Cromwell's Rule¶
Core Idea¶
Never assign a prior probability of exactly 0 or 1 to a contingent proposition, because Bayesian updating multiplies prior by likelihood and zero is absorbing: a credence pinned at the boundary is evidence-sterile, unmovable by any observation.
How would you explain it like I'm…
You Might Be Wrong
Never be 100 percent totally sure that something is true or that it's impossible. If you decide for certain a box is empty and lock your mind shut, then even when someone shakes it and you hear something rattle, you'll still say 'empty.' Leaving a tiny 'maybe' open lets you learn when you're wrong. Always think it's possible you might be mistaken.
Keep A Tiny Maybe
Cromwell's Rule says: never give something a chance of exactly zero or exactly 100 percent if it could possibly be true or false. The reason is how learning from evidence works: you update beliefs by combining new clues with what you already believed, and that combining is like multiplying. If you start at zero, multiplying by anything keeps it at zero forever, so no evidence can ever change your mind. Same trap at 100 percent. So keep a tiny sliver of doubt, even about things you're very sure of, so that evidence can still do its job.
Never Zero, Never Certain
Cromwell's Rule is the injunction never to assign a probability of exactly 0 or exactly 1 to something that could in principle be true or false, because Bayesian updating cannot move a belief away from those endpoints. If P(H) = 0, then for any evidence E the updated P(H given E) is still 0, so the claim becomes permanently unfalsifiable; if P(H) = 1, it becomes permanently unrevisable. The deep reason is that updating is multiplicative: Bayes' rule multiplies your prior by how well the evidence fits, and zero kills any product. So a fully closed belief is evidence-sterile, immune to any observation however striking. It's named for Cromwell's 1650 plea, 'think it possible that you may be mistaken,' but the underlying fact holds for any learning system that combines new information with prior commitments. Keep a small but nonzero credence so evidence can act.
Cromwell's Rule is the structural injunction never to assign a prior probability of exactly 0 or exactly 1 to a contingent proposition, because Bayesian updating cannot move a probability away from those endpoints. If P(H) = 0, then for any evidence E the posterior P(H given E) = 0 as well, so the proposition is permanently unfalsifiable from below; if P(H) = 1, it is permanently unrevisable. Closed beliefs are evidence-sterile: no observation, however striking, can disturb them. The rule is named for Cromwell's 1650 plea, 'think it possible that you may be mistaken,' but the underlying fact is a property of any learning system that updates by combining new information with prior commitments. The structural content rests on four commitments: a belief system that updates by combining evidence with priors; an update mechanism that is multiplicative, since Bayes' rule multiplies prior by likelihood and zero kills the product; the fact that any closed commitment short-circuits all subsequent learning about that proposition; and the prescription to maintain a small but nonzero credence in anything that could be true or false so evidence can act. The descriptive core (a multiplicative update has absorbing boundaries at 0 and 1) is pure structure; the prescriptive face ('never assign 0 or 1') is a maxim on top. The rule has a natural dual: never treat one piece of evidence as carrying infinite likelihood ratio, since that pins the posterior at the boundary just as a boundary prior does, and both halves flow from zero being absorbing for multiplication.
Broad Use¶
- Bayesian statistics: a prior with zero mass on a parameter value can never be updated to nonzero mass, so proper priors stay positive across the support.
- Machine learning / NLP: zero-count events yield undefined estimates, so smoothing (Laplace, Good–Turing, Kneser–Ney) keeps unseen events at small but nonzero probability.
- Law: the presumption of innocence is a deliberate nonzero prior, and appeals exist because no finding should be infinitely irrevisable.
- Intelligence analysis: confidence scales rather than "100% certain" assessments implement the rule institutionally.
- Reinforcement learning: epsilon-greedy and Thompson sampling keep nonzero exploration on apparently dominated options.
- Ideology: dogma, conspiracy, and fundamentalism pin a proposition at 1, with predictable evidence-sterility.
Clarity¶
Sharpens the categorical gap between low probability and zero, and between high and certainty — one permits learning, the other forbids it.
Manages Complexity¶
Collapses a long catalogue of "why won't this system learn?" failures into one diagnosis (a boundary commitment) with one family of fixes: floor, smooth, institutionalize revisability.
Abstract Reasoning¶
Treats any failure-to-update as a question about where credence sits rather than about the quality of the evidence; the endpoints 0 and 1 are absorbing states, not extreme confidences.
Knowledge Transfer¶
- Statistics → law: a positive prior and revisable verdicts (appeals as posterior updates).
- Language modeling → policy: smoothing of unseen events becomes contingency planning for unseen scenarios.
- RL → strategy: exploration noise becomes funding long-shot projects at small nonzero levels.
Example¶
A naive Bayes spam classifier that scores a word never seen in spam at probability 0 multiplies the whole document score to zero — Laplace smoothing floors every word at small positive probability so unseen words can no longer sterilize all other evidence.
Relationships to Other Primes¶
Parents (1) — more general patterns this builds on
- Cromwell's Rule is a kind of Bayesian Updating — Cromwell's rule is a specific boundary CONSTRAINT on bayesian_updating: because zero is absorbing for multiplication, contingent priors must stay off 0 and 1 or the update is inert. A corollary of the algebra, a specialization of the general updating mechanism.
Path to root: Cromwell's Rule → Bayesian Updating → Inductive Reasoning
Not to Be Confused With¶
- Cromwell's Rule is not Bayesian Updating because the rule is a specific boundary constraint (keep priors off 0 and 1), whereas Bayesian updating is the general mechanism of revising prior by likelihood.
- Cromwell's Rule is not Falsifiability because the rule is a property of an agent's credence (pinned at the multiplicative boundary), whereas falsifiability is a property of a theory's content; a falsifiable theory held at probability 1 still violates Cromwell.
- Cromwell's Rule is not Epistemic Humility because the rule is a mechanical fact (boundary credence is inert regardless of sincerity), whereas humility is an attitude; the defect lives in the prior, not in temperament.