2.4.1 Carnap’s Desiderata Revisited
In order to construct a satisfactory explication of the above concept, we need to find an ex- plicatum that satisfies Carnap’s four desiderata: similarity to the explicandum, fruitfulness, precision, and simplicity (see Section 2.2 above). We may guarantee that our explication will score well regarding the last two of these requirements in the following way: First, to ensure that our explicatum is precise in the sense that Carnap (1950, p. 3) requires (stated in a “logicomathematical” language so that the explicatum is “given by explicit rules for its use”), I adopt the probability theory as the formal language in which the explicatum will be expressed. The aim of this explication will thus be a probabilistic measure of the degree of explanatory power that a particular hypothesis has relative to a specified set of evidence.10
9This feature of abduction might suggest that explanation is tied essentially to necessity for Peirce. However, elsewhere, Peirce clarifies and weakens this criterion: “to explain a fact is to show that it is a necessary or, at least, a probable result from another fact, known or supposed” (Peirce 1935, 6.606, emphasis mine). See also (Peirce 1958, 7.220).
10One might take issue with my stipulating from the start that the explicatum be probabilistic. There are at least two important reasons why I do this. First, our explicandum seems well-suited for a probabilistic account. Our explicandum pertains to the ability of a hypothesis to increase the degree to which we expect (or ought to expect) some set of evidence to attain. But probabilities are often interpreted as degrees of expectedness (or degrees of rational expectedness). Thus, depending on one’s interpretation of probability, probabilities may bear a prima facie conceptual resemblance to our explicandum.
Second, I ultimately want to say something informative about whether, and to what extent, a hypothesis’s explanatory power relative to some evidence is relevant to that hypothesis’s probability given that evidence. But then I need to bridge the language of explanatory power with the language of the probability theory in some way. An explicatum stated in terms of the probability theory thus provides me with such a bridge.
Note that, by stipulating that the explicatum be probabilistic, I have not guaranteed that there will be anything approaching a satisfying probabilistic explication of explanatory power. It could be that this stipulation leads me down a dead end. Ultimately, in addition to being stated precisely (which is guaranteed by being stated probabilistically), the explicatum needs to be sufficiently similar to the explicandum, fruitful, and simple in order to be satisfactory. I will in fact argue that my explication satisfies all of these desiderata in what follows. This is another place then where the proof is in the pudding. Whether the probability
Our explicatum will thus be stated as a mathematical function. In this case, Carnap’s simplicity desideratum amounts to a requirement that the mathematical form of this function be as simple as possible (Carnap 1950, p. 7). Accordingly, we will simply lay it down as a condition of adequacy for our explication that it be functionally simple (the probabilistic nature of the explicatum and this simplicity condition are both more fully specified in CA 1 of Section2.5.1).
It is much more difficult to ensure that our explicatum be “sufficiently similar to the explicandum.” This is a requirement that we cannot just stipulate from the start. Instead, we will do our best to ensure that our explicatum satisfies this desideratum by keeping the latter in mind in the very construction of our explicatum. More specifically, in order to develop an explicatum that is similar to the explicandum, we take the following steps. First, Section 2.4.2 will propose a set of intuitive conditions that hold true regarding our explicandum. Second, Section 2.5 will propose a set of formal conditions of adequacy for our explicatum that are probabilistic renderings of the intuitive conditions. The intention is that, by requiring that our explicatum satisfy formal versions of our intuitive conditions, we force our explicatum to resemble the explicandum of explanatory power – at least in certain respects. Even after all of this, I will devote Section 2.6 of this chapter along with the entirety of Chapter 3 to defending further the claim that our explication satisfies Carnap’s similarity desideratum.
Finally, with a candidate explicatum constructed in this way, Chapters 4, 5, and 6 will investigate the epistemic implications of explanatory power. In these later chapters, I will argue that our explicatum proves to be quite fruitful to this investigation. In the end then, I eventually argue that our explication is satisfying according to all four of Carnap’s desiderata.
2.4.2 Conditions for an Explication of Explanatory Power
Above, we clarify that our explicandum is a particular sense of “explanatory power.” A hypothesis exhibits such explanatory power with regards to some evidence when it makes that evidence less surprising (or, we can just as well say, more expected). To ensure that
our account of explanatory power is an account of explanatory power in this sense, we require that it agrees with the following condition: a hypothesis has explanatory power over a proposition to the extent that it makes that proposition less surprising.11
This initial condition itself leads to some related, additional conditions for an account of explanatory power. First, just as (positive) explanatory power comes with a decrease in surprise, one might say that a hypothesis has “negative explanatory power” over some proposition to the extent that it makes that proposition more surprising. Recalling an example from the introduction of this dissertation, the hypothesis that my toddler was playing in my office would seem to me to be a powerful explanation of my books being in a disarranged state on my shelves. This makes sense on the current conception of explanatory power as it would be far less surprising that my books are in this state given the truth of this hypothesis. Correspondingly, I would judge the hypothesis that my wife was recently in my office to be a particularly poor explanation of the disarranged state of the books on my shelves. This is because my wife tends to straighten the books on my shelf when she sees them out of order. This hypothesis thus has negative explanatory power; it does negative explanatory work because it makes the disarranged state of the books even more surprising than it already was.
