A Best System Analysis of Special Science Laws
Markus A Schrenk
Postdoctoral Research Fellow, AHRC Metaphysics of Science Project
Birmingham | Bristol | Nottingham
Junior Research Fellow, Worcester College
Oxford
[email protected]
Abstract
This paper explores whether it is possible to reformulate Lewis’s theory of
fundamental laws of nature—his “best system analysis”—in such a way that it becomes a
useful theory for special science laws.
One major step in this enterprise is to make plausible how law candidates within best
system competitions can tolerate exceptions. This is crucial because we expect special
science laws to be so called “ceteris paribus laws”. I attempt to show how it is possible.
Thereby, a solution to the infamous difficulties surrounding the troublesome ceteris paribus
clause will be offered.
The paper is written in an non-reductionist spirit while Lewis’s original idea about laws
is part of his neo-Humean supervenience programme. Therefore, some friction results
when in comes to the status of special science properties. To resolve the tension, the paper
engages into a discussion about the unity or disunity, hierarchy or equality of the sciences
and the naturalness of their properties. (ca 9.750 w, inc fn)
Content
0. Lewis’s Original Theory
1. Challenge (1): Exceptions
1.1 Exceptions to Laws in Lewis’s System?
1.2 Tailoring Lewis’s Theory to the Special Sciences
1.3 Metaphysical Theory and Scientific Practice
1.4 A Theory for the Ceteris Paribus Clause
1.5 A Comparison to Pietroski and Rey
1.6 A Difficulty
2. Challenge (2): Demarcation
3. Challenge (3): The Status of Special Science Properties
3.1 Natural, yet, not Perfectly Natural
3.2 Natural and Perfectly Natural
3.3 Not Natural
4. Summary
5. References
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0. Lewis’s Original Theory
David Lewis has offered an acclaimed analysis of laws of nature: suppose you knew
everything and organised it as simply as possible in various competing deductive systems
that mention perfectly natural properties only. A contingent generalisation is a law of
nature if and only if it appears as an axiom or theorem in the one true deductive system
that achieves a best combination of simplicity, strength, and fit (cf. (Lewis 1973: 72-77),
(Lewis 1999: 39-43, 231-44)). To have strength is to bear a great deal of informational
content about the world; to be simple is to state everything in a concise way, not to be
redundant, etc.; and to fit is (especially for the probabilistic laws) to accord, as much as
possible, with the actual outcomes of world history.1
Here are three examples for possible competing systems. We could have, (i), a gigantic
lookup-table: for each space-time point the table lists the properties instantiated there. This
would be a very strong but not a very simple system and hence probably lose against the
others.2 We could, (ii), have a single line: “all electrons have unit charge”. This is a very
simple but indeed very weak system for which it is not possible to win any competition.
We could, (iii), have present day physics. If we are lucky it is not too bad a system when it
comes to a convincing combination of strength, simplicity and fit. Yet, maybe it will be
superseded at one point by some stronger, simpler, fitter arrangement. In any case, note
that Lewis operates from the view point of an omniscient being: “if we knew everything”
(Lewis 1973: 73). So, even optimistically speaking, at best the ultimate future physics might
come close to the real laws.
Lewis allows both axioms and also theorems of the winning system to be laws. The
best system might, therefore, include laws of the special sciences—chemistry, biology,
geology, medical sciences, maybe psychology—if they follow as theorems. This comes
close to a reductive account of special science laws: they derive from the more fundamental
axioms. However, I do not want to follow this route in this paper. I would like to suggest a
different, still Lewisian, yet, non-reductionist path to define what special science laws are.
Here is the basic idea. Run Lewisian best system competitions as usual but organise
separate competitions for each special science: one competition for chemistry, one for
biology, etc. depending on how far up you are willing to go.3
Pursuing this enterprise we will be confronted with three challenges:
(1) It has to be made plausible how law candidates within best system competitions can
tolerate exceptions—this is crucial because we expect special science laws to be so
called “ceteris paribus laws”.
(2) A criterion of demarcation for the different sciences has to be devised in order to
run best system competitions for these sciences separately.
(3) Various possibilities regarding the status of special science properties have to be
explored.
1
For reasons of simplicity I will, by and large, ignore the parameter of fit (but see fn. 16).
I ignore, for the sake of clarification, that the table would not list generalisations and so have not laws for,
according to Lewis, laws are only the generalisations within the system.
Note also that, although Lewis’s characterisation of laws makes reference to knowledge it does it in such a
way that a metaphysical view results, not an epistemic one: knowing everything = having all the facts laid out.
3
In fact, I will concentrate here on chemistry and biology for two reasons: I do not want to enter the debate
on what counts still as a proper natural science with natural laws and what not. I believe that both biology
and chemistry are fairly uncontroversial candidates. (This is less and less clear when we think of (neuro)psychology, sociology, economics, etc.) My second reason will only become clear later. It has to do with the
naturalness of special science properties.
2
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As for (1), one of the main suggestions of my paper will be that if we make Lewis’s theory
fit for the special sciences we gain, en passant, a coherent interpretation of the ceteris paribus
clauses which have caused much trouble in philosophy of science. Regarding (2) and (3),
we may acquire a deeper understanding of what is at issue when we follow reductionist or,
alternatively, anti-reductionist lines for special science laws and properties.4
1. Challenge (1): Exceptions
1.1 Exceptions to Laws in Lewis’s System?
Lewis’s theory does not straightforwardly allow for laws with exceptions. Yet,
especially the laws of non-fundamental laws like chemistry or biology are said to be ceteris
paribus laws, i.e., laws that are haunted by exceptions and therefore in need of proviso
clauses. And this seems to be the case no matter how far advanced a special science is
because those sciences are concerned with complex objects whose substructure can easily
break due to all sorts of contingent accidents (a tiger might not have stripes because of a
gene defect, haemoglobin might not bind O2 because of a molecular damage, etc.)
