Explaining, understanding and scientific theories

ERIK WEBER

EXPLAINING, UNDERSTANDING AND SCIENTIFIC THEOR/ES

ABSTRACT. One of the functions of scientific knowledge is to provide the theories and laws we need in order to understand the world. My article deals with the epistemic aspect of understanding, i.e., with understanding as unification. The aim is to explicate what we have to do in order to make our scientific knowledge contribute to an increase of the degree to which the particular events we have observed, fit into our world-picture. The analysis contains two parts, First I define the concept of scientific epistemic explanation. Explanations of these type are the appropriate instruments for increasing the degree of unification of the particular events we have observed. In the second, largest part of the article I analyze the construction process of scientific epistemic explanations, focusing on the application of scientific theories.

1. INTRODUCTION

It is widely agreed upon that one of the functions of scientific knowledge is to provide the theories and laws we need in order to understand the world. But there is much controversy about what understanding of the world consists in. I adhere to the view recently defended by Wesley Salmon (1993, p. 15). He claims that there are at least two irreducible kinds of understanding humans search for: on the one hand there is the desire to fit all the phenomena into a unified world picture (epistemic aspect of understanding); on the other hand there is the desire to know how things in the world work, i.e., the desire to know the mechanisms that produce the phenomena we observe (causal aspect of understanding). This article deals with the first aspect of understanding. My starting-point is that humans have a desire to maximise the degree to which the phenomena they have observed, fit into their world-picture. My aim is to analyze how scientific knowledge can be used to meet this desire. I will confine myself to particular events. So I will analyze how our epistemic understanding of particular events may be increased by applying scientific theories and other forms of scientific knowledge.

In general, explanations are the instruments by means of which understanding is obtained. Because of this instrumental relationship, my aim can be divided in two tasks that have to be completed:

Erkenntnis 44: 1-23, 1996. ~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

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(A) Characterizing the types of scientific explanation which are the appropriate instruments for increasing epistemic understanding of particular events.

(B) Providing insight into the ways in which explanations of the type meant in (A) are constructed on the basis of scientific knowledge.

In Section 2, which is devoted to the first task, I develop the concept of scientific epistemic explanation (SE explanation). This concept is inspired by Philip Kitcher's unification account of explanation and by Peter Railton's deductive-nomological-probabilistic model. Sections 3-7 of this article contain an analysis of the construction process of SE explanations (the second task). I focus on SE explanations that are based on qualitative theories. A qualitative theory is a theory which makes predictions about qualitative properties (e.g., phenotypes), while quantitative theories make predictions about quantitative properties (e.g., positions and velocities of material objects).

The construction process of explanations has hardly been studied in the philosophical literature on explanation. Therefore the analysis in Sections 3-7 goes beyond the traditional limits of philosophical studies of scientific explanation. On the contrary, the aim of Section 2 is a familiar one: task (A) and analogous tasks relating to other aspects of understanding (e.g., causal understanding) or other types of phenomena (e.g., laws or regular patterns) constitute the main topics in the philosophical literature on explanation.

2. SCIENTIFIC EPISTEMIC EXPLANATIONS

In Section 2.1 I will introduce two auxiliary concepts: 'probabilistic syllogism' and 'derivation'. In Section 2.2 1 use these concepts to define SE explanations. In Section 2.3 I compare my definition of SE explanations with Philip Kitcher's unification account of explanation and Peter Railton's deductive-nomological-probabilistic model.

2.1.

A probabilistic syllogism consists of two singular sentences and one probability statement. Singular sentences have the form "Object a has property o~ at time t"; their formal representation is o~(a, t). There are two types of probability statements. The statements of the first type have the form "In domain 7 holds: the limit of the relative frequency of class fl in class

equals r", and are written as P.y(fl [ a) = r. 7,/3 and ~ are arbitrary predicates. 7 is called the domain of the probability statement,/3 its object

EXPLAINING, UNDERSTANDING AND SCIENTIFIC THEORIES 3

class, o~ its reference class and r its frequency number. Probability statements of the second type have the form "In domain 7 holds: the limit of the relative frequency of class fl equals r", and are written as P-r(r = r. A probability statement which is necessarily true because of the meaning of the predicates occurring in it, is called analytic. Probabilistic syllogisms are defined as follows:

(PS) (Sa, $2, H) is a probabilistic syllogism for the singular sentence

/3(a, t) if and only if

(1) H is a probability statement of the first type with object class /3, which is not a theorem of probability calculus and which is not analytical,

(2) $1 is a singular sentence in which the property 7 (the property that determines the domain of H) is attributed to (a, t),

and

(3) $2 is a singular sentence in which the property oz (the property that determines the reference class of H) is attributed to (a,t).

A probabilistic syllogism for/3(a, t) has the following form:

7 (a , t ) a ( a , t ) P.y(/3 I a) = r

A derivation of probability statement H consists of (i) a sequence of sentences of which H is the last one, and (ii) a classification for this sequence. The classification describes the inferential characteristics of the sequence: in the classification some of the entries of the sequence (but not H itself) are given the status of premise; for each other entry, the classification contains a statement asserting that it is derivable from one or more preceding entries by means of a rule of inference. Each rule that is invoked must be explicitly mentioned. The rules of inference may belong to propositional logic, predicate logic, probability calculus, differential calculus, etc.

