The Principal Principle

If objective chance and rational credence are genuinely distinct, a question arises that would not otherwise: why should learning a chance constrain belief? Nothing in the axioms connects them — they are two functions that happen to satisfy the same formal conditions. The link must be a normative principle, and Lewis's Principal Principle is the canonical statement of it.

The principle is the hinge of the whole subject. It is what makes chance matter: a chance that placed no constraint on rational belief would be an idle metaphysical posit, no more relevant to inquiry than an undetectable ether. Conversely, credences that ignored known chances would be irrational in a way no coherence requirement could diagnose. The principle is also, on Lewis's own view, close to a definition: chance simply is that feature of the world which plays this role in guiding belief.

The principle

Let $C r$ be a rational initial credence function, $C h$ the chance function, $t$ a time, and $E$ a proposition. Then

$C r (A ∣ C h_{t} (A) = x \land E) = x$

for any $E$ that is admissible at $t$ .

In words: conditional on the chance of $A$ being $x$ , and on any admissible further information, your credence in $A$ should be $x$ . Told that a coin is biased $0.7$ toward heads, believe to degree $0.7$ that it lands heads — regardless of your prior opinions, and regardless of how much other admissible evidence you have.

Two features are worth noting before the difficulties.

It is a constraint on the initial credence function. Lewis states it for a reasonable initial credence, from which an agent's actual credences arise by conditionalization on their total evidence. This matters because it makes the principle a constraint on priors — one of the few available — and so bears directly on the problem of the priors.

It requires no knowledge of why the chance is what it is. The agent need not know the underlying physics. Chance functions as an expert: deference is warranted by its status, not by understanding its grounds. This is why the principle also underwrites the more general phenomenon of deferring to experts and instruments.

Admissibility

The proviso carries the weight, and it is where the principle is least satisfactory.

Without it the principle is plainly false. Let $E$ be "the coin landed tails". Then even knowing the chance of heads is $0.7$ , no rational agent retains credence $0.7$ in heads. The point of admissibility is to exclude information that bypasses the chance — that bears on $A$ otherwise than by bearing on $A$ 's chance.

Lewis's characterisation: $E$ is admissible at $t$ if it affects credence in $A$ only by way of affecting credence about $A$ 's chance. Paradigm admissible evidence includes:

Historical information about times up to $t$ . The whole past is admissible, since chance at $t$ already reflects it.
Information about the chances themselves, including the theory that generates them.

Paradigm inadmissible evidence includes crystal balls, reliable prophecy, time-travellers' reports, and inside knowledge of the outcome.

The difficulty is that no satisfactory general definition has been found. Every proposal either lets in too much or is circular. Defining admissible evidence as evidence that does not bear on $A$ except via its chance uses the notion of chance-mediated relevance, which is close to using the Principal Principle to define its own proviso. Nor is the historical criterion airtight: in a world containing reliable predictors, or backwards causation, or the Humean dependence of present chances on future events, past information can carry inadmissible content.

This is a genuine soft spot rather than a technicality. A principle whose central qualifier resists definition is hard to apply in exactly the contested cases — anthropic reasoning, self-locating belief, quantum measurement — where one most wants guidance.

What the principle does

Its consequences are extensive enough that they are often mistaken for independent principles.

It entails calibration. Known chances fix credences, which is the calibration norm of objective Bayesianism.
It makes statistical inference possible. Because chance constrains credence, observed frequencies are evidence about chances — via Bayes' theorem run backwards from likelihoods that the principle supplies. Without it, data would tell us nothing about chance.
It connects to the limit theorems. Given the principle, the law of large numbers licenses high credence that observed frequencies approximate chances, which is what makes chances ascertainable.
It disciplines the interpretations. Any account of chance must explain why deference to it is rational. This is criterion 6 of the adequacy conditions, and it is where best-system accounts struggle and propensity accounts do well.

The Big Bad Bug

Lewis discovered that his own two commitments — Humean supervenience about chance, and the Principal Principle — are jointly inconsistent. He called the result the Big Bad Bug, and it is the most discussed problem in the area.

On the best-system account, chances supervene on the whole mosaic of events, future included. A chancy system assigns positive probability to many possible futures, and some of those futures, if actual, would make a different system best — and hence would make the chances different from what they are. Call such a future undermining.

Let $F$ be an undermining future with present chance $x > 0$ , and let $T$ be the complete true theory of chance. The Principal Principle requires $C r (F ∣ T) = x > 0$ . But $T$ entails that the chances are as $T$ says, and $F$ entails they are otherwise, so $T$ entails $\neg F$ ; any coherent agent must therefore have $C r (F ∣ T) = 0$ . Contradiction.

The bug is not peculiar to Lewis's formulation. It arises for any account on which chances depend on the totality of events, which is to say for any Humean account.

The New Principle

Hall and Thau proposed, and Lewis accepted in "Humean Supervenience Debugged", a modified principle:

$C r (A ∣ T) = P_{T} (A ∣ T)$

Credence should match the chance conditional on the theory of chance being true, rather than the unconditional chance. Since $P_{T} (\cdot ∣ T)$ assigns zero to undermining futures, the contradiction dissolves.

Three assessments of this move are current, and the choice among them is largely a choice about how much one wants Humeanism to work.

The New Principle is the correct principle all along (Hall). The original was an approximation, adequate when undermining is negligible, and the refinement is a discovery rather than a patch.
The New Principle is a damaging concession. Chance no longer straightforwardly guides belief; one must first condition on a complete theory of chance, which no agent possesses. The principle's practical content is thereby weakened.
The bug refutes Humeanism. If saving the account requires weakening the principle that gives chance its point, the account has failed. Non-Humeans — propensity theorists, primitivists about chance — take this line, since on their views undermining cannot arise: chances are fixed by the present state, and the future cannot alter them.

Other challenges

Inadmissible evidence in practice. Any agent with access to a reliable predictor has inadmissible evidence, and the principle then gives no guidance at all rather than adjusted guidance. A more graded principle would be preferable but has proved hard to formulate.

Self-locating belief. The principle governs credence in ordinary propositions. Where uncertainty is about which agent or time one occupies — Sleeping Beauty, duplication cases, anthropic reasoning — it is unclear how to apply, since chance attaches to events rather than to centred worlds. This is one source of the persistent disagreement in those debates.

Undermining beyond Humeanism. Even non-Humeans face pressure from cases where an agent's credence about the chance and about the outcome are not independent, and the formulation must be stated carefully to avoid trivialising.

Where this sits

The Principal Principle is the bridge between the two halves of this section: it links the metaphysical question about chance to the epistemic question about credence, and without it the two would be unrelated subjects that happen to share a calculus.

Its status is correspondingly contested. Read as constitutive, it defines chance as whatever plays the credence-guiding role, and the Big Bad Bug becomes a decisive objection to any account that cannot satisfy it. Read as regulative, it is a substantive norm that some accounts of chance satisfy better than others, and its failure in exotic cases is a limitation rather than a refutation.

The next page takes up the question the principle presupposes but does not answer: whether there is any objective chance to defer to in a world whose fundamental laws may be deterministic.

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