Damaged objects, however, do not display the regular behaviour they normally would
display.5
The truth about such proviso clauses is, unfortunately, that they cause trouble: many
philosophers claim that no good sense can be made of a statement like, for example, “All
Fs are Gs, ceteris paribus”. Such a phrase, so they say, is either tautologous like “All Fs are
Gs, unless not” or it stands for a proposition like “All Fs which are also… are Gs” the gap
of which we are unable to close. Some theory has to tell us how to deal with these
challenges.
The task for the following section is, first, to modify Lewis’s original theory of
fundamental laws so that exceptions to laws are admissible. I also have to show that the
resulting exception ridden laws are fit for competition. Later, I turn to the infamous cpclause itself.
Judging by some quotes it seems that, for Lewis, each and every exception would
catapult an alleged law immediately out of the realm of candidates for lawhood:
Few would deny that laws of nature, whatever else they may be, are at least
exceptionless regularities. (Lewis 1986: xi)
Admittedly, we do speak of defeasible laws, laws with exceptions, and so forth. But
these, I take it, are rough-and-ready approximations to the real laws. There [sic!] real
laws have no exceptions, and never had any chance of having any. (Lewis 1986: 125)
4
Emma Tobin drew my attention to one further worry: chemical and biological properties might be
dispositional in nature and/or have an essential nature. In that case, the properties’ dispositions and essences
would dictate (entail) a good deal of the laws so that a best system analysis would be superfluous. However, it
might well be that not all properties are dispositional and not all properties have an essential nature so that
not everything is already fixed. So, there might, after all, still be some work left for the system analysis. (Also,
if essences and powers of properties really dictate behaviour and regularities then the best system should
come up with laws that mimic precisely those essences and powers.)
5
Contrast this to the most basic particles of fundamental physics: these particles are not only called basic
because everything else is supposed to be made out of them. They are also called “basic” because they are not
supposed to have a further substructure which could break or be altered.
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But this textual exegesis is incomplete. When considering Lewis's theory of
counterfactual conditionals we get a different picture. That theory is couched in terms of
possible worlds and relies heavily on similarity relations between those worlds. Lewis does
not, however, attribute a special status to the laws of nature in similarity considerations.
While two worlds which have the same laws certainly have a good deal in common it is by
no means a necessary condition for worlds to share laws in order to be judged similar.
I could, if I wished, incorporate [a] special status of laws into my theory imposing the
following constraint on the system of spheres: […] whenever the laws prevailing at i are
violated at a world k but not at a world j, j is closer than k to i. This would mean that
any violating of the laws of i, however slight, would outweigh any amount of difference
from i in respect of particular states of affairs. I have not chosen to impose any such
6
constraint. (Lewis 1973: 72-3)
This is already a promising remark which loosens up the rigidity of laws. However, we
are only half way on the path towards a notion of laws that allows for exceptions. This is
because
the violated laws are not laws of the same world where they are violated. […] I am using
'miracle' [i.e., violation of law; MAS] to express a relation between different worlds. A
miracle at w1, relative to w0, is a violation at w1 of the laws of w0, which are at best the
almost-laws of w1. The laws of w1 itself, if such there be, do not enter into it. (Lewis 1986:
44-45; my italics)
What we still need is a violation of laws at home, i.e., at w0 itself. Here is a beginning:
A localized violation is not the most serious sort of difference of law. The violated
deterministic law has presumably not been replaced by a contrary law. Indeed, a version
of the violated law, complicated and weakened by a clause to permit the one exception,
may still be simple and strong enough to survive as a law. (Lewis 1973: 75; my italics)
Now, consider the following example: suppose there is a miniature possible world w
with a robustly best system, call it S, that consists of only 10 universally quantified
statements (all Fs are Gs, all Hs are Js, etc.) which, together with the initial conditions, truly
describe that world’s billion years history. Now imagine a very similar world w* which only
differs in that at a tiny space-time point <x0, y0, z0, t0> (we are talking about a nanosecond
and dimensions of nanometers) an F is not a G whereas the other myriads of F
instantiations are all also G instances. Isn’t it almost certain that in world w* no other
system but S* is the robustly better system where S* differs from S of w only in that the
universally quantified F-law is altered to !u (Fu " ¬@<x0, y0, z0, t0>u # Gu)? (¬@<x0, y0,
z0, t0>u =def object u is not where the “small, localized, inconspicuous miracle” happens).
It is neither credible that at w* a different system becomes robustly better nor that
suddenly there are draws between S* and other systems merely because of this minor
exception.
Needless to say, this all depends on how extended the violation is: if it is temporally
and spatially very limited then it is easy to imagine that the loss of simplicity, strength and
fit we have to accept when we “complicate and weaken” the law by a clause does not affect
the robustly best position of the system that includes that law. However, the more laws in
an alleged best system are affected by such exceptions, or the more extended the spacetime area is in which violations happen, the less likely it will be that this system is in fact
the best or, indeed, that there is any such best system. Yet, about that fact we do not have
6
Later, he promotes the status of laws a little. However, having the same exceptionless laws is, though of
first importance, still no necessary condition for similarity amongst worlds: “It is of the first importance to
avoid big, widespread, diverse violations of law.” (Lewis 1986: 47)
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to worry too much because it comes down to saying that the more messy the world is the
less likely it is that it is law governed. This has never been subject to doubt.
