Explanation, Understanding, and Belief Revision∗1 Andrés Páez Universidad de los Andes 2 3 4 Contents5 1 Introduction 2336 2 Three Theses about Explanation 2347 3 The Objective Basis of Explanation 2368 3.1 Probability Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2369 3.2 Epistemic Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910 4 Potential Explanations 24111 5 The Epistemic Value of Explanation 24312 6 Conclusion 24913 1 Introduction14 For the longest time, philosophers of science avoided making reference to the notion15 of understanding in their accounts of explanation. Although Hempel, Salmon, and16 other philosophers who wrote about the subject in the 20th century recognized that17 understanding is one of the main goals of science, at the same time they feared that18 any mention of the epistemic states of the individuals involved would compromise19 the objectivity of explanation. Understanding is a pragmatic notion, they argued, and20 ∗Previous versions of this paper were presented at the Biennial Meeting of the Philosophy of Science Association in Vancouver, at the XVI Jornadas de Epistemología e Historia de la Ciencia in Córdoba (Argentina), and at the III Conference of the Latin American Association for Analytic Philosophy in Buenos Aires. I am grateful to the audiences in these venues for their useful comments and questions. 234 Explanation, Understanding, and Belief Revision although a subject worthy of psychological investigation, pragmatics should be kept at21 a safe distance from the universal, epistemological features of explanation. Although22 this attitude towards the notion of understanding has changed in the last decade1, there23 are still many misgivings about using pragmatic notions in the analysis of one of the24 central concepts in the philosophy of science2.25 My main purpose of this paper is to defend the idea that there is a sense in which it26 is meaningful and useful to talk about understanding in an objective sense3, and that to27 characterize this notion it is necessary to formulate an account of scientific explanation28 that makes reference to the doxastic states and epistemic goals of the participants in a29 cognitive enterprise. It is important to clarify at the outset that my goal is not to offer30 a general analysis of the notion of understanding, and that my approach is restricted31 to the understanding of singular facts in well-defined scientific contexts.32 The essay is divided as follows. In the next section I introduce three theses about33 scientific explanation that will serve as the basis for the rest of the discussion. The34 first thesis, which is defended in sections 3 and 4, states that determining the potential35 explanations of a fact is essentially a non-pragmatic matter. This thesis is meant to36 allay the fears of those who see the introduction of pragmatic factors as the beginning37 of the road towards an unbounded relativism. Since the objective basis of explanation38 will be probabilistic, at the beginning of the paper I include a detailed discussion39 about the way in which probability will be used in my account of explanation. The40 second thesis, which is presented in section 5, states that it is possible to determine41 the epistemic value of most potential explanations of a fact, and that such value can42 be established in a non-arbitrary way despite being the result of the evaluation of43 individual researchers. Finally, towards the end of the essay I explain the third thesis,44 which establishes the criteria for the acceptance of an explanation in the corpus of45 beliefs of those researchers involved. These criteria are based on their joint assessment46 of the credibility and epistemic value of potential explanations.47 2 Three Theses about Explanation48 It has often been said that explanation is an interest-relative notion. Different inquir-49 ing agents impose different demands on the information they regard as explanatorily50 valuable. The interest-relativity of explanation has been accounted for in several ways:51 some authors have proposed a contrastive analysis of the explanandum (van Fraassen,52 1See, for example, de Regt (2009), de Regt, Leonelli & Eigner (2009), Fey (2014), Grimm (2008), Khalifa (2012), Kvanvig (2009), and Strevens (2008, 2013). 2See, for example, Trout (2002, 2007) and Craver (2013) for more recent defenses of a purely ontic approach to explanation. 3Objective understanding in this sense will turn out to be the opposite of what de Regt (2009, p. 585) calls "the objectivist view of the relation between explanation and understanding," which he attributes to Hempel and Trout. 2 Three Theses about Explanation 235 1980; Lipton, 2004) or a detailed description of the illocutionary context of an ex-53 planatory speech act (Achinstein, 1983). In my view, the interest-relativity of expla-54 nation has a much deeper origin. It derives from the interest-relativity of inquiry in55 general. Different agents use information for different purposes, and their acceptance56 of new information is directed by their cognitive and practical interests and goals. Far57 from being a superficial characteristic of inquiry, I believe that this is a fundamental58 trait of the acquisition of knowledge in general. The cost and effort that goes into59 obtaining new information makes the beliefs4 that an inquiring agent has accepted a60 valuable asset that must be treated with care. Gratuitous losses must be prevented61 and the agent's acceptance of new information always involves the risk of bringing62 error into his system of beliefs. The risk must always be compensated by an epistemic63 incentive that outweighs the cost.64 One of the biggest epistemic incentives of all is to obtain understanding of a fact.565 But if understanding a given fact fulfills no purpose in the eyes of an inquiring agent,66 he will be more reluctant to incur the risks involved in accepting an explanation of it.67 On the other hand, if understanding a fact fulfills the cognitive interests and goals of68 the agent, but the information explains too much, it might be too good to be true. The69 acceptance of an explanation thus requires a delicate balance between two conflicting70 cognitive goals: the acquisition of valuable explanatory information and the avoidance71 of