RAIMO DEDUCTIVE TUOMELA EXPLANATION SCIENTIFIC OF LAWS* Since the appearance of Hempel’s and Oppenheim’s famous paper (1948) on the logic of scientific explanation a lively discussion has been going on concerning the logical criteria of adequacy for the explanation of singular events (states of affairs, processes, etc.). However, the theoretically much more important topic of the logic of the explanation of scientific laws has been neglected to a great extent in recent discussion. Notable exceptions to this claim are the works by Campbell (1920), Nagel (1961), Bunge (1967), and partly the recent articles by Ackermann (1965), Ackermann and Stenner (1966) and Omer (1970). Furthermore, in the debate between the representatives of the (or a) two-level picture of science (e.g. Hempel, Feigl) and the ‘omnitheoreticians’ (e.g. Feyerabend, Kuhn) the logical aspects of the explanation of (empirical) laws and theories by more developed theories has been treated to some extent. In any case the discussion on the explanation of laws has in general lacked the formal rigour and sophistication characteristic of the discussion of the explanation of singular events. It has often been argued that the notion of (scientific) explanation is a pragmatic notion which does not have a clear-cut unambiguous objective formal structure. Sometimes these arguments purport to show that the relation of explanation holding between an explanans and an explanandum ceases to hold under the substitution of logically equivalent explanantia or explananda. Or then some other commonly accepted formal invariance conditions are claimed to fail (cf. p. 386). Often this type of argumentation is invoked to show the intensional (nontruthfunctional) character of the notion of explanation, which thus - qua intensional - cannot be clarified by an explicate (or explicates) within standard extensional logic. However, as we do not know of any convincing arguments to prove the latter we will proceed under the working hypothesis JournaI of Phikophical Logic 1(1972) 369-392. All R&htx Reserved Copyright 0 1972 by D. Reidel Publishing Company, Dordrecht-Holland 370 RAIMO TUOMELA that for the time being it is meaningful to look for formal criteria that acceptable scientific explanations should satisfy. On the other hand, it seems clear to us that the enormously difficult and complex notion of explanation will have a number of different objectified and idealized explicates - and not only one. This partly follows from the contextual characteristics of ordinary common sense explanations from which the notion or rather the various notions of scientific explanation have been obtained by the processes of abstraction, idealization, schematization, and objectification, and so forth. Common sense explanation seems to be dependent on the properties of the explainer and the explainee as well as on many other aspects of the context. All this is part of what if often meant by referring to explanation as a pragmatic notion. There are a number of pragmatic features which cannot be abstracted away in a full-blown theory of scientific explanation without too great a philosophical loss. Lacking a developed adequate pragmatics for the philosophy of science - which would be needed - something like Kuhnian paradigms presently seem to provide the best pragmatic background frame for a theory of scientific explanation. Indeed, the general framework within which the problems in the present paper can be embedded is briefly this. Consider a fixed Kuhnian paradigm and within it a research programme incorporating a temporal sequence of theories T,, T, ,..., T,, T,,, ,.... We shall assume that the paradigm determines which (essentially general) statements qualify as scientific laws and which (sets of) statements represent acceptable theories. Furthermore, it will be assumed that the paradigm specifies the philosophical purposes of a scientific explanation. Until these conditions are fulfilled it does not seem meaningful to ask for a clarification of the logical and methodological features of scientific explanations. Within the present frame one can start asking various questions pertaining to the growth of science as seen from the point of view of the present paradigm. For instance, what kind of philosophically, methodologically, and logically interesting transitions from some T. to T,+, can be found? In the present paper we shall consider only one special type of scientific growth occurring within this scheme, viz. that where T,+, (deductively) explains and thus supersedes Tn. Here T,, can be an empirical law or theory and T,+, a more comprehensive theory often introducing new explanatory ideas. DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 371 It has often been contended that in actual science a superseding theory T n+i almost never deductively implies T,. Rather T,+ i implies a theory or law T,,* which can be considered an approximation of T. and which is at least initially in experimental agreement with T.. (The explanation of Kepler’s laws by Newtonian mechanics is often mentioned as an example at this point.) Another way of putting this is to say that T,,, explains T, by correcting it at the same time. Undoubtedly this kind of correction process often takes place in science. Nevertheless, one may argue that this does not basically contradict the very idea of deductive explanation. For one may always argue that what one really aims at in this situation is an explanation and deeper understanding of the (extralinguistic) regularity allegedly described by T.. However, in the process of explanation it is found that Tn* rather than T, seems to be a correct description of this regularity. If this characterization of the situation is accepted there is nothing seriously wrong with deductive explanation, not at least on this point, even if deductive explanation alone does not then seem capable of accounting for transitions within actual scientific research programmes. (It should be remarked here that our notions of paradigm and research programme are to be understood in such a way that really revolutionary transitions always involve abandoning a paradigm and a research