THE ORDERED PAIR THEORY

Let me turn from van Fraassen's theory to one that I elaborate in my book The Nature of Explanation. Here I don't plan to present the theory in detail but only to say enough about it to show that it is pragmatic and to argue the advantages of a pragmatic account.

As did Sylvain Bromberger in a seminal essay in 1965, I begin with the concept of an explaining act. The explanation-sentences of concern to me are ones of the form

(6) S explains q by uttering u,

where q is the indirect form of a question Q. Simplifying my view, such sentences are true iff S utters u with the intention of rendering q understandable by producing the knowledge that u expresses a cor­rect answer to the question Q. To develop this one needs to talk about the concept of understanding, which I will not attempt to do here. (See chapter 6) In any case there is no need to do so, for explanation­sentences of the form “S explains q by uttering u” are clearly pragmatic in the Hempelian sense. Such sentences make essential reference to an explainer.

The second stage in my theory consists in an attempt to provide a def­inition of an explanation itself—i.e., the product of an act of explaining or at least of a potential act of explaining. For certain reasons which we need not explore here I say that an explanation of q can be construed as an ordered pair whose first member is a proposition or set of propositions that constitutes an answer to Q, and whose second member is a type of explaining act, namely, explaining q. So, e.g., if Newton explains why the tides occur by saying that they occur because of the gravitational pull of the moon, then his explanation—whether good or bad, right or wrong— can be construed as the ordered pair

(7) (The tides occur because of the gravitational pull of the moon; explaining why the tides occur).

The second member of this pair invokes the concept of a type of explain­ing act, to which the account briefly summarized before is applicable.

The latter point is important for the issue of the pragmatic character of explanation, so let me take it just a bit further. Let's consider explanation­sentences of the form

E is an explanation of q,

where there is no implication regarding E's goodness or correctness. On the ordered pair theory, the following is a set of truth-conditions for sen­tences of this form:

(i) Q is (what I call) a content-question (see chapter 1, section 9);

(ii) E is an ordered pair whose first member is (what I call) a complete content-giving proposition with respect to Q and whose second member is the act-type explaining q.

Do these truth-conditions contain terms for an explainer or audience or any terms defined by reference to these? They do not do so explicitly.

Yet there is something different about this case and the ones Hempel may have in mind. For although a term for an explainer is invoked, the truth-value of sentences of the form “E is an explanation of q” will not vary with who, if anyone, is giving or receiving the explanation E mentioned in the explanation-sentence. Earlier I characterized an explanation-sentence as pragmatic if it contains terms for a (particular or type of) explainer or audience or if its truth-conditions contain such terms or others defined using such terms. We might now introduce a second condition, and say that the truth-value of explanation-sentences of that form can vary with a change in the person giving or receiving the explanation mentioned or referred to in the explanation sentence. If both of these conditions are satisfied, let us say that the explanation-sentence is strongly pragmatic. If only the first is satisfied, the explanation-sentence is weakly pragmatic. By this criterion, sentences of the form

S explains q by uttering u

are strongly pragmatic. (Such sentences contain a term for an explainer, and their truth-value can vary with a change in explainer.) On the or­dered pair theory, sentences of the form

E is an explanation of q

are only weakly pragmatic. Truth-conditions for sentences of this form (according to the ordered pair theory) invoke a term for a type of ex­plainer, one who explains q, but the truth-value of sentences of this form does not vary with any change in who is giving E as an explanation of q, or to whom.

I am inclined to think that when Hempel uses the term “pragmatic” he has in mind “strongly pragmatic,” and that he would not object too strenu­ously to a “weakly pragmatic” concept of explanation, since the latter can be “objective.” But this is speculation on my part.

Let me turn to another, perhaps more important, concept for which the ordered pair theory offers an account, namely, that of a “good explana­tion.” Are sentences of the form “E is a good explanation of q” pragmatic in either sense?

