Explanation, Causality, and Counterfactuals Author(s): Evan K. Jobe Source: Philosophy of Science, Vol. 52, No. 3, (Sep., 1985), pp. 357-389 Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: http://www.jstor.org/stable/187708 Accessed: 06/05/2008 17:40 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ucpress. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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EXPLANATION, CAUSALITY, AND COUNTERFACTUALS* EVAN K. JOBE Department of Philosophy Texas Tech University The aim of this paper is to develop an adequate version of the D-N theory of explanation for particular events and to show how the resulting D-N model can be used as a tool in articulating a regularity theory of causation and an analysis of the truthconditions for counterfactualconditionals. Starting with a basic model that is largely the product of other workers in this field, two new restrictions are formulated in order to construct a version of D-N explanation that does not yield the counterintuitive results that have plagued all earlier versions. An additional condition is then developed that is indispensable for utilizing the D-N model as a tool in formulating a regularity theory of causation. Finally, it is shown how a suitable model of potential D-N explanation facilitates the formulation of a theory of counterfactual conditionals.
The D-N theory of explanation is widely considered to be defunct, and a major reason for this verdict surely lies in the fact that a satisfactory version of the theory has not yet been articulated. The prevailing view may even be that the lack of such a theory does not much matter. Since the heart of this paper is an attempt to develop an adequate D-N theory of the explanationof particularevents, it is perhaps well to indicate briefly at the outset why I think such a theory worth developing. It is true that scientists, especially those working in the more theoretically oriented fields, do not spend much time in devising explanations for particular events. This fact is, however, quite compatible with the idea that science should be able, in principle, to explain such events, at least in a wide range of cases. And if this is so, it is surely important to know just what such explanations in their ideally complete form would involve. I see the D-N theory as an attempt to satisfy this concern. In actual practice, explanations tend to be highly elliptical and frequently take the form of citing causes for the events to be explained. An adequate theory of explanation should therefore not be viewed as a rival to a causal approach to explanation but rather should elucidate such an approach. In this paper I try to show how a theory of D-N explanation furnishes a basis for a roughly adequate version of the regularity theory of causation.
*Received January 1984; revised July 1984. Philosophyof Science, 52 (1985) pp. 357-389. CopyrightC 1985 by the Philosophyof ScienceAssociation.
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Finally, it has been widely held that laws "support"counterfactual conditionals, but the precise mechanism of that support has remained somewhat mysterious. I shall show how the D-N model can be used as a tool in developing an analysis of the truth conditions for counterfactuals. 1. A Basic Model of D-N Explanation. The general notion of a D-N explanation is vague. Before attempting improvements through the introduction of new restrictions, it is essential to specify the basic model on which the restrictions are to be imposed. David Kaplan's work on S-explanation (1961), particularly as it has been clarified and extended by Brian Cupples (1977, 1980), furnishes a model of D-N explanation that is at least secure against the kinds of trivialization that were originally noted by Hempel and Oppenheim (1948), and later probingly discussed by Eberle, Kaplan, and Montague (1961). For the basic model of a D-N explanation of a partiuclar event I shall adopt a strengthened version of the model developed by Cupples. The present model, which differs from that of Cupples in ways to be explained below, may be set forth in three steps, as follows: A. Potential direct D-N explanans. The ordered couple of sentences (T, C) is a potential direct D-N explanans for the singular sentence E if and only if (1) T is a conjunction each of whose conjuncts is either a scientific law or a general analytic truth. (2) C is a singular sentence that is logically compatible with all scientific laws and all analytic truths. (3) E is a disjunction of basic sentences. (4) The conjunction of T and C logically implies E. (5) There is a disjunctive normal form of C such that none of the disjuncts of that form logically implies E and such that no conjunction of any of those disjuncts with an analytic sentence logically implies E. B. Potential D-N explanans. (T, C) is a potential D-N explanans for the singular sentence E if and only if (1) (T, C) is a potential direct D-N explanans for each of a set of singular sentences {E1, . . ., Enl} (2) {E1, . . EnJ logically implies E. C. True D-N explanans. (T, C) is a true D-N explanans for the singular sentence E if and only if (1) (T, C) is a potential D-N explanans for E. (2) C is true. I follow the writers cited above in assuming that we are working within a system of first-order logic without identity; my use of 'logically im-
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plies', 'logically compatible', etc., is to be understood accordingly. I follow them also in using the term 'singular sentence' to refer to sentences without variables, 'basic sentence' to refer to singular sentences involving no sentential connectives other than negation, and the terms 'conjunction' and 'disjunction' so as to include the limiting cases in which there is only one conjunct or disjunct. In the sequel I shall use the term 'analytically implies' in accordance with the following definition: A sentence P analytically implies a sentence Q if and only if there is an analytic sentence S such that the conjunction of P and S logically implies Q. Given this definition, condition A(S) above can be more succinctly stated as A. (5') There is a disjunctive normal form of C such that none of the disjuncts of that form analytically implies E. Also I shall say that a sentence P is analytically compatible with a sentence Q if and only if P does not analytically imply the negation of Q. And so condition A(2) can be stated as A. (2') C is a singular sentence that is analytically compatible with all scientific laws. The essential differences between the model articulated here and the model developed by Cupples (1977, 1980) are reflected in conditions A(1), A(2), and A(5) above. In contrast to the provisions of A(1), Cupples requires merely that T be an essentially generalized sentence, namely, a sentence that consists of quantifiers followed by an expression without quantifiersand that is not logically equivalent to a singular sentence. One problem I find with this is that it permits T to be a sentence that, even if true, would be only "accidentally true" and not a law of nature, for example, the sentence 'All the chairs in room 101 are wooden'. A further restriction that Cupples places on true D-N explanantia does suffice to rule out many sentences of this type from true explanations, but it does not rule out accidentally true generalizations having no reference to particular physical objects, for example, 'All gold lumps are less than a mile thick'. Even if this sentence is not true, there are surely countless similar ones which are true but still unsuitable for playing the role of laws in scientific explanations. Another problem with such a weaker requirement is that it permits T to be merely an analytic sentence such as 'All red things are colored'. There is nothing in any of the requirements imposed by Cupples to prevent such sentences from serving as the sole generalizations in either potential or true explanantia. In the present model, requirements A(1), A(4), and A(5) jointly require that T contain at least one scientific law essential for the deduction. The need to ensure a certain kind of relevance for laws that is met by the present model but not by earlier models can be illustrated by the fol-
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lowing example. Suppose we wish to explain why it is the case that a certain piece of paper (call it 'a') changed color at a particulartime. (The fact is that a, a previously white piece of writing paper, was dipped into some red ink.) I shall assume it is a scientific law that untreated litmus paper immersed in acid at a given time changes color at that time and that it is analytic that anything which turns red at a given time changes color at that time. But it would be a very bad explanation of the event of a's changing color if one were merely to cite these two generalities and add that either a was an untreated piece of litmus paper that was immersedin acid or else it turnedred. The following symbolization, which suppresses explicit reference to times, illustrates the pattern: T: (x)(Ax -> Cx) & (x)(Rx -3 Cx) C: Aa v Ra E: Ca By requiring that C have a disjunctive normal form none of whose disjuncts logically implies E Cupples rules out one kind of nomic irrelevance in explanations, but he does not avoid the problem presented by this example, in which one of the disjuncts analytically but not logically implies E. This is accoinplished by the stronger condition A(5) of the present model. Finally, condition A(2) above goes beyond the requirements Cupples imposes on potential explanantia in that it requires more than the logical compatibility of C with T itself. What it further ensures, essentially, is that the state of affairs described by C be a physically possible one. This feature brings the model of potential explanation more closely into accord with the notion of potential explanation employed in actual practice. For, in surveying various potential explanations for a given event one normally intends to consider only those which have a chance of being true in the actual world-those therefore that are analytically compatible with the actual laws of nature. I close this section with some more general comments. (1) In referring above to the explanation of "a particularevent" I intended to imply primarily that the explanandum is a singular sentence having essential reference to at least one physical object and a time. Whether in such a case one prefers to speak of explaining an event, a fact, a state of affairs, or even a sentence seems for many purposes to be of no great consequence. (2) My claim of adequacy for the theory here being developed is that it typically succeeds in distinguishing D-N explanations that are scientifically correct from those that are not. It is not designed to distinguish, among correct explanations, those that are more satisfying or illuminating from those that are less so. (3) As is usual in investigations of this kind,
