Inductivism is the doctrine that scientific theories are obtained by inductive inference from experience, i.e. inductive generalization is a method of both discovering and justifying theories. A different conception of inference was imposed in the nineteenth century by the introduction of explanatory theories that go beyond the surface layers of phenomena: Dalton's theory of atoms, Young's wave theory of light, Maxwell's
electromagnetic field theory, Boltzmann's statistical mechanics, Schleiden's cell theory, Darwin's theory of evolution (see Laudan 1981b). The hypothetico-deductive or HD model, as masterfully formulated by William Whewell (1840), takes theories to be bold and ‘happy guesses’ or hypotheses which are tested by checking the truth of their observable predictions (cf. Popper 1959; Hempel 1965; Niiniluoto 1984). A good theoretical hypothesis should explain the old evidence and entail novel predictions.
Induction is then the end p.174
‘converse’ of deduction: successful explanation and prediction confirms or corroborates the hypothesis, while a negative test result refutes the theory by modus tollens (see Fig.
21).
Even though the HD model has to be complemented in many directions—it does not say enough about scientific discovery (cf. Whewell 1840; Kleiner 1993), confirmation, and refutation (cf. Lakatos and Musgrave 1970)—it still gives us a good basic framework for discussing rival philosophical views about the problem of theory-choice.
The empiricists typically formulate the problem of theory-choice by assuming that the available total evidence e includes a finite number of actual observations and
measurements. Then it is clear that hypothetical laws and theories in science transcend evidence at least in three ways. First, they are universal or statistical generalizations about potentially infinite populations (hence, Hume's problem of induction). Secondly, they are lawlike rather than accidental generalizations (hence, the problem of
counterfactuality). Thirdly, they contain theoretical terms not reducible to the observational language (hence, the problem of theoreticity).
In historical sciences, such as cosmology, geology, zoology, palaeontology, and most of the humanities, the hypotheses typically concern some singular event in the past. Such hypotheses also transcend the evidence, consisting of the present traces and causal influences of the past, but they can be indirectly tested by the HD model.
Fig. 21. The Hypothetico-Deductive Model Of Science
Thus, in normal cases there will be a great number of different rival theories, which nevertheless are empirically equivalent. This formulation seems to imply, as Quine (1960) has claimed, that theory-choice in science is underdetermined by data.15
There are many objections to this simple empiricist picture of theorychoice. Let us say that two theories T and T′ are empirically equivalent if and only if EC(T)=EC(T′). This guarantees, assuming a neutral observational language L
To close the gap between empirical evidence and theories, Quine argues that theory-choice has to appeal to the criterion of simplicity. Many sociologists of science take advantage of this thesis and suggest that the choice of theories has to be based on or even appeal to extra-scientific social factors (cf. Ch. 9).
0 for T and T′, that theories T and T′ have the same deductive connections to empirical statements. As Laudan (1990a) argues, this does not imply that they are equally well confirmed by the evidence e in L 0
In the same way, the testability of a theory postulating theoretical entities requires that the truth of this theory should make a difference, at least with some probability, to some observable phenomena. Testing theories may thus be ‘hypothetico-inductive’ rather than hypothetico-deductive (cf. Niiniluoto and Tuomela 1973).
This is correct, for the reason that confirmation depends essentially on the probabilistic . relations between theory and evidence. This is seen in a striking way in the case of
statistical hypotheses: they usually do not deductively entail any empirical statements, but give them conditional probabilities (e.g. a normal distribution of errors of measurement for a given unknown quantity). Nevertheless, there are cognitive—even empirical—
criteria for making rational choices between such ‘empirically equivalent’ hypotheses, if they give different probabilities to different potential observations.
It should be emphasized that the scientists need not believe or accept one of the available hypotheses in each situation. Among the options of a cognitive problem there is always the suspension of judgement—technically speaking, this amounts to the acceptance of the disjunction of all rival hypotheses as the strongest conclusion warranted by the evidence
(Levi 1967). In other words, if the evidence is not strong enough, the scientists may suspend judgement and look for more evidence, instead of adopting extra-scientific criteria of acceptance.
Conflicting theories, if interpreted realistically, cannot be true at the same time. But fortunately we do not have to consider at the same time
end p.176
all logically possible theories; our choice is limited to the relevant ones which are able to solve the initial problem of explanation. For some problems it is difficult to find even one satisfactory theory. Moreover, there may be situations where all the relevant rival theories that are not yet excluded by the problem situation and empirical evidence are truthlike. In this case, the underdetermination argument would fail to block the tentative inference from data to a truthlike theory (cf. Niiniluoto 1989; MacIntosh 1994). But it is not clear that this would be always the case: for example, Cushing (1994) considers the possibility that the standard Copen-hagen interpretation of quantum mechanics and Bohm's non-local hidden variable theory of quantum mechanics, which have entirely different ontologies, are empirically equivalent.