error.72 The account of explanation that I present in this paper takes into account the dif-73 ference between information and informational value, between the informational con-74 tent of an explanation and the epistemic value of that content. When an agent seeks75 to expand his beliefs, his interest is restricted to information that promotes his cog-76 nitive goals or that is relevant to the problems he is trying to solve. In Catherine77 Elgin's words, "truth does not always enhance understanding. An irrelevant truth is78 epistemically inert" (1996, p. 124). I will argue that the goal of an inquiring agent is79 not just to find factually accurate explanations; it is to find explanations that are both80 factually accurate and epistemically valuable. This idea is captured by the following81 three theses:82 1. Whether a piece of information is a potential explanation of the fact that P is83 mostly a non-pragmatic matter.84 4In this paper beliefs should be understood as an agent's epistemic commitments, in the sense of Levi (1980). Some authors, such as Cohen (1989), use the term 'acceptance' for such attitudes, reserving the term 'belief' for involuntary epistemic states, akin to feelings. There is an extensive literature on the distinction between acceptance and belief (e.g., Engel, 2000; Cresto, 2009, among many others), but I cannot discuss the issue in this essay. 5Understanding laws and regularities is, of course, an equivalent or even greater epistemic incentive. The account presented here is restricted to the explanation of singular facts because the well-known objections against the explanation of laws require an entirely different analysis, one that most likely will not be probabilistic. 236 Explanation, Understanding, and Belief Revision 2. It is possible to determine the objective epistemic value of a subset of all the85 potential explanations of the fact that P .86 3. In trying to understand the fact that P , an inquiring agent should only accept87 the potential explanations with positive objective epistemic value.88 In the rest of the paper I discuss and defend each of these three theses.89 3 The Objective Basis of Explanation90 In this section and the next I defend the first of the three theses stated above, namely,91 that determining the potential explanations of a given fact is mostly a non-pragmatic92 matter. My basic contention is that an explanation of a singular fact should provide the93 information required to integrate the explanandum into an agent's cognitive system.94 An explanation should provide some of the factors that contributed to make P a fact,95 and some of the obstacles that could have, but did not prevent it from being one. With-96 out such information, P will describe a brute fact, isolated from the rest of the agent's97 beliefs about the world. Probability sentences are the connecting tissue of an agent's98 corpus of beliefs. The influence of the preventing and contributing factors is captured99 by probability sentences of the form p(P |Q) > p(P | ∼ Q) and p(P |Q) < p(P | ∼ Q)100 that indicate that the fact that Q is statistically relevant to the explanandum6.101 The notion of statistical or probabilistic relevance has been used by many au-102 thors in the analysis of explanation. The best-known examples are Hempel's (1965)103 I-S model, Salmon's (1971, 1984) S-R model, Railton's (1978) D-N-P model, and104 Fetzer's (1974) causal-relevance model. All of these accounts consider precise prob-105 ability values to be an essential part of an explanation. In contrast, I will argue that106 reference to probability values is largely unnecessary. Probability values have descrip-107 tive, predictive, and evidential value, but not explanatory value.108 3.1 Probability Values109 Probability values are thought to be important for two different reasons. If a statistical110 explanation is conceived of as an inductive argument, as it was in Hempel's original111 Inductive-Statistical model, the degree of expectation that a body of evidence confers112 upon a given event must be very high. Thus the value of the inductive probability113 must be kept in check to make sure it does not fall below a certain threshold as inquiry114 proceeds. On the other hand, if a statistical explanation is understood as an objective115 6Many authors have used probabilities to model the epistemic states of researchers (e.g. Boutilier, 1995; van Fraassen, 1995; Halpern, 2003; van Benthem, 2003; Arló-Costa & Parikh, 2005). My account uses probability sentences to model the doxastic basis of an explanation, but an agent's epistemic states should not be understood to be probabilistic. 3.1 Probability Values 237 account of the stochastic process involved, as it is in Salmon's and Railton's models,116 it is crucial to avoid the attribution of false probability values to the probabilistic laws.117 In response to criticism by Jeffrey (1971), Hempel (2001) gave up the high prob-118 ability requirement, together with the claim that the explanans of an I-S explanation119 should show that the phenomenon described by the explanandum sentence was to be120 expected. Without this claim, however, the first reason to attribute any importance to121 probability values disappears. If the explanans is not supposed to justify our expecta-122 tions that the explanandum will occur, there is no need to make sure that the value of123 the probability remains over a certain threshold.124 Before we can evaluate the second reason why probability values are deemed to125 be explanatory, we must take a closer look at the logical structure of statistical expla-126 nations. One of the features of probability theory is that it does not have a weakening127 principle. A sound inductive argument that strongly supports its conclusion can be128 transformed into one that strongly undermines its conclusion with the insertion of129 additional true premises. An individual event can be referred to different reference130 classes, and the probability of the property associated with the event can vary consid-131 erably from one class to another. Hence, a body of evidence may confer a high degree132 of expectation upon a given event, while another body of evidence may confer a very133 low degree of expectation upon the same event. This is the problem that Hempel called134 