Given the above, we may also say that a hypothesis lacks all (positive or negative) explanatory power whatever relative to some given proposition if the latter is neither more nor less surprising in light of that hypothesis. The perceived motions of the planet Uranus, for example, are less surprising in light of the hypothesized existence of Neptune, but they are not any more or less surprising given that my two year old was playing in my office yesterday. The latter hypothesis is simply “explanatorily irrelevant” to the explanandum in question. Notice that this notion of negative explanatory power, as defined above, differs from that of explanatory irrelevance. A hypothesis that makes the evidence even more surprising than it already was is explanatorily inferior to one that is just irrelevant to the evidence. This is
11There are two senses in which the notion of explanatory power described in this condition is allowed to be more general than that suggested by Peirce’s description of abduction above: first, a hypothesis may provide a powerful explanation of a surprising proposition, in our sense, and still not render it a matter of course; i.e., a hypothesis may make a proposition much less surprising while still not making it unsurprising. Second, our sense of explanatory power does not suggest that a proposition must be surprising in order to be explained; a hypothesis may make a proposition much less surprising (or more expected) even if the latter is not very surprising to begin with.
because there is more explanatory work to be done in light of the former, but not in light of the latter.
Insofar as a hypothesis has positive explanatory power over a proposition to the extent that it renders the latter unsurprising, one might additionally conclude that a hypothesis provides a maximally powerful explanation of some proposition just when it would lead one to expect that proposition to be true with certainty; this occurs when the hypothesis implies the truth of that proposition. On the other hand, a maximally poor explanation of some known proposition is one that renders the latter maximally surprising, and this occurs when the hypothesis implies that the proposition in question is false.
Finally, the less surprising a proposition’s truth is in light of a hypothesis, the more surprising is its falsity. Given the above, this means that the more explanatory power a hypothesis has over a proposition, the less it has over the proposition’s negation – my toddler’s playing in my office is a powerful explanation of my books being disarranged to the extent that the same hypothesis would be a poor explanation of my books being in neat order. To summarize then, focusing on our sense of “explanatory power” as decrease in surprise, all of the following are natural, compelling conditions required for any account of explanatory power:
Condition 1: A hypothesis has positive explanatory power over a proposition to the extent that it decreases the degree to which that proposition is surprising (i.e., increases the degree to which we expect that proposition to be true).
Condition 2: A hypothesis has negative explanatory power over a proposition to the extent that it increases the degree to which that proposition is surprising.
Condition 3: A hypothesis has no explanatory power over (i.e., is explanatorily irrelevant to) a proposition if and only if the latter is neither more nor less surprising in light of that hypothesis.
Condition 4: A hypothesis has maximal explanatory power over a proposition (i.e., is a maximally good explanation) if and only if it leads us to expect with certainty that the proposition is true.
Condition 5: A hypothesis has minimal explanatory power over a proposition (i.e., is a maximally poor explanation) if and only if it leads us to expect with certainty that the
proposition is false.
Condition 6: The more explanatory power a hypothesis has relative to a proposition, the less it has relative to the negation of that proposition.
Before moving on to our attempt to construct an explicatum from these conditions, it is worth making a few clarifications. The first comes by way of a reminder: Recall that this account is not intended to reveal the conditions under which a hypothesis provides an explanation of some explanandum (that is, after all, the aim of a metaphysical account of ex- planation rather than an epistemologically motivated account of explanatory power); rather, the goal here is ultimately to explicate the strength or power of a potential explanation. In other words, the explication aimed at in this chapter has, as its target concept, the ex- planatory power of a hypothesis relative to some evidence, given that the former provides a potential explanation of the latter. Accordingly, we restrict ourselves in presenting our condi- tions of adequacy to speaking of theories that do in fact provide potential explanations of the explanandum in question. Thus, it is no counterexample to Condition 1 and Condition 4, for example, to point out that any proposition will render itself maximally unsurprising. Given any proposition, that same proposition is indeed maximally unsurprising. However, this does not thereby make any proposition a maximally powerful explanation of itself. Such an untoward conclusion is precluded by the fact that a proposition simply cannot provide a potential explanation of itself (i.e., it cannot stand in the explanatory relation to itself).
Second, I take no position here on whether the explication given in this chapter captures the notion of explanatory power generally; it is consistent with this account that there be other senses of explanatory power that do not fit the account provided here.12 On the
other hand, the account given in this dissertation does claim to capture one familiar and epistemically compelling sense of explanatory power commonly, if not always, invoked when humans reason explanatorily. The central, defining feature of explanatory power, in this sense, is the notion that a hypothesis has explanatory power over some proposition to the
12As a possible example,Salmon(1970),Jeffrey(1969), andGreeno(1970) all argue that there is a sense in which a hypothesis may be said to have positive explanatory power over some explanandum so long as that hypothesis and explanandum are statistically relevant to one another, regardless of whether they are negatively or positively statistically relevant. As will become clear, insofar as there truly is such a notion of explanatory power, it must be distinct from the one that we have in mind.
extent that it alleviates our surprise in that proposition’s truth.
Finally, regarding the distinction – made in Section1.2– between analyses of explanation and epistemic accounts of explanatory power, it is worth pointing out the following. While this explication of explanatory power does not rule out any particular metaphysical account of explanation (after all, there may be other senses of explanatory power than the one ana- lyzed in this dissertation), it does seem to fit better with some more than others. Without going into much detail, the idea that a hypothesis has explanatory power to the extent that it makes the explanandum less surprising (more expected) seems to fit especially well with the Deductive-Nomological and Inductive-Statistical accounts (Hempel 1965) and necessity accounts (Glymour 1980) of explanation. These have in common that they explicitly analyze explanation in such a way that a hypothesis that is judged to be explanatory of some ex- planandum will necessarily increase the degree to which we expect that explanandum. This notion of explanatory power also seems quite compatible with causal-mechanical accounts of explanation (Salmon 1984, Machamer et al. 2000) given the fact that causal strength is plausibly measured in terms of positive statistical relevance (Fitelson and Hitchcock 2011) (and this will be the same basic approach taken to measuring explanatory power below).13