programme associated with it. Hence we do not have to take stand to questions of meaning variance, etc., which seem to involve a change of paradigm.) Apart from some comments in the final section we shall in this paper concentrate on the logical and methodological rather than on the epistemological and metaphysical aspects of scientific explanation. Our starting point will be to regard explanation as an informative or informationproviding argument. Unless considerably explicated and developed this idea is of course platitudinous. Explanation of laws can and has been regarded, for instance, as aiming at finding hidden causes for observable phenomena, or alternatively a deeper description of reality. Both of these basic types of explanation give us reasons for believing in the laws to be explained. At the same time it seems that an explanation (such as each of the above kinds) always carries a proper amount of relevant information concerning the explanandum and that this piece of information indeed constitutes the grounds for our believing in the explanandum. In 372 RAIMO TUOMELA this paper we shall mainly discuss information as measured by the logical (and especially quantificational) strength of statements. The sense or senses of information due to the introduction of new explanatory and perhaps ontologically more basic theoretical ideas will be emphasized, too. II 1. Our main task in this paper is to examine some of the logical and methodological conditions of adequacy to be satisfied by deductive explanations of laws. Among other things we shall make an attempt to give a set (or actually several alternative sets) of necessary and sufficient logical conditions for an explanatory relation to hold between a theory and an empirical (or observational) generalization. Among theattempts to give exact formal criteria of adequacy for deductive explanation, only those by Ackermann (1965), Ackermann and Stenner (1966), and Omer (1970) were also concerned with the explanation of general explananda in addition to singular ones. Ackerman& (1965) paper has been criticized and shown to be faulty in Ackermann and Stenner (1966). The latter paper again has been criticized by Omer (1970) and by Kgsbauer (reported in Stegmtiller, (1969). Omer also presented a model of explanation of his own based on general information-theoretic considerations. We shall start by a concise description of Omer’s model of explanation. It will be followed by a criticism. Finally we shall try to amend Omer’s model in some respects and discuss the implications of our amended and enriched models. The final amended model to be presented is intended to apply to the explanation of laws only. Whether or not it can easily be applied to the explanation of singular explananda we shall not discuss here in any detail. Hence there is no need to go through the whole welter of counterexamples to various explicates of the deductive-nomological model of explanation, as these counterexamples have generally been restricted to the explanation of singular explananda. Let us thus go on to a review of Omer’s model of explanation with whose main ideas we in general sympathize. Omer starts by applying to scientific explanation one of Grice’s principles concerning the informativeness of assertive discourse. This gives the first general criterion for explanation (see Omer, 1970, p. 419): DEDUCTIVE RI EXPLANATION SCIENTIFIC LAWS 373 In the explanans no sentence which is less informative in the topic should be given when it is possible to give a more informative one. Next Omer states as ‘an important RIS OF subcase’ of this general requirement: No sentence in the explanans should be of less informative content than the explanandum. The explanandum-sentence is considered by Omer to be a true statement ‘on the topic’. R, then indeed follows from RI with the additional premise that it is possible to give an explanans more informative than the explanandum. One important question left open by the above requirements is how to measure the information content of a sentence. Omer uses essentially the logical content explicate due to Carnap’s theory of semantic information. According to it, the content of a statement is identified with its logical strength (and formally measured as one minus the logical probability of the statement). However, following Omer, we do not below utilize the metrical cant-measure at all. The only information-theoretic notion concretely needed in Omer’s account is that of noncomparability of statements. We shall say that two statements. p and q are noncomparable with respect to their information content if and only if not l-p 2 q and not F q up. It is important to notice what is involved in this notion of noncomparability. Two statements are noncomparable exactly when each of them includes some content-elements not included in the other one. Thus it is quite possible that they contain common content-elements. Indeed, unless this were the case within deductive explanation, the requirements concerning the noncomparability of the elements of the explanans and the explanandum to be considered below could not be reconciled with the idea of the explanans logically implying the explanandum. (Consider e.g. Craig’s interpolation theorem from which it follows that there has to be some amount of common content between two statements one of which implies the other.) As a side remark it can be pointed out here that within nondeductive explanation, measures of transmitted information (the inf-, cant- and 374 RAIMO TUOMBLA entropy-measures) have been used to measure the goodness of an explanation. The more an explanans transmits or conveys information concerning the explanandum, the better is the explanation. However, in this sense all deductive explanations are maximally good. For if the explanans logically implies the explanandum, our coming to know the explanans completely relieves us of our agnosticism or uncertainty (these notions understood in the sense of the semantic theory of information) concerning the explanandum. Thus within deductive explanation this kind of information-theoretic explicate of explanation is of no help. Let us now go