the ambiguity of I-S explanation.135 Hempel's partial solution to the problem is the requirement of maximal specificity.136 The requirement states that an acceptable statistical explanation should be based "on137 a statistical probability statement pertaining to the narrowest reference class of which,138 according to our total information, the particular occurrence under consideration is a139 member" (1965, p. 398). The requirement does not completely eliminate the ambi-140 guity because the narrowest reference class can only be determined in the light of our141 current knowledge. It does not guarantee that there are no unknown statistical gener-142 alizations that can be used to construct a rival argument. In fact, Hempel claimed that143 "the concept of statistical explanation for particular events is essentially relative to a144 given knowledge situation as represented by a set K of accepted sentences" (p. 402,145 emphasis kept).146 Salmon (1971) showed that the requirement of maximal specificity failed to rule147 out counterexamples in which irrelevant information finds its way into the explana-148 tion. But his main reason to reject Hempel's solution to the problem was his strong149 conviction that the appropriate reference class for a statistical explanation is one that150 is objectively homogeneous, not one that is epistemically homogeneous.151 The notion of an objective homogenous reference class amounts to this: For any152 given reference classA, and for any given property C, there is, in principle, a partition153 of that class into two subclasses A ∧ C and A∧ ∼ C. A property C is statistically154 relevant to a property B within A if and only if p(B|A ∧ C) 6= p(B|A). Using von155 Mises's concept of place selection, Salmon defines a homogeneous reference class as156 238 Explanation, Understanding, and Belief Revision follows:157 If every property [C1, C2, C3, . . . , Cn] that determines a place selection158 is statistically irrelevant to B in A, I shall say that A is a homogeneous159 reference class for B. A reference class is homogeneous if there is no160 way, even in principle, to effect a statistically relevant partition without161 already knowing which elements have the attribute in question and which162 do not (1971, p. 43).163 Salmon then replaces Hempel's requirement of maximal specificity for the refer-164 ence class rule: "Choose the broadest homogeneous reference class to which the sin-165 gle event belongs" (p. 43). This characterization of statistical explanations is supposed166 to avoid any epistemic relativity because any statement of the form p(G|F ) = r that167 meets the homogeneity condition must be regarded as a fundamental statistical law of168 nature. Its reference class cannot be further specified, not because we do not know169 how to make a further relevant partition, but because in principle it is impossible to170 make a further relevant partition.171 Salmon then defines a statistical explanation as follows. If we want to know why a172 member of the class A has the property B, the answer will be a S-R explanation that173 consists of: (i) the prior probability that a member of the classAwill have the property174 B : p(B|A) = r, (ii) a partition into homogeneous cells with respect to the property175 in question: A ∧ C1, A ∧ C2, etc., (iii) the posterior probabilities of the property in176 cells of the partition p(B|A∧C1) = r1, p(B|A∧C2) = r2, etc., and (iv) a statement177 of the location of the individual in question in a particular cell of the partition: "a is a178 member of A ∧ Ck" (pp. 76-77).179 Salmon explicitly requires the use of probability values in providing an explana-180 tion. The use of probability values stems from the fact that the S-Rmodel is at bottom181 a covering-law model. Since any statement of the form p(G|F ) = r that meets the182 homogeneity condition must be regarded as a fundamental statistical law of nature,183 each of the probability sentences in the explanans of a S-R explanation is a law of184 nature. And since the factive condition on explanation demands that every element in185 an explanation must be true, the probability assigned to the explanandum by each of186 these probability sentences must be the right one.187 To see how restrictive this requirement is, consider the following example pro-188 vided by Humphreys:189 If a man dies from lung cancer, having been a heavy smoker, omitting190 from a probabilistic explanation any of the following minor relevant fac-191 tors will result in a false probability claim: cosmic radiation from Alpha192 Centauri, particles from a chimney in Salem, Oregon, and a smoke-filled193 room he entered briefly at the Democratic convention eight years ago. It194 is good to be strict in matters of completeness, but not to the point of195 absurdity (1989, p. 111).196 3.2 Epistemic Relativity 239 Humphreys argues that if one insists in providing the exact probability of the ex-197 planandum as part of the truth conditions of an explanation, it will be impossible to198 distinguish between a complete explanation and a true explanation. The omission of199 absurdly small probabilistically relevant factors, known or unknown, will result in a200 false explanation.201 How can a true but incomplete statistical explanation be provided? Humphreys202 argues that instead of focusing on probability values, we should focus on causal rel-203 evance. An explanation should provide one or more of the factors that are causally204 relevant to an explanandum, and a factor is causally relevant if it changes the propen-205 sity for an outcome. His strategy has the advantage that it makes it possible to offer a206 true explanation of an event by providing a contributing or a counteracting cause even207 in cases where the other factors are not known and the true probability value cannot208 be calculated.209 3.2 Epistemic Relativity210 Although Humphreys' approach offers an appropriate formal basis for providing a sta-211 tistical explanation, there is an obvious objection. As the many versions of Simpson's212 Paradox illustrate, one or more of the factors that the agent is unaware of can turn a213 contributing cause into a counteracting cause, or vice versa. Humphreys' response to214 this objection is puzzling. He says: "Of course, epistemically, we