Different evaluations of explanations are possible depending on what ends are to be achieved. The ends might be purely universal ones, e.g., the achievement of truth, empirical adequacy, simplicity, unification, and so on. Or they might be more contextual. The end I am particularly con­cerned with is one that, by the definition given in the first part of the theory, an explainer has when he performs an act of explaining q, that is, rendering q understandable (in some appropriate way) by producing the knowledge of the answer one gives that it is a correct answer to Q. An evaluation with this end in view will take into account both universal and contextual criteria. Very roughly, E will be a good explanation for an ex­plainer to give in explaining q to an audience if E is capable of rendering q understandable in an appropriate way to that audience by producing the knowledge of the answer to Q that it supplies that it is correct, or if it is reasonable for the explainer to believe that this obtains.

In the case of such evaluations, which I call “illocutionary,” sentences of the form “E is a good explanation of q” will be construed as elliptical for “E is a good explanation for an explainer to give in explaining q to an au­dience.” Explanation-sentences of the latter form are strongly pragmatic. They contain terms for an explainer and audience, and the truth-value of sentences of this form can vary with a change in explainer or audience.

Now I am not claiming that illocutionary evaluations are the only pos­sible ones (see the Addendum). I do insist that they are important, that they are frequently given, and that using them, by contrast to nonillocu- tionary, nonpragmatic evaluations, will enable us to see why certain scien­tific explanations are generally judged better than others. Let me illustrate this by invoking a simple example, Rutherford's 1911 explanation of the results of scattering experiments involving alpha particles.

In experiments published in 1909 Geiger and Marsden showed that when alpha particles are directed at a thin metal foil most of them go through the foil with small angles of deflection, but some are scattered through an angle of more than 90°, thus emerging on the side of incidence. In order to explain these surprising results Ernest Rutherford proposed a new theory of the structure of the atom. He assumed that an atom con­tains a positive charge that is not evenly distributed but is concentrated in a nucleus whose volume is small compared to that of the atom. He also assumed that the positively charged nucleus is surrounded by a com­pensating charge of moving electrons. Finally, he assumed that each scat­tering was the result of a single encounter between an alpha particle and a foil atom.

Is Rutherford’s explanation of the scattering results a good one? If we evaluate it in a nonillocutionary way using only criteria that are nonprag­matic, it would, I suppose, get a mixed review. True, it derives the wide scattering angles in a precise way from lawlike, quantitative assumptions; it appeals to microentities; and it offers a cause of the scattering—all of which physicists and philosophers of science tend to regard with favor. But, as later developments in physics show, it is only a crude approxima­tion to what actually occurs in the foil atoms. And it introduces a concep­tion of the atom as involving moving electrons that is incompatible with classical electrodynamics. (Moving electrons should radiate energy and collapse into the nucleus, which clearly they do not.) Furthermore, if we use only nonpragmatic criteria in our evaluation, we will have a difficult time seeing why Rutherford’s explanation is better than certain others we might construct that are clearly inferior.

Consider, e.g., the following quantitative hypothesis that Geiger and Marsden could have constructed from their experiments without appeal to Rutherford’s theory. (I’ll call it the G-M hypothesis.)

The G-M hypothesis: When alpha particles are directed at thin metal foils the atoms comprising the foils cause the alpha particles to be scattered at various angles in accordance with the formula

N = Qnt(ze)²E²/4r²(MV²)²sin⁴ 6/2

(N is the number of alpha particles deflected at angle 6, Q is the total number of alpha particles incident on the foil, n the number of atoms per unit volume in the foil, t the thickness of the foil, z the atomic number of the metal of the foil, e the elementary unit of charge, E the charge of the alpha particle, r the distance from the foil to the detection screen, and M and V the mass and velocity of the incident alpha particle.)

From the G-M hypothesis, together with information about a particular experimental setup indicating the number of alpha particles, the thick­ness and the atomic number of the foil material, and so forth, the number of alpha particles scattered at various angles, including large ones, can be described in a precise way, using lawlike, quantitative assumptions. Moreover, this explanation is unifying in the sense that it permits the derivation of several different results obtained in the experiments of Gieger and Marsdon. (For example, it permits a derivation of the fact that the number of alpha particles scattered through a given angle is directly proportional to the thickness of the scattering foil, and that the number is inversely proportional to the square of the energy of the alpha particles.) The explanation is causal in the sense that the G-M hypothesis contains a description of something that causes the scattering, namely, the presence of the atoms in the metal foil. And in so doing it appeals to microentities. Yet I think it would be regarded as vastly inferior to Rutherford's explana­tion. But objective, nonpragmatic values such as derivability from quanti­tative laws, unification, causation, and microentities will not by themselves tell us why Rutherford's explanation is a good one by contrast with the G-M hypothesis. Rutherford's explanation is good, or is as good as it is, not simply because it answers a causal question about the scattering in a quantitative way at a unifying microlevel, but because it does so at the subatomic level of matter in a way that physicists at the time were interested in understanding the scattering.