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the present analysis is intended to apply directly to explanations expressed in a formal language but also indirectly to explanations in a natural language. It thus presupposes a correspondence between the languages that amounts to a kind of reconstruction or regimentation of at least a portion of the natural language. For the present analysis I shall assume that such a reconstruction will include the formulation of a set of standard definitions and other meaning postulates whose logical consequences will constitute the set of analytic sentences, so that the analytic sentences of the formal language can be assumed to form a definite set. While the selection of a particular list of meaning postulates is to some extent arbitrary, such a list can surely be more or less defensible. No particular selection is here presupposed, however, and in choosing examples of analytic sentences I attempt to cite only such as would be derivable from almost any defensible list. 2. The Problem of Nomic Disconnection. The basic model succeeds in ruling out a certain kind of purported explanation in which the role played by laws, though essential for the deduction, is still not properly relevant to the explanandum. This kind of irrelevance was illustrated by the example given above. The model does not, however, preclude another kind of nomic irrelevance that I shall refer to as "nomic disconnectedness." Examples of purportedexplanations exhibiting this kind of defect have occasionally been noted in the literature(for example, see McCarthy 1977, p. 165), but no analysis of the problem has been forthcoming. A variety of examples can be constructed, but the following three are perhaps sufficiently representative. (1) Suppose that we have a true D-N explanation, conforming to the basic model, of why a certain gasoline tank (call it 'a') exploded at 3:00 P.M. on a certain day. Suppose also that a certain bomb (call it 'b') exploded at 4:00 P.M. on the same day. Now, I assume it is an analytic truth that anything that explodes an hour later than a thing exploding at 3:00 P.M. explodes at 4:00 P.M. Let us augment the T-component of our original explanation by adding this truth as a conjunct, and let us augment the C-component by conjoining to it the true singular sentence 'b exploded an hour later than a'. We have thereby constructed a true D-N explanans for the sentence 'b exploded at 4:00 P.M.' Such an explanation of the explosion of b is, of course, absurd. (2) Suppose that we have three coil-type springs, a, b, and c, of the same construction, and arranged so that weights can be suspended from their lower ends. Suppose that the situation at a certain time is that a supportsthree pounds, b supportsthree pounds, and c supportstwo pounds; also that a and c are at a temperature of 400?F, and that b has a tem-
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peratureof 300?F. Under these conditions we might say informally that a is longer than b because it has a higher temperature, that b is longer than c because it supports a greater weight (a factor overcoming the effect of its somewhat lower temperature), and that a is longer than c because it supports a greater weight. Now, we should be able to construct a D-N explanation for the fact that a is longer than b by making crucial use of the fact that a has a higher temperaturethan b. Suppose now that we conjoin to the T-component of the explanans the analytic truth that, for every x, y, z, if x is longer than y and y is longer than z, then x is longer than z, and that we conjoin to the C-component the sentence 'b is longer than c'. We now have a full D-N explanans for the fact that a is longer than c. Such an explanation is quite counterintuitive, since it makes crucial use of the fact that a is hotter than b, a fact that in this situation should surely be explanatorily irrelevant for the fact that a is longer than c. (3) Assuming that it is a scientific law that aqueous solutions of copper sulphate are blue, one might explain why a certain liquid sample (call it 'a') is blue by citing this law and pointing out that a is an aqueous solution of copper sulphate. Admittedly, this is not a particularly illuminating explanation, but in some contexts no deeper explanation would be desired. Suppose, however, that after having furnished this explanation we proceed to explain why a certain fountain pen (call it 'b') is blue by augmenting the earlier explanans as follows: we add as a T-conjunct the analytic truth that anything that is the same color as something blue is blue, and we add as a C-conjunct the sentence 'a is the same color as b'. The full explanation of why b is blue can be set forth, using obvious abbreviations, as follows: (A) T: (x)(Ax -- Bx) & (x)(y)[(Bx & Cxy)
->
By]
C: Aa & Cab E: Bb Again we have a deeply counterintuitive explanation. Surely, neither the fact that a is an aqueous solution of copper sulphate nor the fact that a happens to be the same color as b should play an essential role in explaining why b is blue. The simplicity of this last example makes it a useful paradigm for the kind of purportedexplanation we are considering, and figuring out how it goes wrong may provide a key to the solution of the general problem. Note that the scientific law used here is of a kind that, for any given physical object, furnishes a link, so to speak, between one property of that object and another property of that same object. It is not a law that makes a connection between one object's having a certain property and
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another object's having a certain property. The explanation as a whole does make such a connection, but it does so only by employing a principle other than a scientific law. The basic intuition here is that the explanation goes wrong because the actual law is applied to object a but not to object b, whereas scientific explanation of why a given object has a certain property requires that laws be brought to bear on the object itself. Explanations having this kind of defect will be said to be "nomically disconnected." We have here another kind of nomic irrelevance in which a law, though essential to the deduction, does not properly connect up with the explanandum. Using terms to be defined later, I shall say that the explanandum of example (A) "refers intrinsically to" physical object b, because, roughly, it says something about b that does not involve objects other than b. The explanans, on the other hand, refers intrinsically to a and to the pair (a, b) as such, but not to b. There is therefore a "referential shift" involved in the transition from the singular sentence C to the singular sentence E. This shift is logically warranted by the sentence T, which is therefore a "referentiallydisconnective" sentence. Roughly, it is because the source of the disconnective nature of T lies in its analytic component rather than its nomic component that the explanation as a whole is not "nomically connected," hence is "nomically disconnected." Similarly, the explanation sketched in (2) above involves a referential shift in that the explanandumrefers intrinsically to the pair of springs (a, c) while the C-clause does not, although the latter does have intrinsic reference to the pair (b, c). Again, since it is only with the aid of the disconnective analytic sentence that the shift is warranted, the explanation is nomically disconnected. It can easily be seen that the example described in (1) above is also nomically disconnected. While the diagnosis just presented has intuitive plausibility, there is certainly a need to state these ideas more precisely and with sufficient generality to apply to all defective cases of this kind. The first candidate for clarification would seem to be the notion of "intrinsic reference." Intuitively, the idea of referring intrinsically to a particularobject is that of saying something about the object as such and apart from its relations to objects distinct from it. I assume that the domain of our concern here is that of physical objects, or more generally, spatiotemporalobjects. (This includes times, since a time (instant or interval) can straightforwardlybe regarded as a cross-section of the spacetime world taken orthogonally to the time axis of the inertial frame concerned. It does not include abstract objects-properties, classes, numbers, etc. Hereafter the term 'object' may be taken to mean 'physical object or time'.) The problem then is to state the conditions under which a singular sentence may be said to refer intrinsically to a given object, a given pair of objects, etc. As a start,
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note that a basic sentence need not have intrinsic reference at all. A sentence such as 'a is hot at t' and its negation presumably refer intrinsically to the pair (a, t), but the sentence 'a is a widow at t' surely does not. In this last case one might think the reason is that the sentence implicitly refers to an object distinct from both a and t. But that is a bit too strong, for to say that a is a widow at t is not to say anything about any other particular object. It seems clear, however, that the sentence does analytically imply that there exists (tenselessly) such an object, and that is surely sufficient for the lack of intrinsic reference. It is worth noting that since the sentence does not have intrinsic reference to (a, t) it is possible to construct a satisfactory explanans for it which also lacks such reference. For example, suppose we already have a satisfactory explanation of the death of an individual b at time t, an individual to whom a was married at that time. Now, if we appropriatelyconjoin to our explanans the singular sentence 'b was a man' and 'a was marriedto b at t', together with an analytic sentence to the effect that anyone married to a man at the time of his death is a widow, we would seem to have constructed a satisfactory explanation of why a is a widow at t. The explanation is nomically connected in spite of its formal resemblance to example (A) above. A different sort of case is presented by a sentence such as 'a has a speed of 5 meters/second'. This can hardly be considered as referring intrinsically to a, but the sentence does not merely imply the existence of some object with respect to which the speed of a is 5 meters/second. It is ratherthe case that the locution is not a true-or-false sentence at all unless it is elliptical for some sentence such as 'a has a speed of 5 meters/ second with respect to b'. So, while the sentence does not refer intrinsically to a, it does have intrinsic reference to the pair (a, b). It would be well to require that the sentences to which our analysis directly applies, namely, sentences of the formal language, be expressively complete and not elliptical or context dependent. The notion of intrinsic reference used above appears to be captured for atomic and basic sentences by the following definitions: I. Intrinsic reference of atomic sentences. An atomic sentence refers intrinsically to an n-tuple of objects (a,, . . . , a,) if and only if it contains individual constants referring to all and only the objects a,, . . . , a,, and does not analytically imply the existence of an object other than the objects a,, . . . , a,. II. Intrinsic reference of basic sentences. A basic sentence refers intrinsically to an n-tuple of objects (a,, . . . , a,) if and only if it is an atomic sentence referring intrinsically to (al, . . ., an) or it is the negation of such a sentence.
