Given that we can’t avoid the problems of GNS by simply replacing it with another view of reduction, these problem have to be addressed. This is the task of the present section.
Problem 1 The objection that GNS is based on the syntactic view of theories and therefore untenable is mistaken. Although Nagel was a proponent of the syntactic view, there is no textual (or other) evidence that he took the syntactic view to be an essential part of his model of reduction; and Schaffner makes no assumptions about the correct analysis of theories when presenting his theory of reduction. This is for good reasons, because the syntactic view is unnecessary to get GNS off the ground, as is clear from the above examples: Neither did we present a first-order formulation of the theories, nor did we even mention a bifurcation of the vocabulary into theoretical and observational terms. Where first order logic is too weak, we can replace it with any formal system that is strong enough to do what we need it to do. The bifurcation of the vocabulary plays no role at all.
Problem 2. Feyerabend’s criticism is that reduction is impossible because, in order to associate two terms with each other, they must have the same meaning, which, however, is never the case if the terms occur in two different theories. Whether this argument is cogent depends on what one means by ‘meaning’. Feyerabend associates the meaning of a term with the role the term plays in a theoretical framework. Thus, the meaning of the term ‘temperature’ as it occurs in thermodynamics is determined by everything we say about temperature in the language of thermodynamics. Given this conception of meaning, it is clear that terms occurring in different theories must have different meanings. But when meaning is framed in this way, meaning equivalence is immaterial to reduction; what matters is whether the properties that the terms in the bridge laws refer to stand in a relevant relation to each other. Feyerabend’s imposition that only terms with the same meaning can be associated with each other is unmotivated, unnecessary, and foreign to GNS.Footnote 13
Problem 3. What is the status of bridge laws? The first two options Nagel considers are meaning equivalence and convention. These can be discarded. That bridge laws cannot be claims of meaning equivalence follows from our discussion of Problem 2. Neither can they be mere conventions. Conventions are arbitrary and all that matters is that they be respected after a choice has been made. We can choose to drive on the right or on the left hand side of the road; neither choice is better, or more justified, than the other. What matters is that everybody respects the choice once it has been made by the group. Bridge laws are not like that. Clearly, there is right and wrong in theoretical association. It is true that the temperature of a gas is proportional to 〈Ekin 〉, but it is false that it is proportional to 〈Ekin 〉2. Furthermore, often a process of painstaking research was necessary to make such associations. That does not sit well with an understanding of bridge laws as conventions.
For this reason, bridge laws are factual claims. This, however, leaves open the question whether bridge laws express mere correlations (or Humean regularities), nomic connections involving certain necessities, identity statements, or yet other metaphysical relations. There is a strong push in the literature to first come to a general answer to this question and then settle for identity.
To assess this tendency, we have to distinguish between two different kinds of bridge laws: The first kind associates basic entities of T
P
and T
F
with each other; they identify, for instance, light and electromagnetic radiation, electric currents and the flow of electrons, and gases and swarms of atoms (see, for instance, Sklar 1967, p. 120). We refer to this kind of bridge laws as entity association laws. The second kind of bridge laws enter the scene once the basic entities of T
P
and T
F
are associated with each other and then assert that the T
P
-properties of a system stand in a relevant relation to the T
F
-properties of that system, and that the magnitudes of these properties stand in a relevant functional relationship. Let us call these property association laws.
Entity association laws are different from property association laws both in content and in origin. Entity association laws indeed express identities: gases are swarms of molecules, genes are strings of amino acids, etc. The same does not hold for property association laws; these laws can, but need not express identities. We will argue for this claim shortly. The second difference is that, while property association laws are external to T
F
, entity association laws are internal to T
F
. It is the basic posit of the wave theory of light that light is an electromagnetic wave; it is the basic posit of the kinetic theory of gases that gases are swarms of atoms; and it is the basic posit of statistical mechanics that the systems within the scope of thermodynamics have a molecular constitution and that the behaviour of molecules is governed by the laws of mechanics.Footnote 14 Entity association laws can, of course, be false; but if they are, it is the reducing theory that is false. By contrast, property association laws are external to T
F
. For instance, there is nothing in the kinetic theory of gases per se that tells us to associate mean kinetic energy with temperature. This raises questions both about their content and form.
