Read This First
If this page feels abrupt, start here
These links provide the wider frame, earlier distinction, or branch map that makes the current page easier to enter.
-
Absolute Certainty
Start here if the current page feels compressed: Absolute Certainty gives the broader frame before the argument narrows into the present pressure.
-
Epistemology Branch Guide
If this page feels abrupt, start with the Epistemology branch guide so the wider map is visible before the close reading begins.
Read This Next
If the page clicked, continue here
These are not just nearby pages. They are the strongest next moves if you want the pressure of this page to keep unfolding.
Prompt 1: Provide an full elaboration on Cromwell’s Rule defined below
Cromwell's Rule: why absolute certainty almost never belongs in the model
This section is worth asking because it changes what the reader should compare next. The point is to make Cromwell’s Rule more investigable, not merely more impressive-sounding.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
The pedagogical payoff is practical. After this section, the reader should be better able to explain Cromwell’s Rule in plain language, identify a likely misuse of it, and say what further evidence or argument would actually move the view.
Wikipedia “ Cromwell’s rule, named by statistician Dennis Lindley, states that the use of prior probabilities of 1 (“the event will definitely occur”) or 0 (“the event will definitely not occur”) should be avoided, except when applied to statements that are logically true or false, such as 2 + 2 equaling 4.”
Prior Probabilities These are the probabilities assigned to events before observing any new data. They reflect our initial state of belief or knowledge.
Bayesian Updating This is the process of adjusting our prior probabilities based on new evidence to obtain posterior probabilities.
Inflexibility to New Evidence If a prior probability is set to 0 or 1, no amount of new evidence can change this belief. This goes against the fundamental Bayesian principle of updating beliefs with evidence.
Practical Considerations In the real world, few events are absolutely certain or impossible. Allowing for some degree of uncertainty (i.e., probabilities between 0 and 1) is more practical and realistic.
Logical Certainty The only exceptions to this rule are statements that are logically true or false (e.g., mathematical truths like 2 + 2 = 4).
Flexibility in Belief Revision By avoiding 0 or 1, we remain open to updating our beliefs in light of new evidence, fostering a more flexible and adaptive mindset.
Avoiding Dogmatism This rule helps prevent dogmatic thinking, where beliefs are held rigidly regardless of contrary evidence.
Encouraging Rationality It promotes rational decision-making by ensuring that our beliefs can be modified based on evidence, which is central to Bayesian reasoning.
Scientific Inquiry Scientists often adopt Cromwell’s Rule to maintain an open-minded approach to hypotheses, allowing them to be updated or discarded based on experimental data.
Risk Assessment In fields like finance and engineering, where decisions are based on probabilistic models, avoiding absolute priors helps in making more robust and adaptable predictions.
Decision Theory In decision-making processes, especially under uncertainty, this rule ensures that decisions remain flexible and responsive to new information.
Scientific Hypotheses Suppose a scientist is testing a new hypothesis. Assigning a prior probability of 0 to the hypothesis would mean they are certain it is false, regardless of any evidence that may arise. By assigning a prior probability slightly above 0, the scientist remains open to considering evidence that might support the hypothesis.
Medical Diagnosis In a medical context, if a doctor assigned a prior probability of 0 to a rare disease for a patient, no amount of symptoms or test results suggesting the disease could change the diagnosis. By assigning a low but non-zero prior probability, the doctor stays open to adjusting the diagnosis based on new information.
Limited Knowledge Our knowledge about the world is often incomplete. New evidence or data can emerge, changing the likelihood of an event. Assigning a 0 probability excludes the possibility of encountering such evidence entirely.
More Nuanced Inferences By using probabilities between 0 and 1, we allow the data we collect to have a greater influence on our final conclusions (posterior probabilities). This leads to more flexible and adaptable inferences.
Exceptions The rule makes an exception for statements that are logically true or false, such as mathematical identities (e.g., 2 + 2 = 4) or logical tautologies (e.g., “If it is raining, then it is raining”). In these cases, it is appropriate to assign a prior probability of 1 or 0, respectively, because the truth or falsity of the statement is known with absolute certainty.