Perhaps the strongest counter-argument is the following: the under-determination argument presupposes that there is a clear, context-independent distinction between empirical and non-empirical. This is questioned by the thesis of the theory-ladenness of observations. What counts as observational evidence depends on the available
instruments and on the accepted theoretical background assumptions. There is no unique and fixed ‘upper limit’ of the observable (cf. Newton-Smith 1981; Churchland and Hooker 1985; Laudan 1990a).
Indeed, if we do not endorse naive empiricism, then we have to acknowledge that rival hypotheses in science are usually evaluated relative to some background assumptions which may include tentatively accepted theories. Thus, even when T1 and T2 are
empirically equivalent as such, their relations to the empirical evidence may be different in the light of the background assumptions.
The important insight that theory-choice takes place in a context of background
assumptions was expressed by the concepts of paradigm and disciplinary matrix (Kuhn), research programme (Lakatos), and research tradition (Laudan). The grounds for
choosing between theories may include, besides empirical evidence, background theories, regularity assumptions, conceptual frameworks, exemplars from earlier research, and axiological principles.
The underdetermination argument can of course be repeated on the level of research programmes. This is what Kuhn did in his claim that the choice between paradigms is not dictated by ‘logic and experiment’, but involves persuasive argumentation, conversions, and gestalt switches.16
end p.177
Lakatos's interesting suggestion was that it is possible rationally to appraise research programmes by their rate of progress (cf. Laudan 1977). Such programmes are like vehicles of transportation that carry the scientists forward in their competition for new
results. Even though there is always the possibility that a programme is only temporarily running out of steam, and will be recovered by new improvements, most scientists are opportunists who jump to the most rapidly advancing research programme. In the light of this metaphor, it is important that the concept of scientific progress can be understood in methodological terms appropriate to scientific realism (see Section 6.6).
We are now back in the axiological question of Section 6.2: what are the aims or desiderata of scientific enquiry? Kuhn's (1977) own favourite list includes accuracy, consistency, scope, simplicity, and fruitfulness. Even though truth is not included here, it might be the case that such desiderata in fact have interconnections with each other, and there may be hidden links with truth after all (cf. Section 6.4). For example, accuracy and scope seem to be indicators of the two main ingredients of the concept of truth-likeness:
truth and content. A good strategy for methodology would be to study methodological rules which conditionally assume these desiderata as the aims of science. Here the
optimization model of theory-choice, illustrated in Sections 6.1 and 6.4 with the concepts of truth, information, and systematic power, turns out to be a very flexible conceptual tool.
For example, it can be shown that the realist effort in truth-seeking is compatible with the idea that, in the context of applied research (cf. Niiniluoto 1993), the choice between hypotheses may be influenced by practical, extra-scientific interests. In other words, the decision-theoretical framework is able to conceptualize situations where both epistemic and practical utilities are involved and interact. This can happen in cases where some predictive model is intended to constitute the basis of some action or policy. A good example is given by Helen Longino (1989). Linear and quadratic models have been proposed for measuring the health risks of radiation. The loss (negative utility) of a mistaken model could be equated with its distance from truth, if the problem is purely theoretical and belongs to basic research. However, if the safety standards are adopted by implementing the model in practice for the public and the workers in nuclear facilities, then it is safer to overestimate the health risks than to underestimate them. Hence, the practical interest of protecting people from radiation justifies a loss function that gives higher penalties for too low risk estimates. This kind of case is especially interesting, since the same loss function can reflect the ‘pure’ cognitive value (i.e. distance from the truth) and weight it with a pragmatic human interest.
end p.178
The Quinean conclusion that simplicity has to be used as a criterion of theory-choice is not adequate, since it gives a simplified picture of scientific enquiry. But, of course, there are methodological situations where simplicity may play an important role. The classical question, ever since the days of Ptolemy (cf. Section 5.4), has been the relation of simplicity and truth.
Hans Reichenbach (1938) made a distinction between descriptive and inductive simplicity (cf. Niiniluoto 1994e). The former concerns choice between theories which are ‘logically equivalent, i.e. correspond in all observable facts’ (p. 374), the latter theories ‘which are equivalent in respect to all observed facts, but which are not equivalent in respect to predictions’ (p. 376). According to Reichenbach, preference for descriptive simplicity is a matter of convenience, while inductive simplicity indicates higher probability and truth.
