Laws of Nature, Exceptions and Tropes
in : Philosophia
scientiae 7 (2), 2003, p. 189-219.
Max Kistler
Université Paris
X-Nanterre
et
Institut Jean Nicod
1 bis, avenue de
Lowendal
75007 Paris
Tel bureau 01 53 59 32 80
Fax 01 53 59 32 99
e-mail: kistler@ehess.fr
Résumé
Je propose une théorie réaliste des lois, en
termes de “tropes” (ou instances de propriétés), qui évite les problèmes de
“l’analyse du meilleur système” et le “problème de l’inférence” qui se pose au
réalisme des universaux. J’analyse le concept de situation exceptionnelle,
caractérisée comme une situation dans laquelle un objet particulier satisfait
l’antécédent mais non le conséquent de la régularité associée à la loi, sans
pour autant réfuter la loi. Pour tenir compte de cette possibilité, il faut
concevoir les propriétés mises en relation par une loi, comme dispositionnelles
et non nécessairement manifestes.
Abstract
I propose a
realist theory of laws formulated in terms of tropes (or property instances)
that avoids both the problems of the "best-systems-analysis" and the
"inference problem" of realism of universals. I analyze the concept
of an exceptional situation, characterized as a situation in which a particular
object satisfies the antecedent but not the consequent of the regularity
associated with a law, without thereby falsifying that law. To take this
possibility into account, the properties linked by a law must be conceived as
dispositional and not necessarily manifest.
1. Introduction
Realist and anti-realist conceptions of laws of nature have at least one
point in common : They both consider it to be a necessary condition for the
existence of a law that there be some universal correlation between states of
affairs which can be formulated by a universally quantified material
implication of the form[1]
(1) .
or, if two objects are involved :
(1a) .
Among other things, realists and anti-realists with respect to laws
diverge in their explanation of what makes one universally quantified
implication express (or be entailed by[2]) a law of nature while another taking exactly the same form expresses
only an accidental correlation. To take Goodman's famous example,
(2) "All coins in my pocket are made of silver"
can be expressed by the same formula - namely (1) - as
(3) "All electrons attract positive charges",
yet only the latter expresses a law.
I shall begin this paper with a list of desiderata which a satisfactory
theory of laws of nature should be able to meet. The search for a realist
conception is motivated by some important problems faced by regularity
theories, and in particular by the so-called "best system" analysis
of laws, which John Earman ([Earman 1984], [Earman 1993]) has called, in honour
of its principal defenders, the "Mill-Ramsey-Lewis view".
Then I turn to the analysis of laws which [Dretske 1977], [Tooley 1977]
and [Armstron 1983] have proposed in the framework of realism of universals.
While being able to solve the problems faced by the best system analysis,
realists of universals face a different and serious difficulty which [van
Fraassen 1989] has called the "inference problem". The crucial claim
made by realists of universals is that laws are relations between universals.
But to make good their explanation of how such laws produce regularities on the
level of concrete particulars exemplifying those universals, they have to
postulate a relation that spans the gap separating a relation between
universals and states of affairs involving particulars. The inference problem
arises because this postulate calls for justification on an even higher
ontological level, and thus leads to an infinite regress.
This is motivation enough to look for an alternative account which is
realist but avoids universals. Realism with respect to laws is the view that
the reality of laws is independent of our epistemic attitudes, in particular
our actual and future theories aiming at identifying them. According to
realism, laws are discovered rather than invented, and there may be laws we
perpetually ignore, and laws about which we are wrong. Now realism about
universals is not the only form of realism in this sense. I propose to
elaborate the idea of conceiving laws as relations between tropes. Tropes can
be characterised as abstract particulars, or as instantiated properties.
Then, I introduce the concept of an exceptional situation, characterized
as a situation in which some particular satisfies the antecedent but not the
consequent of the regularity associated with a law, without thereby falsifying
that law. These "exceptions" constitute a serious problem for all
theories which hold either that a law is
a universal correlation between particulars (as regularity theories do) or at
least entails such a universal
correlation (as realism of universals and one version of the trope theory [Fuhrmann
1991] do).
I examine some strategies (none of which takes tropes into
consideration) which have been proposed in order to cope with the difficulty
posed by exceptional or "abnormal" situations. [Cartwright 1983] and [Schiffer
1991] take the position that law statements expressing a regularity which
admits of exceptions are false. [Armstrong 1983] and [Pietroski & Rey 1995]
argue that the problem of exceptional situations characterises a specific class
of laws, called respectively "oaken laws" and "ceteris paribus laws", and try to
interpret those laws as having a special, and complex, logical form different
from (1) (or 1a). However, this is misleading insofar as the problem concerns
almost all laws, even of physics. As I try to show, the idea of solving the
problem by attributing to laws a new and complex logical structure leads to
unacceptable consequences.
My proposal for a solution to the problem posed to theories of laws by
exceptional situations is the following. It is not the logical form of laws but
their subject matter that must be changed. In realist terminology, the idea is
that the properties linked by a law are dispositional rather than manifest.
This conception enables us to explain why there are situations in which a
particular manifests the antecedent but not the consequent of a valid law.
Manifest properties are produced by the interaction of many dispositions so
that a disposition produced by one property according to a certain law may be
hindered from manifesting itself by a stronger disposition produced by another
property by virtue of the same or a different law. This idea can be spelled out
in both of the two main rival realist conceptions, in terms of universals and
in terms of tropes. However, the trope theory of laws is preferable because it
has the explanatory virtues of realism without being, as is Realism of
universals, victim of the inference problem[3].
2. The best systems analysis and the motivation for a realist account
What makes (2) and (3) statements of a fundamentally different type? The
following features constitute a list of explananda for any satisfactory theory
of laws. In statements expressing a law of nature - as (3) does - , but not in
those expressing a mere coincidence - as (2) does -
1. the predicates F and G which refer to the properties linked by the
law occupy opaque positions.
2. Law statements may serve as premises in scientific explanations and predictions;
3. can be confirmed by
situations in which both F and G are exemplified.
4. Laws as (3), but not universal coincidences as (2) confer modal force on the implication of the
consequent by the antecedent;
5. can contribute to determine the truth-value of counterfactual conditionals.
The numerous efforts of explaining the difference between lawful and
accidental universal correlations make up an important literature in philosophy
of science and metaphysics. The antirealist strategy is to look for the
difference by analysing the logical role a statement plays in the framework of
a theory (see, e.g., [Braithwaite 1958]). Statement (3) e.g. can be derived
from definitions and principles of physical theory whereas (2) cannot[4].
According to a widely accepted non-realist account of laws, namely the
one Earman ([Earman 1984, 229f.], [Earman 1993, 416]) has called the
"Mill-Ramsey-Lewis theory", (causal) laws are the "consequences
of those propositions which we should take as axioms if we knew everything and
organised it as simply as possible in a deductive system" [Ramsey 1929,
150]. In Lewis' words, a (contingent) "generalisation is a law of nature
if and only if it appears as a theorem (or axiom) in each of the true deductive
systems that achieves a best combination of simplicity and strength" [Lewis
1973, 73]. The fact that the best-system analysis invokes only criteria of
coherence and avoids postulating special nomological facts over and above the
set of empirically accessible particular facts, has the following implausible
consequences which constitute arguments against this account[5].
First, there is a subjective element in the establishing of the balance
between maximal simplicity and maximal strength. There seems to be no way of
justifying a decision between a rationalist view giving more weight to
simplicity and an empiricist view giving more weight to strength. But if this
is so, there may not be any unique ideal theory which is objectively privileged over its competitors.
Second[6], it is one of the aims of the construction of scientific theories to
find out which predicates denote natural properties, and to justify the
rejection of artificial properties like the one denoted by "grue"[7]. Thus, a scientific theory cannot rely on this distinction from the
beginning. Now it may turn out that the use of some such grue-like predicate
simplifies a given theory without diminishing its strength. In that case the
Ramsey-Lewis approach is condemned to accept the conclusion that the grue-like
predicate denotes a natural property after all[8].
