Chance and Explanation
We routinely explain by citing probabilities. The sample decayed because the isotope has a short half-life; the patient recovered because the drug is effective in most cases; the gas expanded because expansion is overwhelmingly likely. Whether these are genuine explanations, and if so what makes them so, is a harder question than the frequency of such talk suggests.
The difficulty is structural. A deductive explanation shows why the outcome had to occur. A probabilistic explanation cannot do this: it is consistent with the outcome's not occurring, and — in the awkward cases — it is consistent with the outcome's having been very unlikely. If explanation is a matter of showing that the event was to be expected, then improbable events cannot be explained. Yet improbable events happen constantly, and we explain them.
The covering-law model and its failure
Hempel's deductive-nomological model treats explanation as derivation: an event is explained by exhibiting laws and initial conditions from which its occurrence follows. His inductive-statistical model extends this to probabilistic cases, requiring that the explanans confer high probability on the explanandum. To explain a recovery, cite the treatment and the statistical law that the treatment usually works.
The high-probability requirement fails in both directions, and the failures have shaped everything since.
High probability is not sufficient. The classic counterexample: taking birth-control pills is highly correlated with not becoming pregnant, so "John took birth-control pills" confers high probability on "John did not become pregnant". This is not an explanation. Nor is a barometer's fall an explanation of the storm, though it makes it highly probable. What is missing is a causal connection: the probability is high, but for the wrong reason.
High probability is not necessary. Paresis occurs only in untreated syphilitics but develops in only a small minority of them. Asked why this patient has paresis, the answer is that he had untreated syphilis — the only available explanation, and manifestly the right one, despite conferring low probability. Similarly, a low-probability radioactive decay is explained by the isotope's instability even though the decay was unlikely in any given interval.
Hempel also faced the problem of ambiguity: the same event can be given statistical explanations with conflicting probabilities depending on which reference class is used, and nothing in the model selects one. This is the reference-class problem arriving in the theory of explanation.
Statistical relevance
Salmon's response was to abandon high probability altogether. What matters is not that the explanans makes the outcome likely but that it is statistically relevant — that it changes the probability:
A factor explains by making a difference, whether it raises the probability to or from to . The paresis case is handled immediately: untreated syphilis is enormously relevant to paresis even though the absolute probability stays low.
The model requires partitioning the reference class into cells that are homogeneous — containing no further statistically relevant subdivisions — and an explanation consists in locating the event in the right cell and giving the probability distribution across the partition.
The residual problem is that statistical relevance is symmetric and undirected, so it cannot by itself distinguish cause from effect or either from a common cause. The barometer is statistically relevant to the storm. Salmon himself came to regard relevance as insufficient and moved to a causal-mechanical account, on which explanation requires exhibiting the processes and interactions producing the outcome — which is to say that the probabilities are evidence of the explanatory relation rather than constitutive of it. This anticipates the causation folder's conclusion that statistical dependence underdetermines causal structure.
Do chances explain at all?
A more radical question. Grant that the chance of decay in this interval is . Does citing explain why this atom decayed now?
There is a real case for saying no. The chance is the same for every atom of the isotope, and most did not decay. A factor common to decayers and non-decayers alike cannot explain the difference between them, and on standard contrastive accounts explanation is a matter of accounting for why this rather than that. If the chance is genuinely irreducible, there is no fact that makes the difference — that is what irreducible chance means — and so there is nothing to cite.
Three responses:
- Chances explain the pattern, not the instance. The half-life explains why roughly half the sample decayed in the period. Individual decays are not explicable, and this is a discovery about the world rather than a defect in the theory of explanation. Many find this the honest answer for genuinely indeterministic events.
- Explanation need not be contrastive. Citing the chance tells us the event was a manifestation of a known propensity operating in a known way, which is genuine understanding even without contrast. On this view we should weaken the demand rather than deny the explanation.
- Explanation comes in degrees. A chance of explains better than a chance of , without either being no explanation at all. This preserves the intuition that likelier outcomes are better explained while allowing improbable ones to be explained at all.
What the interpretation contributes
The available answers depend on what chances are, and this is one of the places where the interpretive dispute has clear downstream consequences.
- On a propensity account, chances are physical tendencies, so citing one identifies a real feature of the system that produced the outcome. Explanation is straightforward, and this is among propensity's best arguments.
- On a best-system account, chances are terms in the best summary of the total pattern of events. Citing a chance to explain an event then means citing a summary that includes that event among its data. Whether this is viciously circular or merely the ordinary situation of explanation-by-subsumption is disputed, and it is the same objection Humeans face about laws generally — treated in laws of nature.
- On a frequency account, matters are worse: the frequency is a fact about the class, and explaining a member by citing a property of the class it belongs to is at best indirect.
- On a subjectivist account, chances do not explain because there are none; probabilistic "explanations" are really summaries of our epistemic position.
Typicality explanations
A distinct explanatory strategy, important in statistical mechanics and treated fully under typicality, deserves flagging here because it is often mistaken for a probabilistic explanation.
Why does a gas released into a chamber expand? The typicality answer is not that expansion is probable but that the initial conditions leading to expansion are overwhelmingly typical in the space of conditions compatible with the initial macrostate — the non-expanding ones form a set of vanishingly small measure. The explanation is that the actual initial condition was not special, and no probability is assigned to any individual event.
This is a different explanatory form: it explains by showing the outcome required nothing exceptional, rather than by assigning it high probability. Its cost is that "typical" is relative to a measure, and justifying the measure is exactly the difficulty models and applications identified.
Where this sits
Probabilistic explanation is the place where the theory of chance meets the theory of explanation, and neither discipline gets what it wants. The high-probability requirement fails; statistical relevance is undirected; and whether an irreducible chance can explain an individual outcome depends on contested views about contrastive explanation.
The stable results are worth separating from the disputed ones. It is settled that probability alone does not suffice for explanation, since the barometer and birth-control cases show correlation without explanatory force — so an explanatory probabilistic claim carries causal or nomic commitments beyond the numbers. It is disputed whether individual chancy events are explicable at all.
The next page turns to a structural feature of chance that the Principal Principle presupposed without examining: that chances are indexed to times, and change as history unfolds. That raises the question of what happens to the chance of an event once it has already occurred.