# Hempel – Laws and Their Role in Scientific Explanation

Carl Hempel – Laws and Their Role in Scientific Explanation

Carl Hempel – Laws and Their Role in Scientific Explanation1

5.1 Two basic requirements for scientific explanations

• The aim of the natural sciences is explanation insight rather than fact gathering. • Man’s concern for understanding is demonstrated by myths, metaphors, anthropomorphising, invocation of occult forces, God’s inscrutable plans or fate. • Such “answers”, while psychologically satisfying, are not scientifically adequate as they fall short of the two primary scientific requirements – explanatory relevance and testability. • Example of Francesco Sizi’s rejection of his contemporary Galileo’s discovery of the moons of Jupiter – a pseudo-logical argument for why there could only be seven heavenly bodies based on irrelevant “facts”, analogies and anthopocentric assumptions. • Contrast this with the scientific explanation of the rainbow. Even if we’d never seen one, the scientific account would be good grounds for their existence. This satisfies the criterion of explanatory relevance – that the phenomenon was to be expected under the relevant circumstances. • Relevance is necessary but not sufficient for explanation. We also need to know not just what happens, but why. • Empirical testability is our second requirement for explanation. Example of treating gravity as universal affinity analogous to love – no test implications, in contrast with the rainbow example. • Relevant explanations are testable, but not vice-versa.

5.2 Deductive-nomological explanation

• Example of a scientific explanation (the variability with altitude of the height of mercury in a Torricelli apparatus). What is explained depends both on general laws expressing uniform empirical connections and particular facts. The effects are as they are because of particular laws of nature applying to particular circumstances, and are therefore to be expected. • What is to be explained is the explanandum; the explanation is the explanans. • Example : image formation by reflection in a spherical mirror. Explanandum is 1/u + 1/v = 2/r. Explanans is based on rectilinear propagation of light, geometry of spheres and the basic laws of reflection, from which the explanandum is deduced. • The standard for a deductive-nomological (D-N) explanation is :-

Explanans L1, L2, … Ln, (Laws) C1, C2, …Cm (particular Circumstances) Explanandum E

• The laws invoked in scientific explanation are called covering laws and the explanation subsumes the explanandum under these laws. • The explanandum may be a phenomenon taking place at a particular place and time (the height of a mercury column), a general natural phenomenon (rainbows), uniformities expressed in an empirical law (Keppler’s laws). The explanans will

1 Chapter 5 of Philosophy of Natural Science [email protected] Page 1 of 6

Carl Hempel – Laws and Their Role in Scientific Explanation

include reference to laws of broader scope (Newton’s laws). Explanations of empirical laws rely on theoretical principles that make reference to structures and processes underlying the uniformities in question. • D-N explanations exhibit explanatory relevance in the strongest possible sense, offering logically conclusive grounds for occurrence of the explanandum. Testability is also satisfied as the explanans tells us under which conditions to expect the explanandum phenomenon. • The scientific explanations that follow the D-N model most closely are those using mathematical demonstrations from covering laws and initial conditions – as in the discovery of Neptune based on the expectations arising from anomalies (based on Newton’s laws) in the motions of Uranus that allowed the position and mass of the supposed perturbing body to be calculated. • Often, a D-N explanation omits to mention the covering laws, as in explaining the pavement’s remaining free of slush because of the application of salt. The explanans omits mention of the law that salt lowers the freezing point of water, and also conditions, such as that the temperature wasn’t so low as to make the salt ineffective. • Similar elliptical explanations of childbed fever. • General laws are presupposed when we invoke causation in the explanans. Same cause, same effect. Whenever an event of kind F (the cause) occurs, it is accompanied by an event of kind G (the effect). • The fact that an explanation relies on general laws doesn’t always mean that it depended on their discovery. The discovery may only be of a fact that relies on already-known laws to achieve the status of an explanans. Otherwise, both facts and laws may be known, and all that was required was the logical demonstration. • We cannot tell what kind of discovery is required from the problem itself. Irregularities in Mercury’s orbit did not succumb to the same sort of explanation as those of Uranus, as the proposed planet Vulcan was not discovered, but required a more radical explanation in terms of the new system of laws of general relativity.

5.3 Universal laws and accidental generalisations

• The laws, Li, provide the link by which particular circumstances, the Cj, explain a given event. Or, where the explanandum is itself a uniformity, they explain it as a special case of more comprehensive uniformities. • As distinct from laws of probabilistic form, to be discussed later, the laws employed by D-N explanation assert uniform, exceptionless connections given the specified conditions. • Various examples of statements of universal form (eg. gas laws). Most of the laws of natural science are quantitative, asserting specific mathematical relationships between different quantitative characteristics of physical systems. • We only talk of laws if we have evidence to assume their truth. However, this truth has to be within certain limitations of approximation or circumstance, or few of our laws would count as such. • Not all true statements of the form “whenever conditions of kind F pertain, those of kind G pertain as well” are laws. There are accidental generalisations, such as “all rocks in this box contain iron” or “all bodies of pure gold weigh less than 100,000 kilograms”. [email protected] Page 2 of 6

