In 1650, Oliver Cromwell wrote to the General Assembly of the Church of Scotland: "I beseech you, in the bowels of Christ, think it possible that you may be mistaken." Three centuries later, the statistician Dennis Lindley elevated this plea into a formal principle of Bayesian epistemology. Cromwell's Rule states that no rational agent should assign probability exactly 0 or exactly 1 to any contingent proposition—any claim whose truth is not guaranteed by logic alone.

This prohibition may appear to be mere epistemic humility repackaged in probabilistic language. It is considerably more than that. The foundations of Cromwell's Rule are strictly mathematical, not merely normative. Within the standard Bayesian updating framework, absolute certainty operates as an absorbing state in the formal sense. Once a credence reaches 0 or 1, no finite body of evidence—no matter how surprising or overwhelming—can dislodge it through conditionalization. Certainty becomes epistemically irreversible, permanently impervious to any data the world might subsequently provide.

This article examines the formal architecture underlying this prohibition. We begin with the mathematical mechanics of conditionalization rigidity—why Bayes' theorem renders extreme credences permanently fixed. We then address a deeper challenge: whether regularity, the requirement that every logical possibility receive positive probability, is even achievable in infinite domains, and whether infinitesimal probabilities offer a viable resolution. Finally, we distinguish formal certainty from the practical near-certainty that rigorous inquiry permits, providing precise guidance for rational agents navigating conditions of very high confidence.

The mathematical case begins with Bayes' theorem. For any hypothesis H and evidence E where P(E) > 0, the posterior credence is given by P(H|E) = P(E|H) · P(H) / P(E). Suppose a rational agent assigns P(H) = 0. The numerator becomes P(E|H) · 0 = 0, regardless of the likelihood P(E|H). The posterior P(H|E) therefore equals zero—identically, unconditionally, for any evidence E whatsoever.

The symmetric case holds for maximal certainty. If P(H) = 1, then P(¬H) = 0. By identical reasoning, P(¬H|E) = 0 for all admissible evidence E, which entails P(H|E) = 1 regardless of observation. The agent's credence is permanently locked at unity. This is not a contingent feature of particular likelihood functions or evidence streams. It is a structural property of conditionalization itself.

In the theory of stochastic processes, a state from which no transition is possible is called an absorbing state. Credences of 0 and 1 are precisely absorbing states within Bayesian dynamics. They represent epistemic fixed points—positions from which no rational revision can proceed under the standard update rule. An agent at credence 0 or 1 has implicitly declared that no conceivable observation bears evidential relevance to the hypothesis in question.

Consider the implications for scientific reasoning. A researcher who assigns P(H) = 0 to the hypothesis that a particular physical constant varies across cosmic epochs has immunized that hypothesis against any empirical challenge. No anomalous measurement, however replicated or precise, can raise the credence above zero through conditionalization. The prior has effectively placed the hypothesis outside the scope of evidence entirely.

This property generalizes across every domain where belief revision matters—science, law, religion, moral reasoning. Assigning an extreme credence is not merely expressing strong confidence. It is a formal declaration that one's belief state on that proposition is, in principle, permanently closed to revision. The mathematics here does not prescribe humility as a virtue. It diagnoses a structural consequence: certainty, once conferred, admits no rational retraction.

Takeaway

Once a credence of 0 or 1 is assigned, Bayes' theorem guarantees that no evidence can modify it. Absolute certainty is not strong belief—it is the permanent termination of rational inquiry.

Cromwell's Rule naturally motivates a stronger formal requirement: regularity. A probability function P is regular if and only if P(H) > 0 for every logically possible proposition H—equivalently, P(H) = 0 only when H is a logical contradiction. Regularity would elevate Cromwell's Rule from a methodological norm to a theorem of the probability calculus. But this elegant requirement encounters severe technical obstacles when the space of possibilities is infinite.

The central difficulty is countable additivity. Standard Kolmogorovian probability requires that for countably many mutually exclusive events, the probability of their union equals the sum of their individual probabilities. Consider a countably infinite partition {H₁, H₂, H₃, ...}. If each hypothesis in the partition receives some positive real probability ε > 0, the sum diverges—violating the requirement that total probability equals 1. Regularity is therefore mathematically unachievable in countably infinite domains under real-valued, countably additive probability.