can never know for215 certain that such confounding factors do not exist, but that is an entirely separate mat-216 ter, although regrettably relative frequentists have often failed to separate epistemic217 aspects of probabilistic causality from ontic aspects" (p. 114).218 It seems to me that it is Humphreys who is guilty of not keeping epistemic and219 ontic matters in separate baskets. If Salmon's model is too demanding, as Humphreys220 maintains, it is because we can never know if we have met all the conditions that it221 imposes on explanation. But Humphreys' account suffers from a similar problem. In222 order for something to be a contributing or a counteracting cause in Humphreys' sense,223 there cannot be any further factor, known or unknown, that will invert the influence224 of these causes on the explanandum, or that will neutralize them altogether. Thus225 an agent who offers a causal statistical explanation will always have to relativize the226 explanation to a knowledge situation.227 The accounts offered by Salmon and Humphreys avoid the epistemic relativity of228 statistical explanation by introducing a condition that effectively rules out the possibil-229 ity that a bona fide statistical explanation will be defeated by a rival statistical claim.230 But the cost of avoiding the epistemic relativity of explanation is to render useless their231 accounts of explanation. It is hard to see how such a relativization can be eliminated if232 we want to provide a coherent picture of the role of explanation in inquiry. If we adopt233 the view that epistemic relativity is an unacceptable feature of explanation, we will be234 forced to conclude that there has never been a genuine scientific explanation in the235 240 Explanation, Understanding, and Belief Revision history of science. Furthermore, we lose one of the main incentives for any scientific236 inquiry. Why would anyone want to incur the cost and effort involved in searching237 for explanations if the results cannot be assumed to be true in future decisions and238 deliberations? In Isaac Levi's words,239 If inquiry cannot be motivated by a concern to remove doubt, what is240 its rationale? If we cannot incorporate the solutions we come close to241 establishing into the evidence and background information for future in-242 vestigations, why should we care that we come close? The truth of the243 well-established conjecture remains an open question and a legitimate is-244 sue for future investigation. Inquiry never settles anything and, hence,245 inquiry–even inquiry into a specific problem–never legitimately termi-246 nates because the matter is settled but only, so it seems, because the in-247 vestigators are tired or bored or have run out of funds. No matter how248 minute a question might be, if inquiry into that question is free of costs,249 it should go on forever (1991, p. 2).250 The reference to a specific epistemic context in the characterization of explanation251 is clearly a departure from tradition. Many philosophers have claimed that pragmatic252 elements have no place in the study of explanation. They recognize that there are in-253 teresting issues associated with the process of providing an explanation in an actual254 context, and their intention is not to belittle their importance. But the concept of ex-255 planation that they characterize is, in Hempel's words, "a concept which is abstracted,256 as it were, from the pragmatic one, and which does not require relativization with257 respect to questioning individuals any more than does the concept of mathematical258 proof" (1965, p. 426). The same general idea is defended by many other philosophers259 of science.260 Michael Friedman has pointed out that there is a certain equivocation about the261 term 'pragmatic'. 'Pragmatic' can mean roughly the same as 'psychological', i.e.,262 having to do with the thoughts, beliefs, attitudes, etc. of individuals. But 'pragmatic'263 can also be synonymous with 'subjective'. In the latter sense, a pragmatic notion must264 always be relativized to a particular individual. Friedman's claim is that "a concept265 can be pragmatic in the first sense without being pragmatic in the second." Further on266 he explains: "I don't see why there can't be an objective or rational sense of 'scientific267 understanding', a sense on which what is scientifically comprehensible is constant for268 a relatively large class of people" (1974, p. 8).269 The traditional avoidance of any pragmatic element in a theory of explanation can270 thus be evaluated in two different ways. If one takes 'pragmatic' to mean the same as271 'subjective', the insistence in providing a non-pragmatic analysis of explanation, i.e.,272 an analysis that does not depend on the idiosyncrasies of the individuals involved, is273 perfectly justified. But if 'pragmatic' is interpreted in Friedman's first sense, there is274 no reason why an analysis of the concept of explanation should not make reference to275 4 Potential Explanations 241 the epistemic states of the individuals involved in a cognitive project.276 I believe that we should take Friedman's suggestion seriously and explore the pos-277 sibility of characterizing, in logically precise terms, a notion of explanation that is both278 objective and pragmatic, that does not depend on the idiosyncrasies of the individuals279 involved but that regards their epistemic states, their shared commitments, and their280 cognitive interests and goals as a fundamental part of the analysis. The concept of281 explanation will still be an "abstraction", in Hempel's sense, but an abstraction based282 on the decisions that take place when a group of inquiring agents rationally accept283 explanatory information. The resulting concept will be a hybrid, a combination of the284 formal, semantic, and pragmatic dimensions of explanation.285 4 Potential Explanations286 The epistemological framework for the account of explanation that I will present is287 Isaac Levi's version of the belief-doubt model first proposed by Peirce (1877)7. Ac-288 cording to the belief-doubt model, an inquiring agent presupposes that everything he is289 currently committed to fully believing is true. This does not mean that truth or falsity290 is relative to what the agent believes. But the agent's