As an example we consider a derivation of a probability statement relating to the Tay-Sachs disease (also called "infantile amaurotic idiocy"). In children suffering from this disease, an accumulation of complex lipids in the brains, milt and/or liver causes blindness (amaurosis) and idiocy imme- diately after birth; almost all children die within 10 years. We consider the claim that, in the domain of human beings (/ /) , the probability of an individual suffering from the Tay-Sachs disease, given that one of his parents has genotype [[ and the other genotype Ii, is 0. This probability statement may be formally represented as PH(T-S ] [[ • Ii) = 0. The following

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sequence and classification jointly constitute a derivation of this statement:

Sequence of sentences:

(1) (2)

(3)

(4)

(5) (6) (7)

(8) (9) (10)

Each human belongs to one of the genotypes I I , I i or ii.

ii individuals suffer from the Tay-Sachs disease, I I and I i individuals do not.

For any individual w and any alleles X and Y: if w has XY, then the probability that w transmits X to any one of its offspring is 0.5.

For any individual w and any allele X: if w has X X , then the probability that w transmits X to any one of its offspring is 1.

For all types of cross C: PH(T-S ]C) = Pu(i i] C).

PH(T-S ] I I x Ii) = PH(ii ] I I x ii).

For all crosses o f k i n d X X x X Y : P ( X X I X X x X Y ) = 0.5

and P ( X Y I X X x X Y ) = 0.5.

P M I I I I I x f i ) = P H ( I i I I I x I i ) = 0.5

PMiiJ I I x [i) = o

P. (T-S I I I x I 0 = 0

Classification (1)-(4) are premises. (5) is derivable from (1) and (2) by means of probability calculus. (6) is an instantiation of (5). (7) is derivable from (3) and (4) by means of probability calculus. (8) is an instantiation of (7). (9) is derivable from (8) by means of probability calculus. (10) is derivable from (6) and (9) by substitution.

2.2.

Each individual accepts some singular sentences as true. "Accepted" and "rejected" are the two general epistemic statuses a person may assign to a singular sentence (note that these statuses are not exhaustive: X may suspend judgment). Each accepted singular sentence has one or more specific epistemic statuses. Each such status reflects one reason why the individual accepts the sentence. In my definition of SE explanations, I will use one such status: "empirically founded". A person has to give the specific status empirically founded to a singular sentence if and only if his own observations contain sufficient evidence for it or he has good reasons to believe that someone else has gathered sufficient observational evidence for it. Assigning the status "empirically founded" to a singular sentence


is sufficient but not necessary for accepting it: a person may have another reason for accepting a singular sentence. In that case, the sentence has a different specific status.

Scientific entities (theories, phenomenological laws, probability distri- butions, etc.) may be subjected to empirical tests (e.g., a X2-test). On the basis of the results of these tests, we can decide whether or not to assign the status "empirically adequate" to an entity. By assigning the status "empirically adequate" we express the fact that we regard it as reliable for making predictions in certain contexts. In my definition of SE explanations, I will assume that each individual has a world-picture. The world-picture of an individual X consists of (i) a part of the scientific entities which X regards as empirically adequate, (ii) a set of ontological assumptions, and (iii) an epistemological model of man as a cognitive system. The world-picture of X at t is written as Wx#. As the world-picture contains only a part of the scientific entities which X regards as empirically adequate, one may wonder how the elements of the world-picture are selected. To answer this question, I introduce the concept of a derivation pattern. Derivations as I have defined them in Section 2.1 are a specific type of argument. Derivation patterns constitute a subclass of what Philip Kitcher (1981, pp. 515-517 and 1989, pp. 432-433) calls argument patterns. According to Kitcher, an argument pattern is a triple of (i) a sequence of schematic sentences, (ii) a set of sets of filling instructions, and (iii) a classification. A schematic sentence is an expression obtained by replacing some, but not necessarily all, the nonlogical expressions in a sentence with dummy letters. The filling instructions are directions for replacing the dummy letters. An argument pattern contains one set of filling instructions for each entry of the sequence of schematic sentences. A classification describes the inferential characteristics of a sequence of schematic sentences (cfr. Section 2.1). I define derivation patterns as argument patterns that, if we execute the filling instructions, result in a derivation. Each scientific entity provides us with a set of derivation patterns. Whether or not an entity belongs to our world-view depends on the unifying power of the set of derivation patterns it provides: our world-view contains only entities which provide a set of derivation patterns with a high unifying power. I rely on Kitcher for an implementation of the concept of unifying power (1981, pp. 518-519 and 1989, pp. 433-435). So the unifying power of a set of derivation patterns varies directly with the stringency of the patterns in the set. A derivation pattern is more stringent than another if the former sets conditions on its instantiations that are more difficult to satisfy than those set by the latter. The unifying power of a set of derivation patterns also varies directly with

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the degree to which its elements are similar. Finally, it varies inversely with the number of patterns in the set.

We now have all the elements we need to define the concept of scientific epistemic explanation:

(SEE) Let(S1, $2, H) be a probabilistic syllogism for the singular sentence SE, and D a derivation of H. Then (($1, $2, H), SE, D) constitutes a scientific epistemic explanation of SE for individual X at time t if and only if

(1) X at t assigns the status "empirically founded" to SE,

(2) X at t assigns the status "empirically founded" to $1,

(3) X at t accepts the validity of the inference rules which are used in the classification of D,

(4) X at t accepts the derivability claims made in the classification of D, and

(5) the premises of D are (parts of) scientific entities that X regards as empirically adequate at t; more precisely, they are members of Wx,t or parts of such members.

(6) X at t assigns the status "empirically founded" to $2.

