reductionists maintain that pains exist and are different
from corresponding physiological states. Some naturalists employ
various notions of dependence to capture the priority of the natural
while acknowledging a plurality of layers of reality. The debate thus
centered on questions of *existence* and, often, various
*dependence* relations, ranging from emergence to ruthless
reduction. It appears, then, that the reductionist about the mental
maintains that the mental *reductively depends* on the
physiological. 



By these lights, it may seem plausible to reconstruct reduction claims
in terms of ground (if we accept that grounding talk is fruitful).
Above, we have noted that the primitive concept of ground may be
introduced by way of examples, or by lists of paradigmatic dependence
claims. Theories of reduction may offer a viable interpretation of at
least some of these dependence claims. Hence, it should not come as a
surprise that some maintain that redution is sufficient for ground, so
that if *a* reduces to *b*, then *b* grounds
*a* (Rosen 2010, 124f., Schaffer 2009, 378, van Riel 2014). In
a similar spirit, one may suggest that grounding talk offers a
fruitful reconstruction of the idea that some things do not *exist
over and above* other things, an idea frequently invoked in the
reductio debate (cf. Wilson 2014 for an extensive discussion). It
seems that on this conception, reduction turns out to be a particular
variant of grund, perhaps among others.


 Second, conceptions of reduction in the philosophy of science,
which draw on explanatory notions, are committed to the assumption
that there is a tight connection between reduction and ground,
conditional on the assumption that explanatory vocabulary (interpreted
non-causally) does in fact introduce a notion of ground, as some
grounding theorists would have it. A concept of ground is often
introduced by reference to explanatory connectives, such as
'because', and 'in virtue of'. On this
interpretation, there is a close link to conceptions of reduction in
the philosophy of science. Within this strand of the reduction debate,
reduction has been conceived of as an explanatory connection (see
above). Of course, a robust metaphysical interpretation of explanatory
links in terms of ground stands in stark contrast to some standard
conceptions of explanation within the philosophy of science. Realists
have mainly focused on causal explanation (Salmon 1989); only
recently, philosophers of science have turned to non-causal,
mechanistic dependence (cf. Machamer et al. 2000, Craver 2005,
2007). Others have taken explanation to be primarily an epistemic or
pragmatic affair (Kitcher 1989, Achinstein 1983, van Fraassen
1980). Nevertheless, it is worth noting that it is an open question
whether (non-causal) explanation can be construed pragmatically or
epistemically, or whether it requires a metaphysically robust
interpretation. If it turns out that explanation does require a
metaphysically robust interpretation, the fact that reduction has been
construed as an explanatory notion in the philosophy of science may be
seen as an indicator that claims of reduction are, properly construed,
claims of ground.


#### 5.3.3 A possible problem



Yet, there is a problem. Above, it has been pointed out that claims of
reduction are often taken to imply corresponding identity claims.
Recall Identity Reduction (4.5.1): 



> 
> 
> *Identity reduction:*
> *P*1 reduces to *P*2 only if
> *P*1 = *P*2.
> 
> 
> 



As a consequence, reduction is taken to have the "structural
features of identity", such as reflexivity and symmetry, as
Trogdon maintains (Trogdon 2013, similarly: Audi 2012), but also
intensionality. Then, it is not clear how reduction could possibly
exhibit the structural features of ground - irreflexivity,
asymmetry, and hyper-intensionality. (This observation seems to
parallel the tension between supervenience and dependence noted
above.) Philosophers have reacted in different ways. Some take these
observations at face value and conclude that reduction is distinct
from ground (Audi 2012, Trogdon 2013). Others have argued that it
would be short-sighted to construe reduction in this way. From
*Identity Reduction*, it simply does not follow that reduction
has the structural features of identity. Recall the puzzle of
reduction discussed above: Although reduction requires identity,
identity is not sufficient for reduction. In fact, if *a*
reduces to *b*, it follows, by the asymmetry of reduction, that
*b* does not reduce to *a*. For this and similary
reasons, it has been suggested that reduction generates
hyper-intensional contexts, and that reduction statements are
sensitive to the modes of presentation expressed by the terms flanking
the reduction predicate (Jenkins 2011, van Riel 2013, 2014). On this
view, reduction may turn out to be a variant of ground.



It is noteworthy that this debate turns on fundamental questions
regarding the nature of ground. If reduction is a variant of ground,
and if 'reduces to' generates hyper-intensional contexts
in the way just sketched, then ground does not link
non-representational objects only. Put differently, the grounding
structure of the world would then depend, at least in part, on
features of *representations* of the world. Fine (2001, 2012)
and DasGupta (2014) take ground to be concerned with truths or finely
individuated facts, rather than properties or fundamental or composed
objects. On the other hand, this observation has led some philosophers
to assume that reduction, ontologically construed, should not be
regarded as a version of metaphysical ground (Audi 2012). 