Importance of the Rule Cromwell’s Rule is particularly relevant in the context of Bayesian inference, where prior probabilities are combined with likelihood functions to calculate posterior probabilities. If a prior probability of 0 or 1 is used, it can lead to inappropriate or misleading conclusions, as the posterior probability will always be 0 or 1, regardless of the evidence.
- Cromwell’s Rule: An Elaboration: Cromwell’s Rule, named after the English statesman Oliver Cromwell and introduced by statistician Dennis Lindley, is a principle in Bayesian statistics.
- Understanding Cromwell’s Rule: In Bayesian statistics, prior probabilities represent our initial beliefs about the likelihood of events before considering new evidence.
- Examples: Cromwell’s Rule is a critical principle in Bayesian statistics, emphasizing the importance of keeping prior probabilities between 0 and 1 unless dealing with logically certain statements.
- Belief calibration: Cromwell’s Rule concerns how strongly the available evidence warrants belief, disbelief, or suspension of judgment.
- Evidence standard: Support, counterevidence, and merely persuasive appearances have to be kept distinct.
Prompt 2: Elaborate on how an absolute credence of 0 or 1 entails epistemic blindness.
What changes once we define The Role of Credence in Epistemic Processes more carefully
This section is worth asking because it changes what the reader should compare next. The point is to make Cromwell’s Rule more investigable, not merely more impressive-sounding.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
The pedagogical payoff is practical. After this section, the reader should be better able to explain Cromwell’s Rule in plain language, identify a likely misuse of it, and say what further evidence or argument would actually move the view.
Definition Epistemic blindness is a state where one is unable to see or consider alternative possibilities due to rigid belief structures.
Consequences This rigidity can lead to a failure to recognize new information, resulting in a static and potentially flawed understanding of the world.
Non-absolute Credence A credence between 0 and 1 indicates a belief that is open to revision based on new evidence.
Absolute Credence A credence of exactly 0 or 1 signifies absolute certainty or impossibility, leaving no room for change.
Inflexibility to New Evidence Bayesian Updating: Bayesian reasoning relies on updating our beliefs in light of new evidence. If we start with a credence of 0 or 1, our belief system becomes impervious to change, as no amount of evidence can alter these extreme probabilities. Learning from Evidence: When we assign a non-absolute credence, we acknowledge that our current understanding may be incomplete or flawed, and we remain open to learning from new evidence.
Bayesian Updating Bayesian reasoning relies on updating our beliefs in light of new evidence. If we start with a credence of 0 or 1, our belief system becomes impervious to change, as no amount of evidence can alter these extreme probabilities.
Learning from Evidence When we assign a non-absolute credence, we acknowledge that our current understanding may be incomplete or flawed, and we remain open to learning from new evidence.
Dogmatic Belief Systems Resisting Change: Absolute credence fosters dogmatism, where beliefs are held rigidly despite contradictory evidence. This resistance to change can lead to outdated or incorrect views. Closed-mindedness: By refusing to consider alternative possibilities, individuals with absolute credence can become closed-minded, dismissing new information that could refine or correct their beliefs.
Resisting Change Absolute credence fosters dogmatism, where beliefs are held rigidly despite contradictory evidence. This resistance to change can lead to outdated or incorrect views.
Closed-mindedness By refusing to consider alternative possibilities, individuals with absolute credence can become closed-minded, dismissing new information that could refine or correct their beliefs.
Undermining Rational Decision-Making Rationality and Evidence: Rational decision-making involves weighing evidence and updating beliefs accordingly. Absolute credence undermines this process by precluding any consideration of new or conflicting evidence. Error in Judgment: Decisions based on absolute credence can be flawed, as they ignore the probabilistic nature of most real-world events and the potential for error in our initial assessments.
Rationality and Evidence Rational decision-making involves weighing evidence and updating beliefs accordingly. Absolute credence undermines this process by precluding any consideration of new or conflicting evidence.
Error in Judgment Decisions based on absolute credence can be flawed, as they ignore the probabilistic nature of most real-world events and the potential for error in our initial assessments.