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One might note that it is a consequence of these definitions that the sentence 'a is identical to b', if true, has intrinsic reference to object a. Otherwise it does not refer intrinsically to object a or to object b, but only to the pair of objects (a, b). Turning now to the problem of stating the conditions for intrinsic reference of singular sentences in general, we must proceed carefully. For intrinsic reference to object a it would not suffice merely to require that the singular sentence in question contain an essential occurrence of a basic sentence having intrinsic reference to a. For, under this condition, if 'Fa' has intrinsic reference to a then so does 'Fa v Fb'. This not only seems intuitively wrong but also has the catastrophic consequence that even certain logically valid inferences would involve a "referentialshift"-in this case, for example, the inference from 'Fb' to 'Fa v Fb'. And it would not do to make it a necessary condition for intrinsic reference to a that the singular sentence logically imply a basic sentence having intrinsic reference to a. For, if 'Fa' and 'Ga' each has intrinsic reference to a, then surely 'Fa v Ga' does also, in spite of the fact that it implies no basic sentence of the requisite kind. A middle way, which appears to be the right one, is embodied in the following definition: III. Intrinsic reference of singular sentences. A singular sentence P refers intrinsically to an n-tuple of objects (a,, . . . , an) if and only if every disjunctive normal form of P is such that each disjunct of that form logically implies a basic sentence which refers intrinsically to the n-tuple (a,, . . . , a). (The present wording of this definition is designed to meet an objection raised by an anonymous referee of this journal.) Up to this point I have used the term 'referential shift' in connection with inferences of a certain kind, but it is convenient to focus attention on the corresponding conditional sentences, as is done in the following definition: IV. Conditionals involving a referential shift. A conditional P -> Q is a conditional involving a referential shift (a CRF) if and only if for some n-tuple of objects (a1, . . . , an), Q refers intrinsically to (a1, . . . , an) but P does not. We saw earlier that the source of nomic disconnection in explanations appeared to be a certain class of sentences, sentences that are not only analytic but also what I have called "referentially disconnective." The problem now is to demarcate the class of referentially disconnective sentences so that they, or at least such of them as are analytic, can be effectively banned from explanations. Each of the referentially disconnective sentences encountered above could be described as the universal
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generalization of a CRF, but any sentences (with trivial exceptions to be noted later) that logically imply a CRF must be counted as referentially disconnective. Such sentences, however, do not exhaust the class of sentences that can serve as a source of nomic disconnectedness in explanations. That this is so can be seen by modifying example (A) above so as to obtain the following explanation in which the new basic sentence 'Fb' (say, 'b is a fountain pen') is assumed to have intrinsic reference to b: (B) T: (x)(Ax -* Bx) & (x)(y)[(Bx & Cxy & Fy)
-*
By]
C: Aa & Cab & Fb E: Bb This explanation is no less absurd than (A), but the second conjunct of T here does not logically imply a CRF. The strongest conditional that it does imply is one such as '(Ba & Cab & Fb) -* Bb', which does not involve a referential shift but is merely a logical consequence of a conditional that does. Clearly, any analytic CRF can be padded in this manner to produce a conditional whose universal generalization, for example, can serve as a source of nomic disconnection. It appears that we need to count as referentially disconnective not only sentences that logically imply CRF's but also (with trivial exceptions) those that logically imply conditionals that are logical consequences of CRF's. This point is further illustratedby another example obtained by modifying example (A) above in a different way so as to obtain the following explanation in which the new basic sentence 'Ha' (e.g., 'a is hot') does not refer intrinsically to b: (C) T: (x)(Ax -> Bx) & (x)(y)[(Bx & Cxy) -> (By v Hx)] C: Aa & Cab E: Bb v Ha Again, this explanation is no less absurd than (A). The source of nomic disconnection, however, is not a sentence that implies a CRF but one that implies a conditional that is a logical consequence of a CRF, namely, the conditional '(Ba & Cab) -* (Bb v Ha)'. The sum of all these considerations suggests the following definition: V. Referentially disconnective sentences. A sentence is referentially disconnective if and only if it logically implies a logically contingent conditional P -> Q (without logically implying -P and Q is a CRF or is a without logically implying Q) such that P logical consequence of a CRF.
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The parenthetical clause is inserted to exclude the sort of "trivial exceptions" referred to above. It might appear that the problem of nomic disconnection can now be solved merely by stipulating that the T-component of a potential direct explanans not be a referentially disconnective sentence. The trouble with this move is that there appear to be genuine laws that are referentially disconnective. For example, it would seem to be a law that for every x, y, z, if x is a neutral particle dividing exclusively into y and z, and y is positively charged, then z is negatively charged. It might be questioned whether such laws are fit to play a valuable role in explanations, but there is nothing to suggest that the use of such laws results in absurd rather than merely unilluminating explanations. It would perhaps be rash to ban them altogether. But if the proposal just considered is too strong, the alternativeof merely banning such disconnective sentences as are analytic is too weak. This can be seen by referring again to example (A), where one could replace T with its nonanalytic logical consequence '(x)(y)[(Ax & Cxy) -- By]' without appreciably altering the nature of the explanation. The new T, being a logical consequence of a scientific law and a purely general analytic sentence is surely also to be counted as a law. Since it is a law, the fact that it is referentially disconnective is not in itself sufficient reason for rejecting it, but the fact that it inherits, so to speak, its disconnective character from an analytic sentence must be. These considerations suggest the following definition: VI. Nomically connected explanantia. A potential direct D-N explanans is nomically connected if and only if no conjunct of T is either a referentially disconnective analytic sentence or is a referentially disconnective law that is a logical consequence of the conjunction of a nondisconnective law and a disconnective analytic sentence. (The definitions of nomically connected potential D-N explanantia and of nomically connected true D-N explanantia are obtainable in an obvious way from this definition and the conditions of the basic model of D-N explanation.) In completely banning certain kinds of disconnective sentences this definition is somewhat stronger than is strictly necessary, since such sentences could have innocuously been permitted as conjuncts of T when not required for the deduction. But the prohibition of such superfluous components is also surely harmless. 3. The Problem of Directionality. If in accordance with the basic D-N model one can explain why a particular match lights, given that it was struck, was dry, etc., then one can also explain why a similar match is not dry, given that it was struck, did not light, etc. The fact that the
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usual versions of the D-N model authorize such counterintuitive explanations is sometimes called "the asymmetry problem," and is perhaps too well known to be in need of further illustration. The problem is that one cannot usually shuffle various kinds of facts around in an explanation simply in accordance with the logic of the deduction. I shall call this the problem of "directionality." In an earlier paper (Jobe 1976) I referred to explanations that were satisfactory in this respect as "admissible." I now prefer to use the more specific term "directionallyadmissible," since there are various grounds on which an explanation might be judged "inadmissible" in any ordinary sense of that term. As the example just cited shows, the problem is not solely connected, as is sometimes thought (e.g., Kitcher 1981, p. 522), with the fact that some scientific laws have the form of generalized biconditionals. In my 1976 I presented a diagnosis of the problem and a general solution involving a new restriction on D-N explanations which was formulated in terms of a notion of "explanatory dependence." But it was soon shown by Clark Glymour (1978) that the proposed restriction was subject to counterexamples. It can now be seen that one defect of my proposal lay in the crucial definition of explanatory dependence. In its simplest form the definition reads: "A sentence P is explanatorily dependent on a sentence Q if and only if there are D-N explanations of Q that do not involve P; but every D-N explanation of P involves Q." (It should be noted that P and Q are here assumed to be true, as the whole discussion concerned only true explanations.) This definition makes the defined relationship depend on the existence or nonexistence of certain "D-N explanations" quite without regard to whether the "explanations" in question are thernselves trivial or absurd in ways having nothing to do with the problem under investigation. In order to be relevant in a definition of this kind a D-N explanation should surely not only meet the conditions of an adequate basic model such as the one presented earlier but it should also be immune to the kind of absurdity generated by nomic disconnectedness. And it turns out that replacing 'D-N explanations' with 'nomically connected D-N explanations' in the definitions of explanatory dependence in my (1976) suffices to block Glymour's counterexaniple. This last concerned explaining the fact that a flagpole had a certain height by making crucial use of the fact that the pole was casting a shadow of such and such a length at a certain time. My proposed restriction was supposed to rule out this explanation on the grounds that the fact that the shadow had such and such a length was "explanatorily dependent on" the fact that the flagpole had a certain height. Glymour refuted this claim by sketching an explanation of the length of the shadow without making any kind of reference to the height of the pole. His crucial move was to refer to another flagpole having the same height and shadow length as
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the first. If now one sketches a D-N explanans for the length of the second pole's shadow in terms of the second pole's height, etc., and then adds to this explanans the fact that the shadows of the two poles have the same length, one obtains a sketch of an explanans for the length of the first pole's shadow (Glymour 1978, p. 119). This shows that it is not true that every D-N explanation of the length of the first pole's shadow involves the first pole's height. But it is clear that this last explanation sketch is not a sketch of a nomically connected explanation, for the explanandum has intrinsic reference to the first pole and a time t whereas the only sentence in the explanans mentioning the first pole is a sentence expressing the fact that the length of the shadow of the second pole is the same as the length of the shadow of the first pole at time t. Therefore, if the definitions of explanatory dependence in my (1976) are rephrased in the way suggested above, the existence of the sort of "explanation" sketched by Glymour becomes irrelevant. It turns out, however, that my earlier definition of explanatory dependence is defective in a quite different way. In dealing with the standard examples of the directionality problem, one's focus is on explanations involving what Hempel has called "laws of coexistence" (Hempel 1965, p. 352), hence on a kind of explanation in which all the singular sentences refer to a common time. (I shall call such explanations "cotemporal.") But in turning to the task of framing a general definition of explanatory dependence it is essential not to ignore-as I did-the obvious fact that not all explanations are of the cotemporal type. I was only recently made aware of my oversight through a counterexample devised by an anonymous referee of this journal. The counterexample, which concerns a pendulum whose rising temperatureresults in an increase in length, consists in an explanation of the period of the pendulum at a given time, using as the sole initial condition the period at an earlier time. The requisite law connecting the two periods is secured by the fact that there is not only a nomic link between the value of the period at each time and the corresponding length at that time but there is also a nomic link between the earlier and later lengths-this being the law of expansion for the material of which the pendulum is made. Now, there is nothing peculiar about this example, and indeed it appears that for any two kinds of parameter it is typically possible to construct a "historical" explanation of either without reference to the other-a fact which shows the utter inadequacy of my earlier definition of explanatory dependence. But, while it is certainly not the case that every explanation of the period of a pendulum involves its length, it is plausible to suppose that every cotemporal (nomically connected) explanation of the period does so. And, quite generally, the dependency relationships of a given sentence do seem to be specifiable in terms of those (nomically connected) explanantia that are
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cotemporal with it. One might well wonder why the concept of cotemporal explanation turns out to play such a crucial role, but any answer at this point could only be speculative. I would suggest, however, that this outcome should not be particularly surprising. After all, explanatory dependence presumably reflects an aspect of the nomological structure of the world, and we have every reason to believe that the basic laws of nature are laws of coexistence. It appears then that my (1976), when revised along the lines indicated above, may provide a satisfactory solution to the problem of directionality for the class of true D-N explanations. In order to conform to the pattern of the present analysis and for reasons that will become apparent later, it is desirable to deal with merely potential explanations as well. This cannot be done simply by inserting the word 'potential' as a modifier for 'explanation' in the various formulations. Consider, for example, the relation of explanatory dependence. It seems doubtful that there is any pair of sentences (P, Q) such that every potential (cotemporal, nomically connected) explanation of P involves Q. In an actual situation such as one in which a voltage of ten volts is applied to a resistor (call it 'a') of five ohms, resulting in a current of two amperes, the domain of true explanations for the fact that there are two amperes flowing through a is severely limited in that any true explanation would have to be physically compatible with actual facts, such as, that a does have a resistance of five ohms. (In saying that two sentences are "physically compatible," I mean that the body of scientific laws does not analytically imply the negation of their conjunction; and in saying that an explanation is "physically compatible with" a particular sentence, I mean that the explanans (T, C) of the explanation is such that C is physically compatible with that sentence.) On the other hand, the number of potential explanations for the two amperes flowing through a is infinite, since by Ohm's law such an explanation could be based on any pair of values of voltage and resistance such that the voltage across a is twice the resistance of a. Any serviceable definition of explanatory dependence in terms of potential explanations must therefore be relativized to a particular situation, as will be done below. (Note, however, that according to the definitions utilized in the present analysis, any D-N explanans is a potential D-N explanans, and so the term 'potential' here becomes redundant.) Finally, the definition of explanatory dependence can be simplified by deleting an unneeded clause, as will be done below. In the light of the foregoing considerations we have the following definitions: I. Involvement of sentences by explanantia. A D-N explanans (T, C) involves a sentence P if and only if C analytically implies P.