Carl Hempel – Laws and Their Role in Scientific Explanation

• So, being a true statement of universal form is a necessary but not sufficient condition for being a scientific law. So, what is the distinguishing feature ? • The important difference, noted by Nelson Goodman2 is that a law can support counterfactual conditional. We can say what would have happened if certain conditions had applied (but didn’t), whereas we can’t in the case of accidental generalisations (if we had put another rock in the box, this doesn’t imply that it would therefore have contained iron). Similarly, laws support subjective conditionals (“if A should happen, then so would B”) whereas accidental generalisations do not. • A closely-related difference is that laws provide explanations, whereas accidental generalisations do not. • Can the distinction be that laws refer to generalisations over a potentially infinite set, whereas accidental generalisations cover only finite sets (eg. {Rocki}) where the generalisation is short-hand for a finite conjunction (eg. Rock1 contains iron & Rock2 contains iron & …. ) ? Hempel thinks this is suggestive but inadequate, as the accidental set is not specifically enumerated and could even be infinite. • Additionally, a statement of universal form can be a law even if it applies to no instances (eg. if a body were of a certain mass, it’s gravitational field would be ..). • An accidental statement of universal form (eg. “all bodies of pure gold weigh less than 100,000 Kg”) cannot be used to make counterfactual or subjunctive conditionals (eg. “you can’t fuse two bodies of 60,000 Kg to form one of 120,000 Kg”). • What counts as a law depends in part on the physical theories of the time. This is not to say that we cannot have laws without theory (eg. Keppler’s laws were treated as such before Newton supplied the theory), but if the generalisation rules out certain occurrences that are allowed by present theory, as in the “gold” example above, it will not be treated as a law.

5.4 Probabilistic explanation : fundamentals

• Not all scientific explanations are in the form of universal laws; some, like exposure to a contagious disease being given as an explanation of why someone has a disease, are of probabilistic form and are known as probabilistic laws. • So, such a probabilistic law is “exposure to a contagious disease results in contagion with high probability”. Combined with the circumstance that the person in question was exposed to the particular disease, the law forms the explanans. Note that the explanans does not imply the explanandum with deductive certainty, as for D-N explanations, but only with high3 probability. • The form of the argument is therefore very similar to that of D-N explanations :-

Explanans L1, L2, … Ln, (probabilistic Laws) C1, C2, …Cm (particular Circumstances) Explanandum E

The double-line means “makes more or less probable”, as distinct from the single-line of deductive validity in the D-N schema.

2 Chapter 1 of Fact, Fiction and Forecast (The Problem of Counterfactual Conditionals) 3 Or, I would say, even with low probability ! [email protected] Page 3 of 6

Carl Hempel – Laws and Their Role in Scientific Explanation

• The criterion of explanatory relevance in the probabilistic explanation is met by the conclusion of the argument being a “practical certainty”.

5.5 Statistical probabilities and probabilistic laws

• The two differentiating features of probabilistic explanations (as against deductive-nomological) are the invocation of probabilistic laws and the probabilistic implication connecting explanans with explanandum. • Hempel describes the standard sampling with replacement of coloured balls from an urn as an example of a random process or experiment U, each drawing being one performance of U and the colour of the ball drawn the result or outcome of that performance. • Hempel now introduces the probabilities of Urn-drawing P(W,U) = 0.6, of coin- tossing P(H, C) = 0.5 and die-rolling P(1, D) = 1/6. • What do the probability statements mean ? The classical view is that they represent the ratio of the favourable to total possible outcomes of the experiment. • As it stands, this won’t do : all outcomes must be equipossible or equiprobable. • Define these terms is difficult but irrelevant, as (in the case of a loaded die, or the decay of radioactive atoms) we can still assign probabilities to events where the concept of equiprobability doesn’t arise. • Hempel expounds the determination of probabilities by experiment in terms of relative frequency. • Hempel suggests the relative frequency approach is the way to go even in the “fair coin” case. The equiprobable notion in this case is just a heuristic device for guessing at the relative frequency to be found by experiment. Hempel rejects symmetry considerations as suggesting that the equiprobability of outcomes in the fair die or coin cases are self-evident truths. His grounds are that, at the sub- atomic level for example, parity violation shows that symmetry considerations are not a prior truths. The same goes for Bose-Einstein and Fermi-Dirac statistics, which have different assumptions about probability distributions4. • While probabilities represent relative frequencies5, actual experiments give different relative frequencies, though these vary less and less as the number of outcomes in the experiment increases. So, if we have a random experiment R, whose possible outcomes are, O1, O2, … On, then successive performances of R give the Oi in an irregular manner, but the relative frequencies converge to the limits p(Oi, R). • This can be characterised statistically, where p(O, R) = r means that in a long sequence of performances of a random experiment R, the proportion of cases of outcome O is almost certain to be close to r. • Hempel points out the difference between statistical probability and inductive or logical probability. The latter is a quantitative relation between statements such that c(H, K) = r asserts that a hypothesis expressed by the statement H is supported or made probable (“confirmed”) to degree r by evidence (“knowledge“) expressed by the statement K. Statistical probability is a quantitative relation between certain kinds of events, as recently described. • Both concepts share the same mathematical characteristics :-