According to this definition, a SE explanation is a triple of a probabilistic syllogism, a proposition stating the event to be explained (SE) and a derivation of the probability statement used in the probabilistic syllogism. The first condition ensures that such triples are not called explanations if they are used to predict an event. Condition 3 relates to X 's opinion on the validity of the inference rules, while condition 4 relates to X 's opinion on whether the rules are correctly applied. In our Tay-Sachs example the premises of the derivation collectively constitute a scientific theory which may be called "the genetics of the Tay-Sachs disease". If this theory is an element of Wx,t the derivation is useful for X: he can construct SE explanations with it. Condition 6 is not entirely accurate: it is too restrictive. In Section 4.3, when I have defined what qualitative theories are, I will weaken it.

Constructing a SE explanation amounts to showing to which degree the explanandum event could have been expected on the basis of a scientific entity which belongs to Wx,t. As the members of Wxr are selected by means of criteria that relate to unification (world-pictures are always unified), we might conclude that SE explanations are the appropriate instruments for increasing the degree to which the particular events we have observed, fit into our world-picture. I think this conclusion is correct, but only for particular events of a qualitative nature. If we want to analyze how


quantitative properties of objects can be fitted into our world-picture, we need a concept of explanation which, unlike my concept of SE explanation, can cope with approximations. A more extensive discussion of the relation between SE explanations and epistemic understanding can be found in my article 'Unification: What Is It, How Do We Reach It, Why Do We Want It?'.

2.3.

Though my concept of SE explanation is inspired by Kitcher's unification account, there are crucial differences. In the introduction of this article, I said that I adhere to the view that understanding has at least two aspects. Unlike Kitcher, I do not regard epistemic conceptions of explanation (like his unification account and my SE explanations) as rivals for causal conceptions. In my view, epistemic explanations are the instruments by means of which unification is achieved, while causal explanations are explanations that can provide causal understanding. Both types of explanation need to be analyzed. As a consequence of this view, the asymmetry problem disappears: epistemic explanations do not show the asymmetry which is typical of causal explanations.

Apart from this dispute concerning the nature of understanding, my concept of SE explanation differs in two crucial respects from Kitcher's account:

(1) An explanation as defined by Kitcher is always a deductive argument. Kitcher regards his deductive concept of explanation as an explica- tum of what ideal explanations are. Kitcher does not maintain that constructing an explanation (in his sense) is necessary for an increase of epistemic understanding. However, he does not describe the non-ideal types of explanation that are also useful with respect to unification. My concept allows to put forward an ideal: SE explanations in which r = 1 establish that an event fits completely into a world-picture. But it also takes into account the non-ideal instruments, as non-deductive SE explanations with r between 0 and 1 show that the explanandum event fits to some extent into Wx,t . As a consequence, my concept of SE explanation provides a more complete answer to the question of how unification is reached.

(2) My concept of SE explanation puts restrictions on the structure of explanatory arguments: a SE explanation must contain a probabilistic syllogism and a derivation of the probability statement which is used in the syllogism. In Kitcher's account, "argument" is a primitive concept. He holds that all explanations are arguments (not vice versa), but does not analyze the structure of arguments. An example of a structural restriction Kitcher could have imposed (in addition to his requirements concern-

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ing unifying power) is the requirement that an argument can only be an explanation of/3a if it contains (i) a deductive-nomological explanation of /3a (i.e., a set of singular premises c q a , . . . , c~a and a specific law (Vx)[(c~lx & o~2x & . . . & anX) D fix]), and (ii) a derivation showing that the specific law used in the DN explanation is deducible from a set of gen- erally accepted scientific laws and theories. Because my concept is richer it provides more insight into the way unification is brought about.

Formulated in the notation I have used above, Peter Railton's scheme for the structure of DNP explanations (1981, p. 236) is:

(a) A theoretical derivation of the probability statement in (b).

(b) P.~(/3) = r

(c) 7(a, t) (d) The probability of a having/3 at t is r.

(e) /3(a, t)

When this scheme is compared with the structure of SE explanations, two differences can be noticed. Firstly, I omit (d) because statements of this form are superfluous: the value r is already given in (b). The second dif- ference is that, while I use probability statements of the first type to define SE explanations, and consequently require that a probabilistic syllogism contains two singular statements, Railton uses probability statements of the second type. I think my option has a pragmatic advantage, which will become clear in Section 3: we can use 7 to represent explanatory facts that belong to our background knowledge, while c~ can be used to represent newly discovered explanatory facts.

Besides the pragmatic advantage, the definition of explanation I have proposed has two more fundamental advantages. In Railton's account, "derivation" is an unanalyzed primitive concept. He suggests that theo: retical derivations are not just deductions from higher-level theories and facts: they also have to elucidate the mechanisms that are at work (1981, p. 242). But he does not analyze the structure of derivations. Furthermore, the idea of unification is not captured by the DNP model, because Railton does not use concepts like derivation pattern or argument pattern.

3. A METHOD FOR CONSTRUCTING SE EXPLANATIONS

In the remainder of this article I will develop a method for constructing SE explanations which rests on two main ideas. The first idea is that, at the beginning of the process, we choose a scientific entity T (or a small set of such entities) which belongs to our world-picture, and a set of


background knowledge B. T is the element (or set of elements) of our world-picture (e.g., a theory or a phenomenological law) which we select as the main instrument(s) for constructing the explanation. B consists of a set of statements about singular facts we regard as relevant.