#### 5.3.4 Reduction without grounding



Of course, there may be very general reasons to reject grounding talk
altogether (see the discussion in the entry Metaphysical Grounding).
In the present context, one criticism deserves close attention. Wilson
(2014) maintains that reference to a primitive metaphysical relation
in articulations of various dependence claims does not constitute
philosophical progress. Many of the various detailed explications of
different types of dependence, including mereological dependence,
supervenience, and the relation between determinables and
determinates, promise illumination, whereas grounding talk is
ultimately obscure, mainly for two reasons. First,
'ground' is supposed to be primitive. Second, its
structural features of a partial order - ground is irreflexive,
asymmetric and transitive - are too permissive and too narrow at
the same time, to single out a particular relation of metaphysical
dependence, says Wilson. It is too permissive because temporal order
and causation may have the same structural features (Wilson 2014,
569). It may be too narrow because, for instance, mixed cases of
dependence may violate transitivity (Schaffer 2012, Wilson 2014). So,
even if there is prima facie evidence that reduction and ground are
closely linked, it is an open question whether shifting attention from
theories of reduction to the metaphysics of ground will prove
fruitful. 


### 5.4 Problems for reductionism and the relevance of the notion of reduction



Scientific reduction became an important topic in the philosophy of
science within the context of a general interest in the unity of
science, and it was inspired by specific alleged cases of successful
reductions. The most prominent argument against reductionism stems
from the observation that straightforward reductions hardly ever
occur. Hence, reductionism cannot be regarded as yielding a coherent
picture of what actually goes on in science. As long as reductionism
is supposed to be more than a purely metaphysical position and is
intended to say something significant about scientific change or
norms, the value and relevance of the notion of reduction seems to
depend in part upon how well the reductionist positions fit the facts,
which their critics argue they do not (see, for example, Sarkar 1992;
Scerri & McIntyre 1997).



Proponents of reduction have responded in different ways. Some have
suggested more flexible and liberal interpretations of reduction
(Wimsatt 1976; Schaffner 1993; Dizadji-Bahmani et al. 2010), to ensure
that the term 'reduction' still serves a purpose in
describing actual scientific practice. A case in point is
Schaffner's suggestion (Schaffner 2006&2012) that we should
reconsider the relevance of merely partial reductions or of scientific
explanations, broadly construed, that are reductionist in spirit,
despite the fact that they do not comply with more traditional
standards of full-fledged theory-reductions (see also Winther
2009).



Others have opted for a more radical stance: Bickle has recently
suggested that philosophical theorizing about reduction should best be
construed as reconstructing whatever cases scientists judge to be
reductions (Bickle 2008).



A much more modest position consists in the idea that, though
straightforward reductions might be rare, reductionism might play a
role as a regulative ideal (Schaffner 1993); the notion of reduction
can, on this view, be used to characterize one extreme of a spectrum
of possible relations between different stages of scientific
developments. It comes with a specific normative power: If a reduction
can be effected, it *should* be effected.



In addition, the notion of reduction obviously plays an important role
in the characterization of non-reductive physicalism; a careful
investigation of what reductionism consists in is required to
adequately determine in what respects, if any, a theory of mind could
be coherently both *non*-reductivist and materialist about the
mental.



Finally, the concept obviously plays an important role in the field of
metaphysics which often extends far into the realm of possibilities.
Even if reductionism should fail to apply to our world and our
scientific history, it might be of interest to consider what reality
would be like in other worlds where reductionism does hold, worlds
similar to ours in containing many different theories or conceptual
frameworks but ones tied together by reductive links.



The hope that the actual progress of science can be successfully
described in terms of reduction has vanished. Concepts of reduction,
except for those re-adapted to actual scientific practice, are
nowadays studied primarily in the more theoretically oriented parts of
the metaphysics of science.



Still, the question of what the concept of reduction amounts to can be
satisfactorily answered even if no theory actually reduces to any
other theory and no property reduces to any other property. Answering
this question may serve two closely related purposes. First, a precise
characterization of what reduction is supposed to consist in will help
judging whether or not a given theory, such as folk psychology, or a
given property, such as being in pain, actually reduce to their neural
counterparts. Secondly, an appropriate definition of reduction is
called for when it comes to evaluating anti-reductionist
positions.



As we have seen, reduction has been conceived of as a relation that
primarily holds between theories, properties, substances, concepts, or
events. It has been described as an explanatory relation that can be
cashed out in terms of derivation, and it could be tied to mechanistic
or ontological dependence, supervenience and identity. It has been
argued that theory-reduction is prior to ontological reduction; and it
has been argued that ontological reduction is more fundamental than
theory reduction. Reduction has been couched in structuralist,
empiricist and functionalist frameworks, and it has been one of the
core notions of theories defending one form or another of scientific
unification. Most philosophers have abandoned this latter view. But
even if the outlook for strong general reductionism, a form of
reductionism that amounts to scientific as well as ontological unity
might be dim, exploring the question of what it takes for a property
or theory to reduce to another property or theory will help not only
understand what the search for unity consists in, a search that
dominated large parts of the philosophy of science in the last
century; it will also help judging and understanding its rivals, such
as non-reductive physicalism, versions of dualism, or even the
recently defended rainforest pluralism (Wimsatt 1994). If we know the
conditions imposed by the reduction relation on pairs of theories or
properties, we will also know what these anti-reductionist theories
are bound to deny.