On the quality of auxiliary assumptions

In a recent contribution to this journal, David Trafimow and Joshua Uhalt (2015) argue that the alleged tradeoff between two desiderata of scientific theories, namely explanatory breadth and predictive power, is less clear-cut than commonly assumed. Briefly, proponents of the tradeoff claim that as the predictions you derive from your theory gain in precision, which is needed to make them testable, you lose in explanatory breadth. Trafimow and Uhalt’s argument relies on the necessity of auxiliary assumptions in making predictions. To make his prediction of where (what is now called) “Halley’s Comet” would appear in the sky, Halley had to rely on more than just Newtonian theory, which, after all, does not even mention the comet. It is only the combination of Newton’s theory and certain auxiliary assumptions, for example about the comet’s mass and its previous position that gets us the desired prediction. Moreover, this example also serves as a counterexample to the tradeoff, since it combines the impressive explanatory breadth of Newtonian theory with very precise predictions. Trafimow and Uhalt extend the point to psychology, demonstrating that psychological theories with an impressive explanatory breadth, such as Skinner’s behaviorism and psychodynamic theory, can yield precise predictions, given the right auxiliary assumptions. Trafimow and Uhalt conclude: “When combined with high quality auxiliary assumptions, theories with impressive explanatory breadth can … make sufficiently precise predictions” (2015, p. 837).

Trafimow and Uhalt’s (2015) discussion is valuable for a number of reasons. First, as they themselves show with a wealth of examples, the alleged tradeoff is well-entrenched in the literature. Second, there will be few philosophers or scientists willing to deny that explanatory breadth and predictive power are indeed desiderata of scientific theories, associated as they are with the respective virtues of unification and falsifiability. This shows that the tradeoff, if it exists, is undesirable—in other words, that there is something substantive at stake. Finally, it is always intellectually wholesome to highlight the indispensability of auxiliary assumptions in making predictions. To be sure, this general point is widely recognized in philosophy of science, but in discussing the merits of particular theories it is all too easy to forget that theories are never tested in isolation. ¹

All this being said, there is one important omission in the article: the notion of quality of auxiliary assumptions is left unspecified. ² This is unfortunate, since it is precisely this notion that opens the possibility of assumptions that defy the tradeoff. Apparently, there is some property of auxiliary assumptions that, when present to higher degrees, allows for more precise predictions—thus, one requirement for a candidate notion is that it is gradual. What could it be? Since Trafimow and Uhalt themselves do not provide a definite answer to this question, it is to a certain extent up to the reader to figure out what it is about auxiliary assumptions that can boost predictive power. In the remainder of this comment, I will review three possibilities, namely observability, level of detail, and precision. All three proposals are found wanting. I end with an appeal to philosophers and theoretically minded psychologists to come up with alternatives.

So in virtue of what quality of the auxiliary assumptions can we derive testable predictions from our theories? One candidate is observability. Maybe the more our auxiliary hypotheses make references to entities, properties, and processes that are observable, the more they allow for testable predictions. This seems plausible when we consider the case of Halley’s Comet again. Newtonian theory includes all kind of non-observational terms (e.g., “mass”), which are redefined in observational terms by Halley’s auxiliary assumptions. Indeed, Trafimow and Uhalt themselves stress that such assumptions “enable the researcher to bridge the gap between the non-observational terms in theories and observational terms in empirical hypotheses” (2015, p. 835). Conceptually at least, it does make sense to talk about a degree of observability, so this is one candidate for the notion of quality.

Another possibility is detail. The idea would be that the more detailed our auxiliary assumptions are, the more they allow us to make testable predictions. Detail is certainly a gradual notion. In the case of Halley’s comet, the auxiliary assumptions do exhibit considerable detail (precise starting point of the comet, its mass, etc.). In a sense, Halley came to his prediction because his auxiliary assumptions “plugged in” the details about the comet into the general equations of Newtonian theory.

Finally, one might think that increasing the precision of auxiliary assumptions is what leads to increased predictive power. Indeed, one can easily see in the Halley case, that a more precise calculation of the mass, initial position, etc. of the comet would lead to a more accurate prediction. Although as I said earlier, in their 2015 article, Trafimow and Uhalt do not specify what they mean by quality of the auxiliary assumptions, I suspect this is what they have in mind. One indication for this is that in discussing the Halley example, they say that the quality of auxiliary assumptions can increase due to technological advances (2015, p. 835). In another article, Trafimow and Earp discuss the difference between using a stopwatch and a beam to measure walking speed in priming experiments (Earp & Trafimow, 2015, p. 621). The decision to switch from stopwatch to beamer is likely motivated by the auxiliary assumption that one should use the method of measuring time that is least susceptible to human idiosyncrasies. That is, the level of precision is higher. ³

However, none of these candidates will work. The first two proposals run aground when we consider that many of the auxiliary assumptions that are instrumental in allowing us to infer predictions from our theories are idealizations and, strictly speaking, false. We assume an ecosystem to be self-contained, planes to be frictionless, markets to be in perfect equilibrium, populations to be infinite, people to maximize their profit, etc. All these assumptions can be used to make predictions. It has been recognized in the literature that there are different kinds of idealizations (cf. Weisberg, 2007), and indeed it is easy to find examples that rule out observability and detail as candidates.