Reichenbach's concept of equivalence is ambiguous. It may mean logical equivalence (including intertranslatable theories like matrix and wave formulations of quantum mechanics) and empirical equivalence (in the sense defined above). This gives us two formulations of the Principle of Descriptive Simplicity:
• (DS 1
•
)
• (DS 2
• )
As logical equivalence entails empirical equivalence, DS 2 entails DS 1
The first of these principles, DS
, but not vice versa.
1 , is relatively uncontroversial as a practical rule. If two theories are known to be equivalent, there cannot be any purely cognitive differences between them: they make the same claims about reality, they have the same truth value, they are equally accurate relative to observations, and equally worthy of rational belief or acceptance as true. However, greater simplicity may amount to an enormous advantage in cognitive fruitfulness and practical economy: a simple formulation of a theory is often easy to work with, to employ in the quest of further knowledge, or to apply for practical purposes (e.g. in calculation of predictions, teaching of students, programming of computers, etc.). Hence, principle DS 1
On the other hand, the stronger principle DS
is a good rule for choosing among the equivalent formulations of our theory that one which is most convenient for our current purposes.
2
end p.179
is problematic, since two empirically equivalent theories may have non-equivalent theoretical parts.
If we accept theoretical realism, which allows that theoretical terms are interpreted and theoretical statements have a truth value, then two empirically equivalent theories may have different truth values. Thus, DS 2
For example, Heinrich Hertz required in Die Principien der Mechanik (1894) that a physical theory should be admissible (i.e. logically consistent), correct (i.e. agree with all phenomena), and appropriate (i.e. distinct and simple). (See Hertz 1956.) To explain these notions, let us form an equivalence class of consistent theories which are empirically equivalent to each other (e.g. all theories that have the same empirical
consequences as Maxwell's equations). Then a member of this class is distinct, if it makes explicit the ‘essential relations’ of its objects; it is simple, if it does not contain any
‘superfluous or empty relations’. This amounts to a clear formulation of DS expresses the idea that theoretical truth is
irrelevant—or at least less important than simplicity. This view is a traditional element of empiricist instrumentalism.
2
The step from DS
. According to Mach, Hertz's criterion of appropriateness ‘coincides’ with his criterion of the
economy of thought (Mach 1960: 318).
1 to DS 2 could be justified by the Thesis of Translatability, advocated by Logical Positivism. This ‘dogma of empiricism’ (Quine 1953) claims that all terms and statements in science can be reduced to the observational language by means of explicit definitions (cf. Section 5.1). Therefore, it would imply that equivalence and
empirical equivalence coincide. But I do not see that Reichenbach could accept such a defence of DS 2 , given his early rejection of Carnap's phenomenalism. In any case, it should not be accepted by scientific realists. Hence, the only legitimate applications of descriptive simplicity concern cases covered by DS 1
The idea of inductive simplicity also has many different interpretations. First, it may concern what Reichenbach called the context of discovery, or what others have called the pursuit of a theory: a simple hypothesis is chosen as the object of further enquiry, testing, and elaboration. The principle
, where the compared theories are fully equivalent, not merely empirically equivalent.
• (IS 1
• )
can be defended in terms of economy, projection ‘along the lines of least resistance’, effective search strategy, quick testability, and ‘rapid strategy of progress’ (Rescher 1990). Indeed, Rescher extends this ‘simpler-models-first-heuristics’ for the whole context of scientific enquiry.
A simple theory, he says, ‘lightens the burden of cognitive effort’. If a simple solution accommodates the data at hand, ‘there is . . . no good reason for turning elsewhere’. We
‘opt for simplicity’ not because it is ‘truth-indicative’, but because it is ‘teleologically cost-effective for the most effective realization of the goals of inquiry’ (ibid. 3–5).
The economic defence of simplicity may be justifiable in terms of the decision-making strategy that Herbert Simon has called satisficing. (Giere (1988) has argued that scientists in fact are satisficers, rather than optimizers.) Given some criteria, fix a minimum level that a satisfactory or ‘good enough’ solution has to reach. Then—instead of trying to find the optimal solution among all possible solutions—you should accept the first minimally satisfactory solution that you hit upon. In the context of science, this means that we cannot and need not always try to generate and consider all possible hypotheses, but rather we accept for pursuit the first satisfactory hypothesis we are able to find. This is usually the simplest solution to our problem. It seems to me that satisficing can be viewed as a special case of optimization, where waste of time and money is included as a loss in the decision problem (Niiniluoto 1994e).