The second idea is that of gradually implementing the general scheme of probabilistic syllogisms. As noticed in Section 2.1, definition (PS) implies that the general scheme for a probabilistic syllogism for/3(a, t) is:

7 ( a , t ) a ( a , t ) P~(/3 ] a) = r

By a gradual implementation of this scheme I mean the consecutive replacement of the schematic letters (oz,/3, 3' and r) with a predicate or a number. More precisely, we alternately create weak and strong restrictions on admissible instances of the scheme, and materialise strong restrictions. A weak restriction limits the predicates or numbers we are allowed to use to replace a schematic letter of the scheme, but leaves at least two possibilities; a strong restriction excludes all possibilities but one. If an inquirer actually replaces a schematic letter with its unique instantiation prescribed by a strong restriction, we say that he has materialised this strong restriction. The moves in which an inquirer materialises strong restrictions will be called constructive moves. An individual who constructs a SE explanation can create (weak or strong) restrictions in four different ways: by means of an interrogative move, by means of a derivative move, by means of a decisive move or by means of a combinative move. In an interrogative move, we ask a question to Nature, i.e., try to find an answer to a question by means of observation or experiment. In a derivative move we try to derive an answer to a question from information that is available to us (T, B, results of interrogative moves). In a decisive move, we make a decision which does not depend on an answer of Nature or on the result of a derivation. The way in which the results of these moves (the result of the observation or experiment, the statement that is derived or the decision that is made) create restrictions will be clarified in the example in Section 6. In a combinative move, we create a new restriction by combining existing ones. The most important function of combinative moves is to create strong restrictions by combining previously obtained weak ones.

The core of my method is the following general procedure, in which the ideas described in the previous paragraphs are incorporated:

(P) (1) In the general PS-scheme, replace /3 with the property attributed to (a, t) in the explanandum E.

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(2) Choose a scientific entity T (belonging to Wi,t) by means

of which you want to explain E.

(3) Determine the content of B.

(4) Materialise all strong restrictions that have been obtained. If this results in a probabilistic syllogism, go to instruction 7; otherwise go to 5.

(5) Choose one of the following options:

(a) Formulate a question to be asked to Nature; make observations or perform experiments in order to obtain an answer.

(b) Formulate a question that expresses a derivation goal and try to construct an adequate derivation.

(c) Formulate a question that expresses a decision problem and make a decision.

(d) Adjust the content of B.

(6) Try to create new restrictions out of existing ones; then go back to 4.

(7) Check whether the singular sentences of the probabilistic syllogism have the epistemic status required by the definition of SE explanations. If they have, stop; otherwise go back to 5.

Ad 1: I will call every partial implementation of the general PS-scheme a schematic probabilistic syllogism (schematic PS). The first move results in a schematic PS which materialises the strong restriction imposed by the explanandum. Ad 4: The first part of this instruction means that the inquirer has to execute as many constructive moves as possible. Ad 5: Questions to be asked to Nature, questions that express a decision problem and questions that express a derivation goal are respectively questions which the inquirer intends to answer by means of an interrogative move, a decisive move and a derivative move. Option (a) leads to an interrogative move, option (b) to a derivative move and option (c) to a decisive move. The result of option (d) will be called an adjustive move. It is possible that in the course of the construction process the inquirer ascertains that some part of his background knowledge, which he considered irrelevant at the beginning and therefore was not included in B in step 3, is needed to construct the explanation. So we have to admit that I adjusts the content of B. Ad 6: This instruction leads to combinative moves.

EXPLAINING, UNDERSTANDING AND SCIENTIFIC THEORIES 1 1

Ad 7: If an inquirer executes (P) he either obtains a probabilistic syllogism or has to interrupt the procedure by violating the second part of instruction 4. If he obtains a syllogism, the singular sentences usually have the appropriate epistemic statuses, because they belong to the background knowledge of the inquirer or are obtained in interrogative moves. If we encounter an exception, instruction 7 prescribes to go back to instruction 5 in order to obtain the knowledge that is required to give the elements the appropriate statuses.

If an inquirer executes (P) without interrupting it (by violating the second part of instruction 4 or 7), he can easily compose a SE explanation for his explanandum. The explanation consists of the probabilistic syllogism he has obtained, a description of the explanandum (SE), and a record of the derivative move by means of which the probability statement of this syllogism was obtained. Consequently, (P) is a procedure for constructing SE explanations.

(P) is the core of my account of the construction process of SE explanations, not the whole story. According to the method I propose, procedure (P) must be applied in combination with three units of background information. This background information is situated on a meta-level: it tells us how scientific entities have to be applied. It must not be confused with the background knowledge B. The first unit contains rules for selecting the scientific entity T on which the explanation will based (cfr. instruction 2 of the procedure). The second unit contains information which helps the inquirer to formulate useful derivation goals, useful decision problems and relevant questions to be asked to Nature (cfr. instruction 5). The third unit contains information which helps the inquirer to construct adequate derivations (cfr. option (c) in instruction 5). In the subsequent sections I discuss the content of the three units. I start with a definition of qualitative theories (Section 4). In Section 5 I clarify the content of the first unit by focusing on the part that relates to qualitative theories. The first unit is a universal one: whenever an inquirer executes (P) the content of the first unit of background information is the same. The content of the second and third unit of background information is not universal. The content of the second unit depends on the type of scientific entity that is chosen in step 2. In Section 6 1 clarify the content of the second unit if the inquirer tries to construct a SE explanation on the basis of a qualitative theory. The content of the third unit can type-specific (like the content of the second one) or entity-specific. It will be discussed in Section 7.

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4. QUALITATIVE THEORIES

In Section 4.1 I develop my definition of qualitative theories, while 4.2 contains an example. In Section 4.3 I reconsider condition 6 of (SEE), as was announced in Section 2.

4.1.

A family is a series of properties which are defined in such way that an object cannot possess more than one of these properties at the same time. When D is a set of objects, the family G1. . .Gn is called characteristic for D if and only if each element of D at each time necessarily possesses one of the properties of the family. A family is called experimental if there exists an experiment by means of which it can be determined which of the component properties an object possesses. I will use these definitions to introduce the concepts of potential qualitative theory, probability model and, finally, qualitative theory.