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II. Cotemporality of explanans. A D-N explanans (T, C) is cotemporal with a sentence P if and only if C and P contain essential reference to precisely the same time or times. III. Dependency-relevant explanantia. A D-N explanans (T, C) is dependency-relevant (d-relevant) for a sentence P if and only if (T, C) is nomically connected and is cotemporal with P. IV. Explanatory dependence of sentences. Relative to a sentence C, a sentence P is explanatorily dependent on a sentence Q (1) at order 1 if and only if, within the class of explanantia that are d-relevant for P and physically compatible with C, every explanans for P involves Q; (2) at order 2 if and only if, within the class of explanantia that are d-relevant for P and physically compatible with C, every explanans for P either involves Q or involves a sentence R such that every explanans for R involves P or Q; (3) at order 3 if and only if, within the class of explanantia that are d-relevant for P and physically compatible with C, every explanans for P either involves Q or involves a sentence R such that every explanans for R (a) involves P or Q, or (b) involves a sentence S such that every explanans for S involves P or Q or R. The definition extends analogously to the higher orders (see Jobe 1976, p. 547). All of the disjunctions in the definition are to be construed in the inclusive sense. If I were now to proceed as in my 1976 I would state the criterion for directional admissibility as follows: A potential direct D-N explanans (T, C) for a sentence E is directionally admissible if and only if (T, C) does not involve a sentence P such that, relative to C, P is explanatorily dependent on E at some order n. That this will not suffice can be seen from the fact that one can construct a D-N explanation of why a flagpole has a certain mass by utilizing specific facts concerning its cylindrical shape, its cross-sectional area, its iron composition, the angle of the sun, and the length of the pole's shadow. Such an explanation, while unacceptable, does not violate the criterion and is thus not "overtly inadmissible." It may, however, be said to be "covertly inadmissible," since it is equivalent to a chain of explanations of which one is overtly inadmissible. This can be seen by noting that we might first use the facts concerning shadow length and sun angle to (illegitimately) explain the height of the pole and then combine this result with the remaining facts (together with the law that iron has such and such density and the law relating volume to crosssection and length) to explain the mass. While in actual practice covertly inadmissible explanations would seem to be rather easily recognizable it
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may be desirable to state a formal criterion that excludes them, as will be done below. It is now possible to state the formulations pertaining to directional admissibility that are appropriatefor the present analysis. I. Overt directional inadmissibility. A potential direct D-N explanans (T, C) for a sentence E is overtly directionally inadmissible for E if and only if (T, C) involves a sentence P such that, relative to C, P is explanatorily dependent on E at some order n. II. Covert directional inadmissibility. A potential direct D-N explanans (T, C) for a sentence E is covertly directionally inadmissible for E if and only if (T, C) is not overtly directionally inadmissible for E and there exist sentences C' and E' such that (1) C logically implies C'. (2) (T, C) is a potential direct D-N explanans for E'. (3) (T, C' & E') is a potential direct D-N explanans for E. (4) Either (T, C) is overtly directionally inadmissible for E' or (T, C' & E') is overtly directionally inadmissible for E. III. Directional admissibility of potential direct D-N explanantia. A potential direct D-N explanans is directionally admissible if and only if it is neither overtly nor covertly directionally inadmissible. (The definitions of directionally admissible potential D-N explanantia and of directionally admissible true D-N explanantia are obtainable in an obvious way from this definition and the conditions of the basic model of D-N explanation.) The definition of covert directional inadmissibility formulated above is actually unnecessarily strong in that it rules out not only seriously defective explanantia but also some which violate it only because they contain certain superfluous clauses. Since such inflated cases would not occur in actual practice anyway it seems harmless to exclude them.
4. ExplanatoryEfficiency and Temporal Orthodoxy. For any explanandum E, if an explanans (T, C) meets all of the conditions thus far formulated then so does the explanans (T, C & C'), provided only that the singular sentence C' is sufficiently irrelevant to C and E so as not to make the explanation run afoul of any of those conditions-and in any given case there will clearly be countless such sentences. Furthermore, adding a dispensable conjunct is only one way of padding the singular clause of an explanation, for the original explanans above could just as well have had its singular clause replaced by C' & (C' -> C). In either case we have an explanation in which, relative to the laws employed, the singular component is unnecessarily strong. I shall say that such explanations are not "efficient." Examples of a somewhat different kind of explanatoryinefficiency have
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been cited by Wesley Salmon in his 1970. One of the examples, which he attributes to Henry Kyburg, goes as follows: "This sample of table salt dissolves in water, for it has had a dissolving spell cast upon it, and all samples of table salt that have had dissolving spells cast upon them dissolve in water" (Salmon 1970, p. 177). Actually, the wording of this explanation makes it sound stranger than it really is, for the phrasing suggests that the only initial condition appealed to is that of the dissolving spell having been cast. Restating the explanation so as to make it more clearly comformable to the D-N model, we would get something like the following: "a dissolves in water, because a is a sample of table salt and a has had a dissolving spell cast upon it, and all samples of table salt that have had dissolving spells cast upon them dissolve in water." In this and the other examples cited by Salmon the singular component is not unnecessarily strong relative to the law actually used, but it is unnecessarily strong relative to a logically related law that might have been employed. If a sentence '(x)(Ax -> Bx)' is a law, then presumably so is its weaker logical consequence '(x) [(Ax & Cx) -> Bx] -provided, of course, that the latter is also nonanalytic and purely general. In constructing an explanation, one thus typically has a choice between using a stronger law together with a particular singular sentence or a weaker law with a correspondingly stronger singular sentence. By making the latter choice, one is appealing to an unnecessarily strong set of particularcircumstances to explain a given event, and in this sense the explanation can be said to be "inefficient." Inefficient explanations are often potentially misleading and thus pragmatically inappropriate,but they do not seem to be scientifically defective in any way. Moreover, the pragmatic considerations tend to be relative to the background knowledge of the audience addressed. Consider the following informal explanation: "This compass needle was deflected because it was brought near a copper wire carrying a heavy current." Since the explanation is elliptical the exact nature of its inefficiency is not clear, but the mention of the fact that the conductor of the current is a copper wire is clearly superfluous and for some audiences could actually be misleading. Although inefficient explanations are not scientifically incorrect it is nevertheless true that one does not, for the most part, knowingly present an inefficient explanation when one is in a position to offer an efficient one. Similarly, one would not ordinarily assert the statement 'Alice stole the watch or Betty did' when one is in a position to assert 'Alice stole the watch' -although the former statement would also be true. In the presentationof explanations there appears to be operative something analogous to a feature of conversational practice that H. P. Grice (1967) has called "conversational implicature." In actual practice, of course, the
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avoidance of misleading inefficient explanations is generally quite easy and more or less automatic. It might be well to point out that inefficient explanations are sometimes quite as acceptable as their efficient counterparts. Consider, for example, a case of causal overdeterminationsuch as that of a metal rod that is both heated and subjected to a stretching force. An explanation of the merely qualitative fact that the rod expanded at a certain time could be based on either the heating or the stretching force. An explanation citing both factors would thus be inefficient but practically without any potential to mislead. And an anonymous referee of this journal has noted the sort of case presented by the following explanations, where 'Ax' abbreviates 'x is an aqueous solution of copper sulphate' and 'Bx' abbreviates 'x is blue': (1) T: (x)(Ax -* Bx)
(2) T: (x)(Ax -> Bx)
C: Aa
C: Aa v Ab
E: Ba v Bb
E: Ba v Bb
To provide a possible context for these explanations, suppose that a and b are chemical samples about which we know nothing except that they are mutually inert aqueous solutions that were inadvertently dumped into a jar of distilled water last night. Noting the bluish color of the jar's contents this morning we conclude that at least one of the samples was blue. This fact could be explained, as in (2), if we later learned that at least one of the samples was an aqueous solution of copper sulphate without learning the nature of either individually. But explanation (1), if we were in a position to use it, would be quite as acceptable, in spite of the unnecessarily strong singular component which renders it inefficient. Since inefficient explanations pose no significant problem, either theoretically or practically, there is no need to introduce a restriction prohibiting them. It turns out, however, that the notion of an "efficient" explanation is a useful one, as will be seen in the next section, and it is therefore desirable that a precise definition of the concept be available. This is accomplished by the following: Efficient explanations. A D-N explanans (T, C) for a singular sentence E is efficient if and only if it is not the case that there is a DN explanans (T', C') for E such that C logically implies (but is not equivalent to) C' and such that T' logically implies T. Another scientifically correct but potentially misleading kind of explanation results when one employs a T-component which, relative to the C component, is unnecessarilystrong-as when T contains superfluouslaws. No one seems to have considered this sort of thing a potential problem,