back to Omer’s general requirements for deductive explanation. In terms of logical content we now have (Omer, 1970, p. 422): In the explanans no sentence in the topic which is of less logical content should be given when it is possible to give a sentence with more logical content. No sentence in the explanans should be of less logical content than the explanandum. The content of the requirement R; (over and above that of R;,) can be illustrated by the following example. Suppose the statements p v r and p v -r are proposed to be included in an explanans. Now jointly they are equivalent to p but both of them are individually of less logical content than p as p implies them. Thus, according to R; instead of p v r and p v -r we should use p in the explanans. Next consider the following example where T is a theory and L a law: T TDL L (Notice that here the second premise is logically equivalent with -T v L, with clearly follows logically from L.) The requirement R;, clearly blocks, among other things, this kind of trivial deductions from being explanations. But it does not block the following paradoxical case given in Hempel and Oppenheim (1948) p. 273) : B K ii DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 375 In this self-explanation K could stand for Kepler’s laws and B for Boyle’s law. Apparently RiS has to be strengthened. Omer also comes to this conclusion, although on the basis of examples with singular explananda. Omer now proposes (Omer, 1970, p. 423): The explanandum should be noncomparable sentences of the explanans. with any of the But what is meant by a sentence here? Is the conjunction of all the sentences occurring in the explanans a sentence? If so, then certainly R& is not acceptable as it is clearly in blatant contradiction with the basic requirement that the explanans should logically imply the explanandum. Apparently some clarification is needed. This may be what Omer thinks, too, as he proceeds to give a more technical equivalent to RI;,, even if he does not motivate his move. Before giving the final version of the noncomparability requirement we shall have to introduce some technical apparatus. Let us consider some scientific language 9 with specified logical and extralogical vocabulary, well formed formulas, logical axioms and rules of inference. In the manner of Ackermann and Stenner (1966, p. 169) we now define the notions of a sequence of truth functional components and a set of ultimate sentential conjuncts of a given formula (sentence) T. A sequence of statemental well formed formulas (W,, WZ, . . . . W,,) of 9 is a sequence of truth functional components of T if and only if T may be built up from the sequence by the formation rules of $P, such that each member of the sequence is used exactly once in the application of the rules in question. The WI’s are thus to be construed as tokens. Thus, for instance, ((x) F(x), (x) F(x)) and not ((x) F(x)) is a sequence of truth functional components of the formula (x) F(X) & (x) F(x). The formation rules of 2 naturally have to be specified in order to see the exact meaning of the notion of a sequence of truth functional components of a theory finitely axiomatized by a sentence T. A set of ultimate sentential conjuncts Tc of a sentence T is any set whose members are the well formed formulas of a longest sequence (W,, W,, . . ., W,> of truth functional components of T such that T and W1 & WZ BE 8~. . . & W, are logically equivalent. If T is a set of sentences then the set Tc of ultimate conjuncts of T is the union of the sets of ultimate sentential conjuncts of each member of T. We may notice here that although by 376 RAIMO TUOMELA definition the Tc-sets of two logically equivalent theories are logically equivalent they need not be the same. Now we are ready to state the final version of the noncomparability requirement for a Tc of a theory T constituting an explanans (cf. Omer, 1970, p. 424): For any Tci in Tc, Tc, is noncomparable dum. with the explanan- The requirement R;I:. seems to avoid at least the paradoxical examples of self-explanation concerned with singular explananda to be found in the literature. Furthermore, it seems to do the same in the case of the explanation of laws as well (cf. the example by Hempel and Oppenheim given above). Next we shall state some necessary conditions for an explicate of deduo tive scientific explanation which also Omer accepts (although in a somewhat different formulation). Let us write E(L, T) for ‘L is potentially explained by T’. Here T and L are statements of our scientific language _Lp.(Sometimes we shall also denote by T and L sets of statements, which does not make any essential difference as long as these sets are finite.) As above, Tc is a set of ultimate sentential conjuncts of T. For simplicity, in the sequel we shall be concerned with E(L, Tc) rather than E(L, T). (Recall that Tc and T are logically equivalent. When speaking of E (L, Tc) Tc is to be understood as the conjunction of its members rather than a set, which makes no essential difference here.) Now we are ready to give some conditions for an explanatory relation E(L, Tc) to hold between Tc and L: E(L, Tc) satisfies the logical criteria of adequacy for deductive explanation only if (1) (2) (3) (4) {L, Tc) is consistent; Tc!