A potential qualitative theory consists of (i) a set of theoretical principles, (ii) a characterisation of an underlying logic, and (iii) a specification of an intended domain, consequent family and antecedent family. The theoretical principles consist of two subsets: fundamental laws and auxiliary hypotheses. This distinction will be clarified in the example below. Theoretical principles and auxiliary hypotheses must not be regarded as empirical statements: they are recipes for deriving probability statements. The underlying logic of a theory may be probability calculus, predicate logic, differential calculus, etc. A characterisation consists in a set of axioms and/or primitive rules of inference. The intended domain of a theory is a set of objects. The intended consequent family must be an experimental family which is characteristic for the intended domain. The antecedent family must be non-experimental and also characteristic for the intended domain. Potential qualitative theories will be formally represented by an expression of the form (P~, L~, D~, G1. . .Gn, F1. . .F~}, where P~ is the set of theoretical principles, L~ the underlying logic, D~ the intended domain, G1.. .G~ the intended consequent family and F1...Fro the intended antecedent family.

Probability models are defined as follows:

(PM) Consider a domain of objects D and two families characteristic of D, F1 . . .F~ and G1...G~. A series of m • n probability statements


PD(G1 ] F1) = Pll PD(G2 ]F1) = P21 . . . PD(Gn2 ] F1) = P~I

FD(G1 I Fra) m Plra PD(G2 ] Fra) = P2ra "'" r D ( G n [ Fra) = Pnra

is called a probability model with domain D, consequent family Ga...Gn and antecedent family F1...F,~ if and only if for every Fi holds: ~ j Pji = 1

A potential qualitative theory (P~, L~, D~, G1.. .G~, F1...F,~ ), is a qualitative theory if and only if from L~ and P~ a probability model with domain D~, consequent family G1.. .G~ and antecedent family/ ;1. . .F,~ can be derived. We say that a probability model is derivable from L~ and P~ if and only if for each statement of the model there exists a derivation (in the sense defined in Section 2.1) of it in which (i) the premises are members of P~, (ii) the inference rules that are invoked belong to L~, and (iii) the derivability claims made in the classification are correct.

Each qualitative theory specifies a probability model; it is in this sense that the fundamental laws and auxiliary hypotheses (P~) are recipes for deriving probability statements. Probability models must not be interpreted realistically: they are descriptions of the structure of idealized systems. What we claim by putting forward a qualitative theory with domain D, is that certain real systems (viz. finite subsets of D) have a structure (viz. the occurrence of certain relative frequencies) which is similar to the structure described in the probability model. So if the antecedent and consequent family are respectively F Fm and G Gn the theory claims that in finite subsets of D (which constitute real systems) the properties Fl...Fm and G Gn are so distributed that the real relative frequencies approximate the values P Pmll we find in the probability model which is derivable from P and La

4.2.

To illustrate definition (QLT), I will discuss a theory which I call the Mendelian theory of the human ABO blood group system. The domain of this theory is the set H of all human beings. The phenotypes of the ABO blood group system (the blood groups A, B, AB and O) constitute the consequent family. Each member of H has exactly one of these phenotypes, so they constitute a family which is characteristic of H. Since the blood group of an individual can be determined by means of a serological test, this family is experimental. The formal representation of the blood groups will be BA, BB, BAB and B

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In order to construct the antecedent family we distinguish three genes (IA, IB and I ). We assume that in each somatic cell of each human being, there is a pair of chromosomes of which each element contains exactly one of these genes. This assumption entails that each member of H possesses exactly one of the following genotypes: IAIA, IAIB, IAI, IBIB, IBI and I I . It also implies that 21 types of crosses can be distinguished and that each member of H belongs to exactly one of these types. Examples of these types are IAIAxIAIA and IAIBXIBI. The individuals belonging to category [AIAxlAIA are those who descend from parents who both have genotype IAIA. The individuals belonging tot category IAIBxlBI descend from a parent with genotype IAIB and a parent with genotype [BI~ The types of crosses constitute the antecedent family of the theory, and will be formally represented by C1...C21.

The underlying logic of the theory is probability calculus. The fundamental laws of the theory are a reformulation of Mendel's principle of segregation:

(1) For any individual w and for any alleles X and Y, if w has X Y, the probability that a particular one of w's offspring will inherit X is 0.5.

(2) For any individual w and for any allele X, if w has X X , the probability that a particular one of w's offspring will inherit X is 1.

The theory contains seven auxiliary hypotheses. The first is that each member of H has exactly one of the genotypes IA1 A, fAIB, [A[O, 1BIB, [B[O, and [~176 The other auxiliary hypotheses relate genotypes with blood groups: all humans with genotype 1A[ A have blood group A; all humans with genotype IA1 B have blood group AB; all humans with genotype IAI ~ have blood group A; all humans with genotype 1B[ B have blood group B; all humans with genotype IB1 ~ have blood group B; finally, all humans with genotype [~176 have blood group O. The cri- teflon for distinguishing fundamental laws from auxiliary hypotheses is that, while the former occur in a large number of theories belonging to the family of theories called "Mendelian genetics", the latter are typical of the Mendelian theory of the ABO blood group system. Note that the first auxiliary hypothesis is a statement about unobservables, while the others link unobservable properties to observable ones. So auxiliary hypotheses are not always bridge principles linking unobservable to observable properties.

The Mendelian theory of the human ABO blood group system sketched in the preceding paragraphs can be formally represented as (PABo, LABO,


tI, BA.. .Bo, C1...C21). By putting forward this theory, we claim that, in finite sets of human beings, the ABO blood groups and 21 types of cross are distributed so that the real relative frequencies approximate the 84 values we can derive from the fundamental laws and auxiliary hypotheses by means of probability calculus.