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and I have not chosen a label for explanations of this kind. In actual practice such explanations are also easily avoided. While requiring explanations to be efficient is clearly not in order, a much better case could be made for requiring them to be "temporally orthodox," a notion that may be defined as follows: Temporally orthodox explanations. An explanans (T, C) for a sentence E is temporally orthodox if and only if C contains no essential reference to a time that is later than all times to which E contains essential reference. Imposing such a requirement would accord with our common-sense notions, and it seems practically certain that no explanations offered in actual practice would thereby be made illegitimate. Still, there is some reason to doubt that temporal orthodoxy should be considered part of the very concept of explanation. The possible existence of faster-than-light particles (tachyons) brings with it the possibility of backwards causation, as Paul Fitzgerald has shown in his 1971. Confronted with a situation in which a tachyonic message was received prior to its transmission we would surely not hesitate to use the transmission as a crucial factor in explaining the reception. It seems reasonable to require that our model of explanationbe applicablein all physically possible situations, not merely in actual ones, and so the inclusion of a requirement of temporal orthodoxy is arguably too strong. Nevertheless, it is undisputable that the direction of time is pragmatically of great importance, and this is reflected in what is at least a strong prejudice in favor of temporal orthodoxy. It appears that this prejudice could conceivably be overcome only in a situation, such as that of the tachyonic message, in which the temporally unorthodox explanation is vastly simpler than any available explanations of the orthodox kind. Our operational attitude in this matter could be captured, if only imprecisely, by the requirement that explanations be "temporally admissible," which may be defined as follows: Temporally admissible explanations. An explanans (T, C) for a sentence E is temporally admissible if and only if either (T, C) is temporally orthodox or else (T, C) is temporally unorthodox but there is no comparably simple explanans for E that is temporally orthodox. I shall leave it to the reader to decide what, if any, temporal restriction should be imposed on D-N explanations. Whatever the choice, it can be assumed to qualify implicitly all subsequent references to D-N explanations in the applications to which I shall now turn. 5. Explanation and Causality. I should now like to show how the D-N model can be applied in the analysis of the actual causal relation,
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the relation referred to in such statements as 'Striking the match caused it to light'. (The furtherproblems involved in analyzing statements about potential causes and statements of generic causality such as 'Smoking causes cancer', while not without interest, presumably pose no additional difficulties of a fundamental nature.) What has become known as the "regularitytheory" of causation surely provides a welcome alternative to those approaches which simply take as primitive and irreducible either the causal relation itself or some closely related concept that is equally mysterious-approaches which are tantamountto having no theory at all. Unfortunately there appears to be no reasonably precise version of the regularity theory that is not obviously unsatisfactory, and some of the problems are closely related to those that have afflicted the D-N theory. The relevance of the D-N model to the task of articulating a regularity theory of causation was essentially envisaged by Hempel and Oppenheim in their classic study (1948, p. 139) and by Hempel in his more recent work (1965, pp. 347-54). Such an application of the model is prima facie quite straightforward:the T-component of the explanans plays the role of the "regularities," and the C-component describes a "cause" or set of "causal factors" for the event described by the explanandum. An important presupposition here is that laws are simply regularities, that they describe no relation between properties that is not expressible by generalized conditionals. This implies that whatever differentiates laws of nature from accidental generalizations is not their intrinsic content but rather certain extrinsic facts such as how they fit into the total system of empirical truths. Such a view of the nature of laws has received some needed clarification and, I think, gained enhanced plausibility from a brief discussion by David Lewis (1973, pp. 73-74). I shall assume for the present discussion that this approach to laws is essentially correct. But granted the identification of laws with regularities, can we also identify an arbitrary conjunct of the C-component of an explanans with a description of a "partialcause" or "causal factor"? Clearly not if the explanans is nomically disconnected: it would not do to say that the fact that a certain sample of copper sulphate was blue was a causal factor for a certain fountain pen's being blue. And clearly not if the explanans is directionally inadmissible: we would not want to say that the length of the shadow cast by a flagpole at a certain time was a causal factor for its having such and such a height at that time. Finally, there is a problem connected with inefficiency: it is certainly not the case that a dissolving spell's having been cast on a certain sample (as in Salmon's example) is causally related to the sample's solubility. Aside from such considerations the application of any kind of D-N model to the analysis of ordinarycausal talk requires some care. Consider that, where a and b are such things as events, facts, or conditions, we
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are faced with such varied locutions as 'a was a causal factor for b', 'a was a cause of b', 'a caused b', and 'a was the cause of b'. While the first two of these seem at least roughly synonymous, as do the third and fourth, the conditions for asserting either of the latter pair are surely somehow strongerthan those for asserting the former pair. The difference seems to be pragmatic in nature and therefore also context dependent. For example, we would normally say that the presence of atmospheric oxygen is a causal factor but not "the cause" for the lighting of a particular match struck in the open air. But consider a bored astronaut in an evacuated chamber who happens to be idly scratching a match against a frictional surface when someone inadvertantly opens a valve letting air into the chamber. In this case we would probably be quite willing to say that the arrival of the atmospheric oxygen caused the match to light. The pragmaticconsiderations that incline us to select a particularcausal factor as "the cause" of an event are perhaps not so very mysterious, but they clearly lie beyond the purview of a regularity theory of causation. So it seems that a total analysis of a locution of the form 'the cause of a was b' requires a certain division of labor. What we can expect from a regularity theory is a formulation of the conditions under which one event, fact, or condition is a causal factor for another. I take it that this is the problem of crucial philosophical interest, and it is the problem to which a D-N theory of explanation is directly relevant. While our ordinary causal talk refers to events, facts, conditions, etc., I assume that little loss of philosophical interest would be incurred by a convenient retreat to talk of sentences instead. I shall therefore simply speak of one sentence as being a causal factor for another. It appears, in fact, that a causal factor in this sense will always be a conjunction of basic sentences. For, while we certainly sometimes speak of the cause of a particularevent as being either this or that, I take it that we never think of a single cause as being essentially disjunctive in nature. But perhaps not just any such conjunction that might properly occur in a D-N explanans is suitable for playing the role of a causal factor, as will be seen below. Before turning to this matter, however, I should like to consider whether the class of laws utilized needs also to be restricted in some way. It is often thought that in formulating a regularity theory of causation it is essential to restrict the body of laws so as to exclude laws of coexistence. Thus Hempel has noted that one would not ordinarily say that a pendulum's having such and such a period is caused by its having a certain length, and he attributes this to the fact that the relevant law is one of coexistence (1965, p. 352). It seems doubtful, however, that this is the reason, for one might very well say that an increase in the length at a certain time caused (or was the cause of) an in5rease in the periodand this statement is supportable by the very same law. And it would
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perhaps not be too odd to speak of the length as a causal factor for the period. The pragmatic considerations that govern our causal talk leave the boundaries vague, and where a clear-cut rationale is lacking it seems best to opt for a broad construal of causal language. To do otherwise is likely only to invite the intrusion of prejudices of an a prioristic kind, as Michael Scriven has argued in his 1975. I think a stronger case can be made for restricting the class of basic sentences that are to be considered suitable to play a role in singular sentences that count as either causes or effects. Perhaps the matter can best be approached on the side of effects or explananda. To begin, the class of sentences for which scientific explanations are actually desired is probably narrower than is suggested by most discussions of the D-N model of explanation. Questions such as 'Why is that thing an emerald?' and 'Why is that thing a digital computer?' would probably never be understoodin actual practice as requests for scientific explanation. A reasonable answer in each case would indicate ratherhow the thing in question meets the criteria for being such and such a kind of thing, and no reference to scientific laws would be relevant. Indeed, sentences such as 'a is an emerald' and 'a is a digital computer' are reasonably construable only as making tenseless statements and are therefore not normally really synonymous with 'a is now an emerald' and 'a is now a digital computer'. It is only the latter sentences, referring to both a physical object and a time, that at least formally qualify as falling within the intended domain of D-N explananda indicated earlier. But while these last sentences might be broadly referred to as "event sentences," and thus nominally admissible as explananda, there is certainly something odd about them, as there is with many other sentences such as 'a is a typewriter at time t', 'a is a tiger at time t', etc. The oddness is doubtless related to the fact that it seems sensible to speak of an object as having a certain property at a certain time only if it would also be sensible to speak of the object as not having the property at some time. In cases in which the property is that of being of a certain natural kind or of a certain kind of artifact it is usually not at all clear how an individual that is of that kind could fail to be of that kind and still be the same individual. And perhaps there are still other kinds of properties that an individual object could not sensibly be said to have at one time but not at another. In any case-and in spite of the evident vagueness involved-I am inclined to speak of a class of basic sentences to be referred to as "proper event sentences" in which the predicate involved is connected with a property of the temporally variable kind. I assume that anything that could, even broadly speaking, be called an "effect" would correspond to a sentence (or conjunction of sentences) of this kind. As far as I can see, this restriction applies to causal factors as well. D-N explanantia may, of course, contain singular sen-