-L; some Tci in Tc is a universal law; for any Tci in Tc, TCi is noncomparable with L. In view of what was said in Section I of this paper, (1) seems acceptable. Requirement (2) is obvious. As to (3) we shall here require of a law only the syntactic feature that it essentially contains at least one universal DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 377 quantifier, and no fuller treatment of lawlikeness, beyond giving this necessary condition, is attempted at in this paper. Requirement (4) is just R;:. Notice that in this model L can be either a singular or a general statement. Can the conjunction of the conditions (l)-(4) be regarded as a sufficient condition of explanation as well? It seems that the resulting model of explanation is too wide. For if we have found an explanans for a statement, say G(a), within this model, the same explanans will also be allowed to be an explanans for all the disjunctions in which G(u) occurs as a disjunct and which do not contradict condition (4). In accordance with Omer we are here inclined to consider this consequence unacceptable (or at least undesirable). How can we exclude such counterintuitive explanations? Let us first consider Omer’s own argumentation. (See Omer, 1970, p. 425) First he argues that the phrase ‘in the topic’ of the original R, secures that all additional pieces of information are relevant information. This seems acceptable, though vague. But then he claims that RI is to be interpreted or qualified so as to rule out all redundant information (i.e. redundant over and above what is needed for T to imply L). This requirement, however, is in clear contradiction with R; (supposedly equivalent with RI). Indeed, as RI clearly seems to call for maximally informative explanantia, Omer on the contrary is here led to look - ceteris paribus - for minimally informative explanantia. This is seen from the following, which - apart from an index correction - is Omer’s last condition (Omer, 1970), p. 426): (5) It is not possible to find sentences Si, . . . . S, (r > 1) such that for some TCj, . . ., Tc, (n 2 1) : TCj &.a. & TC”tSi &.a. 8~ S, not S, & . . . & S,~TCj &... &Tc, Tc,tL where Tc, is the result of the replacement of TCj, ..*, Tc, by Si, ..., S, in Tc. Omer now claims that the conditions (l)-(5) are necessary and sufficient for deductive explanation. Condition (5) requires that within Omer’s model of explanation we 378 RAIMO TUOMELA choose as the explanans a minimally strong set of statements. To explain something is then to give a specific (minimal) amount of (proper) information. Omer claims with examples that his model is free from the paradoxes found against the previous models of deductive explanation. (See Omer, 1970, pp. 422f., 427-432 for the discussion of the arguments by Hempel and Oppenheim, 1948; Eberle et al., 1961; Kaplan, 1961; Kim 1963; Ackermann, 1965; Ackermann and Stenner, 1966.) We shall not comment upon them here directly, partly because the paradoxes deal only with singular explananda. Furthermore, we shall below show that there are so obvious flaws in Omer’s model that they lead to the rejection of it in its present form. 2. Let us now proceed to a criticism of Omer’s model which will lead us to two amended versions. Let us consider the following paradigm case for the explanation of singular explananda: (4 (F(x) = G(x)) F(a) G(a) Does it satisfy Omer’s conditions? Somewhat it surprisingly it does not, for it does not satisfy condition (5). Contrary to Omer’s claim that conformity to the above condition (5) assures the relevance of the law more than ever, our explanans would here become simply (G(a)}. But this explanans obviously contradicts both condition (3) and (4). In other words, Omer’s model of explanation is clearly inconsistent. An apparent and trivial flaw in Omer’s model seems to be this. A requirement to the effect that the statements S 1,. . ., S, should be (universal) laws whenever the statements TCj, . . . , Tc, are, seems to be missing from condition (5). But this correction does not yet suffice to establish a consistent model of explanation. For the amended condition (5) will still always lead to a contradiction in those cases where the explanandum is a law statement. That this is the case is seen by taking simply the explanandum as the only Si. This makes the amended condition (5) true but it falsifies (4). To block this we have to insert a conditional consistency requirement. Thus we propose the following amendation of (5): DEDUCTIVE (5’) EXPLANATION OF SCIENTIFIC LAWS 379 It is not possible without contradicting any of the previous conditions to find sentences Si, . .., S, (r> 1) at least some of which are essentially universal such that for some TCj, ,.. Tc,, (n>l): Tcj &... & Tc,l-S, &... &S, not Si &... & S,FTCj &... & Tc, Tc,FL where Tc, is the result of the replacement of TCj, . . . . Tc, by Si, . . ., S, in Tc. The model satisfying (l), (2), (3), (4), and (5’) (call it Omer’s corrected model) really seems to be a ‘minimal law’ model according to which an explanation provides a minimal amount of specific information. To illustrate, if we have a singular explanandum G(a) v H(u) and the initial condition statement F(a), then we have to choose in the explanans the law (x) (F(X) 3 G(x) v H(x)) and not, say, (x) (F(X) 3 F(x)). The latter law is stronger than the former, and it, together with F(u), suffices to imply G(u)v H(u). This model also seems to be a possible candidate for the deductive explanation of scientific laws. As we noticed, neither the model (l)-(5) nor the present version (l)-(5’) satisfies the general information theoretic principle R;; and they satisfy the original version Rr only if it is interpreted in a rather queer way which is indeed the opposite of our (and Omer’s) original reading of it. We suggest that the present model should rather be considered as satisfying something like the following strong modification of R1 (actually compatible with what Omer says on p. 425 of his paper): RF The explanans should contain redundant information is it relevant. only if The notions of information and redundant information are to be understood in the previous information theoretic sense in which information content and logical strength go together. The problem left then is what is to be understood by relevancy. One possibility is that formalized by (5’). But it is not the only possible explicate. Consider the following two proposed explanations for the same singular explanandum : 380 RAIMO (4 (*I (Y) (F(x, F(a, TUOMELA Y) = G(x, YN Tc: Tclf b) G(a, b) (Y) @(a, Y) = G(a, YN (**I Tc:* Tcf* F(a, b) W, b) The more general proposal (*) is not accepted as an explanation by Omer’s corrected model, but (**) is. Now one may argue that there is a sense of explanation in which (*) counts as an explanation of G(a, b), perhaps even as a much better one than (**). In this sense of explanation the information provided by the additional quantificational generality of the law Tc: over and above the generality of the law Tc:* may be considered relevant redundant information (in the sense of RF). (Cf. e.g. Hintikka, 1970, for an argument concerning the increase of ‘surface’ information with quantificational depth, and cf. Hintikka and Tuomela, 1970, for an argument concerning how observational information content increases with quantificational depth due to the introduction of new explanatory concepts.) Let us thus accept that there is a sense of information in which the increase in quantificational generality provides relevant information. How can this observation be taken into account in a more technical fashion? We propose the following. Let us write p iq when q is deducible from p by using sentential logic only. Now (5’) is to be replaced by: (5”) It is not possible, without contradicting any of the previous conditions for explanation, to find sentences S i, ...r S, (r> 1) at least some of which are essentially universal such that for some Tcj, . . . . Tc, (n>, 1): Tc, &... & TC,~Si not Si &... & S,tTCj &... 8r S, &... & Tc, Tc,tL where Tc, is the result of the replacement of TCj, . . . , Tc, by Si, . . .. S, in Tc. The resulting model of deductive explanation thus explicitly defines the DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 381 relation E(L, Tc) by the conditions (I), (2), (3), (4), and (5”). This model accepts both (*) and (**) as explanations. In principle the explananda L can again be either singular or general. To get a simple example with a general explanandum we can change (**) by adding a third argument to the two-place predicates F and G and by universally quantifying over the new argument (say z). The resulting explanation (call it (“)) qualifies in our present model but not in Omer’s corrected model. The same holds true for the analogue (‘) of (*). The only explanatory argument which in the present frame would be accepted both by our present model and Omer’s corrected model is one in which only z is quantified. (Call this explanation (“).) We propose that the conditions (1)-(5”) are to be accepted as necessary and sufficient for determining the logical structure of the explanation of (at least) singular explananda. (Below we shall still require something more of good explanations of scientific ~uws.) It should be noticed that the conditions (1)-(5”) are not restricted to any specified formal language, as most of the previous models of explanation have been. Therefore it should be widely applicable. In addition, we have not made any philosophical restrictions as to the substantial content of explanations, but have restricted our analysis to the purely formal aspects of explanation. 3. At this point we shall recall one of our claims from Section I of this paper. There we mentioned that particularly in the explanation of laws one introduces theories which contain new explanatory concepts. To make justice to this we shall assume that we are dealing with an interpreted scientific language whose extra-logical vocabulary is dichotomized into a set 1 of observational (or descriptive, or old) predicates and a set p of theoretical (or explanatory, or new) predicates. The only thing we essentially require of this dichotomy here is that the members of p cannot in aZZcontexts be used in direct reporting. Therefore they have to be connected by correspondence rules to the members of L, at least in some contexts, to be methodologically useful (and perhaps they are not even fully intelligible without that). We shall below assume that our explanandum laws contain only members of A, and that the explanantia always contain at least some formulas in which members of p occur. In the explanation of a law L(A) the explanans will be a theory axiomatized by a sentence T(1 u p) (or, alternatively, a Tc(3, u ,u) correspon- 382 RAIMO TUOMELA ding to T(3,up)). One can now distinguish the following three parts in the theory: (1) the Campbellian part (or core theory) which consists of those statements of T(Jup) which are solely in the vocabulary ~1; (2) the observational content of T(1 u ,u) which consists of those statements of T(nup) which contain members of 1 only; (3) the correspondence rules of T(J u p) which consists of all the other statements of T(Jup) and are in the vocabulary Iup. Let us call the deductively closed sets comprising the components (l), (2), and (3) by H(P), WI, and C(n u p), respectively. For first-order languages, the decomposition of a sentence T(1up) can be accomplished by means of, for instance, the reduction operation defined in Hintikka and Tuomela (1970). More generally, Craig’s general replacement method shows how this can be done for deductive systems in the case of almost any formal system (cf. Craig, 1956). (We shall below denote by Cn(T(lup)) (= H(,u)uO(~)UC(JU~)) the deductive closure of T(lup), and by the Craigian p-reduct of a deductive system we shall mean the recursively axiomatizable part of Cn(TLup)) solely in the vocabulary PJ When we below investigate how a theory T(rZup) explains a law L(A) this problem can thus also be considered as how a core theory Hb) conjoined with some suitable correspondence rules C(n up) explains a law L(A) belonging to O(A). As we have repeatedly pointed out, there are many ways to explicate the notion of explanation. Furthermore, one may argue about which properties of the relation E(L, Tc) are analytic and which are synthetic. We shall not take a definite standpoint to this here but, rather, we shall liberally try to take into account all the logically and methodologically important features of E(L, Tc). In the model of explanation satisfying (l), (2), (3), (4), and (5”) a number of important aspects of explanation (e.g. proper informativeness, prohibition of completely circular explanation, etc.) are already incorporated. But there are still at least two logically explicable properties to be considered (nonlogical ones will be discussed later). These are the requirements of observational creativity of the explanans over the explanandum law and the theoretical noncreativity of the explanans with respect to its Campbellian part. DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 383 As to the first of these let us quote Nagel. “At least one of the premises in the explanation of