Third[9], the best deductive system has grown out of a systematisation of
observed facts and regularities. What reason do we have to think that this
system can cover counterfactual situations, in particular possible situations
in which there are novel properties which have not been encountered in the
course of establishing the "best deductive system", as could happen
for artificial chemical elements or new biological species?
Fourth, the Ramsey-Lewis account makes coherence alone determine the
truth value of law statements. This has the undesirable consequence that if
science ends up with more than one system of equal maximal simplicity and
strength which do not have any axioms in common, this implies that there
aren't, after all, any fundamental laws of nature.
Fifth, the Ramsey-Lewis account excludes on a priori grounds the
possibility of a law 1) whose statement cannot be deduced from other axioms or
theorems already established and thus has to be itself considered as an axiom
and 2) which is instantiated very rarely in the actual history of the universe.
Such a law would - by property (1) - diminish the simplicity of the system, but
by property (2) would enhance its strength only very weakly. On the
Ramsey-Lewis account, such a regularity couldn't be a law, but a contingent
fact to be included in the “initial conditions”. Yet it seems implausible to
exclude the possibility that it be a law in spite of its properties (1) and
(2), on the basis of purely a priori considerations.
These aren't of course knockdown arguments against the best systems analysis[10]. Nevertheless, I think they are powerful enough to motivate a serious
attempt to construct a realist analysis which doesn't face problems of this
kind.
3. The realist account in terms of universals and the inference problem
Within a realist framework, the difference between (2) and (3) is
objective and independent of our knowledge about the regularities involved, and
a fortiori independent of our
theories involving them. The realist takes it that this objective difference
requires an explanation which is equally objective, i.e. makes no mention of
our epistemic attitudes towards the regularities to be explained. He has to
choose between two main options: Realism of universals and realism of tropes.
The realist explanation put forward by [Dretske 1977], [Tooley 1977] and
[Armstrong 1983] posits a relation between universals which is then used to
explain their universal coinstantiation. (2) differs from (3) in that (3), but
not (2) is backed by "a relationship between the universal properties
F-ness and G-ness" [Dretske 1977, 252]. Armstrong expresses this
"backing" with the following formula:
(4) "" [Armstrong 1983, 85].
Arguably, the realist of universals can explain all the explananda on
our list. In formula (4) the positions occupied by F and G on the right-hand
side of the main implication are transparent whereas those on the left-hand
side are opaque for F and G appear there as individual constants referring to
the universals F and G, and not as predicates as they do on the right-hand side
(This was our requirement # 1, see above).
The realist of universals argues that the postulate of a relation
between universals permits him to explain
why it is possible to explain why all Fs are G in case the correlation is
nomic, and why no such explanation is available in case the correlation is, on
the contrary, accidental. This explanation is provided by the main implication
in (4). There is no corresponding implication of (1) by something else in the
case of accidental uniformities (# 2). (Similarly for confirmability, # 3).
Furthermore, not only does the law N(F,G), i.e. the relation between
universals, (materially) imply that each object x which possesses F also
possesses G, but that it must. The universal generalisations backed by laws,
and only they, have the modal force of necessity[11] (# 4).
This is also what explains the fact that a law covers counterfactual
situations. Understood as a relation between universals, a law holds in all
possible worlds sharing the laws of the actual world[12], i.e. differing from it only in contingent facts. These are precisely
the worlds we have to look at in order to evaluate counterfactuals (# 5).[13]
The realist position which explains laws in terms of relations between
universals faces an important problem which van Fraassen has called the
"problem of inference" and which leads the realist of universals to
an infinite regress of the type of Plato's "Third Man"[14]. It arises from the need to justify the realist's explanation of how a
law implies a universal correlation between the instances of the properties it
links, i.e. the first implication in Armstrong's formula (4). How can it be
that a relation between two objects of second order, namely the universal
properties F and G, can entail an implication relation between the states of
affairs Fx and Gx which have the first order object x as a component? From a
logical point of view, such an implication cannot be justified. To justify the
(first) implication in (4) it is necessary to postulate a further relation
whose relata are the law N(F,G) and the individual objects x. If one conceives
of the relation between the universals F and G as a second order relation, this
new relation would have to be at least of third order. But then the problem of
justifying how a relation of third order can have logical consequences on the
level of first and second order relations arises anew, which is the beginning
of an infinite regress. Furthermore, it is fatal for the realist because it
cancels the explanatory advantage which he claims the theory in terms of
universals provides, and which constitutes the major motivation for the
explanation of the laws of nature in these terms.
If the law-founding relationship N(F,G) relating the universal
properties F and G does not provide the explanation which motivated its
postulation in the first place - i.e. if the pretended explanation is shown to
be logically invalid - the realist conception of laws in terms of universals
loses its major justification.
4. Exceptional situations
Realist theories of laws share with regularity theories the following
problem. The validity of most laws – with the possible exception of fundamental
physical laws - is compatible with the existence of exceptional situations[15]. Such situations constitute a problem for all theories which either
identify a law with a universal generalisation of the form (1) or (1a)(as the
regularity theory does) or hold that the former at least implies the latter (as
the Realist of universals does). The fact that they imply (1) makes these
accounts incapable of explaining why exceptional situations do not refute the
laws with respect to which they are exceptional.
The laws with the simplest logical structure are those expressing a
relation between different properties of one and the same object.
Let, e.g., Pl be the predicate "is a pendulum of length
l"[16] and Fl be the
predicate "has a period of ". ("g" designates the gravitational
acceleration on the surface of the Earth.) Let L(P) be the law that all Pl
are Fl. Then if L(P) was equivalent to the universal generalisation
U(P)
where the domain of x is the set of all concrete objects, one could
build the following deductive argument (I) : let b be a pendulum of length l.
According to U(P) all such pendulums have a period T(l).
(I)
But there is another well warranted generalisation which when added to
the major premise U(P), leads to a contradiction even if the first premise is
true.
Let A, said of an object satisfying Pl, mean that its maximum
angle is 45°[17].
Let L(Angle) be a law according to which pendulums of length l which
swing with a maximum angle of 45° have a period . Thus, this law implies that pendulums of length l
satisfying A do not satisfy Fl. The corresponding generalisation
could be written :
U(A) : .
But now we can build the following argument (II) which is valid just as
(I).
(II)
On the assumption that the particular premise is true, it follows that
at least one of U(P) and U(A) must be false. In this case it seems clear that
the blame should be put on U(P) which is too general as it stands because
pendulums with large amplitudes falsify it. Other situations which are
exceptional with respect to this law arise: 1. If a pendulum doesn't swing in
empty space, but is subject to frictional forces, 2. if a pendulum is made of
iron, and a magnet is in its proximity.
Similar arguments can be constructed for most laws, both in the special
sciences and in physics. Nevertheless, a situation like the one described is
not taken to falsify the law L(P) (nor the law L(A)). Thus, it seems desirable
to find a way of interpreting the relation between a law and the situations it
commands which allows us to stick to the truth of the law, notwithstanding the
existence of situations like the one described.
We can do this by distinguishing between manifest and dispositional
properties[18]. Manifest properties are causally efficacious: the falling of a fragile
vase as well as its hitting the ground are manifest in this sense. Their being
manifest makes them directly observable. However, the vase can possess the
property of being fragile without ever falling or breaking, and without
manifesting itself in any way by being causally responsible for anything. Something
can have a dispositional property at a time when this property is not causally
efficacious in any way. It is controversial whether dispositional properties are
sometimes efficacious, in bringing about their manifestations, e.g., whether
the vase’s fragility causally contributes to its breaking when the vase hits
the ground[19]. However, what makes a property like fragility dispositional is that it
is possible that at some times, it does not exert any causal influence.
Let us then conceive of the law L(P) as linking the property Pl
to a dispositional property whereas the corresponding universal generalisation
U(P) bears on manifest properties. This implies putting some distance between
the law and its impact on concrete particulars exemplifying their antecedent.