Scientific Rigor Hypothesis Testing: Scientists often begin with hypotheses that have non-absolute credence. This openness allows them to update their hypotheses based on experimental results. An absolute credence of 0 in a hypothesis would mean no evidence, however compelling, could ever support it. Scientific Progress: Scientific advancements depend on the ability to revise theories. Epistemic blindness resulting from absolute credence stifles this progress.
Hypothesis Testing Scientists often begin with hypotheses that have non-absolute credence. This openness allows them to update their hypotheses based on experimental results. An absolute credence of 0 in a hypothesis would mean no evidence, however compelling, could ever support it.
Scientific Progress Scientific advancements depend on the ability to revise theories. Epistemic blindness resulting from absolute credence stifles this progress.
Medical Diagnosis Diagnostic Flexibility: Doctors must remain open to revising diagnoses based on patient symptoms and test results. Assigning a prior probability of 0 to a rare disease can lead to misdiagnosis, as no amount of evidence will prompt the doctor to consider that disease. Patient Outcomes: Flexible credence ensures better patient outcomes by allowing doctors to consider and incorporate new evidence into their diagnoses.
Diagnostic Flexibility Doctors must remain open to revising diagnoses based on patient symptoms and test results. Assigning a prior probability of 0 to a rare disease can lead to misdiagnosis, as no amount of evidence will prompt the doctor to consider that disease.
- The Role of Credence in Epistemic Processes: Credence refers to the degree of belief or probability that we assign to a proposition.
- Examples Illustrating Epistemic Blindness: Assigning an absolute credence of 0 or 1 to any proposition leads to epistemic blindness by rendering our belief systems inflexible and closed to new evidence.
- Borderline case: The reader should be able to say what would make the claim merely plausible rather than justified.
- Objection test: A strong section names the best reason a careful critic would withhold assent.
- Calibration test: The answer should distinguish certainty, high confidence, tentative belief, and responsible agnosticism.
Prompt 3: Create a hypothetical discussion between an epistemologist who has just introduced Cromwell’s Rule and a student who feels the absolute credences of 0 or 1 are sometimes justified.
What changes once we define Epistemologist more carefully
“Today, I’d like to introduce you to Cromwell’s Rule. This principle, named after Oliver Cromwell, suggests that in probabilistic reasoning, we should never assign a probability of 0 or 1 to any event unless it is logically impossible or a certain truth, respectively. This is.
This section should give the reader a usable epistemic lever: what would support the central claim, what would count against it, and what would make suspension of judgment more rational than either assent or denial. The point is not to make Cromwell’s Rule tidy; it is to help the reader notice the difference between having a belief, having a reason, and having enough reason.
Epistemologist So, as we discussed, Cromwell’s Rule advises against assigning absolute probabilities of 0 or 1 unless dealing with logical truths. Does this resonate with everyone?
Student Professor, I understand the value of considering uncertainty, but there are cases where absolute certainty seems justified. Isn’t it perfectly reasonable to assign a 1 probability to the statement “I am currently sitting in this chair?”
Epistemologist That’s an interesting point. Tell me more about why you think so.
Student Well, I can feel the chair beneath me, and I’m actively perceiving the classroom. To deny this absolute certainty feels illogical.
Epistemologist That’s a common intuition. But have you considered the possibility of being deceived? Perhaps it’s an elaborate dream or a simulated reality. While highly unlikely, we can’t definitively rule it out.
Student But that’s just a philosophical thought experiment, right? In the real world, such possibilities are negligible.
Epistemologist True, but the point is that absolute certainty requires complete knowledge, which we seldom possess. Cromwell’s Rule encourages acknowledging this limitation. Even in your chair example, there’s a tiny sliver of uncertainty, however improbable.
Student I see your point, but assigning a very high probability, close to 1, achieves the same outcome for most purposes, doesn’t it?
Epistemologist It can seem that way. But there’s a subtle difference. When you assign a 0.99 probability instead of 1, you leave a sliver of space for new information. Imagine if, say, tomorrow you woke up with amnesia. You wouldn’t be certain about anything, including whether you sat in that chair yesterday.