judgments of truth are relative291 to what he believes. If the agent is concerned with establishing true explanations of292 phenomena, his decision to accept an explanation can only be made relative to the293 judgments of truth available to him.294 To claim that an inquiring agent presupposes that everything he is currently com-295 mitted to fully believing is true is not to say that he cannot change his mind. Certainty296 or full belief does not entail incorrigibility. Levi explains the claim thus: "To regard297 some proposition as certainly true and as settled is to rule out its falsity as a serious298 possibility for the time being. . . . But from this it does not follow that good reasons will299 not become available in the future for a change of mind and for calling into question300 what is currently considered to be true" (1991, p. 3). Peirce puts it more graphically:301 "The scientific spirit requires a man to be at all times ready to dump his whole cartload302 of beliefs, the moment experience is against them" (1931, p. 55).303 An inquiring agent has no doubt that all the sentences in his corpus of beliefs are304 true. Nonetheless, he does not regard all of the facts stated by these sentences as being305 equally well understood. The degree to which an agent understands the fact expressed306 by a sentence P will depend on how well integrated P is to the agent's cognitive sys-307 tem. It will not depend on how much support it has or on how epistemically entrenched308 7Although my account of explanation uses Levi's belief revision theory as theoretical framework, it must be pointed out that Levi does not agree with my approach (personal communication). The main reason is that Levi believes that all explanations with probabilistic premises presuppose a D-N explanation stated in dispositional terms. Furthermore, Levi associates statistical explanations with the elimination of surprise and an increase in the expectation of the occurrence of the explanandum (Levi, 1988, 1995). The account of explanation that I present here does not entail those two consequences. 242 Explanation, Understanding, and Belief Revision it is. On the one hand, if a sentence has been accepted in his corpus of beliefs, it is309 judged to be true and no further argument is necessary. On the other hand, poorly un-310 derstood phenomena can be highly epistemically entrenched, and completely useless311 facts can be very well understood.312 According to the belief-doubt model, an inquiring agent's judgments of truth are313 always relative to what he is currently committed to fully believing. Thus, an agent's314 decision to accept an explanation can only be made relative to the judgments of truth315 available to him. Naturally such decisions will lack any sort of objectivity. An agent316 who wants to claim objectivity for the explanations that he accepts must first make sure317 that the explanation is consistent with K, the set of beliefs that represents the shared318 agreement between the members of a community of experts. More technically, the319 states of belief of the individual experts can be partially ordered in a manner satisfying320 the requirements of a Boolean algebra. In consequence, it will be possible to form the321 meet of their individual states, i.e., the strongest common consequence of all their322 states of belief (Levi, 1991, p. 13).323 Let P be a sentence in K. A set of sentences E is a potential explanation of the324 fact stated by P relative to K just in case the following conditions are fulfilled:325 (i) K ∪ E is consistent.326 (ii) E 6⊂ K.327 (iii) There is a sentence Q such that Q ∈ E.328 (iv) Either p(P |Q) > p(P | ∼ Q) ∈ E or p(P |Q) < p(P | ∼ Q) ∈ E.329 (v) There is no R ∈ K such that p(P |Q&R) = p(P | ∼ Q&R).330 (vi) P and Q are logically independent.331 (vii) Nothing else is an element of E.332 The first condition states that a potential explanation must be consistent with the333 corpus of beliefs in which the explanandum is accepted. The second condition states334 that the potential explanation cannot be already accepted in K. The third condition335 says that the potential explanation must include a singular sentence Q that describes336 a potentially relevant factor. The fourth condition states that Q is positively or nega-337 tively relevant to the fact that P . The fifth condition guarantees that P and Q will not338 be spuriously correlated, as far as we know. Condition (vi) guarantees that P will not339 explain itself. It also prevents the inclusion of trivial cases in which p(P |Q) = 1 be-340 cause P ` Q. A potential explanation is thus a set containing a singular sentence that341 describes a fact, and a probability sentence that states the potential statistical relevance342 of that fact to the explanandum.343 5 The Epistemic Value of Explanation 243 Using this definition of a potential explanation, we can now characterize the no-344 tion of an explanation space. An explanation space can be understood as the set of345 sentences that contains all the potential explanations of P , regardless of whether the346 inquirers are aware of them or not.347 (EP ) For every sentence P in K, there is a set {E1, E2, . . . , Ek} such that Ei is an348 element of the set iff it is a potential explanation of P . The set, denoted EP , is349 the explanation space of P .350 The explanation space will contain logically equivalent and empirically equivalent351 potential explanations. On the one hand, if E1 = {Q, p(P |Q) > p(P | ∼ Q)} and352 E2 = {R, p(P |R) > p(P |R)}, where Q and R are logically equivalent, then E1353 and E2 are logically equivalent potential explanations. If an agent accepts E1, she is354 thereby committed to E2. On the other hand, if Q and R contain coextensive singu-355 lar terms or predicates that occupy the same places in Q and R, E1 and E2 will be356 empirically equivalent potential explanations. However, the explanatory value and the357 credibility of E1 and E2 will not be assessed in the same way unless the agents who358 assess them are aware that the singular terms or predicates are coextensive8.359 5 The Epistemic Value of Explanation360 Consistency with K, the set of beliefs that represents the shared agreement between361 the members of a learning community, is not enough to guarantee the objectivity of