This brings my assessment of Lewis's original best system to a conclusion. The answer
to the question whether his theory can allow for laws with exceptions is “Yes”.7
1.2 Tailoring Lewis’s Theory to the Special Sciences
Now, if that works even for Lewis’s original theory targeting fundamental laws then it
should, in principle, also work for special science laws. I repeat the basic idea: accept law
candidates as rivals in Lewisian best system competitions as usual but organise separate
competitions for each non-fundamental science: one competition for chemistry, one for
biology. It is necessary to separate the realms because otherwise biological systems would
probably lose out against their physical competitors.8 Each realm will provide its very own
general statements, that is, law candidates. However, we cannot expect these special science
systems to be as neat as the systems for the fundamental level are (or, to be more cautious,
as neat as we expect the system for the fundamental laws to be). The best system for
biology might contain law candidates about, for example, tigers and their stripes which
have exceptions for albino tigers. The hope is that a system containing such law candidates
can still win the best system competition (within biology, within chemistry) because it is
unlikely that there are other, more advanced systems: no matter how hard we try there are
no systems strong and simple enough with only strict regularities in biology and chemistry.9
We need to spell out now how exceptions should be registered in special science
generalisations, for there are differences to exceptions to fundamental laws. The structure I
have suggested for fundamental laws facing miracles, !u (Fu " ¬@(x0, y0, z0, t0)u # Gu), is
not useful because the exceptions we have to expect in non-fundamental laws are rarely
restricted to certain space time regions.10 Rather, they are often bound to certain
individuals: take, for example, Bino the albino tiger who has no stripes. However, it seems
possible to exclude those individuals in the antecedents just in the same way in which we
exclude the small miracles in the case of fundamental laws.11 Yet, even if we accept this
possibility a new twofold difficulty appears.
First, for us, the exclusion lists of individuals in the antecedents of law candidates
would most probably be unmanageably long. Also, many if not most exceptions are
unknown to us. Therefore, second, we do not in fact find law statements of that kind in
actual scientific practice. Rather, the lawlike statements we come across in chemistry,
biology, etc. often bear (implicit) proviso clauses like ceteris paribus but they do not explicitly
list exceptional cases. Hence, my suggestion to characterise non-fundamental laws via
Lewis’s idea plus lists of exceptions in antecedents seems to distort not only what is the
7
For a more detailed version of this argument see (Braddon-Mitchell 2002), (Schrenk 2007: chapter 2.1&2.2).
But see my later remarks on that issue in §3.
9 There is a justified anxiety here that too many law candidates have too many exceptions in any possible
(biology, chemistry) system so that, as a consequence, many systems are roughly equally good and there is no
robustly best system. I think we have to bite this bullet and acknowledge that if it turns out to be true for a
science then that science has no laws (but mere helpful generalisations).
10
However, there are such cases: black swans seem to be a phenomenon endemic to the space-time region
called “Australia”.
11
I ignore the possible allegation that law statements ought not to refer to individuals but must contain
perfectly universal predicates only (but see (Schrenk 2007: 46-48) for a detailed discussion). In §1.3 we get rid
of the reference to individuals anyway.
8
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actual practice in science but also what is possible in scientific endeavour. How do we get
from the armchair into the lab?
1.3 Metaphysical Theory and Scientific Practice
I will now show that there is actually no major problem. I even claim that we can solve
some theoretical problems ceteris paribus clauses in law statements usually cause. This is a
pleasant consequence of a Lewis system applied to the special sciences.
For a start note that when dealing with Lewis systems we are always supposed to be
operating from a heavenly or, at least, metaphysical perspective: “if we knew everything
and organized it as simply as possible in a deductive system” (Lewis 1973: 73). To run the
best system competition we can, so to speak, employ an omniscient being. She will have no
problem comparing systems including law candidates with long lists of exceptional
individuals excluded in their antecedent. More importantly, she will have the complete
world history in front of her eyes so that she should also know all exceptions to all
regularities.12 In comparing systems she will, hopefully, come up with one robustly best
system. If not, not: as with fundamental sciences, the world—or the aspects of the world
we are dealing with—could be too messy to have any laws.
After her job is done (with success I assume) we ask our divine helper to hand over the
list of laws. However, and this is the fundamental clue which will help us to interpret
proviso clauses, not without asking her to perform some cosmetic surgery on the candidate
laws: delete all the exclusion phrases listing exceptional individuals from the laws’
antecedents and attach, instead, the clause “ceteris paribus” to these law statements. This
cosmetic operation is less superficial than it might seem: first, it translates the lengthy law
statements that have participated in the contest into the law statements typical for the
special science: law statements with proviso clauses. Hence, we have arrived from the
armchair to the lab. There is another positive aspect of this step: it seems to be a rather
contingent matter which individuals are exceptions to non-fundamental laws (think of the
albino tiger who has a gene defect caused by exposure to random x-rays). Hence, with the
introduction of the ceteris paribus clause we remove the reference to contingent matters of
fact from the law statement as well. There is an even greater benefit to be gained from this
genesis of our laws: a theory for these ceteris paribus clauses. Let me explain.
1.4 A Theory for the Ceteris Paribus Clause
The proviso clause serves as a reminder that the law, in its virgin form, listed
exceptions in its antecedent when competing together with other laws for the best system
status (balancing strength, simplicity, and fit ideally). That is, the proviso refers to, or,
better, stands for the long exclusion list the original competing law statement incorporated.
Also, and this is important to keep in mind, it reminds us of the fact that the original
statement describes a good chunk of world history even though it excludes many
individuals. It has precisely been selected (together with its peers) because of its strength.
Imagine that all the facts of the respective special science are somehow presented by dots
on an x-y-plane. The goddess’ task is, in this picture, to engage into a curve fitting exercise.
12
By “all regularities” I really mean all and not only those which might, after the competition, belong to the
best system if such there is.
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The bundle of curves which captures most dots will come out as the laws. Yet, we cannot
expect that all dots are captured.13
The present proposal is that the analysis or interpretation of provisos in proviso laws
should be something along the following lines:
“Fs are Gs, ceteris paribus” is, say, a biological law iff
if we knew everything about living creatures and organised it as simply as possible in
various competing deductive systems then “All Fs are Gs except for the individuals x,
y, z, ...” (with all exceptions explicitly mentioned) would appear as an axiom or theorem
in the one true deductive system (within biological systems) that achieves a best
combination of simplicity, strength, and fit.