a given law will meet the following requirements: when conjoined with suitable additional assumptions the premise should be capable of explaining other laws than the given one; on the other hand it should not in turn be possible to explain the premise with the help of the given law even when those additional assumptions are adjoined to the law” (Nagel, 1961), p. 36). We shall accept a strengthened form of this condition, and its formalized version will appear below as condition (6) among our additional desiderata for explanation. The theoretical noncreativity condition has been formulated by Nagel as follows: “the introduction of new correspondence rules does not change either the formal structure or the intended meaning of the theory, though new rules may enlarge the theory’s range of application” (Nagel, 1961, p. 102. Also see Campbell, 1920, p. 133). In Hintikka (1972) this condition was dubbed the theoretical adhockery condition, as it requires that no ad hoc theoretical principles, but only (or at most) new correspondence rules, should be added when applying the theory to explain new laws. However, this condition cannot be applied unqualified to all explanation as clearly the core theory H(p) must be allowed to grow if this growth takes place ‘in a fruitful way’. We shall not include the theoretical noncreativity (or adhockery) requirement among the conditions of our model proper but as a separate condition ((7) below) to be applied when the above qualification can be accepted. (After this accounting for this requirement, all of Nagel’s logical conditions of adequacy for the explanation of laws are taken case of by our conditions; cf. Nagel, 1961, pp. 33 ff, and p. 102.) Now our final model of explanation of scientific laws can be stated in full as follows. As before, let thus T be a statement. Tc a set of ultimate sentential components of T (or actually a conjunction of components in the context E(L, Tc)), and L a statement to be explained. Then we say that the relation E(L, Tc) satisfies the ZogicaZconditions of adequacy for the deductive explanation of scientific laws if and only if (1) (2) (3’) {L, Tc} is consistent; TcbL; Tc contains universal formulas at least some of which contain members of ~1and L contains only general statements in A; 384 (4) (5”) RAIMO TUOMELA for any TCi in Tc, Tc, is noncomparable with L; it is not possible, without contradicting any of the previous conditions for explanation, to find sentences Si, . . ., S, (r > 1) at least some of which are essentially universal such that for some TCj,...,TCn(n>l): TCj &... &Tc”~S~ &... &S, not Si &..v & S,FTCj &... & Tc~ Tc,l-L where Tc, is the result of the replacement of TCj, . . ., Tc, by S*,..., S, in Tc. As an additional desideratum we consider: (6) If Cn(Tc)=H(p)uO(A)uC(Aup), then there is a L’#L(LEO@), L’EO’(2)) such that not l- LD L’ and such that E(L’, Tc’) for some Tc’ (of a T’) for which Cn(Tc’)= =H’QuO’(A) uC’(Jup) and HcH’; in addition, not {L}u(H’-H)u(O’ -O)u(C’-C)t-H. (I owe Prof. W. Stegmtiller my present formulation of the last part of condition (6).) The optional theoretical adhockery condition, to be applied if no fruitful growth is present, is: (7) If H@) is an explanatory-core theory, and if, for some L, E(L, Tc) with Cn(Tc)=H,(~)uO(~)uC(~u&), then the difference H-H, = 0. (H, is the Craigian p-reduct of Cn (Tc)). It can be noticed that none of the above conditions is restricted to a specific formal system (such as first-order predicate calculus). Therefore they should have a wide area of application. Let us call a model of deductive explanation of laws for which (l), (2), (3’), (4), and (5”) hold the DEL-model. If it is not required that theoretical concepts be essential for deductive explanation our old (3) can be substituted for (3’) in the above model (and the appropriate notational changes can be made elsewhere) and we get the model discussed in subsection 11.2. Call it the weak DEL-model. (An analogous remark holds for the situation where one is not willing to dichotomize the extralogical concepts in the manner assumed here.) Earlier we described three simple law-explanations (‘), (“), and (“‘) DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 385 which all qualified as explanations within the weak DEL-model. If, in addition, the three-place predicate F is theoretical, all these explanations are accepted by the DEGmodel, too. Explanations (‘) and (“) carry more relevant and deeper (quantificational) information than (“‘). One consequence of this is that they in addition satisfy (6) which (“‘) fails to satisfy. (Notice that the fate of condition (7) cannot be determined before fixing an initial core theory H@).) As an additional example, consider the following simple chain explanation : (4 (G, (4 = F(x)) (4 (F(x) = Gz(4) (4 (GI (4 = G, (4) where Iz= { G1, G,} and p = {F}. Here G, and G2 might be observable symptoms of a virus F.) This explanation is accepted by the DEL-model. But its shallowness is reflected by the fact that it does not satisfy (6). As a scientific illustration of the DEL-model we can consider, for instance, the explanation of the well-known probability matching law. This law is explained by the linear learning theory of Bush and Mosteller. But it is also explainable by the general stimulus sampling theory of Suppes and Estes which in turn explains the linear learning theory as its special case. Let us next state some formal properties of E(L, Tc) in the DEL-model. Most of them will also apply to the weak DEL-model, as can easily be seen. First, it is easy to verify that in the DEL-model (a) (b> (c) E(L, Tc) is asymmetric E(L, Tc) is irreflexive E(L, Tc) is not transitive. In the weak DEL-model, E(L, Tc) is easily seen to be irreflexive and neither symmetric nor transitive. To see that it is not transitive assume that E(L, Tc) and E(Tc, Tc’) for some L, Tc, Tc’. Then we may possibly have I-Tc’~zTc,, though not I-Tc] DL, for some TC’i ETc’, Tcj ETC. Thus even if not bTcj 3 L it may happen that tTc’* 2 L. Hence condition (4) fails to hold. But still E(L, Tc) is not intransitive in the weak DEL-model. 