Problem 4. The question about the content of property association bridge laws is best discussed in the context of arguments against reduction based on MR. Unlike entity association laws, which clearly have to be identities, property association laws could, at least in principle, also be mere regularities, lawlike connections, or express yet another relation. However, there is a long tradition of arguing that all bridge laws have to establish identities. Hence, property association laws have to establish identities between properties because everything less than identity is insufficient for a genuine reduction.Footnote 15
As per the first argument, the driving force behind the requirement that bridge laws express identities is the view that, for a reduction to be successful, it has to be shown that T
P
-properties are nothing over and above T
F
-properties. We believe this to be mistaken. Whether or not the establishment of strict identities is a desideratum for a reduction depends on what one wants a reduction to achieve. If metaphysical parsimony or the defence of physicalism are one’s primary goals, then identity may well be essential (although, even then, less than identity might be sufficient; we return to this issue when discussing explanation). But in science neither of these are very high on the agenda. Reductions are desirable first and foremost for two other reasons: consistency and confirmation. That is, T
F
and T
P
have to be consistent, and evidence confirming T
F
also has to confirm T
P
and vice versa. Further items can be added to this list, explanation being the most obvious addition (the condition that T
F
explain T
P
, we come back to this below). However, these additions are not essential: Reductions that achieve nothing but consistency and confirmation are bona fide reductions. These aims, and this is the crucial point, can be achieved without bridge laws being identity statements. In fact, mere de facto correlations between properties are all that is required for the needs of reduction, and we can remain agnostic about the question of whether bridge laws express anything beyond mere correlation.
Let us discuss consistency and confirmation in more detail. No rational person should hold contradictory beliefs. Hence, given two (self-) consistent theories T
1 and T
2, these ought to be consistent with each other (T
1 and T
2 are required to be consistent because no one should hold an inconsistent theory to begin with). If the two theories use completely different languages and are about a different target domain, then this requirement is satisfied trivially; there does not seem to be a problem about the consistency of algebraic quantum field theory and costly signaling theory in evolutionary biology. Things become more involved if the two theories’ target domains are identical (or have significant overlap), in which case consistency does not come for free (i.e. not merely as a result of the theories not sharing any non-logical vocabulary). Theories like SM contain what we have above called entity association laws and so SM and TD are not consistent merely on the grounds that they use different vocabulary; they make claims about the same systems and the question arises whether these claims are consistent with each other.Footnote 16 Establishing a reductive relation between SM and TD ensures the consistency and hence co-tenability of the two accounts, because, trivially, if one consistent theory can be deduced from another consistent theory the two are consistent.Footnote 17 All that is needed for such a deduction is that there be conditionals saying ‘for all x, x is t
F
if and only if it is t
P
’.Footnote 18 It simply does not matter whether this conditional expresses an identity, a nomic necessity or a mere de facto correlation; all we need for the deduction is that whenever t
F
applies, then t
P
applies.
Next in line is confirmation. Consider again two theories whose target domains are identical (or have significant overlap). We then would expect evidence confirming one theory to also confirm the other theory, and we expect confirmation to ‘travel’ both ways (though not necessarily with the same strength). This, however, can happen only if the two theories are connected to one another, and the connection postulated by GNS fits the bill.Footnote 19 Assume, first, that we have evidence supporting \(T^{*}_{P}\) and the bridge laws. On GNS, this theory is a deductive consequence of T
F
(plus auxiliary assumptions) and the bridge laws, and on every credible account of confirmation, a general theory receives some boost in confirmation if one of its consequence bears out (although different accounts of confirmation analyse the basic idea in different ways). Conversely, if we have evidence supporting T
F
and the bride laws, then \(T^{*}_{P}\) receives confirmatory support because a deductive consequence of a hypothesis inherits the confirmation of the hypothesis itself. As in the case of consistency, all that matters for confirmation is that there be sentences connecting terms from one theory to terms of the other so that the deduction becomes possible, but it is immaterial to the deduction whether these sentences express mere Humean regularities or some strong metaphysical relation. So, again, no commitment to an identity reading of bridge laws is forced upon us.
The second argument is that reduction is incompatible with there being a diverse set of realisers for one T
P
-property: There must be something that binds together, or unifies, all the realisers or a T
P
-property over and above merely being realisers of that particular T
P
-property. This demand is unjustified. In fact, the second argument is just the identity view in disguise. While it admits that there can be different realisers, it requires that they all share something in common and then the implicit assumption is that what T
P
-property is really reduced to is this common feature. We have already argued that identity is unnecessary for reduction, and so we also reject this argument. There simply is no reason to think that, say, ‘temperature’ for gas being co-extensional with mean kinetic energy precludes it from being co-extensional with a completely different micro-property in other systems.