The novelty of this proposal is that it does not aim to define provisos solely in a law
immanent way. Rather, provisos gain a holistic aspect transcending the isolated law:
statements bearing provisos are abbreviations of those ideal statements (including the
exceptions) which are part of the robustly best system. It is the membership in the best
system that makes the proviso clause acceptable. In a radical interpretation of this move a
vice is turned into a virtue: the ceteris paribus tag can now be interpreted not only as a
weakening of a law but rather as a knighting of a merely almost general statement that has
the honour to be included in the best system. “Ravens are black” is a mere factual and also
false general statement. “Ravens are black, ceteris paribus” shall, however, indicate both that
there are exceptions but foremost that it is a law belonging to the best system to describe
the biological world.14
1.5 A Comparison to Pietroski and Rey
The theory offered approaches the issue of cp-laws from a metaphysical perspective.
There are other theories which focus on epistemic aspects (such as predictions and
explanations) of laws with provisos. One such theory of ceteris paribus clauses has,
justifiably, some fame in the literature: a proviso clause is, in Pietroski and Rey’s view
(Pietroski & Rey 1995), a promise to the effect that if a prediction with the help of a
proviso law should fail a scientific explanation can be given of what went wrong. In their
metaphorical words, Pietroski and Rey treat these clauses as
'cheques' written on the banks of independent theories. (Pietroski & Rey 1995: 82).
These cheques represent a 'promise' to the effect that all [exceptional] instances of the
putative law in question can be explained by citing factors that are […] independent of
that law. (Pietroski & Rey 1995: 89)
Unfortunately, in a paper entitled “Ceteris paribus, there is no Problem of Provisos”
Earman and Roberts found a counterexample to Pietroski and Rey's promising account.
They showed that “All spheres conduct electricity” comes out as a ceteris paribus law in
13
I borrow the curve fitting metaphor from Robert Kowalenko.
One might start to wonder at this point how regularities with many exceptions are to be distinguished from
probabilistic laws. Shouldn’t we interpret “All Fs are Gs except for the individuals x, y, z, ...” as a probabilistic
law of kind “P(G|F) = n%” rather than as “All Fs are Gs, ceteris paribus”? The criteria simplicity, strength, and
fit will actually help to decide: a probabilistic law is simpler than a law listing exceptions (especially if there are
many). When it comes to strength laws with exceptions win: they, unlike the probabilistic laws, tell us exactly
where the consequent of the law is not instantiated. When it comes to fit, again the cp-law candidate is ahead
because it, listing its exceptions, has 100% fit whereas the probabilistic law’s probability might divert from the
actual frequency. For more on that matter see (Schrenk 2007: 88-90).
14
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Pietroski and Rey's analysis. Let, in “cp: !x(Fx $ Gx)”, Fx mean “x is spherical” and Gx
“x is electrically conductive”. Now, some fact explains why a certain x is not conductive,
for example, because x has a certain unfavourable molecular structure. Molecular structures
are explanatory independent from things being spherical and, hence, we have an
independent factor which explains ¬Gx.
If Pietroski and Rey's proposal were correct, then it would follow that ceteris paribus, all
spherical bodies conduct electricity. (Earman & Roberts 1999: 453)
More generally, whenever any object's failure to exhibit property G can be explained by
anything independent of whether the object exhibits property F, then Pietroski and
Rey's proposal implies that ceteris paribus, anything with property F also has property G.
(Earman & Roberts 1999: 453-4)
Do we have to give up Pietroski and Rey's account? I think not and Earman and
Roberts themselves point in the direction of where to look for a remedy. They claim that
Pietroski and Rey's account fails
because [it] does not guarantee that A is in any way relevant to B, which surely must be
the case if cp: (A $ B) is a law of nature. Perhaps Pietroski and Rey's proposal could
be modified to remedy this defect. But we do not see how to do this other than by
requiring that the antecedent of the law be relevant to its consequent, in a previously
understood sense of “relevant”. (Earman & Roberts 1999: 453-4)
Now, if we merge Pietroski and Rey’s epistemic theory with the metaphysical idea of
best system analyses for special sciences then counterexamples like Earman and Roberts'
pseudo sphere-law do not succeed anymore because they fail to meet the precondition to
be a member of the best system. The previously understood sense of being ‘relevant’ can
be derived from membership in the best system: F is relevant for G iff the respective
generalisation is a member of the best system.
Similarly, the best system account also explains why some absurd ceteris paribus laws that
have been invented (independently of considerations relating to Pietroski and Rey’s theory
but related) to expose the difficulties proviso clauses pose do not really count as laws.
Take, for example, the unpleasant “cp, eating cheese causes excess gas production, stomach
aches and diarrhoea”. Although it is true that cheese could cause these symptoms if
someone has a lactose intolerance this is not the norm. Now, we can rely upon the quality
of the system of laws which should, on the basis of the lack of strength and simplicity, not
except the above weird law candidate but rather its complement: “Cp, eating cheese
nurtures”. That is, nonsense ceteris paribus laws are dropped because of the simplicity
constraint: “eating cheese causes excess gas production, stomach aches and diarrhoea”
needs to list far more counterexamples than “eating cheese nurtures” and is therefore far
less simple and not a member of a best system.
1.6 A Difficulty
You might say that, at best, ceteris paribus clauses would be justified in the way I have
endorsed if we had the best system available. Yet, neither are we omniscient nor can we
expect any divine help. The question is, hence, whether the ceteris paribus clauses we attach
to our actual law candidates are equally justified for we might attach these clauses to
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candidates which are no members of the ultimate best system.15 It could even happen that
we temporarily use nonsense candidates like the absurd cheese law.