an362 explanation. The objectivity of our conjectures lies, as Popper correctly points out, "in363 the fact that they can be intersubjectively tested" (1959, p. 44). The intersubjective test364 that an explanation must pass is the evaluation of its credibility and of its explanatory365 value in the eyes of the experts.366 Suppose a group of inquirers–a community of experts in the field–wants to con-367 sider the adoption of an explanation. To do so, they must first adopt a belief state368 K representing the shared agreement between them. Such a belief state will be the369 strongest common consequence of all their states of belief. Obviously, such a state370 will contain more than just singular sentences representing facts and probability sen-371 tences. It will also include sentences that state which are the most relevant problems372 in the field, what type of experiments and observations are considered more reliable,373 in addition to basic methodological and reasoning principles.374 Once the members of the community of experts have accepted a common corpus375 K, they must take it as the basis for establishing a set of potential explanations of the376 problem at hand, For example, suppose a group of inquirers are trying to establish377 why P . They must initially agree on a set of ground facts and low-level hypotheses.378 8Condition (iv) also introduces an element of epistemic relativity because the non-existence of a screening off factor can only be guaranteed relative to K. 244 Explanation, Understanding, and Belief Revision Statistical data and the chronology of the explanandum will be easy to agree upon.379 The explanation of some aspects of the phenomenon can be non-controversially ac-380 cepted, while the explanation of others will be a matter of heated debate. After the381 inquirers have agreed on a common corpus of beliefs K, they can put together a set382 of explanatory options, denoted OP , which will include all the factors consistent with383 K that might explain P and that have been identified by the inquirers. At this stage384 of inquiry it does not matter whether the potential explanations are uncontroversial or385 completely outlandish, as long as they are somehow relevant to the problem at hand386 and consistent with K, that is, if they fulfill the requirements to be in EP .387 It is possible for a group of agents to share the same information and yet disagree388 about the degree of belief or credal probability that they assign to the information in389 the set of explanatory options. Since the agents do not want to beg the question by390 assigning the highest marks to their favorite explanations, they must adopt a com-391 mon credal probability measure. A common strategy to eliminate the conflict be-392 tween different credal probability distributions is to represent the shared agreement as393 the weighted average of the distributions in conflict. The resulting credal probability394 function C determines the objective risk of error incurred in accepting a potential ex-395 planation in OP . Let Ei be the conjunction of the elements of a potential explanation396 Ei in OP , i.e., the conjunction of a singular sentence and a probability sentence. For397 every potential explanation Ei, the risk of error is 1− C(Ei).398 On the other hand, different inquirers will disagree in their assessment of the im-399 portance of the explanations contained in the set of explanatory options. Despite these400 differences, there must be a minimal objective criterion to measure the explanatory401 value of any potential explanation. That criterion is the new information carried by402 the potential explanation, which, following Levi, I identify with its logical strength.403 The set of potential expansions of a belief setK can be partially ordered by a classical404 consequence relation. The set is a Boolean algebra in which the minimum is K and405 the maximum is the inconsistent state. If a probability function M is defined over406 this set, and if the only element that has probability zero is the inconsistent state, po-407 tential expansions of K will strictly increase in probability with a decrease in logical408 strength. When the M -function is defined over the set of potential explanations of409 interest to the inquirer, we obtain a measure of the informational content of the po-410 tential explanations in OP . The measure of the informational content of a potential411 explanation Ei, denoted Cont(Ei), is 1−M(Ei).412 The informational content of a potential explanation is the first objective criterion413 that should be used in assessing the explanatory value of the elements of OP . The414 evaluation of their explanatory value is subject to the following weak monotonicity415 requirement (WMR):416 (WMR) If a potential explanation E1 in OP carries at least as much in-417 formation as another potential explanation E2 in OP , E1 carries at least418 as much explanatory value as E2.419 5 The Epistemic Value of Explanation 245 Not all potential explanations of the fact that P are comparable in terms of logical420 content. Since the community of experts wants to consider all the explanations avail-421 able to them, they might invoke further criteria in order to complete the quasi-ordering422 imposed by the weak monotonicity requirement. In order to assess the explanatory423 value of the remaining elements of OP , they can evaluate if they have certain proper-424 ties that are considered explanatorily virtuous.425 There are several explanatory virtues mentioned in the philosophical literature.426 Friedman (1974) and Kitcher (1989), for example, argue that explanations improve427 our understanding through the unification of our knowledge. Explanations that reduce428 the number of independent assumptions we have to make about the world are to be429 preferred to those that do not. This suggests that the potential explanations in OP430 could be ordered according to some set of rules that determines their unifying power.431 The problem is that neither Friedman nor Kitcher have provided an account that432 can be applied to explanations generally. Friedman's original argument was intended433 as an account of the explanation of scientific laws. Friedman argued, for example,434 that the kinetic theory of gases is explanatory because it unified different laws and435 properties of gases that were previously disconnected. Friedman's only attempt to436 formalize and generalize the idea of explanation as unification was incisively criticized437 by Kitcher (1976) and Salmon (1989).438 But Kitcher's account is no more helpful that Friedman's. According to Kitcher,439 the explanatory worth of candidates cannot be assessed individually. In his view, a440 successful explanation earns that name because it belongs to the explanatory store, a441 set that contains those derivations that collectively provide the best systematization of442 our beliefs. 