The third argument from MR is that bridge laws cannot be genuine laws where multiply realisable properties are involved because multiply realisable properties require disjunctive bridge laws but genuine laws of nature cannot be disjunctive. It is hard to see why this should be so, and we can only share Sober’s ‘sense of incomprehension and mystery’ at why the word ‘or’ should undermine the aims of reduction (1999, p. 553). First, as Sober points out, it is not clear where to draw the line between disjunctive and non-disjunctive laws, since what is non-disjunctive in one formulation could turn out to be disjunctive in another one and vice versa. Second, even if it is true that ‘proper’ laws of nature (whatever these are) cannot be disjunctive, there is no need for bridge laws to be laws of nature in that sense. Bridge laws can be of a different kind and have to satisfy less stringent demands than other laws of nature. All we require from bridge laws is that they serve the purposes of reduction (which, on our view, are consistency and confirmation), and disjunctions pose no problem for these (even if they are open-ended). Third, it is not clear why laws of nature cannot have a disjunctive form. What seems to lie in the background are worries concerning natural kinds and spurious confirmation. But it is not clear whether these worries are conclusive, and the burden of proof lies with those who argue against disjunctive laws.Footnote 20
The last argument is that MR undercuts the explanatory power of reductions. We want to resist this argument for two reasons. First, rife doctrine nonewithstanding, reductions do not ipso facto have to double as explanations. The two core aims of reduction— consistency and confirmation—can be had without adding further items to the list, and reductions are desirable even if they do not serve any other purposes. Explanation, in particular, is nice to have where it can be had, but it is not a sine qua non of reduction.Footnote 21
Second, it is not clear to us why MR should undercut reductive explanation. Kim (2008, p. 94) characterises a reductive explanation as one that shows that a particular T
F
phenomenon constitutes ‘an underlying mechanism’ whose ‘operation’ yields a T
P
phenomenon and which makes the T
P
phenomenon ‘intelligible in the light of the underlying phenomena and mechanisms’. It is not clear why MR should undercut reductive explanations in this sense. We explain why gases have temperature by appeal to the dynamical properties of its constituents. If this explanation is successful, then it is so irrespective of whether other kinds of systems can have temperature, too. Assume that gases were the only kind of objects that had temperature, and that we had a successful explanation of why gases have temperature in terms of the molecular motion of gas molecules. Why would this explanation no longer be an explanation once we realise that other systems also have temperature? There is no reason to believe that what used to be an explanation suddenly loses its status as an explanation. It has just become a more local explanation, because it does not cover all cases of temperature, but local explanations are still explanations.
Problem 5 How do we establish bridge laws? The alleged problem is that we cannot test them independently. In fact, it is not the case, as Nagel seems to suggest, that we start with T
F
, then write down a bridge law (which we know to be correct!), and finally deduce \(T^{*}_{P}\). Rather, what happens is that we begin with T
F
and T
P
and then try to find bridge laws that (modulo small corrections) make T
P
derivable from T
F
(cf. Ager et al. 1974, pp. 119–122). So the correct analysis of how the two theories relate should be
-
Premise 1: T
F
-
Premise 2: T
P
-
——————————-
-
Conclusion: bridge law
In the above example, it is not the Boyle–Charles Law that we derive from the kinetic theory plus a bridge law (Eq. 5); it is the bridge law that is derived from the Boyle–Charles Law and the kinetic theory.
We agree with this point, but deny that it is a problem for GNS. In fact, this is just an instance of the Duhem problem: We are often unable to confirm hypotheses independently because we can only put entire packages (consisting of theories and auxiliary assumptions) to test. That the Duhem problem crops up in Nagelian reduction is hardly a cause for celebration, but given that this is a widespread problem in many (if not all) parts of science, it hardly is a reason to give up Nagelian reduction. As is well known, there is no royal route around the problem and arguments vary from case to case. So the conclusion to be drawn from this is simply that, in any given case of a purported reduction, we have to think carefully about what evidential support we have for the bridge laws we use. Sometimes we may take the bridge law seriously because we have good evidence for both T
F
and T
P
, and the reduction is sufficiently smooth.Footnote 22 In other cases, we may have other reasons to take the bridge laws seriously. Asking for a universal account of evidential support for bridge laws is a mistaken demand, and not one the GNS has to meet.
Problem 6 Let us begin with the second problem, namely that GNS is too liberal. GNS, so the objection goes, allows for auxiliary assumptions that are so strong that they are doing all the work, and, in fact, render T
F
itself an idle wheel. Yet, it still forces us to say that T
P
has been reduced to T
F
, which is implausible. This is a fair concern, but not one that poses an insurmountable problem. Our proposal is to impose the following two conditions on auxiliary assumptions: First, T
F
must be used in the deduction of \(T^{*}_{P}\); that is, \(T^{*}_{P}\) must not follow from the auxiliary assumptions alone. We call this the condition of non-redundancy. Second, the auxiliary assumptions must belong to the paradigm of T
F
; i.e. auxiliary assumptions cannot be foreign to the conceptual apparatus of T
F
. This is the condition of immanence. These two restrictions successfully undercut spurious reductions.