If the preference for simplicity is applied in the cases where a theory is accepted for practical purposes, e.g. for prediction or action, the following special case is obtained:
• (IS 2
• )
But this is clearly problematic, as Shrader-Frechette (1990) has illustrated with examples of applied science in public policy. Preference for ontological simplicity, she points out, may lead to ‘dangerous consequences’. Thus, a rule such as IS 2 is unsatisfactory, since it does not take into account the losses and risks of alternative choices (as the Bayesian decision theory does), and it does not allow for the possibility of withholding the decision and searching for better solutions or more data.
On the other hand, simplicity as manageability or applicability has a role in applied science. If a theory is defined by equations which cannot be solved for the relevant case, it is a normal practice to introduce some ‘simplifications’ or ‘approximations’ in the theory—even when we know that this will lead us away from truth (cf. Section 5.3). If we have to choose between a calculated prediction or no prediction at all, simplicity may be favoured even at the expense of accuracy and truth (Niiniluoto 1984: 262).
Reichenbach's own interpretation of inductive simplicity—unlike the economic formulations IS 1 and IS 2
end p.181 —applies to the context of cognitive justification:
• (IS 3
• )
In Reichenbach's view, the simplest theory is the most probable candidate for truth, and it will also ‘furnish the best predictions’.
Assuming that the simplicity S(h) of a hypothesis h, or its complexity K(h), can be measured, it could be included as an additional factor in the formula of expected utility U(h/e) of h given evidence e (cf. Levi 1967). It might be used also as a secondary factor which distinguishes hypotheses with the same expected utility U(h/e). But in these cases it is not clear that the chosen hypothesis would generally be the ‘best candidate for truth’.
However, appeal to simplicity as an additional factor in theory-choice may be justified in basic science at least in those cases where the form of a quantitative law can be derived from already accepted theories: in such cases the improved accuracy of a complex function does not ‘pay’, if the law cannot be incorporated within the established theoretical framework.
Eino Kaila defined in 1935 the relative simplicity of a theory h as the ratio between the multitude of empirical data derivable from e and the number of logically independent basis assumptions of h (see Kaila 1939; 1979). Thus, the relative simplicity RS(h, e) of h given e is defined by
• (19)
•
This measure, Kaila observed, is proportional to the ‘explanatory power’ of a theory. (He operated with a deductive or non-probabilistic notion of systematic power.) And if it could be exactly defined and measured, relative simplicity RS(h, e) would also define the
‘inductive probability’ of h on e.
An interesting feature of Kaila's concept of relative simplicity is that its application as a decision rule immediately justifies Reichenbach's Principle of Descriptive Simplicity, i.e.
DS 1 and DS 2
Kaila's definition is essentially equivalent to the concept of explanatory unification of Friedman (1974) and Kitcher (1989), and explanatory coherence of Thagard (1992).
These notions elaborate the idea that a good theory should contain powerful but simple . If theories h and h′ are equivalent or empirically equivalent, then (at least in Kaila's sense) syst(h, e)=syst(h′, e), and formula (19) recommends us to prefer h to h′ if and only if K(h)<K(h′) or S(h)>S(h′).
explanatory schemata. If logical strength were the only desideratum, it would be easy to satisfy by making the theory more and more complex—and eventually a logical
contradiction would entail every statement; therefore it seems natural to opt for strong but still simple theories.
Kaila's proposal is also related to Laudan's (1977) definition of scientific progress in terms of the difference between the empirical problem-solving capacity of a theory (cf.
syst(h, e)) minus the ‘conceptual problems’ of h (a end p.182
variant of K(h)). Where Kaila uses the ratio of two factors, Laudan employs their difference, but I do not see that this makes an important distinction between their approaches. (For a similar case, see (10) and (11).)
The concept of relative simplicity has further an interesting connection to measures of beauty in ‘algorithmic aesthetics’. Stiny and Gips (1978) propose that the aesthetic value of a text x (sequence, figure, picture, composition, etc.) is defined by L(x)/K(x), where L(x) is the length of x and K(x) is the Kolmogorov complexity of x. This is directly analogous to (19). However, this is not a good definition of beauty, since the aesthetic
The concept of relative simplicity has further an interesting connection to measures of beauty in ‘algorithmic aesthetics’. Stiny and Gips (1978) propose that the aesthetic value of a text x (sequence, figure, picture, composition, etc.) is defined by L(x)/K(x), where L(x) is the length of x and K(x) is the Kolmogorov complexity of x. This is directly analogous to (19). However, this is not a good definition of beauty, since the aesthetic