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tences of other kinds-for example, the fact that a is a copper sulphate solution could figure in an explanation of why a is blue. But we should, I think, be reluctant to say that a's being copper sulphate caused it to be blue. In the formulations to come I shall therefore require that both effects and causal factors be composed of proper event sentences. Readers who regard such a restriction as unwarrantedmay choose to ignore it or reinterpret it in their own way. A final general problem that the regularity theory must face is that the language of ordinarycausal discourse often fails to connect up in the right way with the more precise language in terms of which scientific laws may properly be expressed. An ordinary description of an event, being relatively vague, may be implied by a large number of more precise scientific descriptions without implying any one of them. For this reason the problem concerns causal factors rather than effects. We often have occasion to say such things as 'That stone's hitting the window caused it to break', but there is of course no scientific law to the effect that whenever a stone hits a window the window breaks. The approach to this problem that I favor is essentially that of Donald Davidson (1963, pp. 697-98) and is based on the notion of event redescription. That is, a given actual event can surely be described in a variety of ways, some being more precise or informative than others. Thus, if a certain traffic light changed its electromagnetic frequency emission pattern in a certain precisely describable way at time t one could, in principle furnish a description of the event in the technical language of physics. But the event might, as a matter of fact, also be correctly describable as the event of the light's changing from red to green at time t. And the same event might be described even less informatively as the light's changing color at time t. Whether a pair of such descriptions actually describe the same event (as opposed to whether they could describe the same event) cannot usually be determined merely by an analysis of the descriptions involved and apartfrom relevant information about the actual situation. In the case of description pairs that do in fact describe the same event I shall say that the corresponding sentences (e.g., the sentences 'the light changed color at time t' and 'the light changed from red to green at time t') are "descriptively substitutible" for each other. The formulation of a more precise account of what is involved in the redescription of events is of course a project of considerable interest quite aside from its relevance to a theory of causation. I assume that a version of the regularity theory may be considered roughly satisfactory even though it employs this notion which still awaits final clarification. These preliminaries out of the way, I shall proceed to define what I call a "precise causal factor" for a sentence and then go on to try to state truth conditions for causal factors in general. Since the concern is with
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actual causation, the basic concept for the following formulations will be that of a true D-N explanans. I. Conjunctive D-N explanans. A D-N explanans (T, C) for a singular sentence E is a conjunctive explanans for E if and only if C is a conjunction of basic sentences. It should be noted that a restriction to conjunctive explanantia involves no essential loss of explanatory possibilities, for David Kaplan has shown (1961, p. 433) that if a singular sentence E can be given a true D-N explanation in terms of a theory T then this can be accomplished by an explanans (T, C) in which T is the theory in question and C is a conjunction of basic sentences. II. Precise basic causalfactor. A sentence B is a precise basic causal factor for a sentence E if and only if B and E are proper event sentences and B is a conjunct of C in a true conjunctive D-N explanans (T, C) for E, where (T, C) is nomically connected, directionally admissible, and efficient. I trust that the need for the last clause in this definition is made evident by earlier examples of explanations that were nomically disconnected, directionally inadmissible, or inefficient. III. Precise causalfactor. A sentence C is a precise causal factor for a sentence E if and only if C is a conjunction of precise basic causal factors for E. Here, as earlier, the term 'conjunction' is used so as to include the limiting case in which there is only one conjunct. It is apparent from these definitions that precise causal factors must be sentences that are either expressed in the language of the relevant scientific laws or at least analytically imply sentences that are so expressed. As we have seen, the language of ordinary causal discourse often fails to satisfy either of these conditions. Hence, one final formulation is needed to extend the theory to causal factors in general. IV. Causalfactor. A sentence C is a causal factor for a sentence E if and only if C is either a precise causal factor for E or C is descriptively substitutable for a sentence C' such that C' is a precise causal factor for E. 6. Explanation and Counterfactuals. For a final application of some of the ideas developed earlier I shall turn to the problem of formulating the truth conditions for counterfactual conditionals. My primary concern will be with what I shall call "regular counterfactuals." It is these that
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have been the main focus of a vast literature, and other kinds of counterfactuals that I shall mention in passing do not, I believe, raise additional problems of a fundamental kind. Regular counterfactuals are statements such as 'If that rod had been heated it would have expanded', 'If Harrywere here now Betty would be happy', and 'If Joe had had a wife he would not have been a bachelor'. They may be characterized as singular subjunctive conditionals in which the apodosis involves 'would' or 'would have' as opposed to other auxiliaries such as 'would still', 'would have to', 'could not have' etc. In speaking of the "antecedent" and "consequent" of a subjunctive conditional I refer to the sentences that constitute, respectively, the antecedent and consequent of the corresponding indicative conditional. One who asserts a counterfactual usually assumes that the antecedent is false (but see Burks 1977, pp. 438-39) and frequently assumes that the consequent is also false (but see Parry 1957, p. 93). At any rate, one never asserts a regular counterfactual when one positively believes either the antecedent or consequent to be true. (When one believes the consequent to be true the appropriate form is what is sometimes called a "semifactual":'If that sample had been hexed it would still have dissolved'.) But since the truth values of counterfactuals appear to be independent of any such presuppositions, it suffices to identify counterfactuals simply by their form, as indicated above. I shall now proceed to develop what I believe is a satisfactory version of the metalinguistic approach to counterfactuals. The general thesis is that a regular counterfactual is true if and only if it is either "nomically warranted"or "analytically warranted"-aside from a rare type whose warrantis parasiticon other counterfactualshaving warrantsof these kinds. Since the nomically warranted type presents all of the interesting problems I shall first concentrate the discussion solely on counterfactuals of this kind. When is a regular counterfactual having P as antecedent and Q as consequent "nomically warranted?"According to the familiar metalinguistic approach this must at the very least require that there exists a true statement S such that Q analytically follows from P & S conjoined with relevant scientific laws. But, as is well known, such a formulation needs to be tightened up in various ways in order to avoid trivialization. For one thing, it must be the case that at least one scientific law is essential for the deduction of Q. But, as we saw in Part 1 of this paper, a law can be essential for a deduction of basically this kind without thereby acquiring much intuitive relevance to the statement deduced. It would therefore be well to insure a certain minimum nomic relevance by imposing a requirement such as that of A(5) of the basic model of D-N explanation set forth in Part 1 above. For another thing, it is essential that P and S be physically compatible-that P & S be analytically compatible with all scientific laws. Otherwise not only Q but also its negation