Student Okay, that’s a fair point. But couldn’t Cromwell’s Rule make assigning probabilities overly complicated? Isn’t it sometimes simpler to say something is certain or impossible?
Epistemologist Simplicity can be appealing, but it can also be misleading. Cromwell’s Rule encourages a more nuanced approach. It highlights the fact that most knowledge falls on a spectrum of likelihood, not absolute categories.
Student I suppose I can see the value in that, especially for scientific inquiry where new evidence is constantly emerging.
Epistemologist Exactly! By avoiding absolute credences, we stay open to revising our beliefs as we learn more. It’s a more robust and adaptable way of approaching knowledge.
Epistemologist Today we’ll discuss Cromwell’s Rule, which advises against assigning prior probabilities of exactly 0 or 1, except for logically true or false statements.
Student But surely there are some cases where it makes sense to have an absolute credence of 0 or 1? What if I’m absolutely certain about something?
Epistemologist Even if you feel absolutely certain, assigning a probability of 0 or 1 represents a state of complete dismissal or complete commitment that precludes any possibility of revising your belief based on new evidence.
Student But if I know something with 100% certainty, like the fact that I exist, doesn’t that justify a probability of 1?
Epistemologist The problem is that very few propositions outside of logical tautologies or mathematical identities can be known with 100% certainty. There is always some degree of uncertainty in most real-world situations.
- Epistemologist: “Today, I’d like to introduce you to Cromwell’s Rule. This principle, named after Oliver Cromwell, suggests that in probabilistic reasoning, we should never assign a probability of 0 or 1 to any event unless it is logically impossible or a certain truth, respectively. This is.
- Student: “I understand the general idea, but aren’t there situations where assigning an absolute credence of 0 or 1 is actually justified? For example, if I see a fair coin flip result in heads, can’t I assign a probability of 1 to the proposition ‘The coin landed heads’?”
- Epistemologist: “That’s a good point, and it’s important to consider the context of observation. However, even in that scenario, Cromwell’s Rule would encourage you to think about the reliability of your observation. Could there have been an error in your perception? Is it possible, however.
- Student: “But what about mathematical truths? If I say ‘2 + 2 = 4’, isn’t it completely justified to assign a credence of 1 to this statement?”
- Epistemologist: “Yes, mathematical truths are an exception because they are tautological and based on logical certainty. Cromwell’s Rule is more applicable to empirical and contingent propositions where there is room for doubt due to the limitations of our knowledge and potential for new.
- Student: “I see. So, Cromwell’s Rule is more about maintaining a stance of epistemic humility in the face of empirical uncertainty. But doesn’t this make decision-making difficult if we always entertain the possibility of being wrong?”
Prompt 4: What resources related to Cromwell’s Rule are available?
What changes once we define Books more carefully
This section is worth asking because it changes what the reader should compare next. The point is to make Cromwell’s Rule more investigable, not merely more impressive-sounding.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
The pedagogical payoff is practical. After this section, the reader should be better able to explain Cromwell’s Rule in plain language, identify a likely misuse of it, and say what further evidence or argument would actually move the view.
“Probability Theory The Logic of Science” by E.T. Jaynes This book covers Bayesian probability theory extensively and discusses why assigning probabilities of 0 or 1 can be problematic in the context of scientific reasoning.
Stanford Encyclopedia of Philosophy Probability This entry provides a comprehensive overview of probability theory, including discussions on Bayesian probability, which is closely related to Cromwell’s Rule.
LessWrong Cromwell’s Rule An accessible article that explains Cromwell’s Rule in the context of rationality and decision-making, suitable for readers who are new to the concept.
Khan Academy Probability and Statistics While not specific to Cromwell’s Rule, Khan Academy offers foundational courses on probability and statistics, which can help understand the broader principles related to probabilistic reasoning.
YouTube Introduction to Bayesian Inference There are numerous videos explaining Bayesian inference, which often discuss why probabilities should not be absolute and how Cromwell’s Rule fits into this framework.
Coursera Bayesian Statistics: From Concept to Data Analysis This course provides a deeper dive into Bayesian statistics, where Cromwell’s Rule is relevant in avoiding absolute credences.