In other words, it is not possible to directly use the law itself in a deductive
argument (III) analogous to (I). (III) is not valid:
(III)
The task is to conceive of the law in a way which permits us to say that
all the premises of (III) are true although its conclusion may be false - as in
the situation described above. Formally, this is possible if (III) is an
enthymeme. It isn't a valid deduction as long as it isn't completed by
additional premises[20].
Since the existence of exceptional or "abnormal" situations
with respect to a given law L, i.e. situations in which the antecedent of L is
satisfied but not its consequent, but which nevertheless do not falsify L, has
been acknowledged, several proposals have been put forward in order to
accommodate this fact in a coherent general theory of laws. The accounts I
shall consider have one thing in common: they all treat the possibility of
admitting for exceptional situations as a property of a certain type of laws,
called by different authors "special science laws" (Fodor), "ceteris paribus laws" (Cartwright,
Schiffer, Pietroski and Rey) or "oaken laws" (Armstrong). I shall
argue that one should rather conceive of exceptions as situations in which a
law of the simple logical structure (1) or (1a) is at work but doesn't manifest
itself directly.
One strategy that has been proposed in order to deal with the problem of
exceptions is to complete the antecedent of the law statement in a way to
exclude all possible interfering factors. Canfield and Lehrer have called this
the "completeness condition" [Canfield and Lehrer 1961, 207]. The
problem with this idea is that laws completed in this way "are incapable
of being written down explicitly simply because the number of provisos implicit
in any law is indefinitely large" [Giere 1988, 40]. The fact that it is
impossible to state explicitly a list of requirements on a given situation
which would ensure that it obeys a given law has led a number of authors to
think that laws facing this problem are either vacuous ([Giere 1988], [Lange
1993]), unexplanatory if completed with an unspecific ceteris paribus clause [Cartwright 1983], or false if explicitly
stated, in which case the purported law simply does not exist ([Schiffer 1991],
[Earman and Roberts 1999])[21]. As long as one looks for a logical form of laws whose antecedent
contains an explicit specification of the situations in which the consequent
will be realised, it is indeed inevitable to draw one of these conclusions. As
Schiffer puts it, "if it were claimed that some special-science ceteris
paribus sentence expressed a ceteris paribus law, then the challenge would be
[...]: to specify the true proposition expressed by the sentence that would
prove the existence of the law" [Schiffer 1991, 15f.]. Schiffer points out
that there is no specific proposition expressed by a law statement fulfilling
the "completeness condition" because the number of provisos is
indefinitely large. Below, I shall add my own reason for rejecting the idea
that, in order to be equivalent to a true universal statement of type (1), law
statements must or can be supplemented by a completing clause. Yet these
reasons do not justify rejecting the notion of the very existence of laws (nor
of the existence of a restricted subclass of non-strict laws), for there is a
different way to solve the problem of exceptions to valid laws.
Geoffrey Joseph has proposed a substitute for the "completeness
condition" which has often been conceived as a ceteris paribus clause: It is not that other things must be equal in order for a law to be valid, it
is rather, Joseph says, that other constraints have to be absent. Thus, he considers the hypothesis that laws are made
literally true by ceteris absentibus
clauses. "Were it the case that all other factors were absent, then given
certain initial conditions, certain resultant conditions would obtain" [Joseph
1980, 777][22]. However, this proposal faces a serious problem: it requires to
consider the logical structure of law statements to be counterfactual,
which - as Joseph himself points out -
has unacceptable consequences. If the law is only true ceteris absentibus then it is vacuous in the actual world for the
factors stated in the antecedent are never instantiated alone. Furthermore, if
factors F1 and F2 are antecedents of two laws which interfere in the actual
world, then the possible worlds where one law is instantiated are different
from those possible worlds where the other is true. For in no possible world
can F1 and F2 both be exclusively present.
Joseph gives two more reasons for rejecting the ceteris absentibus proposal. First, the possible worlds in which
our laws are instantiated differ from the actual world not only in fact but
also in law. Thus laws are more precisely counterlegals : they are instantiated
in possible worlds with different laws from ours. But this means that we lose
our grip on the identity of the laws we are talking about. What is worse, it
seems incoherent to say of a law L both that it is valid in our world and that
it is instantiated only in worlds whose laws differ from those valid in the
actual world. The basic hypothesis of the ceteris
absentibus account which is that our law statements are not instantiated in
the actual world, seems incompatible with the idea that these laws are
nevertheless valid in the actual world. A further reason to reject the ceteris absentibus proposal is that it
forbids us, on pains of circularity, to analyse counterfactuals in terms of
laws, for on this account laws themselves are counterfactual and even
counterlegal.
The second proposal considered by Joseph is promising, indeed I think
more so than Joseph himself judges it to be. Rather than in changing the
logical form, it consists in changing the subject matter of laws. Each single
field law of physics, e.g., is not intended to accurately describe the actual
trajectory of particles. The subject matter of each of these laws is its own
distinctive field: gravitational, electromagnetic etc. "We must consider
each field law as giving only one component of the total force. [...] Each individual
field law would be interpreted not as a statement linking the sources of the
field to the behaviour of objects subject to the field, but instead as a
statement linking the sources of the field to the field itself, conceived as a
physical entity distinct from the objects upon which it acts" [Joseph
1980, 779]. Joseph then goes on to reject this proposal as well. His reason is
that, as history of science teaches us, the subject matters of the different
laws are not independent. It would be an error to think that it is possible to
correctly describe reality by simply juxtaposing the laws governing the
different kinds of fields and forces generated by them. "The
reconciliation of the theories of mechanics, gravitation, and electromagnetism
was not achieved through any simple manoeuvre such as logically conjoining them
or pooling their respective standard models." [Joseph 1980, 780]. Or, in
other words, "they do not fit together [...] into a single consistent
theory demonstrably possessing a physical model." [Joseph 1980, 780].
To overcome the difficulty pointed out by Joseph, I propose to
distinguish between the concept of, say, electrical charge, as elaborated by
physical theory, and the real property designated by the predicate
"electrical charge". We don't know the property perfectly well as
long as we haven't discovered all the laws governing the behaviour and
interactions of the particles possessing that property. Joseph is right to note
that we can be certain of this ignorance to the extent that different theories
aiming at characterising different properties cannot be satisfied by any single
model. But the following picture is compatible with this ignorance: every
property is characterised[23] by the set of dispositions to
behave and interact, that it bestows on the particulars possessing it. These
dispositions are determined by the laws of nature in which the property
participates[24]. A theory can be seen as providing implicit definitions of its
theoretical predicates. But what these definitions describe is not the real
property itself but a concept intended to match as well as possible with that
property. In general, the match between the concept implicitly defined by a
theory and the property is imperfect. In that case reality is not a model of
the theory.
Drawing the distinction between the concepts elaborated by a theory and
the real properties those concepts are intended to capture, characterises
realism with respect to those properties and the laws governing the
dispositions characterising them (and the particulars possessing the property).
Realism implies that it is possible that we ignore the existence of some
properties (maybe even fundamental ones), and that we are possibly wrong about
the exact nature of those properties we know something of ("the exact
nature" here always means: the set of dispositions regulating the
behaviour and interactions of the particulars possessing the property). In a
realist framework, Joseph's objection would strike only the thesis (which
nobody holds) that science has already reached its endpoint: that we already
perfectly know the laws of nature. More reasonably, our law statements are
hypotheses using concepts and conjecturing relations between them which only
imperfectly correspond to real properties and their relations. Reconciling our
different theories requires modifying them, but this modification may lead to a
convergence of the models of each theory towards a single consistent model,
i.e. reality.