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will be an analytic consequence of P & S conjoined with laws. This second provision corresponds to requirement A(2) of the basic D-N model. Putting all this together it appears that as a minimum we want to require that P & S be related to Q in the way that the singular component of a potential D-N explanans is related to the explanandum in the basic model of D-N explanation. When P & S is so related to Q I shall say that Q is a "nomic consequence" of P & S. It turns out, however, that the relation of nomic consequence is not strong enough to play the key role in the analysis of nomically warranted regular counterfactuals. Consider a certain body of gas a confined to a tank and a certain short time interval t during which the pressure, temperature,and volume of the gas remained constant. Since it is a scientific law that the product of the pressure and volume is proportional to the temperature,the statement 'the volume of a decreased during t' is a nomic consequence of the false statement 'the pressure of a increased during t' taken together with the true statement 'a is a body of gas and the temperatureof a was constant during t'. But we would not be willing to assert that if the pressure of a had increased during t then the volume of a would have decreased during t. Consider also that, speaking of a rocket lifting from a launching pad, we would be quite willing to affirm that if the thrustwere to increase then the acceleration of the rocket would increase, but it would seem somehow wrong to say that if the acceleration of the rocket were to increase then the thrust would increase. Yet, it appears that an increase in thrust is as much a nomic consequence of an increase in acceleration as is an increase in acceleration a nomic consequence of an increase in thrust-both being authorized by Newton's second law of motion. Finally, note that Nelson Goodman, essentially using a relation similar to that of nomic consequence-with the extra aid of a certain exclusive clause to boot-was unable to assign different truth values to the counterfactuals 'If match m had been scratched it would have lighted' and 'If match m had been scratched it would not have been dry'-in a situation in which the match is in fact not scratched, is dry, and does not light (1965, p.14). Goodman considered such difficulties as part of a problem of "cotenability" for which he had no solution. Wilfrid Sellars (1958) has suggested that the problem of cotenability could be circumvented by appealing to the presence or absence of relevant causal relationships. According to Sellars, the crucial fact to note in Goodman's match example is that while it is true that scratching dry matches causes them to light it is false that scratching matches that do not light causes them to become wet. And he concludes that counterfactuals typified by Goodman's example must be supportedby generalizations of a richer kind than those expressible as generalized conditionals (1958, pp. 240ff.). I think Sellars has made a valuable contributionhere but that his suggestion
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is unnecessarily strong in two ways. First, as will be shown later, the key relation need not be a fully causal one-it must merely conform to a directionalityrequirementwhich also governs causal relationships. Second, this conformity can be achieved without appealing to generalizations other than laws of the kind discussed in Part 5 of this paper. As is perhaps obvious by now, what I wish to suggest is that the key relation suitable for the analysis of counterfactuals is not that of nomic consequence but ratherthe relation that holds between P & S and Q when P & S is related to Q in the way that the singular component of a directionally admissible potential D-N explanans is related to the explanandum. I shall express this relationship by saying that Q is a proper nomic consequence of P & S. One might wonder at this point whether the relation of proper nomic consequence should be defined in a still stronger way-whether, for example, it should be made to correspond to a D-N explanation that is not only directionally admissible but also nomically connected or efficient. To see that nomic connectedness should not be required, consider the example of the three springs described in the second paragraphof Part 2 above. Suppose that a is at such a low temperature that it is actually shorter than c. Given this and the other facts of the situation it would seem right to assert 'If a had been hotter than b then a would have been longer than c'-but the corresponding explanation is not nomically connected. To see that efficiency should not be required, consider the counterfactual 'If this sample had been placed in water it would have dissolved'. If this is in fact true then it is also surely true to assert 'If this sample had been hexed and placed in water it would have dissolved'. (Of course, if one knew that the hexing was irrelevant it would not be reasonable to assert the latter counterfactual. As with other kinds of sentences, the truth of a counterfactual is not a sufficient condition for its reasonable assertion in a given knowledge situation-the canons of conversational implicature apply also to counterfactuals.) To sum up, the key relation of proper nomic consequence involves neither nomic connectedness nor efficiency. This is why it is less than causal, since the causal relationshipis based on a D-N model that involves both of these features. Assuming that in order for a regular counterfactual having P as antecedent and Q as consequent to be nomically warrantedit is necessary that there exist a true sentence S such that Q is a proper nomic consequence of P & S, is this also sufficient? Intuitively it is not, for it is surely essential that the circumstances described by S are not such as would be "'undermined"by the realization of the condition described by P. To take a contrived example, suppose that a certain match m meets all those conditions, such as being dry, etc., which together with being scratched have as a proper nomic consequence that m lights. But suppose also that m is
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in the vicinity of an extremely fast-acting sprinkler system that is acoustically triggered and so sensitive that the very first sound of the scratching of m is sufficient to activate it and get m thoroughly wet. It is now clearly false to assert that if m had been scratched it would have lighted. This is a case in which P, though alone compatible with S, is such that when conjoined with a true S' (a complicated sentence describing the sprinkler arrangement)has the negation of S as some kind of "consequence"-and this sort of thing must be ruled out. The crucial question concerns the kind of consequence relation that is relevant here. It would be too strong to require that there be no true sentence S' such that -S is a mere nomic consequence of P & S'. For if Q is a nomic consequence of P & S then it will far too frequently be the case that -S will be a nomic consequence of P & - Q, where - Q is true. (For example, if the lighting of a match is a nomic consequence of its being struck and being dry, then its not being dry is a nomic consequence of its being struck and not lighting.) As later examples will show, this trouble does not arise if we take the relevant relation to be that of proper nomic consequence. These considerations lead to the following formulations: I. Proper nomic consequence. A sentence E is a proper nomic consequence (PN-consequence) of a sentence C if and only if there is a sentence T such that (T, C) is a potential directionally admissible D-N explanans for E. II. Nomically warranted regular counterfactuals. A regular counterfactual having P as antecedent and Q as consequent is nomically warrantedif and only if there is a true sentence S such that Q is a PN-consequence of P & S and there is no true sentence S' such that -S is a PN-consequence of P & S'. In order to illustrate how the proposed criterion differentiates between intuitively correct and intuitively incorrect counterfactuals it would be well to choose a set of examples in which the application of relevant scientific laws is especially clear. Let us suppose then that in a certain laboratorywe have hooked up a rheostat to a source of voltage with the result that a certain amount of electrical current is flowing through the rheostat. For brevity I shall refer to the voltage applied during a certain short time interval as simply "the voltage," the resistance of the rheostat during that interval as "the resistance," and the amount of currentflowing during that same interval as "the current." Consider the following cases. (1) One knows that the voltage, the resistance, and the current were all constant, and one asserts 'If the voltage had increased, the current would have increased'. (2) One knows that the voltage, the resistance, and the current were all constant, and one asserts 'If the voltage had increased, the resistance would have increased'. (3) One knows that the voltage in-
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creased, the resistance increased, and the current remained constant, and one asserts 'If the resistance had remained constant, the current would have increased'. (4) One knows that the voltage increased, the resistance increased, and the current remained constant, and one asserts 'If the resistance had remained constant, the voltage would have remained constant'. Since by Ohm's law the current is at all times equal to the voltage divided by the resistance, it can be seen that in each of these cases the consequent of the counterfactual follows by that law from the antecedent together with some particularfact, in (1) the fact being that the resistance was constant, in (2) that the current was constant, in (3) that the voltage increased, and in (4) that the current remained constant. In spite of this it seems clear that the counterfactuals in (1) and (3) are correct whereas those in (2) and (4) are not. Any satisfactory theory of counterfactuals must account for these judgments. Since only nomic supportis in question here, the present theory will account for these facts if it turns out that (1) and (3) are nomically warrantedwhile (2) and (4) are not. Let us now apply the proposed criterion to each of these cases. For (1) it is first necessary that 'the current increased' be a PN-consequence of 'the voltage increased' conjoined with some true sentence. For the latter we may take the sentence 'the resistance remained constant'. Since we have Ohm's law and any required analytic truths to play the role of T, the only question is whether the corresponding potential D-N explanation is directionally admissible. This will be so unless either 'the voltage increased' or 'the resistance remained constant' is, relative to their conjunction, explanatorily dependent on 'the current increased'. But neither is, since there are clearly explanations for the voltage and for the resistance that are compatible with each other and which do not refer to the current. Second, it is necessary that there be no true sentence which, together with 'the voltage increased', has as a PN-consequence the sentence 'the resistance did not remain constant'. The only plausible candidate for such a sentence would seem to be 'the current remained constant'. But 'the resistancedid not remain constant' is not a PN-consequence of 'the voltage increased and the currentremained constant', because 'the currentremained constant' is, relative to 'the voltage increased' explanatorily dependent on 'the resistance increased' and hence on 'the resistance did not remain constant'. (That this is so is obvious within the domain of ordinary circuit theory involving such matters as Ohm's law and the law relating resistance to the geometry of the conductor and the resistivity of the substance of which it is composed, and I trust it is plausible that it remains so when deeper levels of theory are applied-but I shall not attempt to justify this claim here.) It therefore appears that both clauses of the criterion for nomic support are satisfied by the intuitively acceptable counterfactual of (1).