TED Talks The Power of Believing That You Can Improve by Carol Dweck Although not directly about Cromwell’s Rule, this talk on growth mindset aligns with the idea of remaining open to new evidence and adjusting beliefs accordingly.
Scholarly Articles While there might not be a plethora of articles specifically on Cromwell’s Rule itself (given it’s a principle within a broader field), you can find discussions of it in statistics and decision analysis journals. Look for articles that mention “prior probabilities” or “Bayesian reasoning.” Here’s a starting point, but using academic search engines like JSTOR or Google Scholar might yield more results: [scholarly article on cromwell’s rule]
Statistics Textbooks Many statistics textbooks discuss Bayesian reasoning and assigning probabilities. These sections might mention Cromwell’s Rule as a best practice. Try searching for textbooks with titles like “Introduction to Bayesian Statistics” or “Decision Theory.”
Online Resources Several websites explain Cromwell’s Rule in a clear and concise way. Here are a couple of examples: DeepAI: [Cromwell’s Rule definition ON DeepAI deepai.org] Wikipedia (though brief): [Cromwell’s rule Wikipedia en.wikipedia.org]
DeepAI [Cromwell’s Rule definition ON DeepAI deepai.org]
Wikipedia (though brief) [Cromwell’s rule Wikipedia en.wikipedia.org]
Dennis Lindley The statistician who named Cromwell’s Rule is Dennis Lindley. While finding resources directly on Cromwell’s Rule might be limited, exploring Lindley’s work on Bayesian statistics could provide a deeper understanding of the underlying concepts. You can search for his books or articles on Bayesian reasoning.
- Books: “The Logic of Scientific Discovery” by Karl Popper While not exclusively about Cromwell’s Rule, Popper’s philosophy of science emphasizes the importance of falsifiability and the tentativeness of scientific knowledge, which aligns well with the principles behind Cromwell’s Rule.
- Articles and Papers: “Cromwell’s Rule and the Strong Likelihood Principle” by Michael Goldstein This paper delves into the application of Cromwell’s Rule within Bayesian statistics and discusses its implications for scientific inference.
- Online Resources: This entry provides a comprehensive overview of probability theory, including discussions on Bayesian probability, which is closely related to Cromwell’s Rule.
- Videos and Lectures: Cromwell’s Rule encourages maintaining epistemic humility by avoiding absolute certainties in probabilistic reasoning.
- Belief calibration: Cromwell’s Rule concerns how strongly the available evidence warrants belief, disbelief, or suspension of judgment.
What ties this page together.
The best route is to track how evidence changes credence, how justification differs from psychological comfort, and how skepticism can discipline thought without paralyzing it.
The recurring pressure is false certainty: treating a feeling of obviousness, a social consensus, or a useful assumption as if it had already earned the status of knowledge.
Keep Cromwell’s Rule: An Elaboration, Hypothetical Discussion on Cromwell’s Rule, and Resources on Cromwell’s Rule in the same frame. That is what shows what the page is claiming, where it gets tested, and what would have to change if the claim is right.
Read this page as part of the wider Epistemology branch: the prompts point inward to the topic, but they also point outward to neighboring questions that keep the topic honest.
- Which online resource provides a comprehensive overview of probability theory, including discussions relevant to Cromwell’s Rule?
- What is the name of the principle that discourages assigning probabilities of 0 or 1 (except for logical truths)?
- According to the discussion, why should we avoid assigning absolute probabilities in most situations?
- Which distinction inside Cromwell’s Rule is easiest to miss when the topic is explained too quickly?
- What is the strongest charitable reading of this topic, and what is the strongest criticism?
Deep Understanding Quiz Check your understanding of Cromwell’s Rule
This quiz checks whether the main distinctions and cautions on the page are clear. Choose an answer, read the feedback, and click the question text if you want to reset that item.
Future Branches
Where this page naturally expands
This page belongs inside the wider Epistemology branch and is best read in conversation with neighboring topics. Use the branch guide, concept tags, and reading paths to keep the question moving rather than treating the page as a polite dead end.