'Science supplies us with explanations whose worth cannot be appreciated443 by considering them one-by-one but only by seeing how they form part of a systematic444 picture of the order of nature" (1989, p. 430). The idea that a virtuous explanation445 should have the potential to unify our beliefs is uncontroversial, but no one, to my446 knowledge, has provided a general account of explanation as unification that is not447 restricted to the case of scientific laws or scientific explanatory exemplars.448 Mellor (1995) provides an account of explanatory value that is better suited for449 our purposes.450 Mellor approaches explanation via his theory of causation. The theory requires451 every cause to raise the chances of its effects. That is, a fact C causes a fact E iff452 chC(E) > ch∼C(E). When causes are used in the explanation of a given fact, Mel-453 lor argues that the explanans must necessitate its explanandum, or at least raise its454 probability as much as possible, thereby reducing its chance of not existing. Thus, he455 concludes, "the more C raises E's chance the better it explains it" (p. 77). If we were456 to accept Mellor's idea, it would be possible to order the potential explanations in OP457 according to the difference between chC(E) and ch∼C(E).458 The main problem with Mellor's proposal is that when we examine a genuinely459 stochastic process, the value of the chance that the cause confers on the explanandum460 246 Explanation, Understanding, and Belief Revision will be irrelevant. As Jeffrey has convincingly argued, the information required to461 explain E is the same information used to explain ∼E, regardless of the value of462 the chance. Furthermore, if E is sometimes randomly caused by C and sometimes463 randomly caused by C∗, and chC(E) > chC∗(E), there is no reason to think that C464 is a better explanation than C∗.465 Mellor will respond to the objection by claiming that chances measure possibili-466 ties. "The less possible ∼E is, i.e. the less ch(∼E) is and hence the greatest ch(E)467 is, the closer the fact E is to being necessary. This is the sense in which a cause C468 may explain E better or worse, depending on how close it comes to making E nec-469 essary, i.e. on how much it raises ch(E)" (p. 77). Independently of whether we can470 make sense of such concepts as almost necessary or nearly impossible, it is not clear471 how such notions would enhance our notion of explanation. Probabilities are impor-472 tant in statistical contexts because knowing that C raises the chance of E allows us473 to know what makes E possible, and because the chance that C gives E allows us to474 adjust our expectations ofE's occurrence. But it seems to me that mixing chances and475 possibilities adds nothing to our understanding of why E is a fact.9476 A third candidate for judging the epistemic value of an explanation is Whewell's477 (1837) notion of consilience. Consilience is intended to serve as a measure of how478 much a theory explains, and it can therefore be used to compare the explanatory value479 of two different hypotheses. One hypothesis has more explanatory value than another480 if the former explains more of the evidence than the latter. Thagard (1978) provides481 compelling evidence that this idea is often used by scientists in support of their theo-482 ries. For example, Fresnel defended the wave theory of light by saying that it explained483 the facts of reflection and refraction at least as well as did the particle theory, and that484 there were other facts involving diffraction and polarization that only the wave theory485 could explain. Translated into my account, this means that if Ei raises the probability486 of more facts connected to the explanandum than Ej , then Ei is a better explanation487 than Ej .488 The problem with consilience is that, once again, the account works well in the489 explanation of laws, but it will not work in the explanation of singular facts. Whether490 a given fact explains more aspects connected to the explanandum than another fact is491 hard to say. We would have to define what a fact "connected to the explanandum"492 is, and it is doubtful that a non-pragmatic formalization of this notion can be found.493 Besides, sometimes a theory can explain too much. Lavoisier accused the phlogiston494 theory of this particular crime.495 Are there any other criteria that will allow us to assess the explanatory value of the496 potential explanations in OP ? We still have not examined the values that are usually497 mentioned in the context of theory choice: simplicity, accuracy, fruitfulness, and so498 9In Páez (2013) I offer an exhaustive analysis of the relation between causation and explanation in Mellor's work. 5 The Epistemic Value of Explanation 247 on10. But such an analysis is unnecessary. If the criteria are such that the community of499 experts can agree on their importance and on how they should be applied in particular500 cases, they can be added to the belief state K that represents their shared agreement.501 The agents will then be able to complete, to some degree, the quasi-ordering gener-502 ated by the monotonicity condition with respect to the M -function. But to expect a503 complete agreement in the way that all the agents engaged in common inquiry assess504 the explanatory value of different potential explanation is to expect a heterogeneous505 group of inquirers to agree on what aspects of reality they find interesting or useful.506 If a common decision is required nonetheless, the community of experts can adopt507 the following compromise. The agents must first identify the elements of the set OP508 that can be completely ordered because