By 1911, the time of Rutherford's paper, the atomic theory of matter was widely accepted in physics, as was the idea that the atom itself is not atomic but has an internal structure. The latter idea emerged from the discovery of radioactivity and the electron, and the results of scattering of beta particles by atoms. It was also thought reasonable to suppose that alpha particle scattering was produced by events at the subatomic level. The question was how to work this out quantitatively using some theory about the structure of the atom. About five years before Rutherford's paper, J. J. Thomson had proposed the “plum pudding” model of the atom according to which the positive electricity in the atom is uniformly distributed throughout the atom and the electrons are held stationary in equilibrium positions by the positive charges surrounding them and the repulsion of other electrons. However, it was impossible to derive the wide scattering angles of alpha particles from the Thomson model.

One of the reasons Rutherford's explanation is highly regarded is that it does derive these angles from a model of the internal structure of the atom—which physicists at the time were seeking. And I think that the major reason the G-M explanation would not be so highly regarded— despite the fact that it derives the wide scattering angles from quantita­tive hypotheses—is that it does not give an explanation by appeal to sub­atomic structure. (It simply says that the scattering is produced by atoms, and it provides an empirical formula for the scattering.) But to assess Rutherford's explanation in the manner suggested is to offer an illocutionary evaluation. In the present case we are considering whether Rutherford's explanation (by contrast, say, to G-M) is a good one for Rutherford to have given. To determine this we need to look at the situation of Rutherford and other physicists in 1911. What did they know, and what did they seek to know? Doing this means treating the explanation-sentence “Rutherford's explanation of the alpha scattering is a good one” as strongly pragmatic. We need not, of course, treat it this way only with reference to Rutherford as explainer or a 1911 audience. The explanation-sentence might have a dif­ferent truth-value if construed as elliptical for one making reference to a contemporary explainer and audience.

Now let me offer a conjecture. Suppose, following in the footsteps of Hempel and Salmon, you formulate a set of objective, nonpragmatic cri­teria that you think all scientific explanations must satisfy to be evaluated highly. These criteria will be universal in the sense that they are not to vary from one explanation to the next, but are to be ones applicable to all scientific explanations. They are also universal in the sense that they are not to incorporate specific empirical assumptions or presuppositions that might be made by scientists in one field or context but not another. So they might include the use of laws, causal factors, and quantitative hypotheses, the satisfaction of some criterion of unification or simplicity, and so forth. My conjecture is that whatever set of objective, nonprag­matic, universal criteria you propose you will be able to find or construct counterexamples to it, both as a set of necessary conditions and as a set of sufficient conditions. You will be able to find explanations that you will want to evaluate highly, despite the fact that they violate one or more of your favorite criteria. (Although this is not something I have illustrated here, you will evaluate them highly because they satisfy pragmatic criteria that are appropriate to use in the context of evaluation.) And you will be able to find, or at least construct, explanations (as I tried to do with the G-M explanation) that satisfy your criteria yet would not be highly re­garded. You can emphasize criteria such as the introduction of laws, causal factors, and unification. But unless you say something more specific about the kinds of laws and causal factors to be used, or what is to be unified, you won't find your criteria sufficient to exclude examples you want ex­cluded. But this “something more,” as I tried to illustrate in the Rutherford case, will involve fairly specific empirical assumptions that may be made by certain scientists at certain times but not by others at others times: You want to derive the scattering angles not just from any laws that will do the job, or from any causes no matter how described, but (e.g.) from ones that invoke events occurring within the atom. You desire an explanation that provides unification, but not just any sort of unification. (One that unifies only various results obtained in scattering experiments, as does the G-M hypothesis, may not be of sufficient interest to you.) To determine what this “something more” is requires pragmatic assumptions about the explanatory context.