386 RAIMO TUOMELA Next we observe that: (4 (4 (0 If E(L, Tc) and if for some Tc’, t Tc’xTc (assuming not ~Tc’ I>Tc), then E(L, Tc’), provided every Tci~Tc’ is noncomparable with L. If E(L, Tc) and if for some T’ such that k TET’, T and T’ possess identical sets of ultimate sentential components (i.e. TccTc’), then E(L, Tc’). If E(L, Tc) and for some L, ~LEL’, then E(L’, Tc), provided that, for all Tc, in Tc, Tc, and L’ are noncomparable. It should be noticed, however, that a DEL-explanation is not invariant with respect to the substitution of logically equivalent explanantia or explananda. This lack of linguistic invariance may be taken to reflect the pragmatic nature of our notion of explanation. How you state your explanatory argument makes a difference. For some purposes it may be adequate to define a notion of indirect explanation. We may say that if tT = T’, E(L, Tc) for some Tc of T and Tc#Tc’ (for all Tc’-sets of T’), then T’ explains L indirectly. The properties (d), (e), (f) and (g) below hold for the weak DEL-model as well. It is easily seen that the explanatory relations of the DEL-model and of the weak DEL-model do notsatisfy the following debated subset and conjunction properties : If E(L, Tc) and L’ is a conjunct of L, then E(L’, Tc). (Subset property) If E(L, TC) for all L, of an L=L, &... & Li & . . . &L”, then E(L, Tc). (Conjunction property) This means that the explanandum can be neither weakened nor strengthened without the explanation possibly ceasing to be valid. We have already noticed that in the DEGmodel a law L may have explanantia differing in their quantificational strength (and information content). The DEGmodel does require ‘minimal’ explanation (in Omer’s sense) only within explanans-candidates of the same quantificational strength. Thus we have a tree-structured hierarchy of explanations such that on each level of this quantificational hierarchythe minimalityprinciple DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 387 is satisfied.We can finally notice that the branches of the explanation-tree for an L may be created in diverse logically imcompatible ways: (g) If E(L, Tc) and E(L, Tc’), then it is possible that Tc and Tc’ (and hence the corresponding theories T and T’) are mutually logically incompatible. The explanantia in one and the same branch are of course logically compatible, as the deeper explanantia quantification-theoretically imply the shallower explanantia for L. III We shall conclude this paper by making a few sketchy philosophical and methodological comments concerning the above results and by briefly discussing a few ideal nonlogical conditions of adequacy for the deductive explanation of scientific 1aws.l The various definitions for deductive explanation given in the previous section were only concerned with potential explanation. To make a potential explanation actual some requirements concerning the truth and/or the evidential support of the explanandum and its explanans are needed. One plausible candidate, which we shall adopt, is simply: If E(L, Tc) then L and Tc should be accepted as true. 03) The term ‘true’ of course means something like ‘true in the actual world or ‘true in intended model(s)‘. The notion of acceptance is a notoriously tricky one, and here we shall not attempt to give an analysis of it. (It may be the case that most actually accepted scientific theories are false and even known to be false, but this does not speak against accepting (8) as an ideal for good explanatory theories.) There is, however, one familiar requirement related to how we accept our explanans as true that we want to make more explicit. This is a requirement related to condition (6) and prohibiting what might be called ‘confirmational adhockery’ (cf. Nagel (1961), p. 43). We shall adopt the following weak version of it for our DEL-model: If Cn(Tc(lZup))=H&)uO(ll)uC(IZup) and if E(L, Tc), (9) then Hh) is - through some set C’(A u ,u) of correspondence rules - adequately supported by evidence based on data other than the observational data upon which the acceptance of the explanandum law L is based. 388 RAIMO TUOMELA (Notice that (9) still allows that O(1)= {L}, although it prohibits that 0’ (A) = {L}.) The conditions (8) and (9) are necessary conditions to make the DEL-model for potential deductive explanation a model for actual (in the above sense) deductive explanation as well. As we have seen, the DEL-model (together with conditions (6) and (7)) seems to be capable of giving an adequate account of many troublesome issues in the theory of explanation while it correctly emphasizes some features considered essential for good deductive explanations of scientific laws. Based on our discussion in Section I we found condition (1) - the consistency condition - of the DEL-model acceptable. Condition (2) is obviously also acceptable. Condition (3’) will be commented upon below. Conditions (4) and (5”) were arrived at from general information-theoretic considerations. (4) was found to take care of circular explanations of the most vicious kind. (5”) is supposed to guarantee that an explanation provides a proper amount of relevant (information-theoretically) redundant information. Condition (6) refers to the dynamics of theorizing. According to (6), it is essential that an explanatory theory be at least capable of growth. Condition (9) then actually realizes this possibility by requiring the growth of theory in the evidential sense. It can be noticed here that our condition (6) guarantes against ad hoc,-type and condition (9) against ad hoc,-type of the three kinds of ad hoc-explanations in Lakatos (1970). In addition, our theoretical adhockery condition (7) seems to be a special kind of Lakatos’ ad hoc,-explanations. Condition (3’) serves to indicate our emphasis on the importance of new theoretical ideas (concepts). As Campbell has frequently emphasized, an explanatory theory shall ‘add to our ideas and the ideas which it adds shall be acceptable’ (Campbell (1921), p. 83). Even if our condition (3’) as such says nothing about the difficult and debated notion of ‘acceptability’ of the concepts in ~1,still it - and the DEL-model as a whole - refers to interpretative rather than phenomenological theorizing (cf. Bunge, 