Pietroski and Rey's proposal takes into account the fact that we cannot
explicitly state which situations are regular with respect to a given law and
which are exceptional. But according to these authors, admitting for exceptions
is a property characterising a certain class of laws, called "ceteris paribus laws" which possess
a specific complex structure. In order for such a law to be true, it must
contain a clause quantifying over all possible interfering factors. In each
situation where the consequent isn't exemplified although the antecedent is,
there is some explanation or other, in terms of an interfering factor H
explaining why the consequent isn't exemplified. They give the following
example:
"A chemist holding that cp(PV = nRT) [where "cp" means
"ceteris paribus"], is
committed to the following: if a gas sample G is such that , there are independent factors (e.g. electrical attraction)
that explain why with respect to G.
[...] C- abnormal[25] instances are to be expected. But
such instances must be explicable by citing the factors ignored, else the
putative cp-law is either vacuous or false. [...] If a putative cp-law is such
that all its C-abnormal instances can be explained by citing independent
factors, [....] the cp-law is true. But if there are inexplicable C-abnormal
instances, the putative cp-law is either vacuous or false." [Pietroski and
Rey 1995, 91/2].
This account is not meant to be verificationist. A cp-law[26] can be true even if we never find
the interfering factor in some C-abnormal situation. Such a situation doesn't
objectively falsify the law even if the interfering factor responsible for the
non-satisfaction of the consequent is unknown. As realists, Pietroski and Rey
hold that we can be wrong about whether the law is falsified by such a
situation or not, and that this is an objective difference (about which we may
be ignorant in a given situation).
Pietroski and Rey's account faces the following difficulty: they don't
make it perfectly clear whether their analysis of the cp-clause is intended to
be epistemological or ontological, i.e. whether it expresses the way we
describe, predict and explain situations with the help of laws or whether it
means that the laws themselves have a so much more complex structure than the
simple universal generalisation of the type (1) or (1a).
Pietroski and Rey use the metaphor comparing cp-clauses with
"cheques written on the bank of independent theories" [Pietroski and
Rey 1995, 89]. This suggests an epistemological reading: to make good the
cheque, it must be cashed some day by some actually formulated theory. Given
that they consider the cp-clause as a part of the law itself, the truth
conditions of the law become dependent on explicitly formulated theories. This
interpretation of Pietroski and Rey's proposal would burden them with the
difficulties of nominalistic accounts of laws - in particular the best systems
analysis of laws. However, this is not, if I understand them well, the
interpretation intended by these authors. But if one follows the alternative
ontological interpretation, one arrives at a picture of laws whose consequences
are equally implausible. In fact, on an ontological reading, their analysis of
ceteris paribus laws appears as an elaboration of Armstrong's conception of
oaken laws, and inherits its difficulties.
Just as Pietroski and Rey, Armstrong takes it to be a property of a law to be either strict or cp. He
expresses this by the terminological distinction between "iron laws"
and "oaken laws". A law is iron if there are no exceptions to it,
otherwise it is oaken. Armstrong spells out the concept of an oaken law in the
following formula:
(6) "It is a law that Fs are Gs, except where Fs are Hs, are Js,
and Ks ... and so on for an infinite set of distinct properties."
[Armstrong 1983, 28].
There seem to be two options for interpreting this distinction: either
the question whether a law has exceptions or not is external to it, i.e., with
respect to that law it is just an accidental fact, or it is internal, i.e., a
property of the law itself. On the first alternative, the distinction between
iron and oaken seems ad hoc and
unjustified. If it isn't the laws themselves which somehow permit or prevent
the possibility of exceptional situations, the distinction doesn't really bear
on them, but rather on the accidental fact of which sort are the situations in
which the antecedent of the law happens to have been exemplified so far. Due to
another accident, a given law can switch category. When an exceptional
situation arises for the first time, the law ceases to be iron and becomes
oaken. So let us consider instead that being iron and being oaken are
properties of the law itself.
This leaves us with two major difficulties. Armstrong's conception fails
with respect to its primary aim of finding a criterion justifying the
distinction between the two types of laws. Formula (6) is incapable of
establishing that oaken laws are any different from iron laws. A particular
situation satisfies the antecedent in Armstrong's formula (6) only if it is not
exceptional, namely if the properties H, J, K etc. which are responsible for
eventual exceptions to the law N(F,G) are absent. This means that, by
definition, there cannot be any exceptions to the law thus formulated. In the
end, Armstrong's proposal leads to transforming oaken laws by stipulation into
iron, i.e. exceptionless ones, by integrating into its antecedent the
requirement that potential exception-provoking factors be absent.
Next, Armstrong's proposal faces the difficulty of all ceteris absentibus theories, that it
makes laws expressed by a formula of type (6) vacuous. But there is a another,
stronger argument against Armstrong's strategy to eliminate the problem of
exceptional situations by explicitly mentioning their absence in the
antecedent. The various factors H, J, K, etc. which are potentially responsible
for exceptional situations with respect to the law N(F,G), have this potential
of interference in virtue of other nomic links between H and F, between H and
G, between J and F, and so on. Not only do these factors already form an open
disjunction with an indefinite number of terms, but the laws linking them to F
and G, the antecedent and consequent of the main law N(F,G) are themselves of
the type of (6): They are themselves subject to exceptions. This puts Armstrong
on a slippery slope[27] which leads to depriving the very
notion of a law of its content. For at the end, following the logic of
Armstrong's proposal (6), each single law turns out to contain a large number
of other laws. Appearances to the contrary notwithstanding, the truth
conditions of laws are holistic. But the concept of a law of nature makes sense
only insofar as a law is what determines regularities across the diversity of
situations. If in the limit there is only one law which equally governs all situations, the very distinction
between regularity and diversity loses its content.
Pietroski and Rey's elaboration of Armstrong's proposal in which they
substitute a quantificational structure for Armstrong's points of suspension
"...." has the merit of making this difficulty more clearly visible.
Each law - among those concerned by our problem - contains an existential
quantification running over all "otherwise nomological" [Pietroski
and Rey 1995, 92][28] properties H. Furthermore what is
decisive about these "interfering factors" H is that they can
"explain" the non-occurrence of the consequent of the main law. This
explanatory capacity appears as an unanalysed constituent in Pietroski and
Rey's formula of the logical structure of cp-laws. But, at least if we stick to
a realist interpretation of explanation which seems to be in the spirit of
Pietroski and Rey's account, each potentially interfering factor H will be able
to interfere and its presence will be part of an explanation of non-G in virtue
of other laws, linking H to non-G.
Thus they are, just as Armstrong, at the beginning of a slippery slope: not
only are there infinitely many potentially interfering factors H, but their
potential interference is covered by just as many laws in general different
from the law one started with, and each of those laws will in general also be a
cp-law with another clause quantifying over interfering factors, and so on. In
the end, laws appear to be holistic, in the sense that the truth conditions of
each law depend on those of many, if not all, other laws. A single law cannot
be discovered or stated in isolation from many other laws because its very
content depends on those other laws. I take this consequence to be in conflict
with the notion of a law as an objective feature of nature which is responsible
for uniformity on a background of variety.
Note that this problem is in a sense worse then the one raised by Duhem
and Quine's thesis that law statements cannot be tested independently of the
whole theory they are part of. If Armstrong or Pietroski and Rey are correct,
not only is it impossible to test law statements independently, but their very
content is dependent on many other laws. In other words, not only our epistemic
access to laws, i.e. the testing procedure, is holistic as Duhem and Quine
argue it is, but the very content of the laws is holistic too.
There is a variant of the strategy put forward by Armstrong and
Pietroski and Rey which is a quite familiar move in reaction to the existence
of non-falsifying exceptional situations. Consider once again the pendulum law
L(P).
It is tempting to protect the law from falsification by such situations
by introducing the qualifier "classical" into the law statement, thus
restricting its validity to "classical pendulums" only. All pendulums
not obeying the law count by definition as non-classical pendulums which thus
do not satisfy the antecedent predicate "is a classical pendulum of length
l". This move can be interpreted in two ways: either one considers that it
makes a convention of the validity of
the law, or one considers the word "classical" as an abbreviation for
Armstrong's clause stating that an infinite conjunction of potential sources of
exceptions are absent, or for Pietroski and Rey's existential quantification
over them. It seems to me that both possibilities obscure the difference
between regular and exceptional situations instead of contributing to its
explanation. Making it a convention deprives the law at least partly of its
empirical content while the second option results, as we have seen, in making
the law's truth conditions holistic.