Now let me consider one major objection the nonpragmatist may offer. It is the one mentioned earlier that he might make against van Fraassen. Even if you accept the importance of illocutionary evaluations, the nonpragmatist may say, all this shows is that sentences of the form “E is a good explanation of q” are incomplete. In the case of illocutionary evaluations the view I have espoused completes such sentences by writing “E is a good explanation for an explainer to give in explaining q to an audience”—which makes them strongly pragmatic. But there may be ways to complete such sentences that yield the same evaluations but that are not pragmatic.

Let me use the term “instructions” to refer to a set of rules or guide­lines an explainer may be following when he explains q to an audience, or that an audience may want followed when q is explained to it. In­structions impose conditions on the answer to the explanatory question. They may incorporate very specific empirical conditions assumed by the explainer or audience. (For example, “Describe the structure of the atom in such a way that the interaction between alpha particles and ei­ther positively or negatively charged constituents of the atom produces the scattering.”) They may also incorporate some very general conditions. (“Derive the scattering angles from quantitative laws.”) Suppose that by appeal to a particular explanatory context—by appeal to the knowledge, beliefs, desires, and values of the explainer and audience—we determine that some set of instructions I is an appropriate one for that explainer to follow in explaining q to that audience. (The instructions themselves will not include reference to any explainer or audience.) We can now take the (allegedly) incomplete sentence “E is a good explanation of q” and com­plete it by relativizing it to the instructions I (and perhaps also to some set of beliefs K of explainer and/or audience):

(8) E is a good explanation of q relative to instructions I (and K).

We then supply truth-conditions for sentences of this form which are “objective” and are not relativized to explainer or audience. Here is one possibility:

(9) A sentence of form (8) is true iff

(a) E satisfies instructions I, and E provides a correct answer to question Q; or

(b) Given K, it is probable that (a) obtains.

I don't wish to defend these conditions but only to use them as an example. By our earlier definition, sentences of the form (8) should be neither strongly nor weakly pragmatic. Such sentences contain no terms for an explainer or audience; their truth-conditions (9) do not contain such terms; and their truth-values will not vary with a change in who is explaining or to whom (as long as instructions I are kept the same). So, the nonpragmatist will admit, just as you need to appeal to the context to determine what question Q is being raised, and what beliefs K can be assumed, so you need to appeal to the context to determine what instruc­tions I are to be followed. But once all these things are determined, then the issue of whether E is a good explanation of q relative to I and K is settleable in an objective, nonpragmatic way (by determining, e.g., whether (a) or (b) of (9) is satisfied).

This reply, I suggest, trivializes the nonpragmatist's position with regard to the evaluation of explanations. The aim of nonpragmatists such as Hempel and Salmon is to provide nonpragmatic criteria of evalua­tion—criteria whose applicability does not depend on, or vary with, who is explaining or to whom. What I have called “instructions” are rules that incorporate criteria to be used in evaluating explanations. And the non­pragmatist is now agreeing with me that the applicability or appropriate­ness of some set of instructions will depend upon, and vary with, explainer and audience. But this is too much of an admission. When it comes to evaluating explanations I take the nonpragmatist to be seeking a set of instructions whose appropriateness is not affected by context.

Let me put this in another way. The nonpragmatist should not transform a sentence of the form “E is a good explanation of q” into “E is a good explanation of q relative to instructions I,” but into “E is a good explanation of q relative to appropriate instructions I.” Or better, he should say that sentences of the form “E is a good explanation of q” are true only if there is some set of appropriate instructions that E satisfies. In either case the instructions are to be appropriate ones. And if, as previously, the nonpragma­tist admits that appropriateness always depends, in part, on context, he is in agreement with the pragmatist. If the very definition of “appropriateness” with regard to instructions requires reference to an explainer and audience (see The Nature of Explanation, pp. 112ff), and if the truth-conditions for “E is a good explanation of q” require the satisfaction of appropriate instruc­tion, then “E is a good explanation of q” is strongly pragmatic.

In sum, the situation here is different from that of van Fraassen, who appeals to the context to determine only the question being raised, a set of alternative hypotheses, and the background information. By contrast, the instructions he formulates for evaluating explanations are not pragmatic. Their applicability does not depend on, or vary with, explainer or audience.

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