they are comparable in terms of strength or509 because they can be compared using the criteria to evaluate explanatory value that they510 have incorporated to K. The agents can then agree to disagree about the explanatory511 value of the remaining elements of OP . Let O∗P be a set of explanatory options such512 that O∗P ⊆ OP and such that the M-value of each element of the set is determined.513 Combining the credal probability function C with the M -function defined over the514 elements of O∗P we obtain a value that the community of experts can use to select the515 best explanation of P . I will call this result the objective epistemic value of a potential516 explanation11:517 (OEV) V (Ei) = αC(Ei) + (1− α)Cont(Ei)518 The agents' interest in valuable information should not outweigh the desideratum to519 avoid error; thus α ≥ 0.5. And since the information they seek should not be worth-520 less, 1 > α.521 Now, some researchers will be bolder than others in privileging content over cred-522 ibility, while others will be more cautious and adopt the opposite attitude. Let q be a523 common boldness index, which is the average of their individual boldness indices. If524 q = (1− α)/α , we obtain the following affine transformation of OEV:525 (OEV) V (Ei) = C(Ei)− qM(Ei)526 The experts should reject a potential explanation in O∗P if OEV is negative, remain527 uncommitted if it is 0, and accept it if it is positive. Any potential explanation in O∗P528 with positive objective epistemic value is an objective explanation of P in K. The529 disjunction of all such objective explanations is the objective explanation of P in K:530 (OEP ) The objective explanation of P in K, denoted OEP , is the disjunction of all the531 potential explanations in O∗P with positive objective epistemic value.532 One of the consequences of taking the functions C and M –which represent the533 average credibility and the agreed upon explanatory value, respectively– as a basis534 10There is a vast literature on the epistemic and social values used in science. The compilations by Machamer and Wolters (2004) and Kinkaid, Dupré and Wylie (2007) offer a contemporary perspective on the topic. 11This strategy is similar to the one followed by Levi (1991) to characterize the maximization of the expected epistemic utility obtained by expanding a corpus of beliefs. 248 Explanation, Understanding, and Belief Revision for the analysis of the potential explanations in O∗P is that each individual agent was535 forced to sacrifice his personal evaluation of credibility and value in order to accept536 the verdict of the community of experts. Suppose an agent has accepted a potential537 explanation of P based on his individual assessment of its credibility and explanatory538 value. Now suppose that he submits his "subjective" explanation to the community539 of experts, and the explanation is judged to be maximally credible and maximally540 valuable by the community, thus becoming an objective explanation. Does the agent541 understand more now that his explanation has been certified by others? It seems to542 me that he does not. But if the agent does not obtain more understanding from this543 recognition, why should anyone seek objectivity for an explanation that he or she544 already believes?545 Part of the answer is that the belief-doubt model is not a recipe for dogmatism. A546 seldom-noted fact about inquiry is that most newly suggested explanatory hypotheses547 do not survive the test of intersubjective scrutiny. If the agent is aware of this fact–548 and he should be if he is a responsible inquirer-it would be imprudent for him to549 give his full assent to an explanatory hypothesis that contradicts firmly established550 theories and findings without obtaining at least a partial intersubjective assessment551 of its merit. An agent does not need to fully believe that an explanation is true to552 obtain the understanding that the explanation provides. Any inquirer can explore the553 consequences of a hypothesis by assuming, for the sake of argument, that it is true. If554 the hypothesis is judged to have positive objective epistemic value by a community of555 experts, the inquirer will then be fully justified in giving it his full assent.556 But the question remains. If the agent does not obtain new understanding in the557 approval that he receives from his peers, why should he seek their approval? What558 prevents an agent from individually assessing the credibility and explanatory value of559 a potential explanation, and deciding to fully believe it if his individual understanding560 is thereby increased? In other words, why should objectivity matter? The answer561 is that objectivity itself is a property of information that some agents find valuable562 and some do not. An agent who decides to be a member of a learning community563 does so because he is convinced that his beliefs will be more valuable if they are564 objective. Other agents will find that objectivity adds no value to their corpus of565 beliefs. Just as there is a difference between objective and subjective explanation,566 there is an analogous distinction between objective and subjective understanding. The567 latter is the type of understanding that Hempel (1965) correctly believed should be568 shunned at all costs from an account of scientific explanation. But the reason it should569 be shunned is not that it is an inferior type of understanding. The reason is that the570 members of a scientific community are among the many agents who find objectivity571 valuable. Therefore, an account of scientific explanation should avoid any reference572 to an evaluative process in which the agent shows no concern for the views of others.573 6 Conclusion 249 6 Conclusion574 The belief-doubt model provides an adequate basis for an account of explanation that575 takes into consideration the epistemic value of the information that we acquire through576 inquiry. By including the shared commitments and the cognitive interests and goals577 of the individuals engaged in a cognitive enterprise, we obtain a notion of explanation578 that is objective by any reasonable standard of objectivity, and that clarifies the con-579 nection between explanation and understanding. 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