5. Laws allowing for exceptions
I shall now sketch a theory of laws in terms of tropes. The idea is to
localise the constraint exercised by a law on the situations falling under it,
not at the level of universals nor at the level of concrete objects, but on the
level of the tropes which are the ultimate constituents of concrete particulars[29]. Tropes are properties as they are instantiated by particulars,
"property instances" or, in other words, abstract particulars[30]. I shall treat as equivalent the locutions "the particular a consists of the tropes in set E"
and "the particular a possesses the tropes in set E". Among the
tropes a particular possesses some are manifest and some are dispositional[31]. The first are actually causally efficacious, notably when they are
observed, the second are only potentially efficacious. Laws of nature are
dependence relations among such tropes which are partly dispositional. We can
postulate that laws have the simple logical structure of the universally
quantified material implication (1) or (1a) on the condition that the domain
over which the variable x ranges is fixed in an unusual way. I propose to let
this domain be constituted, not by the set of all concrete particular objects
but by the set of tropes constituting these objects. The material implication
of the consequent by the antecedent means that the disposition referred to by
the consequent of a law is present in every situation in which the antecedent
is present.
Why do we need to assume that some of the tropes related by a law are
dispositional? A disposition cannot be reduced to its manifestations even
though its manifestations are what ultimately justifies postulating its
existence[32]. A disposition doesn't necessarily manifest itself and its not
manifesting itself doesn't mean it isn't present[33], but postulating its existence can explain a manifestation
characteristic of that disposition. The difference between the presence of a
disposition and its manifestation is precisely what permits us to explain how a
situation may be exceptional with respect to a law in that its antecedent is
satisfied but not its consequent, and how the law may still remain valid. If we
conceive of the consequent as designating a dispositional property, we can
explain such situations in the following way: the consequent is satisfied as
well as the antecedent, but it doesn't manifest itself because it is overridden
by other dispositions imposed on the same particular at the same time, due to
other tropes according to the same law, or according to different laws. Given
the possibility of several dispositions competing with each other for
manifestation, we can say that each disposition imposes a constraint on the
manifest properties of the particular at a given time[34]. The possibility of a disposition not to manifest itself because it is
overridden by other dispositions is responsible for exceptional situations.
This happens, e.g., if a pendulum not only contains the trope designated by
"the length of this pendulum", but also the trope "this
pendulum's consisting of iron" when an external magnetic field is present.
The second trope can then be responsible for the pendulum's behaving in an
exceptional way and in particular for its not having the period T which the law
L(P) prescribes it to have. Such situations are not exceptions to the law itself and do not falsify it
because the consequent disposition is present in any case, but they are
exceptions to the generalisation
associated with the law. This is because the generalisation bears over the
manifest properties of the particulars, and an exceptional situation is one in
which the antecedent property is manifest but not the consequent. Universal
material implications can express valid laws having exceptions if one takes the
domain of the variable x in (1) or (4) to range over tropes which are partly
dispositional. It is only if one takes the domain of this variable to be the
set of all concrete objects, that a law statement linking F to G is falsified
by exceptional situations in which a concrete object possesses the property F,
but not the property G.
To speak of "normal and abnormal conditions" instead of
exceptional and regular situations may be misleading because this terminology
wrongly suggests[35] that normal conditions occur more
often than abnormal ones. The relevant distinction is rather between situations
in which the disposition manifests itself (more or less clearly) and others
where it does not. The frequency of situations where a disposition manifests
itself and situations where it doesn't is likely to be the opposite of what
talk of normal conditions suggests: situations in which the pendulum law or the
free fall law manifest themselves (by manifesting a correlation to a very good
approximation of the antecedent property with the consequent) are rare in
nature or even exclusively to be found in laboratory environments[36]. But the point is not simply to reverse the sense of the dependence of
the "normality" of conditions on frequency, but to recognise that the
truth of a law is independent of the frequency of situations in which it
manifests itself.
Thus we can rescue the traditional conviction that laws are
exceptionless. To do this we have to distinguish the law from the corresponding
generalisation bearing on manifest properties. The generalisation has
exceptions, which comes to saying that taken literally, it is false. The law,
as distinct from the generalisation associated with it, may be stated in a
simple logical form - without reference to other laws, possible interfering
factors etc., and nevertheless may be literally true, the truth-maker being a
relation between tropes, the consequent being dispositional and thus not always
manifesting itself.
This proposal could not pretend to solve the problem of exceptions but
only to indicate a direction in which it might be possible to construct such a
solution. To yield a complete solution, the present proposal would have to be
complemented by a theory of dispositional properties which I do not attempt to
provide here. The task of that theory would be to explain how and when a
dispositional property manifests itself, and how exactly several dispositional
properties jointly determine a manifest property. However, I do not think that
the reference to dispositional properties has the consequence that the laws
with a dispositional consequent are immunised against refutation, or
"vacuous", as they are in the traditional cp-account, for the reason
that observation does not directly
allow to decide whether or not the consequent is exemplified in a given test
situation. Dispositional properties are theoretical properties, and the
predicates designating them cannot, as [Carnap 1936] and [Hempel 1965] have
shown, be operationally defined by simple material test conditionals. Whether
or not a theoretical, or not directly observable, property is exemplified by a
given object cannot be judged by a simple observational criterion, but depends
in the end on the application of a - more or less holistic - inference to the
best explanation. Only in a strict verificationist framework does this imply
that dispositional predicates do not have a well determined meaning or do not
designate objective properties.
6. Laws, accidental regularities, and the "inference problem"
It is the fact that the consequent of a true law statement refers to dispositional tropes that allows us to
explain, as we have tried to do in the preceding section, how a law can be
valid although there are many situations that are exceptions to it. But it is
the fact that the consequent of a true law statement refers to (dispositional) tropes that allows us to explain, as I
shall try to show now, all the differences between laws and accidental
universal regularities, and nevertheless to escape the "inference
problem" (introduced in § 3 above). The account of laws sketched in the
preceding section has a realist explanation of the difference between (2) and
(3).
(2) "All coins in my pocket are made of silver"
(3) "All electrons attract positive charges",
In the case of (2) there is no dependence relation between the tropes
referred to by the antecedent and consequent, whereas there is such a nomic
dependence in the second case. The competing realist accounts of [Armstrong 1983]
and [Fuhrmann 1991] also derive
(1) .
by postulating relations on a deeper ontological level, but in doing so
they both fix the domain of the universally quantified variable x in formula
(1) to consist of concrete objects. Neither Armstrong's nor Fuhrmann's theory
does justice to the complexity of the relation between a) relations among the
properties (tropes) themselves and b) the behaviour of the concrete objects
possessing these properties.
Let us have a look at Fuhrmann's proposal of a trope theory of laws. It
turns out that his version cannot overcome the difficulty of accounting for
exceptional situations. Fuhrmann conceives of the nomic relation as a
consequence of a relation between tropes which has the logical structure of the
part-whole relation. Fuhrmann analyses the concept of law in the following way:
" 'All Fs are Gs' expresses a true basic law of nature if and only if the
predicates 'F' and 'G' correspond to resemblance classes of tropes and N(F,G)
is contingently true." [Fuhrmann 1991, 73]. N(F,G), the relation of nomic
necessitation N between resemblance classes of tropes is defined thus:
(5)
"N(F,G) iff " [Fuhrmann 1991, 72].
The crucial relation "" between the tropes x and y "exhibits the
characteristics of a part of relation"
[Fuhrmann 1991, 65].