I openly admit that this is a weak point in the overall idea but let me hint at how one
might be able to dispel the worries: with a bit of luck, our system approximates the divine
ideal but even if there is and always will be some gap between our and the best system
there is a crucial structural similarity between them: simplicity and strength of a coherent
whole are certainly virtues scientists strive for and present theories do, to a great extend,
display these virtues. The hope is, then, that if, in the characterisation given above, we
supplement “the system which best balances strength, simplicity, and fit” with “… that we
currently have” my theory is still sound. The actual system is a mere shadow of the Platonic
ideal but, hey, it’s at least a shadow.
2. Challenge (2): Demarcation
Running competitions separately is not as straightforward as it might seem. We have to
suppose that it is possible to delineate chemistry from physics and biology from both
physics and chemistry to run individual contests for these different sciences. We could, as a
first step towards a separation, demand that for a chemical competition the respective law
candidates shall refer to chemical properties only, for a biology competition to biological
properties only, etc. but this is not feasible for the following reason: the borderlines
between those sciences might well be blurred. Laws of one science could purposefully
quote properties from other sciences. Take, for example, the biological (or medical) rule
that humans cannot survive much longer than ten days without water (H2O + certain
isotonic salts). I therefore propose, as a second step, the following ordering mechanism:
Admit law L into systems which are supposed to participate in competitions for science
X iff (i) L refers to at least one X property and (ii) X is highest up in the hierarchy of the
sciences of which L quotes properties.
The law about humans and their need for water, for example, then qualifies clearly as
biological law because being human is a biological property (and there is no further higher
psychological or economical property).
However, the problem of demarcation has merely been shifted: the categorisation of
law and system candidates has been reduced to property categorisation and it has yet to be seen
how this is possible. Still, progress has been made in an important respect: challenge (2)
(the demarcation of sciences) is now directly related to challenge (3) (the status of special
science properties). As we will see now, we thereby get down to the ground on the basis of
which any system competition eventually runs.
3. Challenge (3): The Status of Special Science Properties
The overall theory presented here is non-reductive in spirit. It treats the special
sciences as grown up theories. However, there’s some friction with Lewis’s original idea: as
I said at the beginning, Lewis allows both axioms and also theorems of the winning
fundamental system to be laws. That means especially that, if they follow as theorems, the
15
This is related to the specific problem of cp-laws mentioned earlier: “All Fs are Gs, cp.” Might just hide a
gap which we can’t fill: “All Fs that are not… are Gs”.
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laws of the special sciences are already captured by the original system. Yet, then, there is
no need for extra competitions as I have envisage them.
In order to assess the prospects of my non-reductive line and the meaningfulness of
separate competitions in the light of Lewis’s reductive attitude I have to shift the focus
from laws to properties. This will reveal certain background assumptions we generally
make about the hierarchy or equality of sciences, their (dis-)unity, and (anti-)reductionism.
The best system is the winner of a contest between systems competing for the highest
balance of simplicity, strength and fit. These systems consist of a number of law candidates
(some systems share law candidates others are completely distinct). The law candidates
themselves make reference to properties and properties are the basic atoms on whose basis
the whole business runs. How so?
It has not been mentioned yet but remember that Lewis’s main goal in presenting his
theory of laws is to follow his Neo-Humean Supervenience Programme where neo-humean
supervenience is
the doctrine that all there is to the world is a vast mosaic of local matters of particular
fact, just one little thing and then another. […] We have geometry: a system of external
relations of spatio-temporal distance between points. […] And at those points we have
local qualities: perfectly natural intrinsic properties. […] For short: we have an
arrangement of qualities. And that is all there is. There is no difference without a
difference in the arrangement of qualities. All else supervenes on that. (Lewis 1986: IXX).
Needless to say, if the arrangement of qualities, i.e., the mosaic of natural properties, is
all there is and all else supervenes on it then laws, too, are supposed to supervene on it.
That they really do is exactly what is supposed to be shown in Lewis’s best system analysis.
For that reason (and another one immediately to emerge) Lewis has, understandably, to
demand that “the primitive vocabulary that appears in the axioms [of the competing
systems; MAS] refer[s] only to perfectly natural properties” (Lewis 1999: 42). Only then can
laws be said to supervene on the arrangement of these properties.
The second motive for the vocabulary choice is, roughly, that if any odd language
(containing absurd gruesome predicates, for example) were allowed for the candidate
systems then comparisons between these systems regarding their simplicity and strength
would be futile because, for example, “a single system has different degrees of simplicity
relative to different linguistic formulations” (Lewis 1999: 42; further reasons are given by
Lewis on the same page).
Lewis mentions elsewhere “that laws and natural properties get discovered together”
(Lewis 1999: 43). This can be easily misunderstood. The statement is solely meant to be a
statement about human epistemology: scientists discover natural properties and the laws
together (if they are lucky enough). It is not a metaphysical statement concerning the best
system competition. It does not mean that laws and perfectly natural properties are jointly
the winners of such a competition. Only the laws are while the properties, in contrast, are
given already before the system analysis starts. They are the inventory nature comes
equipped with and the laws (amongst several other phenomena suspicious to the Humean:
dispositions, causation, etc.) supervene on the mosaic of perfectly natural properties.16
Note, however, that we, as human epistemic subjects, are indeed in the predicament to have to get going
with nothing in our hands: neither the laws nor the natural properties are given to us. It’s almost miraculous
that such a kind of bootstrapping should be possible. We are of course not without any material we can work
with: have must assume that the perfectly natural properties and the causal roles they play somehow leave
their impression on our senses even if not directly.