One response invokes infinitesimal probabilities from non-standard analysis. Hyperreal number systems contain positive quantities smaller than every positive real number. Assigning infinitesimal credence to each element of a countable partition avoids zero-valued credences without generating divergent sums. Formal frameworks developed by Bernstein and Wattenberg, and later by Benci, Horsten, and Wenmackers, demonstrate that hyperreal-valued probability functions can satisfy regularity where standard real-valued functions cannot.

Infinitesimal approaches, however, introduce substantial complications. Hyperreal probability functions are not uniquely determined by the standard constraints—multiple non-equivalent extensions exist, and the choice among them appears arbitrary. Conditionalization becomes technically delicate when both numerator and denominator are infinitesimal, as the standard ratio definition can yield indeterminate forms. Some philosophers further argue that infinitesimal credences are pragmatically indistinguishable from zero, undermining the very motivation for regularity.

This debate illuminates a genuine and unresolved tension in formal epistemology. The demand that every logical possibility receive positive probability and the requirement that probability functions behave well over infinite spaces pull in opposite directions. No existing framework satisfies both desiderata without significant concessions. Rather than a failure of the formal project, this tension constitutes a precise characterization of a deep structural constraint on rational credence functions.

Takeaway

Regularity—giving every logical possibility positive probability—is the natural formalization of Cromwell's Rule, but it provably conflicts with standard probability theory in infinite domains. The tension is mathematically genuine, not merely philosophical.

If formal certainty is forbidden for contingent propositions, what epistemic status should rational agents accord to overwhelmingly well-supported hypotheses? The resolution lies in a distinction that is mathematically precise though easily overlooked: the distinction between credence exactly equal to 1 and credence arbitrarily close to 1. These are not notational variants. They differ fundamentally in their structural properties under conditionalization.

Consider a credence of 1 − 10⁻¹⁰⁰ assigned to some well-confirmed hypothesis. This quantity is, for all practical purposes, indistinguishable from 1. No decision-theoretic calculation would produce different actions. Yet under Bayesian updating, this credence retains a crucial property that credence 1 lacks: revisability. Sufficiently strong counter-evidence can, in principle, shift the posterior downward. The agent has left open—however narrowly—the channel through which new information can flow.

This framework provides formal expression for an intuition with deep philosophical roots. The early modern distinction between moral certainty—confidence sufficient for action—and metaphysical certainty—confidence admitting no possible doubt—maps directly onto the Bayesian distinction. Practical certainty is a property of a credence's magnitude. Formal certainty is a property of its logical structure. The former scales continuously; the latter is a qualitative threshold with irreversible consequences.

The implications extend across domains. A physicist who assigns credence 0.99999 to general relativity acts, in every practical respect, as one who is certain. But her credence structure preserves the capacity for rational revision—she can respond appropriately if persistent gravitational anomalies defy relativistic explanation. Contrast this with an agent who assigns credence 1 to any contingent claim, whether scientific, religious, or political. That agent has not merely expressed extreme confidence. She has structurally excluded the possibility of learning otherwise.

Cromwell's Rule therefore does not counsel skepticism or demand perpetual doubt. It imposes a single structural constraint: never assign a credence that conditionalization cannot subsequently revise. Rational agents may hold beliefs of extraordinary strength. They must never hold beliefs of infinite rigidity. The most epistemically virtuous form of high confidence is one that carries within it an implicit—however infinitesimal—acknowledgment of its own fallibility.

Takeaway

The difference between credence 0.99999 and credence 1 is not a matter of confidence—it is a structural difference in whether your belief remains, in principle, responsive to evidence.

Cromwell's Rule is often presented as a pragmatic heuristic—a reminder toward intellectual humility. The formal analysis reveals something stronger. Within Bayesian epistemology, assigning credence 0 or 1 to any contingent proposition is structurally incompatible with the very mechanism that makes rational belief revision possible. Absolute certainty does not represent the upper bound of confidence. It represents the termination of inquiry.

The difficulties that regularity encounters in infinite domains reinforce rather than weaken this conclusion. They demonstrate that genuine openness to all logical possibilities is mathematically demanding—a formal constraint mirroring the philosophical difficulty of authentic intellectual humility. The tension between regularity and countable additivity precisely delineates what coherent credence functions can achieve.

Between overwhelming confidence and absolute certainty lies a boundary that appears vanishingly thin. Formally, it separates a mind that can still learn from one that cannot. Preserving that boundary is not a limitation on rational commitment—it is the structural precondition for rationality itself.