Fuhrmann doesn't tackle the issue of exceptions explicitly, but his
conception excludes their possibility in principle for if the existence of a
law linking F and G is based on a ""-relation between tropes x and y belonging to the
resemblance classes characterising respectively F and G, then all individuals
containing a trope of type F will necessarily contain a trope of type G. Yet it
avoids the inference problem by conceiving laws as relations between tropes.
Tropes are particulars just as the concrete objects of which they are the
ultimate constituents and whose behaviour they determine in virtue of their
lawful relations to other tropes. Thus the trope theory avoids the ontological
gap leading to the inference problem, i.e. the gap between the law itself and
its manifestation in the regular behaviour of particulars.
If one accepts to supplement the trope theory with the distinction
between a disposition and its manifestation, it seems to inherit the
explanatory virtues of realist theories while avoiding the inference problem
that threatens Realism of universals. As all realist accounts, the trope theory
does not identify a law with an empirical regularity. It explains why some
regularities - like (3) - are nomic with the hypothesis that these regularities
result from the manifestation of a lawful dependence between tropes. Other
regularities - like (2) - are accidental because they are not manifestations of
underlying dependence relations between tropes.
A trope theory doesn't encounter van Fraassen's "inference
problem" because it takes both the entities whose relatedness founds
nomicity and the entities whose behaviour is determined by those nomic
relations as first order particulars[37]. Tropes are abstract
particulars (like: the length of this pendulum) whose relations are called for
to explain the behaviour of concrete
particulars (such as this pendulum).
Finally, let us check whether our proposal meets all of the adequacy
criteria we imposed on any acceptable account of laws (cf. section 2 above).
The referential opacity of law statements is explained by the fact that the
universally quantified variable x in (1) is interpreted as ranging over tropes.
It may be a law that all F are G, and an accidental truth that G and H have the
same extension. Then it is true that all F are H, but it need not be a law. By
the criterion of truth preservation on substitution of coreferential terms, the
statement “it is a law that all F are G” is opaque in the positions occupied by
F and G. Tropes, being properties, can differ even if they are extensionally
equivalent, i.e. if they are contained in exactly the same concrete
particulars. Substituting H for G in the statement “it is a law that all F are
G” does not preserve its truth value if there exists a dependence relation only
between tropes of type F and G, but not between those of type F and H (#1).
As other realist accounts, trope theory explains why laws, but not
accidentally true universal generalisations, may be part of scientific
explanation and prediction (#2), and objects of confirmation (#3). A law can
non-vacuously explain or predict particular manifestations of a given
phenomenological regularity because it differs from that regularity itself. An
accidental generalisation just states a regularity and thus cannot explain that
same regularity. The link between a disposition and its manifestation must
itself be lawful. It is only on this condition that laws as construed by trope
theory will help in explaining and predicting. If laws are relations between
tropes, they can be tested, and our belief in their truth can be justified,
only indirectly, by the predictions which can be derived from them concerning
the properties of concrete objects. But the detailed account of how several
dispositions present at the same space-time point result in some well
determined manifest property must be left to scientific enquiry. In classical
mechanics, for example, the net force acting on a given point is calculated by
vector addition of the component forces. In quantum mechanics, all sources of
interference on the behaviour of a given system are part of the Hamiltonian
describing the state of that system; and perturbation theory permits incorporating
the influence of these factors in the prediction, using the strength of the
factors acting in the situation as an ordering principle.
One item on our list of explananda for any acceptable theory of laws is
perhaps more difficult to interpret. Does our account in terms of tropes
vindicate the intuitive idea that laws confer modal force on their
instantiations (#4) ? The very phenomenon of exceptional situations obliges us
not to interpret this modal force as bearing upon the manifest properties of particulars.
It cannot be necessary that the behaviour of all freely falling bodies obey the
law of free fall because this isn't even actually the case. In other words, the
free fall law doesn't describe the behaviour of falling bodies in all possible
worlds, for the actual world is a counterexample.
But, on the other hand, it is essential to the concept of law that its
domain of validity extend beyond actuality and also cover merely possible
situations. It is only thus that laws can contribute to the determination of
the truth value of counterfactuals (#5).
The solution lies in attributing necessity to the dependence relation
between dispositional tropes. The relation between the manifest properties referred to by the antecedent and the
consequent of a true law statement isn't even that of actual let alone
necessary universal correlation, but the underlying dispositions are linked with necessity. This is what makes laws
relevant to the evaluation of counterfactuals. If these counterfactuals bear on
the manifest behaviour of objects in possible but non actual situations, a law
may not be sufficient as a truthmaker for the counterfactual because the
possible situation may be exceptional, but the law is sufficient as a
truthmaker for counterfactual statements bearing on the presence of
dispositional properties[38].
References
1983 What is a Law of Nature,
1989 Universals. An Opinionated Introduction,
1992 Properties, in
[Mulligan, Kevin (ed.), Language, Truth,
and Ontology,
1997 A World of States of Affairs,
Bacon, John
1995 Universals and Property Instances. The
Alphabet of Being, Aristotelian Society Series, Vol. 15,
Campbell, Keith
1990 Abstract Particulars,
Canfield, John
and Lehrer, Keith
1961 A Note on
Prediction and Deduction, Philosophy of
Science, 28, 204-208.
Carnap, Rudolf
1936
Testability and Meaning, Philosophy of
Science, 3, 420-471.
Cartwright,
1983 How the Laws of Physics Lie,
1989 Nature’s Capacities and their Measurement,
1999 The Dappled World, A Study of the Boundaries
of Science,
Coffa, José A.
1968 Discussion:
Deductive Predictions, Philosophy of
Science 35, 279-283.
Dretske, Fred
1977 Laws of
Nature, Philosophy of Science,
44, 248-268.
Earman John
1984 Laws of
Nature: The Empiricist Challenge, in [Bogdan, Radu J. (ed.), D.M.Armstrong, Profiles, Vol. 4,
1993 In Defence
of Laws: Reflections on Bas van Fraassen's Laws
and Symmetries, Philosophy and
Phenomenological Research,
53, 413-419.
Ellis, Brian, and
Lierse, Caroline
1994
Dispositional Essentialism, Australasian
Journal of Philosophy, 72, 27-45.
Ellis, Brian
2001 Scientific Essentialism,
Fuhrmann, André
1991 Tropes and
Laws, Philosophical Studies,
63, 57-82.
Giere, Ronald N.
1988 Laws, Theories,
and Generalizations, in [Grünbaum, Adolf and Salmon, Wesley (eds.), The Limitations of Deductivism,
Goodman, Nelson
1955 Fact, Fiction, and Forecast,
Hempel,
Carl G.
1965
Empiricist Criteria of Cognitive Significance: Problems and Changes, in [C.G. Hempel, Aspects of Scientific Explanation,
1988 Provisos: A
Problem Concerning the Inferential Function of Scientific Theories, in [Grünbaum,
Adolf and Salmon, Wesley (eds.), The
Limitations of Deductivism,
Joseph, Geoffrey
1980 The Many
Sciences and the One World, Journal of
Philosophy, 77, 773-791.
Kistler, Max
1999 Causalité
et lois de la nature, Mathesis, Paris : Vrin, 1999.
2002 The Causal
Criterion of Reality and the Necessity of Laws of Nature, Metaphysica, 3 (1), 57-86.
forthcoming, a, Le combinatorialisme et le
réalisme nomologique sont-ils compatibles ?, in [Monnoyer, Jean-Maurice (ed.),
La structure du monde : objets,
propriétés, états de choses, Paris : Vrin].
forthcoming, b, L'identité des propriétés et la
nécessité des lois de la nature, Cahiers
de Philosophie de l'Université de Caen, 38/39 : Le réalisme des universaux.
Lakatos, Imre
1970
Falsification and the Methodology of Scientific Research Programmes, in
[Lakatos, Imre, The Methodology of
Scientific Research Programmes, Philosophical Papers, Vol. 1, Worrall, John
and Currie, Gregory (eds.),
Lange, Marc
1993 Natural Laws
and the Problem of Provisos, Erkenntnis,
38, 233-248.