16
- 11 / 16 -
The consequence of all this is for my application of Lewis’s idea to the special sciences
that here, too, the basis on which the competitions run are properties: special science
properties. Yet, what is the status of biological and chemical (etc.) properties? There are
four questions to be asked:
(i)
(ii)
(iii)
(iv)
Does nature come equipped with these properties, i.e., are they natural
properties?
If so, are these properties perfectly natural (fundamental) or only fairly natural
(non-fundamental)?
Do they come sorted in classes (physics/chemistry/biology)?
If so, is there a hierarchy for these property classes?
Together, these questions comprise the third challenge I have introduced in the
introductory passages to the paper. I will discuss three options as answers to the first two
questions (i) & (ii) and within these options give answers to (iii) & (iv). ((i) & (ii) are
conceptually not independent so that a prima facie possible fourth option, non-natural but
fundamental, makes no sense). The different answers will have different consequences for
the prospect of the enterprise of this paper.
3.1 Natural, yet, not Perfectly Natural
Suppose biological and chemical properties are natural, yet, not perfectly natural (=not
fundamental). They do not belong to the basis on which, in Lewisian terms, everything else
supervenes. That is, they are not available as a part of the basis for the original competition.
Still, in some sense, nature comes equipped with them. In what sense? They fall out of the
theorems the best system delivers. These properties are all those which figure in derived
laws that follow from the fundamental axioms of the best system. Consider Lewis:
Laws will tend to be regularities involving natural properties. Fundamental laws, those
that the ideal system takes as axiomatic, must concern perfectly natural properties.
Derived laws that follow straightforwardly also will tend to concern fairly natural
properties. (Lewis 1999: 42)
If this is so, we get both at once out of the original system competition:
biological/chemical properties and their laws. However, in that case there would simply be
no need for further separate competitions. This is no secret and I have, from the start, laid
bare my background assumption that the special sciences are not reducible and that, I shall
add now, so are their properties. In other words, the possibility of biological and chemical
properties as natural, yet, not perfectly natural properties as here discussed is a possibility I
have to excluded (and actually did exclude) by fiat.
To (iii) & (iv). The questions about a categorisation and a hierarchy of properties is
of minor relevance in the reductionist scenario (3.1) here under discussion because a
satisfying answer to categorisation and ordering is only necessary if we need a criterion for
demarcation to run separate competitions. Yet, there are no separate competitions here.
However, questions (iii) and (iv) are interesting in their own right. Question (iii) about
a hierarchy of sciences (or of individual laws) can probably be answered when focussing on
entailment relations. Roughly, the further away a theorem is from the fundamental axioms
- 12 / 16 -
the less basic it is as a law.17 The axioms themselves lead the hierarchy. Note, however, that
this would first of all establish a hierarchy of laws, not of systems or sciences. Whether we
also gain a natural sorting of laws into different special sciences (iv) is questionable. Do
laws with the same distance from the axioms form natural classes? Do these classes
correspond to the special sciences? These are questions I (fortunately) do not have to
answer.
3.2 Natural and Perfectly Natural
Suppose, now, that biological and chemical properties are perfectly natural, that is, both
natural and irreducibly fundamental.18 At a first glimpse, this possibility seems to be more
promising when it comes to the enterprise of this paper. If biological and chemical
properties are indeed properties in their own right then, we might hope, we can
immediately kick of separate competitions for systems containing laws that refer to these
properties. However, something else is true then: in that case there would be phenomena
in the world which are not (or at least not entirely) built on the physical fundament.
Consequently, if the world comes indeed fragmented into different realms which are not
reducible to just one basic fundament then also Lewis’s original best system will include
laws concerning these biological and chemical properties. If it did not, it would very
probably not be the best system after all: it would lose a good deal of strength because the
goings-on of the whole biological and chemical realm, say, would not be captured.
Therefore, it seems yet again that there is no need for separate competitions as I have
suggested.
Luckily, the prospects are not so bleak after all. A closer inspection of Lewis’s
competition will reveal that there are separable parts which we can isolate within that single
one competition. In this scenario, the many rival systems include laws that quote not only
properties from a realm we would have supposed to be the sole fundamental level (particle
physics, say) but also from biology and chemistry. Some systems incorporate laws with
predicates referring to properties from all different realms, some are purer and only talk
physics (or biology, or chemistry).19 As already mentioned, those pure systems (whether
purely chemical, biological or even physical) won’t make the race. They should be weaker
in descriptive power than appropriately mixed systems. In fact, we expect the winner to be
a mix of physics, chemistry and biology.20
However, we should not just applaud the winner and instead also have a look at less
successful systems. Suppose that for this purpose we represent competing systems (i.e.,
bundles of law candidates) as dots on an x-y-plane. Some of these bundles will contain
more biological, some more chemical, some more physical generalisations, others, probably
the most successful ones, are a mix, yet others might contain a completely random and
useless bunch of statements (including statements like “all spheres made of gold are less
than a diameter” and worse). Arrange these systems, that is, their dot-representations, in
17
Or: the more contingent information has to be added to the fundamental axioms in order to make possible
a derivation of a theorem the less basic it is as a law.
18
They will probably not be fundamental in the sense of being a reduction basis for many other higher order
properties.
19
I assume here that such a categorisation into physics, chemistry, and biology can be made so easily. I will
later cast some doubt on this possibility.
20
Note aside that we certainly have to allow the possibility that some laws have exceptions as discussed in §1.
