Chance over Time
Chance is time-indexed. The chance that a coin lands heads is before the toss and, once it has landed heads, . The Principal Principle writes this into its statement as , and the indexing is not a formality: it encodes a substantive asymmetry between past and future that any account of chance must explain.
This page examines that asymmetry — why past events have trivial chances, what happens at the moment of an event, how chance relates to the openness of the future, and the undermining problem that time-dependence creates for Humean accounts.
The past has trivial chances
The governing principle, which Lewis took as near-analytic:
Once an event has occurred, its chance is ; once it has failed to occur, its chance is .
The past is fixed, and chance concerns what is still open. This is why historical information counts as admissible for the Principal Principle: knowing that the coin landed heads yesterday does not conflict with knowing yesterday's chance was , because today's chance is .
The formal expression is that chance evolves by conditioning on history. If is the complete history up to , then — the chance at is the initial chance conditioned on everything that has happened. Chance is thus a martingale: the expected value of a later chance, given the present, equals the present chance. This is the same martingale structure that governs an agent's credences under conditionalization, and the parallel is not accidental — it is the objective counterpart of the reflection principle.
The martingale property has an immediate and useful consequence: you cannot expect the chance to drift in a predictable direction. If you knew the chance of rain would be higher tomorrow than today, it should be higher today already. Any predictable component of the change has already been incorporated.
The transition problem
If chance goes from to when the coin lands, what happens at the landing?
The question is not merely pedantic. If chances change continuously, there must be intermediate moments at which the chance is or — and it is unclear what physical fact those numbers correspond to, since the outcome is either determined by then or not. If chances change discontinuously, we need an account of what triggers the jump, and "the event occurs" is unhelpfully close to circular.
Two treatments are available. On a dynamical view, the chance is given by the physics at each moment: as the coin's trajectory unfolds, the set of outcomes still compatible with its state shrinks, and the chance evolves accordingly — the intermediate values are perfectly meaningful, reflecting how much is still open given the current state. On a coarse-grained view, chances attach to macro-level setups and simply cease to apply once the outcome is settled, with no continuous transition to describe.
The dynamical view is the more satisfying and connects to the method of arbitrary functions: as a deterministic system evolves toward its outcome, the induced probability over outcomes concentrates, giving a principled sense in which intermediate chances are real.
The future and its openness
The asymmetry of chance is often taken to reflect a metaphysical asymmetry between past and future — the past is settled, the future open. That reading connects the present topic to the philosophy of time, and the connection needs care, because the inference is not as direct as it seems.
On a growing-block or presentist view, the asymmetry of chance mirrors an asymmetry in what exists: future events are not there to have determinate truth values, so intermediate chances record genuine openness.
On an eternalist view, all events are equally real, and tomorrow's coin toss already has an outcome in the block. Chance cannot then be a matter of the future's being unwritten. But eternalism is not thereby committed to trivial chances: what the eternalist says is that measures the extent to which the state at , together with the laws, fixes — which is a relation between a time-slice and an event, not a claim about which events exist. An indeterministic eternalist world is one in which earlier slices do not fix later ones, and that is perfectly coherent.
So the temporal asymmetry of chance requires an asymmetry in the laws — in what earlier states determine about later ones — rather than an asymmetry in existence. This matters because the reverse inference is common and invalid: one cannot argue from the reality of chance to the openness of the future, nor from the block universe to determinism.
There is a further complication worth flagging. Chance is asymmetric in time, but fundamental dynamics is largely time-symmetric. If the laws run equally well in both directions, why do chances only concentrate toward the future? The answer generally given appeals to the same boundary condition that grounds the thermodynamic arrow — a low-entropy past — which makes the asymmetry of chance a consequence of the Past Hypothesis rather than a primitive feature.
Undermining
Time-dependence creates the problem for Humean accounts already met under the Principal Principle, and it appears here in its most natural setting.
On the best-system account, chances supervene on the entire mosaic, future included. So the chance at depends on what happens after . But the chance at also assigns positive probability to various futures, some of which — if they occurred — would make a different system best and hence make the chances at different from what they are.
The result is that present chances assign positive probability to their own falsity. There is no contradiction in the chances themselves; the contradiction arises when the Principal Principle is applied, requiring an agent to have positive credence in a proposition their evidence rules out. The New Principle is the standard repair.
Non-Humeans avoid this entirely: if chances are propensities fixed by the present state, the future cannot undermine them, since chances do not depend on it. This is a genuine advantage, and one of the strongest arguments for taking chance as a fundamental feature of the present state rather than as a summary of the whole.
Chances of past events
Two puzzles remain about the principle that past chances are trivial.
Retrospective probability talk. We say things like "there was a chance that plan would fail, and we got lucky". This appears to attribute a non-trivial chance to a past event. The natural reading is that the claim concerns the chance at the earlier time — where precedes the outcome — which is compatible with . Retrospective chance talk is temporally displaced, not a counterexample.
Unknown past events. If a coin was tossed yesterday and I have not seen the result, its chance of heads is or , but my credence should be . This is not a problem but an illustration: it is exactly the chance/credence distinction, and it shows why credence cannot be defined as known chance. The Principal Principle handles the case correctly, because I lack admissible evidence about the present chance, and my credence is properly governed by the earlier chance I do know.
Where this sits
The time-indexing of chance connects this section to the philosophy of time and to the thermodynamic arrow, and it is the source of Humeanism's central difficulty. The stable points: chance is time-relative, evolves by conditioning on history as a martingale, and becomes trivial once outcomes are settled. The contested points: what happens during the transition, whether the asymmetry reflects the structure of time or only of the laws, and whether undermining is a repairable bug or a refutation of Humean chance.
The next page takes up a notion that behaves like chance in explanations but is not one, and which does much of the work in statistical mechanics: typicality.