In every theory, the antecedent family is non-experimental. Therefore, a conventional decision method is needed to determine which of the properties F1. . .F~ an element of D possesses. Without such decision method, the theory does not lead to any predictions, and cannot be tested or put to use. The decision method is conventional because it cannot be tested separately: only by combining a theory and a decision method can we make predictions. The decision method of the theory in our example is pedigree analysis. To determine the type of cross to which an individual il belongs, we have to determine the genotype of i2 and i3 (the parents of il). To determine these genotypes, we determine their phenotype and the phenotype of their parents. In most cases, the genotype can be derived from these data, the fundamental laws and the auxiliary hypotheses. If necessary, additional phenotypical data (of other relatives of i2 or i3) are collected.

4.3.

As I announced in Section 2.2, I will now weaken condition 6 of (SEE). The requirement that $2 must be empirically founded is too restrictive. Either condition 6 or the following condition must be fulfilled in order to have an explanation:

The premises of the derivation are members of the P~ of a qualitative theory; X has applied the decision method associat- ed with this theory and has ascertained that it leads to what is described in the singular sentence $2.

5. BACKGROUND INFORMATION: THE FIRST UNIT

For an inquirer who executes (P), the first problem is the choice of T. My definition of SE explanations entails that the entity T must belong to the inquirer's world-picture. This restriction leads to a first selection rule, which was already incorporated in instruction 2 of the procedure. Further selection rules derive from conditions of relevance. For each type of scientific entity, a set of conditions of relevance can be formulated. Each of these conditions specifies a characteristic an entity must have in order to be useful for constructing the explanation that is sought. The sets

16 ERIK WEBER

of conditions of relevance have common elements, but in general are not identical. The aim of this section to explicate the conditions of relevance for qualitative theories. I assume that my detailed discussion of qualitative theories will also give the reader an idea of what the conditions of relevance for other types of scientific entities might look like. If our world-picture contains more than one entity that satisfies the conditions of relevance, additional selection criteria are needed. I will not discuss them here.

From any qualitative theory, a number of probability statements are derivable. The probability statements are of various types. To illustrate this, we consider an arbitrary qualitative theory. First of all, the ra • n statements of the corresponding probability model will be derivable from this theory. These statements are of the type PD(Gj I Fi) = r. We can also derive m • n statements of the form PD&Fi (Gj) = r. Each statement of the second type is logically equivalent to a member of the probability model. A third type of probability statements are those of the form PD(Gj or Gj, I El) = r , where Gj and Gy are different members of the consequent family. A fourth type is PD(Gj I Fi or Fi, ) = r (i r i'). This classification is not exhaustive: combinations of these types and types similar to the third and fourth with more than two elements of the consequent or antecedent family, are possible.

Though the probability statements that are derivable from a qualitative theory belong to different types, three conditions of derivability can be formulated:

(1) From a qualitative theory T, we can only derive probability statements in which the object class is an element of the consequent family of T or a disjunction of such elements.

(2) From a qualitative theory T, we can only derive probability statements in which the domain or the reference class is a (proper or non-proper) subset of the domain of T.

(3) If the domain of a probability statement H is equal to the domain of a qualitative theory T, then H is derivable from T only if its reference class is an element of the antecedent family of T, or a disjunction of such elements.

Condition 3 will be used in Section 6. The two first conditions of derivability imply the following conditions of relevance for qualitative theories:


(1")

(2*)

If the explanandum is G(a, t), only qualitative theories for which G is an element of the consequent family of the theory, or a disjunction of such elements, are relevant (in the sense that they enable us to derive a probability statement which can be part of a SE explanation of G(a, t)). If the explanandum is G(a, t), only qualitative theories ~vith a domain that contains a are relevant.

That (1 *) follows from (1) is obvious. (2") follows from (2) in the following way: if we are convinced that (a, t) does not belong to the domain of the theory, then for every pair of predicates {a, 7} such that c~ or 7 is a subset of the domain of the theory, we will be convinced that a(a , t) or 7(a, t) is false.

The conditions of relevance for scientific entities of other types are similar to the ones described above: they ensure that the property which is to be explained is among the properties that can occur as object class in probability statements that are derivable from the entity, and they impose restrictions on the domain of the entity.

6. ASKING USEFUL QUESTIONS

The second unit of background information contains knowledge which helps the inquirer to formulate useful questions. The first unit was a universal one: each time we apply (P), the same body of information, containing the various sets of conditions of relevance, is used. The content of the second unit is type-specific: it depends on the type of entity the inquirer chooses in step 2 of the construction process. In Sections 6.1 and 6.2 I clarify what the second unit consists in if the inquirer chooses a qualitative theory: Section 6.1 contains an example, 6.2 the conclusions that can be drawn from it.

6.1.

Imagine an inquirer I who observes that a person a has blood group A at time t. ! wants a SE explanation of this fact and decides to use procedure (P). Then his first move will be:

(1) In the general PS-scheme, I replaces/3 with "has blood group A ~ .

This move results in the following schematic PS:

18 ERIK WEBER

BA is the formal representation of "has a blood group A". We assume that I takes the following decisions:

(2) I decides to construct a SE explanation by means of the theory

( PABO, L ABO, H, B A. . .Bo, C1. . .C21)

(3) I decides that the content of B is "a is at t a human being".