Lewis, David
1973 Counterfactuals,
1994 Humean
Supervenience Debugged, Mind, 103,
473-490.
Lipton, Peter
1999 All Else
Being Equal, Philosophy, 74, 155-168.
Martin, C.B.
1993 Power for
Realists, in [Bacon, John,
Molnar, George
1999 Are
Dispositions Reducible?, Philosophical
Quarterly, 49, 1-17.
Mumford, Stephen
1995 Ellis and Lierse
on Dispositional Essentialism, Australasian
Journal of Philosophy, 73, 606-612.
1998a Dispositions.
1998b Laws of
Nature Outlawed, Dialectica, 52,
83-101.
Pietroski, Paul
and Rey, Georges
1995 When Other
Things Aren't Equal: Saving Ceteris Paribus Laws from Vacuity, British Journal for the Philosophy of Science, 46, 81-110.
Prior, Elizabeth W., Pargetter, Robert and Jackson, Frank
1982 Three Theses about Dispositions, American Philosophical
Quarterly, 19, 251-257.
Ramsey, Frank P.
1929 General
Propositions and Causality, in [Ramsey, Frank P., Philosophical Papers, Mellor, D.H. (ed.),
Roberts, John
1999 "Laws
of Nature" as an Indexical term: A Reinterpretation of Lewis' Best-System
Analysis, Philosophy of Science, 66
(Proceedings), S502-S511.
Schiffer, Stephen
1991 Ceteris
Paribus Laws, Mind, 100, 1-17.
Shoemaker, Sydney
1980 Causality
and Properties, in [Shoemaker,
1998 Causal and
Metaphysical Necessity, Pacific
Philosophical Quarterly, 79, 59-77.
Simons, Peter
1994 Particulars
in Particular Clothing: Three Trope Theories of Substance, Philosophy and Phenomenological Research, 54, 553-575.
Spohn, Wolfgang
1997 Begründungen a
priori - oder: ein frischer Blick auf Dispositionsprädikate, in [Lenzen, Wolfgang (ed.), Das weite Spektrum der analytischen Philosophie:
Festschrift für Franz von Kutschera, Berlin: de Gruyter, 323-345].
Stegmüller, Wolfgang
1966
Explanation, Prediction, Scientific Systematization and Non-explanatory
Information, Ratio, 8, 1-24.
Tooley, Michael
1977 The Nature
of Laws, Canadian Journal of Philosophy,
7 (4), 667-698.
van Fraassen, Bas
1989 Laws and Symmetry,
Williams, Donald
1953 On the
Elements of Being, Review of Metaphysics, 7, 3-18 and 171-192.
Woodward, James
2000 Explanation
and Invariance in the Special Sciences, British Journal for the Philosophy
of Science, 51, 197-254.
2001 Law and
Explanation in Biology: Invariance is the Kind of Stability that Matters, Philosophy
of Science, 68, 1-20.
[1] I follow tradition in attempting to shed light
on the nature of laws in general, by studying the properties of laws which have
a very simple logical form. I thereby sidestep some important issues. To be
able to represent e.g. probabilistic laws, one or both of the predicates F and
G must be interpreted as expressing the objective chance to have a given
property, rather than that property itself. In order to represent conservation
laws, it is necessary to introduce quantification over time. If F is the
property of having a determinate amount of some conserved quantity, the
conservation law for that quantity should guarantee the following
generalization : . In the paper, I shall consider the implications of the fact
that such a generalization may be false if the system is not isolated.
[2] According to realists of universals, such as
Armstrong, laws are not identical to universal correlations; rather, a law entails
a universal correlation. I shall say more about this doctrine below. Before, I
keep speaking of universally quantified statements “expressing” laws.
[3] The present
paper elaborates some aspects of the account of laws I defend in [Kistler 1999].
[4]Antirealists
have recently been led to consider that the very idea of a law of nature bears
a commitment to realism, and thus to claim that science not only discovers no
laws (this is the realist's basic intuition) but that it doesn't even invent or
construct any. See, e.g., [van Fraassen 1989].
[5] They are due to [Armstrong 1983] and [van
Fraassen 1989].
[6] Cf.
[Van Fraassen 1989, 53f.].
[7] This predicate famously invented by Goodman
[Goodman 1955] applies to objects that have been observed before some instant t
if they are green and to other objects if they are blue.
[8] Anti-realists can of course reject this
argument as question-begging and insist that what it is for a property to be
natural, non-grueish, just is playing a successful role in scientific
theory. To avoid begging the question, the argument must be supplied by an
[9] Cf.
[Armstrong 1983, 69], [Van Fraassen 1989, 47].
[10]David
Lewis himself has not been moved by these arguments. Yet he admits having no
solution to the objections attacking the best system-analysis for its
subjective nature. He just tries to put the burden of proof on the objectors:
in order to give their objections strength, they are liable to show that it is
plausible that the possibilities they put forward have a chance of arising -
Lewis doesn't contest that they are logically possible. Awaiting such a demonstration,
he recommends simply to suppose that they don't arise: "If nature is kind
to us, the problem needn't arise" [Lewis 1994, 479]. Otherwise, he says,
"I'd blame the trouble on unkind nature, not on the analysis; and I
suggest we not cross these bridges unless we come to them" [Lewis 1994,
479]. John Roberts has proposed to construe the expression "law of
nature" as token-reflexive, i.e. as an expression whose extension depends
on the speaker using it and the standards the speaker employs in judging what
is the "best system". True, this "indexical best-systems
account" [Roberts 1999, S503] makes the laws independent of any particular standards. However, it seems
to me that his thesis that what is a law depends on the speaker's choice of
standards makes the notion of law relativistic, and thus less objective than Lewis' own"chauvinistic [...] rigidified
best-systems analysis" [Roberts 1999, S503-505] in which what it as law
depends on our standards.
[11]Cf.
Dretske [Dretske 1977, 264). On the realists’ problem of explaining how necessary
relations between particular states of affairs can be implied by contingent second-order
states of affairs (which is what realists as Dretske and [Armstrong 1983] take
laws to be), see [Kistler forthcoming, a].
[12] The thesis that laws themselves are contingent
threatens to make the realist account of how laws ground the truth of
counterfactuals tautologous, because it makes it necessary to add the qualification
that laws hold only in those possible worlds in which they hold. On this count,
it is easier to explain how laws ground the truth of counterfactuals, by
supposing that laws are metaphysically necessary (cf. [Shoemaker 1980],
[Shoemaker 1998], [Ellis 2001]). I have defended the thesis of the necessity of
laws on independent grounds elsewhere [Kistler forthcoming, b], but I cannot
repeat the argument here.
[13]According
to David Lewis, counterfactuals are evaluated with the help of intuitions
concerning the closest possible world W among all those in which, contrary to
what is the case in the actual world, the antecedent is true. The
counterfactual is true if and only if the consequent is true in W. Lewis argues
that W, for a given counterfactual, is in general a world in which a
"small, localized, inconspicuous miracle" [Lewis 1973, 75] has
occurred just before the event described by the antecedent because such a
violation of law allows the class of
particular facts preceding the event
described by the antecedent to resemble exactly their actual counterparts, and
because this greater factual resemblance may often outweigh the perfect lawful
resemblance with the actual world of another non-actual worlds without any
miracles and in which the entire past is factually different from the actual
world. However, it seems to me that, first, every small miracle apt to change a
single fact must involve the violation of many laws, e.g. at least the
violation of both the conservation laws of energy and momentum, and second,
that the very idea of a small miracle puts Lewis before a dilemma: Either a
world with a miracle however small is inconsistent in that its own laws both
hold and do not hold (if they hold, there can be no miracle, and if they do not
hold there can be no miracle either because a miracle violating a given law L
presupposes that L is indeed a law), or the "miraculous" world is
consistent in that the "miracle" is not really a miracle but rather a
lawful event in a world whose laws differ from our actual laws, and therefore
only appears to us as miraculous. But
in that case, we lose the clear intuition that such a world is closer to the
actual than a world with many but only factual differences with respect to
actuality, for the least difference in laws entails many factual divergences
throughout the entire history and future. Furthermore, the idea of judging that
a miraculous world is closest to the actual world creates a difficulty for the
evaluation of counterfactuals which it was its role to contribute to. This is
the problem that we do not seem to have any clear-cut intuitions about
miraculous worlds, i.e. worlds in which our laws do not hold. To judge what
happens in a non-actual world W, our only foothold seems to be the supposition
that the laws of W are the same as our actual laws, otherwise just anything
could happen in W, and there is no intuitive ground for judging that the
consequent of the counterfactual is true in W rather than false (or false
rather than true). This is my reason for saying that only "homonomic"
non-actual worlds can help us evaluate counterfactuals.