1968). The explanation of observational laws by theories introducing new ideas (hidden causes, etc.) is often regarded as ontologically and epistemologitally deeper, and it has been considered to add to our understanding of the world more than mere phenomenological explanation (explaining laws with the help of theories where no new kind of concepts occur). As we noticed, this kind of penetration into deeper levels of reality was in DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 389 part and in addition made possible by the hierarchical organization (tree-structure) of explanations accomplished by condition (5”). (In a model of explanation based only on minimal theory explanations - such as Omer’s corrected model - this would not have been possible.) The additional and optional desideratum - our condition (7) - was introduced to avoid theoretical adhockery. This condition was above also called the theoretical noncreativeness condition as it requires that new correspondence rules added to the core theory to make it explain some new observational laws should be noncreative with respect to their A-content. Hence one way to satisfy this condition is to give noncreative definitions of the observational concepts in terms of the theoretical concepts of the full theory. For instance, explicit definitions of the observables in terms of the theoreticals - rather than vice versa - would be such definitions (cf. Ramsey, 1931; Sellars, 1961; Carnap, 1966; and Hintikka 1972). In this case one can even theoretically eliminate observational or L-concepts in terms of theoretical ones. Theoretical explanation (such as reductive explanation) seen as redescription or reinterpretation can be considered to require ultimately just this. Applied to intertheoretic reductive explanation this is seen to imply that every case of ‘principle-reduction’ can be performed at least as well by accomplishing a ‘concept-reduction’ simultaneously (see Hempel, 1969, for these notions). Interpretative theorizing thus seems ideally to require explicit definitions of observables in terms of theoreticals while phenomenological theorizing seems to imply the reverse at least asymptotically. Nagel has required that in the explanation of laws the explaining theory should be more general than the explanandum law (Nagel, 1961, pp. 37-8). As the generality in question is denied to be proportional to logical strength, the following suggestion seems to be at hand. A sufficient, though perhaps not necessary, condition for arriving at Nagel’s generality would be to define explicitly the concepts occurring in the law in terms of theoreticals such that each definition is a conjunction of at least twotheoreticals. For then one can in an obvious sense say that observable concepts are applications of theoretical concepts and that every theoretical concept is more general and broader in its scope than each observational concept in whose definiens it occurs. Our final remark is concerned with the relation between explanation and various notions of inductive support. It seems that between such 390 RAIMO TUOMBLA notions as explanation, generalization, cormrmation, corroboration, etc., there are many connected features. Indeed, it might be required that theories of explanation and inductive support should always be developed simultaneously. One argument for this arises from an investigation by Smokler (1968). He pointed out that there are two essentially different notions of inductive support, one corresponding to ‘abductive inference’ and the other to ‘enumerative induction’) the latter includes eliminative induction). Within abductive inference it is true to say that evidence supports a hypothesis if the hypothesis explains the evidence. Of the familiar principles of qualitative confirmation abductive inference satisfies the conditions of consistency, nonuuiversalizability, converse entailment, converse consequence and equivalence but it does not satisfy the conditions of entailment and special consequence. On the other hand, enumerative induction makes true the conditions of consistency, nonuniversalizability, entailment, special consequence, and equivalence but it does not make true converse entailment nor converse consequence condition. (See e.g. Smokler, 1968, for an exact formulation of these conditions.) As the reader can easily verify, apart from minor technicalities, the notion of theoretical explanation that our DEL-model makes precise (clearly) belongs to abductive inference and thus not to inductive enumeration. (One might now also argue that our DEL-model also explicates a notion of corroboration peculiar to abductive inference.) Corresponding to Smokler’s notion of enumerative induction we of course also get an implicit definition of a converse relation which might be dubbed inductive systematization or inductive generalization. The picture that emerges from this is that we have a dichotomy of theorizing both within explanation (interpretive versus phenomenological theorizing) and inductive support (abductive inference versus enumerative induction). Interpretive or theoretical explanation demonstrably goes together with abductive inference (corroboration). It can als be argued that some important forms of phenomenological theorizing go together with the above wide notion of enumerative induction and with a high probability-criterion of confirmation (cf. Niiniluoto and Tuomela, 1972; and Tuomela, 1972). McGill University, Montreal2 DEDUCTIVE EXPLANATION OF SCIENTIFIC LAWS 391 NOTES * I want to thank Mrs. Marja-Liisa Kakkuri-Ketonen, Mr. Seppo Miettinen, and Mr. Ilkka Niiniluoto for suggestionsand criticismsconcerning an earlier version of this paper. 1 For a fuller discussionof the methodological and philosophical applications of the DEL-model of explanation seeTuomela (1972), Chapters VII and VIII. a On leave of absencefrom the University of Helsinki, Helsinki, Finland. BIBLIOGRAPHY Ackermann, R., 1965, ‘Deductive Scientific Explanation’, Philosophy of Science 32, 155-67. Ackermann, R. and Sterner, A., 1966,‘A Corrected Model of Explanation’, Philosophy of Science 33, 168-71. 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