Since frictionless planes, self-contained ecosystems, and markets in perfect equilibrium do not exist, it cannot be the observability of the auxiliary assumptions that grants us the predictive power we need to test a theory, if we take for granted that one cannot observe what does not exist. Likewise, level of detail cannot be the answer, since we sometimes deliberately add simplifying assumptions in order to get predictions. For example, the assumption that people maximize their profit can be used to infer predictions from rational choice theory, which is a theory with considerable explanatory breadth. A simple case is the prediction that people will accept any distribution of money proposed by someone else, as long as they get something. We know from the results of game theory experiments, such as the ultimatum game, that this is a simplification, and that people take many more things into account when deciding to reject or accept an offer. Still, it is precisely the fact that we make this simplification that gives rational choice theory its predictive power. Indeed, if we factor in all the reasons people have for rejecting or accepting, the theory becomes intractable.

Finally, when it comes to precision, the situation is more complex. There is certainly a sense in which precision is a default value. If we have more precise measurements available, it seems ludicrous not to use them if our resources permit. ⁴ And indeed, almost trivially, the more precise the values we can assign to the variables in equations, the more precise our predictions are going to be. Yet the problem with this candidate is that it only applies when we consider quantifiable auxiliary assumptions. This is a problem, because qualitative auxiliary assumptions can also yield testable predictions. Thus, using Boyle’s gas law, according to which at constant temperature, the pressure exerted by gas and its volume are inversely proportional, together with the auxiliary assumption that gas consists of freely moving molecules, we can predict that when we heat a gas inside a closed container, the pressure the gas exerts on the walls of the container increases (the molecules bounce against the walls with greater speed). To be sure, if we measure pressure and temperature, and fill in the resulting values for the variables in the equation of Boyle’s law, this gives us a more detailed prediction: for every degree the temperature rises, the pressure will increase with a certain degree. Again, the more precise our measurements, the more precise the prediction. Yet one need not take this quantitative route. If we leave the auxiliary assumptions at a qualitative level, we get the prediction that increased heat will lead to increased pressure. Though it derives from qualitative, hence not precise, auxiliary assumptions, this prediction is nevertheless testable.

This point is especially relevant in the field of psychology. As in physics, psychological theories can often be interpreted both qualitatively and quantitatively. For example, the Weber–Fechner law can be expressed by means of an equation, or by the informal statement that there is a non-linear relation between sensation and intensity of physical stimuli. However, unlike physics, when psychological theories are interpreted quantitatively, the numerical values are often obtained by auxiliary assumptions that are far less precise than those in physics. This is because psychological theories and assumptions often involve reference to mental states, and unfortunately, these are not directly measurable. There is, for example, no device for measuring the mood of participants to psychological experiments; instead, we get scores by letting them fill in questionnaires in which they are asked to rank their feelings, or by judging and ranking their behavior and treating that as a proxy for them being in a certain mood. Think again about the priming experiments. Measuring walking speed with a beamer instead of a stopwatch will get you a more accurate result; but ultimately, the aim is not merely to measure walking speed, but to measure walking speed as an indication of a mental state, for example having a memory of a stereotype (Bargh, Chen, & Burrows, 1996, p. 237). Indeed, the use of the auxiliary assumption that behavior is a reliable proxy for mental constructs is very common in psychology. ⁵ If it is the precision of the auxiliary assumptions that make for the predictive power of theories, so much the worse for psychology.

I hope it is evident to the reader that the aim of this comment is not to present some kind of damning criticism against Trafimow and Uhalt’s (2015) argument. As I have said above, I find their discussion valuable for many reasons. Rather, it is meant as an exhortation to further reflect on how the role of auxiliary assumptions might allow us to escape the tradeoff between explanatory breadth and predictive power. Again, what is it about auxiliary assumptions that allow us to infer testable predictions? Perhaps there are other candidates besides the ones I have considered, or perhaps my rejection of one or more of them is premature. In any case, I sincerely concur with Trafimow and Uhalt’s hope and expectation that “philosophically oriented psychologists will continue to appreciate the general importance of auxiliary assumptions and their relevance to many additional issues in the philosophy of science” (2015, p. 838).