As property BA is an element of the consequent family and a an element of the domain, the theory which i has chosen satisfies the conditions of relevance. Probabilistic syllogisms as defined in 2.1 contain two singular sentences because we want to distinguish between new explanatory facts and explanatory facts that belong to the background knowledge of the inquirer. Because of the convention that 7 represents relevant facts which the inquirer already knows at the moment he starts constructing the explanation, the decision in step 3 creates a strong restriction: 7 must be replaced with "is a human being". In combination with the third condition of derivability (see Section 5) this first restriction entails a second, weak restriction: ~ must be replaced with a predicate referring to one of the 21 types of crosses, or to a disjunction of two or more of these types. The construction process will go on as follows:

(4) The inquirer materialises the strong restriction and thus obtains the following schematic PS:

H(a, t ) a(a,t) PH(BA I a) : r

In his fifth move I must formulate a question. A reasonable thing for I to do is trying to create an additional restriction which, in combination with the weak restriction mentioned above, entails a strong restriction. I can obtain such restriction by choosing one of the available explanatory families and trying to determine to which element of the selected family the object a belongs. The set of explanatory families consists of the antecedent family of the theory and all families characteristic of the domain of the theory that can be formed by means of disjunctions of elements of the antecedent family. So a rational way to continue the construction process is:

(5) I formulates a question which expresses a decision problem:

Q l: Which explanatory family shall I use to construct a SE explanation of BA (a, t)?

We assume that [ decides to use the family C1...C21 (the antecedent family). On this condition, the process goes on as follows:

(6) No combinative moves can be made.


(4 t) No constructive moves can be made. As he has not obtained a probabilistic syllogism, the inquirer decides to go back to instruction 5.

(5Pa) I formulates a question to be asked to Nature:

Q2: To which element of the partition C1,.. . ,6'21 does the object a belong?

(5'b) I looks for an answer to Q2 by making observations and using the decision method of the theory.

When I has obtained an answer to Qa, he must try to make a combinative move. Let's suppose that Nature's answer to Q2 is that a belongs to the category I A I A • I A I B. On this condition, the construction process goes on as follows:

(6 t) I ascertains that, combined with the answer to Q1, the answer to Q2 entails a strong restriction: a = I A [ A • I A I B.

(4") I materialises the restriction obtained in step 6~; he obtains the following schematic PS:

H(a,t) a at t belongs to [AIA • I A I B

Px4(BA I I A I A • I A I B) = r

I has not obtained a probabilistic syllogism: r still is to be replaced with a specific value. He has to go back to instruction 5. Since the only problem left is the replacement of r, I should go on as follows:

(5"a) I formulates a question which expresses a derivation goal:

Q3: In the domain H , what is the long run relative frequency of object having blood group A in the set of objects belonging to cross [ A I A • IAIB?

(5"b) I derives an answer to Q3 from the Mendelian theory of the A B O blood group system.

(6") No combinative moves can be made.

(4 m) s ascertains that there is a strong restriction on the schematic PS obtained in step 6: r = 0.5. He materialises this restriction and obtains a probabilistic syllogism:

H(a,t) a at t belongs to I A I A • IA1 B

PH(BA [ I A I A • [AIB) = 0.5

20 ERIK WEBER

(7) I checks whether the singular sentences of this probabilistic syllogism have the appropriate epistemic status.

As the elements of the probabilistic syllogism possess the appropriate epistemic statuses, I can compose a SE explanation for "a has blood group A". The explanation consists of the probabilistic syllogism obtained in step 4 I", the explanandum and a record of the derivation made in step 5"b.

6.2.

From our example we can derive five items of background information. They constitute the second unit if a qualitative theory is chosen as T. The items are:

(a) If D is the domain of the selected theory, B always contains the statement D(a, t); this is a consequence of the second condition of relevance the selected theory must satisfy. In combination with the convention that 7 will represent relevant facts which belongs to our background knowledge, we can conclude that 7 must be replaced with D or a subset of D. Infor- mation about which member of the antecedent family contains a is usually not in the background knowledge of the inquirer. As 7 must not contain information we gather during the construction process of the explanation, we usually obtain the strong restriction that 7 must be replaced with D. In combination with the third condition of derivability (see Section 5), this restriction entails a weak restriction: c~ must be replaced with a member of the antecedent family of the selected theory, or with a disjunction of such members.

(b) The first restriction can always be materialised. So we can always obtain a schematic PS of the following form:

D(a,t) a ( a , t ) P o ( C t a ) = r

(c) The second restriction is weak and has a heuristic value: an obvious way to limit the possibilities is to ask the question "Which of the available explanatory families shall I use to construct a SE explanation for G(a, t)?". This question expresses a decision problem.

(d) When we have made a decision in response to this question, a second question (to be asked to Nature) arises: "To which element of the selected explanatory family does (a, t) belong?". By answering this question we obtain an additional restriction which, together with the answer to the first question, entails a strong restriction.

(e) If we combine the answers to the questions in (c) and (d), we obtain a strong restriction: o~ = E (I assume that the answer of nature to the


second question is "(a, t) belongs to E"). By materialising this restriction we obtain a schematic PS which has the following form:

D(a,Q E(a,t) P D ( a l E ) = r

(f) The schematic PS we have obtained does not contain a probability statement. Because r is the only schematic letter left, the scheme has a heuristic value: it provokes the question "In the domain D, what is the long run relative frequency of object with property G in the set of object having property E?".

(g) If an answer can be derived to the question in (f), the probability statement has the appropriate epistemic status. Usually D(a, t) and E(a, t) respectively have the status "'empirically founded" and "theoreti- cally founded" (D(a, t) was in our background knowledge, while E(a, t) was obtained in an interrogative move). This means that we can compose a SE explanation.