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The counterfactual in (2) fails to meet the first clause of the criterion, which requires in this case that 'the resistance increased' be a PN-consequence of 'the voltage increased' conjoined with some true sentence. The only plausible candidate for such a sentence would seem to be 'the current remained constant'. But this sentence is, as we saw in the last paragraph,relative to 'the voltage increased', explanatorily dependent on 'the resistance increased'. For the counterfactual of (3) to be nomically supported it is first necessary that 'the currentincreased' be a PN-consequence of 'the resistance remained constant' and some true sentence. And this will be so if we take as our true sentence the sentence 'the voltage increased', since we saw in discussion (1) above that 'the current increased' is a PN-consequence of 'the voltage increased and the resistance remained constant'. The second requirement is that there be no true sentence which, conjoined with 'the resistance remained constant', has as a PN-consequence 'the voltage did not increase'. The only plausible sentence for this role would appear to be 'the current remained constant'. The question then is whether 'the voltage did not increase' is a PN-consequence of 'the resistance remained constant and the currentremained constant'. And it is not, for 'the current remained constant' is, relative to 'the resistance remained constant', explanatorily dependent on 'the voltage remained constant' and hence on 'the voltage did not increase'. (Again, this is obvious within the domain of ordinary circuit theory and surely plausible in general.) There thus appears to be no sentence of the stipulated kind, and so both clauses of the criterion are satisfied by the intuitively acceptable counterfactual in (3). The counterfactual in (4) fails to meet the first clause of the criterion, which requires in this case that 'the voltage remained constant' be a PNconsequence of 'the resistance remained constant' conjoined to some true sentence. The only plausible candidate for such a sentence would seem to be 'the current remained constant'. But it was noted in the last paragraph that 'the current remained constant' is, relative to 'the resistance remained constant', explanatorily dependent on 'the voltage remained constant'. It is perhaps worth noting that many counterfactuals that might pass as reasonable in everyday contexts are literally false and could be considered true only by being viewed as elliptical. Thus, Goodman's well-known example 'If match m had been scratched, it would have lighted' must be considered as elliptical for something like 'If match m had been scratched in such and such a way, it would have lighted', where the asserter presumably has roughly in mind a certain way or range of ways of striking, ways which could in principle be more precisely described. The problem here is that hypothetical events, unlike actual ones, are not subject to
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redescriptionexcept in analytically equivalent terms, and so the antecedent of a counterfactualneeds to be relatively explicit and precise in order for the counterfactual to attain literal truth. Since, however, in everyday life elliptical modes of speech appear to be more the rule than the exception, charity in construing these locutions is prevalent. Before leaving the topic of nomically warrantedcounterfactualsI should like to point out that the present theory does not commit us to any of the counterfactual fallacies discussed by David Lewis (1973, pp. 31-36). The first of these, the fallacy of strengthening the antecedent, is the assumption that a true counterfactual having P as antecedent remains true when P is replaced by P & R. Now, from the fact that Q is a PN-consequence of P and some true S it by no means follows that Q is a PNconsequence of P & R & S-this last might even fail to describe a physically possible situation. And even if Q is a PN-consequence of P & R & S it could well be that -S is a PN-consequence of P & R and a true sentence, so that the second requirement for nomic warrant could be violated. The second fallacy, that of transitivity, sanctions the inference from 'If P were, Q would be' and 'If Q were, R would be' to the conclusion 'If P were, R would be'. But, given that Q is a PN-consequence of P and some true S while R is a PN-consequence of Q and some true T, it need not be that R is a PN-consequence of P and some true sentence. In particular, it does not follow that R is a PN-consequence of P & S & T. And even if R is a PN-consequence of P & S & T it might well be that, say, -T is a PN-consequence of P and some true sentence. The third fallacy, contraposition, assumes the equivalence of 'If P were, Q would be' and 'If Q were not, P would not be'. Now, the fact that Q is a PN-consequence of P and some true S is no reason at all for supposing that -P is a PN-consequence of -Q and S-or, of course, any other true sentence. For example, the required underlying condition of directional admissibility might well hold for one direction but not the other. To continue the development of the full theory of regular counterfactuals we may add the following: III. Analytically warranted regular counterfactuals. A regular counterfactual having P as antecedent and Q as consequent is analytically warranted if and only if P analytically implies Q. Adding the requirement that P not be analytically false would rule out vacuous cases, but I think the usual avoidance of these is a matter of conversational implicature rather than truth value. Lewis (1973, pp. 2425) gives other reasons for counting counterfactuals with "impossible" antecedents as true. One might wonder why III is not constructed as parallel to II, perhaps by simply replacing 'PN-consequence' in the latter with a corresponding "analytic consequence." But this would not do, for
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if Q is an analytic consequence of P & S then -S is an analytic consequence of P & - Q, and so the exclusionary clause would never be satisfied by counterfactuals having false consequents. Well then, perhaps with such a strong relation as analytic consequence the exclusionary clause could just be dropped? No, for then we would authorize both members of such a pair as 'If Bizet and Verdi had been compatriots, Bizet would have been Italian' and 'If Bizet and Verdi had been compatriots, Verdi would have been French', and surely neither of these is acceptable. It thus appears that the notion of analytic warrant must not involve appeal to true auxiliary sentences. Nevertheless there is every reason to think that auxiliary sentences may play the role of "fixed presuppositions," and a counterfactualrelying on such a presupposition, while not to be counted as simply true, might be said to be "true relative to such and such a presupposition." And it is presumably reasonable to assert such a counterfactualprovided that the speaker has a right to the presupposition and it is clear to the audience what the presupposition is. For example, I think it would be reasonable to assert 'If Bizet had been of the nationality of Verdi, he would have been Italian', where the wording strongly suggests that the sentence 'the nationality of Verdi was Italian' is playing the role of a fixed presupposition. Similarly, 'If the UN Building were taller than the Sears Tower it would be taller than the Empire State Building' seems reasonable, since the wording suggests as a presuppositionthe well-known fact that the Sears Tower is taller than the Empire State Building. On the other hand, the literally equivalent counterfactual'If the Sears Tower were less tall than the UN Building, the UN Building would be taller than the Empire State Building' is not at all reasonable, failing to suggest the needed presupposition. We would rather assert 'If the Sears Tower were less tall than the UN Building, the Sears Tower would be less tall than the Empire State Building', where the fixed presupposition is that the UN Building is less tall than the Empire State Building. But since none of these statements are simply true, but only true relative to presuppositions, they all count as false under the present theory. A final, if extremely rare, kind of case must be dealt with for the sake of completeness. Suppose that 'If P were, Q would be' is nomically warranted and 'If R were, S would be' is analytically warranted. Then 'If P or R were, Q or S would be' is neither analytically or nomically warranted (Q v S is not a PN-consequence of P v R, since R analytically implies S) but could hardly fail to be true. I shall call the last counterfactual a "disjunctive merger" of the two simpler ones. (A special rule for corresponding "conjunctive mergers" is not needed, since they are not similarly truthpreserving but must make it either nomically or analytically.) More generally, a counterfactual having P as antecedent and Q as consequent is a disjunctive merger of a set of counterfactuals having ante-
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cedents P1, . . ., P, and, respectively, consequents Ql . . . , Q, if and only if P is logically equivalent to P1 v . . . v P, and Q is logically equivalent to Q1v . . . v Q,. With this definition in hand the final formulation of the theory of regular counterfactuals may be stated: IV. Truth conditions for regular counterfactuals. A regular counterfactual is true if and only if it is nomically warranted or is analytically warrantedor is a disjunctive merger of a set of counterfactuals each of which is either nomically or analytically warranted. REFERENCES Burks, A. (1977), Chance, Cause, Reason. Chicago: The University of Chicago Press. Cupples, B. (1977), "Three Types of Explanation", Philosophy of Science 44: 387-408. . (1980), "Four Types of Explanation", Philosophy of Science 47: 626-29. Davidson, D. (1963), "Actions, Reasons, and Causes", The Journal of Philosophy 60: 685-700. Eberle, R.; Kaplan, D.; and Montague, R. (1961), "Hempel and Oppenheim on Explanation", Philosophy of Science 28: 418-28. Fitzgerald, P. (1971), "Tachyons, Backwards Causation, and Freedom," in PSA 1970: In Memory of Rudolph Carnap, R. Buck and R. Cohen (eds.). Boston Studies in the Philosophy of Science, Vol. 8. Dordrecht: D. Reidel. Glymour, C. (1978), "Two Flagpoles are More Paradoxical Than One", Philosophy of Science 45: 118-19. Goodman, N. (1965), Fact, Fiction, and Forecast. Indianapolis: The Bobbs-Merrill Company. Grice, H. P. (1967), William James Lectures, Harvard University (unpublished). Hempel, C. G. (1965), Aspects of Scientific Explanation. New York: The Free Press. Hempel, C. G., and Oppenheim, P. (1948), "Studies in the Logic of Explanation," Philosophy of Science 15: 135-75. Jobe, E. (1976), "A Puzzle Concerning D-N Explanation", Philosophy of Science 43: 54249. Kaplan, D. (1961), "Explanation Revisited", Philosophy of Science 28: 429-36. Kitcher, P. (1981), "Explanatory Unification", Philosophy of Science 48: 507-31. Lewis, D. (1973), Counterfactuals. Cambridge: Harvard University Press. McCarthy, T. (1977), "On an Aristotelian Model of Scientific Explanation", Philosophy of Science 44: 159-66. Parry, W. T. (1957), "Reexamination of the Problem of Counterfactual Conditionals", The Journal of Philosophy 54: 85-94. Salmon, W. (1970), "Statistical Explanation", in The Nature and Function of Scientific Theories, R. Colodny (ed.). Pittsburgh: University of Pittsburgh Press. Scriven, M. (1975), "Causation as Explanation", Noas 9: 3-15. Sellars, W. (1958), "Counterfactuals, Dispositions, and the Causal Modalities", in Concepts, Theories and the Mind-Body Problem, H. Feigl, M. Scriven, and G. Maxwell (eds.). Minnesota Studies in the Philosophy of Science, vol. 2. Minneapolis: University of Minnesota Press.