- 13 / 16 -
such a way that the more generalisations they share the closer their dots on the x-y-plane,
so that systems of biology and, likewise, systems of chemistry and physics will cluster.21
Assume further that the z-axis represents the degree of balance of the systems’
simplicity, strength, and fit. In other words, we have a function from systems to overall
system-performance along the parameters of simplicity, strength, and fit. Wouldn’t we
expect the following landscape as this function’s graph: within the area of mixed systems
we find a peak that is by far the highest: Lewis’s robustly best system. Let’s call this peak
“Mount Lewis”. Within what we might identify as clusters of pure biological systems and
clusters of pure chemical systems (also within the pure physical ones) there should be local
maxima: islands of high performance. In other words, there will be, next to “Mount
Lewis”, some hills which hopefully coincide with or are very close to our present biological
and chemical sciences.22
As attractive and aesthetic as this picture of a landscape from which we can read off
various sciences’ successful systems might be, it is, no doubt, also quite speculative. If we
are allowed further speculations we could even extend the picture. If we can grant more
and more properties an autonomous status within the class of perfectly natural properties
we could ask whether we get little hills for psychology, sociology, economy etc., as well. We
might also wonder whether there will be some “freak peaks” we did not expect and which
do not coincide with any of our sciences. However, in a less optimistic mood we could also
fear that there are mere shallow plateaus to be found in the country of system competition.
Nature might be too messy in every realm—biology, chemistry, physics—to provide us
with outstanding, robust best systems and laws.23
To (iii) & (iv). Next to these speculations and anxieties some real difficulties await us.
For if we suppose that there are basic properties other than those fundamental physics has
to offer then we might start to wonder what distinguishes them from the physical ones.
That is, if biological or chemical properties do not supervene on physical ones then, it
seems, there is no naturally given distinction and hierarchy of these properties. Worse, it
might not even be possible to sort them into different classes (as has tacitly been assumed
so far).
Fortunately, we do not need there to be such classes or a hierarchy beforehand (before
the best system analysis starts) because we do not run separate competitions. More
importantly, the similarity clusters into which we sort competing systems in order to
discern different parts of the one competition do not need a categorisation either. The
criterion of similarity for systems is merely based on the sharing of properties the systems’
laws refer to, not also on possible categories the properties potentially belong to.
Yet, if “nature is kind to us” (to borrow a phrase from Lewis; (Lewis 1994: 232-3)) and
the similarity clusters on the x-y-plane do in fact show peaks on the z-axis then we can take
these peaks to individuate distinct sciences. In other words, the prospective peaks are our
flagpoles indicating different realms.
21
Note that it is not necessary to know which properties are biological etc. It suffices to put systems closely
together that share quantifications over the same properties no matter which category these properties come
from. In fact, as we will see later, a categorisation might only be possible after the competition.
22
Assuming, off course, that our present research culture with its efforts and achievements is not completely
off the track.
23
Which, one would hope, is somewhat unlikely if the respective properties are really perfectly natural. The
whole next section is written with the hope that nature comes equipped with reasonable properties and
regularities and that our present sciences are not completely off the track to find those. If not, much of what
I am saying is mere fiction.
- 14 / 16 -
Interestingly, two questions remain yet unanswered even if individual peaks are
discernable: first, we do not know which peak represents which science and, second, we
have still no categorisation for properties although there is one for the winning systems
(local maxima). This is because (as mentioned in §2) the law candidates of the local maxima
systems will mention properties from different property categories (if such there are).
Both questions get answers if we manage to establish a hierarchy amongst the different
peaks. The one leading the hierarchy should correspond to physics, the medium one
chemistry, the lowest in rank biology. Further, we can assume that physics quotes physical
properties only, chemistry only chemical and physical ones, biology all three. Now, subtract
the physical properties from the chemical system and you retain the chemical ones, subtract
both the physical and the chemical ones from the biological ones and the biological
properties remain.
As I said, all depends on whether we are able to establish such a hierarchy amongst the
peaks but there is a most natural idea how to this: simply look at the peaks height, for
height is the indication for performance: biology might be relevant only for phenomena on
earth so that the rest of the universe remains undescribed by biology. Hence, it has much
less strength than both chemistry and physics which are both also relevant for Mercury,
Venus, Mars, and the rest of the universe. Hence, the biological peak’s height is
considerably lower. I assume that something similar is true about chemistry and physics.
The second reason for biology’s performance being lower than chemistry’s, chemistry’s
lower than physics’ relates back to the issue of exceptions discussed earlier. The number of
exceptions law candidates have to list will limit the height of their system’s performance,
too. The higher the number of laws with exceptions and the higher the number of total
exceptions within the system24 the less simple the law candidates. Now, if we can assume
that biology has to expect far more exceptions than chemistry, chemistry more than
physics (if there are any in physics at all) then, again, the heights of the respective peaks are
lowered. This underlines again that the altitudes of the local maxima are a good indication
for the standing in the hierarchy.25
There is a further interesting aspect about exceptions. Even a strong believer in antireductionism cannot deny that biological and chemical entities have a physical substructure.
An alteration or destruction of this structure has often massive consequences on the
respective higher level. That is, if the relevant entity still exists after a (maybe radical)
change it most probably won’t exhibit the same qualities and behaviour patterns it
displayed prior to the change (think of molecular changes in a creature’s DNA or some
physical brain damage, for example). Changing the perspective, we can also claim that
exceptions to regularities on the biological and chemical level can be accounted for by
structural changes on the physical level, yet, not vice versa.26 If this is true then the hierarchy
of sciences is affirmed by which science accounts for the exceptions of which other
science. Since physics is to be blamed for the exceptions of both chemistry and biology,
24
Counting exceptions and comparing the numbers might prove to be difficult in the following respect: there
are much more chemical entities than there are biological entities. So, the absolute number of exceptions
might be a deceiving measure. The hope is that it is possible to compare relative numbers somehow.
25
I have mentioned the possibility of “freak peaks” already. Who knows, maybe there are only such bizarre
peaks and none of them coincides remotely with any of our sciences. That is, all I say about hierarchies and
separate peaks is quite right except that the systems thereby established have nothing to do with our known
sciences. That’s no argument against the theory, though. Or, if it is, then it is one against Lewis’s original
system as well for who tells us that present day physics has anything to do with Lewis’s best system?