Let us illustrate this with the example of the Second Law. Trivial self-deduction is ruled out by the first condition: we cannot simply write down the Second Law as an auxiliary assumption and then deduce it. But our two conditions also deal correctly with less trivial cases. Assume for the sake of argument that Boltzmann’s programme has been completed successfully and a derivation of (a close cousin of) the Second Law of TD from the apparatus of Boltzmannian SM and the auxiliary assumption that the system is ergodic has been given. In our view, this would be a successful reduction, because ergodicity is part and parcel of classical mechanics, which is central to Boltzmannian SM. The auxiliary assumption merely restricts the class of allowable phase flows to ones that are ergodic, but it does not introduce anything into the theory that is in principle foreign to it. By contrast, consider the research programme known as stochastic dynamics.Footnote 23 The leading idea of this approach is to replace the Hamiltonian dynamics of the system with an explicitly probabilistic law of evolution. Characteristically, this is done by coarse-graining the phase space and then postulating a probabilistic law describing the transition from one cell of the partition to another one. The Second Law is then derived from this probabilistic dynamics. In our view, this is not a successful reduction of TD to SM, because the Second Law follows from the auxiliary assumptions alone (contra non-triviality), and the probabilistic transition laws are entirely foreign to classical mechanics. Unless one could somehow derive the probabilistic laws from the Hamiltonian equations of motion governing the system, these probability laws violate immanence.
The two conditions also offer a straightforward solution to the first worry: Given that the auxiliary assumptions have to belong to the paradigm of the reducing theory, there is nothing wrong with saying that T
P
has been reduced to T
F
.
Problem 7. The first criticism is that the notion that two theories be analogous to each other seems hopelessly vague and that therefore an account of reduction based on this is a non-starter. At least in the context of GNS, not anything goes, however. There are two conditions that \(T^{*}_{P}\) must satisfy. First, we require that the two theories use the same conceptual machinery: \(T^{*}_{P}\) must share with T
P
all essential terms. Consider again the Second Law. \(T^{*}_{P}\) is couched in the same terms as T
P
, namely entropy, and differs only in how the properties vary, namely that in the former entropy fluctuates. Second, Schaffner (1967, p. 144) requires that \(T^{*}_{P}\) corrects T
P
in the sense that \(T^{*}_{P}\) makes more accurate predictions than T
P
. This is the case in our example because experiments show that entropy fluctuates as predicted by \(T^{*}_{P}\) (and ruled out by T
P
). While Schaffner’s requirement sits well with the example of the Second Law, it may be too restrictive in general. So we propose a slightly weaker requirement, doing the same work without running the risk of ruling bona fide reductions. The requirement is that \(T^{*}_{P}\) be at least equally empirically adequate as T
P
. These two conditions undercut any attempt at playing fast and loose with analogies in such a way as might.
There is a further worry that there is no general characterisation of ‘strongly analogous’, but such a characterisation is an essential part of a workable theory of reduction. Therefore, the criterion that \(T^{*}_{P}\) and T
P
be strongly analogous is empty and GNS is not a definite position at all. We disagree with this conclusion. Being strongly analogous is a contextual relation, and we should not expect there to be a general theory of analogy. Whether or not \(T^{*}_{P}\) is strongly analogous to T
P
has to be decided either in the relevant scientific discipline itself or the special philosophy of it. The above example of the derivation of the Second Law makes this clear. That a close cousin of the Second Law of TD allowing for fluctuations is strongly analogous to the strict Second Law in a way that underwrites reductive claims does not follow from some philosophical theory of analogy; it is the result of a careful analysis of the case at hand. Callender (1999, 2001) has argued, in our view convincingly, that the unrestricted Second Law is too strong, and that we can accept a watered down version without contravening any known empirical fact, which is why we can regard these laws as strongly analogous. Indeed, we should expect the same to be the case with almost every putative case of reduction: it is the particular science at stake that has to provide us with a criterion of relevant similarity in the particular context.
The third worry is that, unless the analogy is identity, T
P
has in fact been replaced rather than reduced, and so we should not longer speak about reduction; in fact, \(T^{*}_{P}\), not T
P
, has been reduced. This is a matter of definition. If the term ‘reduction’ is reserved for cases of exact derivation, then T
P
is not reduced. However, we see no reason to regiment language in this way. As we have just seen, GNS imposes strict conditions and what counts as a strong analogy is by no means arbitrary. As long as it is understood that reduction involves an analogy of this kind, we can see no harm in calling the GNS procedure ‘reduction’.