Similar items of background knowledge, which help the inquirer to exploit the heuristic value of the schematic probabilistic syllogisms and the restrictions, may be formulated for other types of scientific entities.

7. MAKING DERIVATIONS FROM QUALITATIVE THEORIES

The third unit of background information contains knowledge that helps the inquirer to make adequate derivations. I will first clarify what the third unit may consist in if we apply the Mendelian theory of the ABO blood group system. Then I will give a general characterisation of the content of the third unit.

The Mendelian theory of the AB�9 blood group system and the theory used in the Tay-Sachs example belong to a family of theories, Mendelian genetics. Some scientists possess knowledge (on a meta-level) on how theories belonging to this family are to be used to construct derivations. What does this knowledge consist in? In the derivation in the Tay-Sachs example in Section 2.1 I used the statement "For all crosses of kind X X • XY: P ( X X I X X • X Y ) = 0.5 and P ( X Y [ X X • XV) = 0.5." This statement was derived from the fundamental laws of the theory. In a derivation of PH(B n I IAIA X Ia I B) = 0.5 (the probability statement of my example in Section 6) from the theory of the ABO blood group system, the same statement must be used: we need it to derive that PH(IA[ A ] [AIA • IAI B) = 0.5 and PH(IAI B [ IAI A • IAI B) = 0.5. The reason why the same statement is needed is that the same general kind of mating (viz. X X x XY) is involved. All matings between two individuals can be

22 ERIK WEBER

classified into seven general kinds: X X x X X , X X • Y Y , X X • X Y , X X • Y Z, X Y • X Y , X Y • X Z and X Y • W Z. In crosses of the first general kind, two identical homozygous individuals are mated. In crosses of the second general kind, two different homozygous individuals are mated. In the third and fourth kind, a homozygous individual is respectively mated to a heterozygote who carries the gene of the homozygous individual and a heterozygote who does not carry this gene. In the three last kinds, a heterozygous individual is respectively mated to an individual with the same genotype, a heterozygous individual who has one gene in common with the first individual, and a heterozygous individual who has no genes in common. Crosses of last kind require four or more possible genes. Crosses of kind X X • Y Z and X Y • X Z require three possible genes (as e.g., in the A B O blood group system). The four other kinds require only two genes (as e.g., in our Tay-Sachs example). From the fundamental laws of the theory, we can derive seven statements of the type mentioned above, one for each kind of cross. For instance, we can derive "For all crosses of kind X X x X X : P ( X X ] X X • X X ) = 1". In order to make a derivation from a theory belonging the family of Mendelian genetics, we need exactly one of these seven statements.

We now can see what the third unit of background information may consist in with respect to the theory of the A B O blood group system and other theories of the same family. Some inquirers will know the seven statements mentioned above. They also know that they need exactly one of them to make a derivation. Other inquirers will not know these statements before they start constructing an explanation. They probably will not know what to do first when they want to make derivative moves.

In general, the third unit of background information contains knowledge about the structure of the derivations, e.g., about the kind of intermediary results that are to be obtained. This knowledge tells us how to proceed in derivative moves. What distinguishes experienced inquirers (in a certain field or with respect to a specific theory) from unexperienced ones is the content of the third unit of background information: for an unexperienced inquirer it will be (almost) empty.

8. CONCLUSIONS

In Section 2 I have introduced the concept of scientific epistemic explanation. In this concept, I have incorporated the main ideas of Kitcher and Railton, while eliminating some obvious drawbacks of their models. By defining SE explanations, a part of our first task was completed: the type of scientific explanation which is the appropriate instrument for increas-


ing our epistemic understanding of the qualitative particular events we have observed, has been characterized. As already mentioned, quantitative events require a more complex type of explanation.

In Sections 3-7 1 have developed a method for constructing SE explanations. My analysis in these sections is not sufficient to complete our second task (providing insight into the ways in which SE explanations are constructed). In the same way as I have done in Sections 6 and 7, the content of the second and third unit of background information has to be determined for every type of scientific entity that can be used to construct a SE explanation. With respect to the first unit, the other sets of conditions of relevance have to be determined.

My pluralistic view on the nature of understanding implies that tasks analogous to the ones considered here, but relating to other aspects of understanding (e.g., causal understanding) or other types of phenomena (e.g., laws or regular patterns) are equally important. The method I have developed can function as a paradigm for analyses of the construction process of other types of scientific explanation. For each type of scientific explanation, there exists a characteristic general scheme. The construction process of all scientific explanations, not only SE explanations, may be seen as a process in which the corresponding general scheme is gradually implemented by creating restrictions and materialising them.

REFERENCES

Hardin, G.: 1966, Biology. Its Principles andlmplications (Second Edition), W.H. Freedman & Co, San Francisco and London.

Kitcher, P.: 1981, 'Explanatory Unification', Philosophy of Science 48, 507-531. Kitcher, P.: 1989, 'Explanatory Unification and the Causal Structure of the World', in P.

Kitcher and W. Salmon, (eds.), Scientific Explanation, University of Minnesota Press, Minneapolis, pp. 283-306.

Railton, P.: 1981, 'Probability, Explanation and Information', Synthese 48,233-256. Salmon, W.: 1993, 'The Value of Scientific Understanding', Philosophica 51, 9-19. Stansfield, W.: 1983, Theory and Problems of Genetics (Second Edition), McGraw-Hill

Book Company, New York,

Manuscript submitted December 27, 1994 Final version received July 10, 1995

Postdoctoral Fellow of the Belgian NationaI Fund for Scientific Research Universiteit Gent Vakgroep Wijsbegeerte -Moraalwetensch appen Rozier 44 B-9000 Gent Belgium

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Explaining, understanding and scientific theories