[14] Cf. [Van Fraassen 1989, 103].
[15]Cf.
[Canfield and Lehrer 1961, 207], [Lakatos 1970, 98], [Hempel 1988].
[16]In
realist language, we say that Pl denotes the property of being a
pendulum of length l.
[17]The
period of a pendulum diverges from T(l) well below 45°. The criterion for the
quality of the approximation given by L(P) is the quality of the linear
approximation of f(x) = sin(x) by f(x) = x.
[18]Mumford
([Mumford 1998a, chap. 10], [Mumford 1998b]) has presented the hypothesis of
the existence of laws of nature and the hypothesis of the existence of real
dispositions as alternative and rival approaches to the explanation of change.
Similarly, Cartwright thinks that “we must admit capacities”, and that “once we
have them we can do away with laws” [Cartwright 1989, 8]. While I agree with
Mumford and Cartwright that real dispositions (or capacities) are needed to
solve the problem of exceptions, I disagree with his view that dispositions
make laws explanatorily useless. When we explain why a particular electron moved
as it did by reference to the link between its being an electron and its having
the disposition to move in a certain way according to the electric field
present at its location, it is only the generality of the fact that all other
electrons would have the same disposition in that situation, that makes the
account explanatory. [Cartwright 1999] has recently turned to a realist account
of ceteris paribus laws, according to which such laws describe local
regularities that exist in specific circumstances, in particular experimental
arrangements, that she calls “nomological machines”.
[19] Among others, Armstrong ([Armstrong 1968],
[Armstrong 1997]) holds that dispositions (or dispositional properties) can be
causally efficacious, [Prior, Pargetter and Jackson 1982] argue that they
cannot.
[20] It is important to note that the number of
these premises is indefinite. This has, in particular, the consequence that
they can't be explicitly enumerated. As long as one overlooks this fact, one
runs the risk of being "trapped in the idea that the 'ceteris paribus clause' is a premise which is joined conjunctively with the obvious
premises" [Lakatos 1970, 98; Lakatos’ emphasis]. Lakatos detects this
"confusion" in [Canfield et Lehrer 1961] and [Stegmüller 1966] before
he admits that his own phraseology may sometimes generate the same
misunderstanding.
[21] [Earman and Roberts 1999] and [Woodward 2000],
[Woodward 2001] argue that there is no coherent way to construe the notion of a
non-strict or ceteris paribus law, and then point out that there are
ways to make sense of the aim of the special sciences, which do not presuppose
ascribing to them the aim of discovering laws. Rather, according to Woodward,
special sciences, and biology in particular search for “invariant generalizations”
[Woodward 2000, 199], rather than for laws. According to Earman and Roberts,
statements that philosophers have erroneously interpreted as ceteris paribus
laws, are “vague claims” that are formulated before any satisfactory theory of
a domain has been found. They are characteristic of “work-in-progress theories”
[Earman and Roberts 1999, 471].
[22]This
is also Cartwright's interpretation of the content of ceteris paribus statements. She states, e.g., the law of
gravitation as follows : "If there are no forces other than gravitational
forces at work, then two bodies exert a force between each other which varies
inversely as the square of the distance between them, and varies directly as
the products of their masses" [Cartwright 1983, 58].
[23]This
is a neutral term intended to bypass the debate between categoricalism and
dispositionalism (Cf. [Ellis and Lierse 1994], [Mumford 1995], [Armstrong 1997,
chap. 5], [Molnar 1999]). Is the link between the property and the disposition
it bestows on the particular possessing it contingent or necessary? In other
words, does the property simply consist of its dispositions (dispositionalism)
or is it contingently linked to it, contingently that is by laws of nature
which hold in our world but need not hold in every possible world
(categoricalism)?
[24]The
conception defended here takes up suggestions of [Coffa 1968] and [Cartwright
1983]. Coffa proposed to interpret laws (of the type admitting for exceptions)
as "laws of tendency, stating the contribution of a certain factor to a
given process" [Coffa 1968, 281f.]. Yet Coffa underestimates the
complexity of the interactions between different properties and different laws
even if he is right in noting that there are some rare laws which do not allow
for exceptions - as Newton's law stating the equivalence of force and the
product of acceleration and mass - and which can therefore be considered as
equivalent to their associated universal generalisation. Cartwright's
suggestion according to which "the laws we use talk not about what bodies
do, but about the powers they possess" [Cartwright 1983, 61] is also
compatible with the present proposal, as long as one abstracts from the
theoretical commitments this claim is associated with in Cartwright's work, in
particular her opposition between fundamental and phenomenological laws, the
latter but not the former being capable of literal truth and of causal
efficacy, which in turn is used in "effective strategies".
[25]C-abnormal
instances occur in what we have called exceptional situations. [Pietroski and
Rey 1995] give a rather complex formal characterisation of the concept of a
C-abnormal instance, but the basic idea is I hope captured by what I say in the
text.
[26]"cp"
always stands for "ceteris paribus".
[27]Cf.
[Canfield and Lehrer 1961] who propose a similar argument in a nominalistic
framework.
[28]This
qualifier is intended to exclude Goodmanian predicates like "grue"
from those denoting genuine natural properties.
[29]I
cannot enter here into the debate about whether a particular is merely a
bundle, whether it has substance over and above its tropes, or whether the role
played by the traditional substance can be overtaken by a nucleus constituted
by tropes internally related to each other. Cf. [Armstrong 1989], [Armstrong
1997], [Martin 1993], [Simons 1994].
[30]Cf.
[Williams 1953], [
[31] Peter
Lipton [Lipton 1999] has made a similar proposal. However, Lipton does not
specify whether the dispositions that non-strict (or ceteris paribus) laws refer
to, are tropes or universals.
[32]Cf.
[Carnap 1936], [Spohn 1997], [Mumford 1998a].
[33]This
is what Prior et al. call "the distinctness thesis".
Interestingly, they count the existence of what we call exceptional situations
as an argument for that thesis. "Even if there is only one causal basis of
fragility, say bonding a, it may happen that although all fragile objects have
a, some objects that have a are not fragile. This would be the case if there
were an internal structural property S which swamped the effect of having
a." [Prior et al. 1982, 253].
[34]It may
turn out to be possible - but whether or not it actually does is an empirical
issue - to classify these constraints in orthogonal dimensions such that all
the tropes imposing conflicting constraints are represented within the same
dimension. Dispositions belonging to different - orthogonal - dimensions would
not come into conflict with one another. The constraints imposed by the
dispositions belonging to one dimension can then be ordered according to their
magnitude, to explain the resulting manifest property of the particular.
[35]This
interpretation is explicitly intended by [Spohn 1997].
[36] Pietroski and Rey explicitly
reject a probabilistic interpretation of cp-laws for similar reasons. As they
say, "it often turns out that 'ideal' circumstances are not merely rare,
they are nomologically impossible." [Pietroski and Rey 1995, 84].
[37] [Fuhrmann 1991] shows that this is also true
of his version of trope theory.
[38]I am
obliged for helpful comments and discussion to Joan Cullen, Geert Keil, D.H.
Mellor, Nenad Miscevic, Joëlle Proust, and Markus Schrenk.