26
I assume that downwards causation is not possible.
- 15 / 16 -
and chemistry of biology but supposedly not the other way round the natural ordering
mechanism for the sciences is confirmed.
(Next to all the parts I have mainly been focussing on (pure physics, chemistry,
biology) there is still also Mount Lewis (a mixed system in the case under scrutiny) and it is
an interesting question (one I have no space to follow up) whether some (or even all)
axioms and theorems of this robustly highest mountain coincide with the axioms and
theorems of the pure biological or chemical local maxima. If properties of science X really
enjoy their own status then the contribution the respective law candidates make to the
overall performance of a system should not depend on the law candidates of science Y and
Z that are also included in that system. In other words, in that case, the height (or at least
strength) of Mount Lewis should more or less equal the addition of heights (or at least
strengths) of the local maxima peaks. Yet, this is probably too optimistic a picture. But
then again, what would a (massive) divergence indicate? Which system should we trust in
case of conflict? This is a contentious question.)
3.3 Not Natural
We have one option left: suppose biological and chemical properties are not natural.
Rather, assume they are human constructs. If they are the further question whether they
are fundamental (in our reading of fundamental, i.e., perfectly natural) makes no sense.27
Consequently, they do not figure in an original Lewis competition so that we are free to
run separate system competitions. A good start, it seems, but there are a couple of
drawbacks to be dealt with.
Because of the anthropocentric parameters which enter the picture one might claim
that it is questionable whether we will really be dealing with “laws” if separate system
competitions yield any. These alleged laws, so the critique continues, would not be what we
want laws to be: the objective rules nature comes equipped with just by herself; they will
have a human flavour. There is a cheeky response to this allegation: someone who buys
into the option under discussion has given up naturalness anyway. So surely naturalness
cannot be expected to re-enter the stage at some higher level of abstraction. The more
sincere answer is that, indeed, we might want to call the regularities we might gain from
competitions not “laws” but “lawlike rules” in order to distinguish them from the “real”
laws that might exist on the fundamental level for perfectly natural properties. The
mechanism to generate the lawlike rules is all the same.
To (iii) & (iv). How about the separation of properties into chemical and biological
ones and how about the hierarchy? If we already assume that the properties are social
constructs then the categorising and ordering is in our hands as well. It is entirely up to us
which set of properties we hand over to the omniscient being. If she’s also benevolent and
patient we can ask her to run competitions on any basis we like.
The biggest anxiety is, however, that because the properties are human constructs,
tailored to human interests and human intellectual dispositions, they might be a very weak
27
Note that the option of biological and chemical properties being not natural is open even if we believe in
the success of Lewis’s original system and also in his neo-Humean supervenience programme: we can still,
disregarding nature’s own joints, sort the phenomena as we please, if only for the fun of it. How successful
such non-natural categorisations are is, of course, questionable, especially when our goals are scientific
knowledge and/or effective technical applications. It seems that non-natural categorisations which in no way
track the truth will be quite inapt for such purposes.
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and murky bunch to work with. So, when we commission our divine helper to run the
competition she might well not come up with anything like a robustly best system. There
either is no system of regularities at all which we could call a system of laws or there is no
one system which is the winner, rather, there’s a draw between many systems. In the first
case we would probably have to accept that Human conceived properties are not a very
good basis. The second case need not worry us too much. If it is anyway agreed that we are
not dealing with real laws but with lawlike regularities there is no great urge for the
uniqueness of a system of such rules. For sure, we are watering down the initial project, a
theory for special science laws, more and more. Yet, we can blame human nature and not the
idea behind system analysis if nothing better is on offer.
We might be lucky still. Maybe we are, with our constructs, not too far off the track
after all. Lewis’s quote from above (§3.1) continues with a line I had omitted so far:
Regularities concerning unnatural properties may indeed be strictly implied [by the
original system], and should count as derived laws if so. (Lewis 1999: 42; my italics)
Suppose some regularities concerning humanly invented unnatural properties may indeed be
strictly implied by the original best system then, it seems, the constructs are so close to the
truth that their own competition should also yield some other lawlike regularities. (Those
which are shared with the original best system competition as theorems count anyway as
laws for that reason.)
4. Summary
I have introduced a way to characterise special science laws on the basis of Lewis’s best
system analysis. The core idea is to run best system competitions for each special science
separately. I have argued how this theory can provide a solution for the problems proviso
clauses pose for special science law statements. I have discussed why the paper has to
assume that biological and chemical properties have a life of their own either in that they
are perfectly natural properties or in that they are Human constructs taken seriously. I have
discussed the consequences concerning the hierarchy and unity of sciences these
possibilities would have.28
5. References
Braddon-Mitchell, D. 2001. “Lossy Laws”, in: Noûs 35.2, 260-277.
Earman, J. & Roberts, J.T. 1999: “Ceteris paribus, there is no Problem of Provisos”, in:
Synthese 118, 439-478.
Lewis, D. 1973: Counterfactuals. Oxford: OUP.
Lewis, D. 1986: Philosophical Papers II. Oxford: OUP.
Lewis, D. 1999: Papers in Metaphysics and Epistemology. Cambridge: CUP.
Pietroski, P. & Rey, G. 1995: “When other Things aren't equal: Saving Ceteris Paribus
Laws from Vacuity”, in: British Journal of Philosophy 46, 81-110.
Schrenk, M. 2007: The Metaphysics of Ceteris Paribus Laws. Frankfurt: Ontos.
28
Thanks are due to Emma Tobin who has read and commented on a draft of this paper; Stephen Mumford,
Matthew Tugby, and Charlotte Matheson for many valuable discussions; and the audience at the GAP6
conference, Berlin 2006, for their helpful comments on an early version of this paper.