4 I’m not convinced by these examples. The latter one is based on differences about distinguishability of particles. Once this has been decided, symmetry can be applied. 5 According to Hempel … ! [email protected] Page 4 of 6

Carl Hempel – Laws and Their Role in Scientific Explanation

a). 0 ≤ p(O, R) ≤ 1; 0 ≤ c(H, K) ≤ 1 b). If O1 and O2 are mutually exclusive, p(O1or O2, R) = p(O1, R) + p(O2, R) If H1 and H2 are logically exclusive, c(H1or H2, K) = c(H1, K) + c(H2, K) c). p(O or not-O, R) = 1; c(H or not-H, K) = 1; and similarly for other necessary outcomes or logically true hypotheses. • Hempel now considers the logic of determining statistical probabilities by the long-run relative frequencies of the outcomes. Say we have a hypothesis H that p(A, D) = 0.15 is the probability that a roll of a die will yield an ace6. The probability doesn’t deductively imply any experimental outcome. Any divergence from the expected frequency is possible. A frequency in a long series of throws divergent from 0.15 does not logically refute H, as in the case of a black swan refuting the hypothesis that “all swans are white”, nor does one close to 0.15 logically confirm it as H does not claim that the frequency in a long series of throws will definitely be close to H (in the sense that in a N-D law, the explanadum will definitely occur if the explanans does. • While H does not logically preclude the possibility of divergences from 0.15 of the frequencies of aces in long series of throws, it does logically imply that they are statistically highly improbable. In a series of long series (taken as independent) the number of significantly divergent frequencies is low. We can calculate the probabilities particular divergences (Hempel gives examples). If the observed outcomes diverge greatly from these expectations, the hypothesis is likely to be false. The hypothesis is disconfirmed, or its credibility reduced, and is rejected for practical purposes even though it is not logically refuted. In contrast, close agreement with expected frequencies will result in the acceptance of an hypothesis. • We now need two criteria : (a) what divergences from expected frequencies are sufficient grounds for rejecting an hypothesis and (b) how close an agreement is required for acceptance. There is no single answer. Our chosen degree of certainty depends on the circumstances and on the consequences of being wrong – the cost of accepting a wrong hypothesis or rejecting a correct one. • Many important laws and principles in the natural sciences are of probabilistic form. Eg. random radioactive decay formulated in terms of half-lives, or the explanation of classical thermodynamic in terms of statistical mechanics. • Hempel notes an important distinction. All scientific laws are only more or less probable, because the evidence for them is finite and logically inconclusive. However, this doesn’t make all laws probabilistic. The difference between the two forms of laws (N-D and probabilistic) is not in the degree of confirmation but in the logical character of their claims. The N-D laws state that in all cases where conditions of type F are realised, conditions of type G are also realised. In contrast, probabilistic laws only claim that a certain outcome of a random experiment R will occur in a certain percentage of cases. No matter how well or badly laws of the two types are supported, they are of logically different forms. • Laws, whether D-N or probabilistic, are not mere summaries of past experience, but assertions about behaviour past, present, future, subjunctive and counterfactual.

5.6 The inductive character of probabilistic explanation

6 A “1” ? [email protected] Page 5 of 6

Carl Hempel – Laws and Their Role in Scientific Explanation

• Hempel returns to the simplest form of probabilistic explanation :-

Explanans p(O, R) is close to 1 i is a case of R Explanandum i is a case of O

• The actual case in point was :-

Explanans People exposed to measles have a high probability of catching it Jim was exposed to measles Explanandum Jim caught measles

• The high probability that the explanans confers on the explanandum (“how did Jim come to catch measles”) is not a statistical probability for it characterises a relation between sentences rather than events7. Rather, the probability in question is the rational credibility of the explanandum in the light of the explanans, and is a logical or inductive probability8. • In simple cases, the rational credibility will be the same as P(O, R). Where the explanans is more complex, the inductive probability of the explanandum is more difficult to calculate and we may have to resort to vague terms like “provides strong inductive support for”. • We distinguish D-N laws from probabilistic laws by saying that the former “effect a deductive subsumption under laws of universal form” while the latter “effect an inductive subsumption under laws of probabilistic form”. • There is an objection that probabilistic accounts do not really explain an event because the explanans does not logically preclude the non-occurrence of the explanandum. However, since many scientific laws are irreducibly probabilistic in form, we should be satisfied with probabilistic explanations (eg. of radioactive events, where “practical certainties” are reached). • Hempel closes with an exposition of how statistical mechanics explains Graham’s Law (that the diffusion rates of gases are inversely proportional to the square-roots of their molecular weights). He says that the account does genuinely provide an explanation, even though only with “very high probability”, and that this practise of treating the accounts as explanations is followed in standard physics texts.

7 If the question was “what’s the likelihood of getting measles on exposure to the disease”, a statistical probability would be the answer. 8 In the Bayesian sense. [email protected] Page 6 of 6