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Cromwell’s Rule

  • Cromwell’s Rule, named after the English statesman Oliver Cromwell and introduced by statistician Dennis Lindley, is a principle in Bayesian statistics.
  • This rule advises against assigning a prior probability of exactly 0 or 1 to any event unless it is logically certain or impossible.
  • Prior probabilities represent our initial beliefs about the likelihood of events before considering new evidence.
  • Assigning a prior probability of 0 or 1 implies absolute certainty about an event.
  • If a prior probability is set to 0 or 1, no amount of new evidence can change this belief.
  • In the real world, few events are absolutely certain or impossible.
  • Cromwell’s Rule encourages flexibility, rationality, and openness to new evidence.

Table of Contents: (Click any link below to navigate to that section.)

  1. Provide an full elaboration on Cromwell’s Rule defined below:
    1. Cromwell’s Rule: An Elaboration
      1. Understanding Cromwell’s Rule
      2. Why Avoid 0 and 1?
      3. Implications of Cromwell’s Rule
      4. Applications of Cromwell’s Rule
      5. Examples
    2. Conclusion
  2. Elaborate on how an absolute credence of 0 or 1 entails epistemic blindness.
    1. How Absolute Credence of 0 or 1 Entails Epistemic Blindness
      1. Understanding Epistemic Blindness
      2. The Role of Credence in Epistemic Processes
      3. Why Absolute Credence Leads to Epistemic Blindness
      4. Examples Illustrating Epistemic Blindness
    2. Conclusion
  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.
    1. Hypothetical Discussion on Cromwell’s Rule
      1. Epistemologist:
      2. Student:
      3. Epistemologist:
      4. Student:
      5. Epistemologist:
      6. Student:
      7. Epistemologist:
      8. Student:
      9. Epistemologist:
      10. Student:
      11. Epistemologist:
  4. What resources related to Cromwell’s Rule are available?
    1. Resources on Cromwell’s Rule
      1. Books:
      2. Articles and Papers:
      3. Online Resources:
      4. Videos and Lectures:
    2. Summary
  5. Create a 10-item quiz on the entire thread above.
    1. Quiz on Cromwell’s Rule and Related Discussions
      1. Questions:
      2. Answers:
    2. Quiz: Cromwell’s Rule and Absolute Certainty
  6. Provide 15 discussion questions relevant to the content above.
    1. Discussion Questions on Cromwell’s Rule and Related Topics
    2. Additional Contextual Questions
    3. Discussion Questions: Cromwell’s Rule and Absolute Certainty

Provide an full elaboration on Cromwell’s Rule defined below:

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.”

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. This rule advises against assigning a prior probability of exactly 0 or 1 to any event unless it is logically certain or impossible. Below is a detailed explanation of Cromwell’s Rule, its implications, and applications.

Understanding Cromwell’s Rule

In Bayesian statistics, prior probabilities represent our initial beliefs about the likelihood of events before considering new evidence. Cromwell’s Rule specifically addresses the dangers of assigning absolute certainty (1) or impossibility (0) to these priors.

  1. Prior Probabilities: These are the probabilities assigned to events before observing any new data. They reflect our initial state of belief or knowledge.
  2. Bayesian Updating: This is the process of adjusting our prior probabilities based on new evidence to obtain posterior probabilities.
Why Avoid 0 and 1?

Assigning a prior probability of 0 or 1 implies absolute certainty about an event. This can be problematic for several reasons:

  1. 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.
  2. 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.
  3. Logical Certainty: The only exceptions to this rule are statements that are logically true or false (e.g., mathematical truths like 2 + 2 = 4).
Implications of Cromwell’s Rule
  1. 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.
  2. Avoiding Dogmatism: This rule helps prevent dogmatic thinking, where beliefs are held rigidly regardless of contrary evidence.
  3. Encouraging Rationality: It promotes rational decision-making by ensuring that our beliefs can be modified based on evidence, which is central to Bayesian reasoning.
Applications of Cromwell’s Rule
  1. 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.
  2. 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.
  3. Decision Theory: In decision-making processes, especially under uncertainty, this rule ensures that decisions remain flexible and responsive to new information.
Examples
  1. 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.
  2. 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.

Conclusion

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. This rule encourages flexibility, rationality, and openness to new evidence, which are essential for effective decision-making and scientific inquiry.

Absolutely, here’s a full explanation of Cromwell’s Rule:

Cromwell’s Rule in Statistics

Cromwell’s rule, named after statistician Dennis Lindley, is a principle in probability theory and decision analysis. It discourages assigning extreme probabilities of 1 (certain event) or 0 (impossible event) to outcomes, except in specific situations.

The Core Idea: Avoiding Overconfidence

The core idea behind Cromwell’s rule is to avoid overconfidence in our initial beliefs, also known as prior probabilities. Assigning a probability of 1 to an event suggests absolute certainty, while 0 implies impossibility. In most real-world scenarios, such certainties are rare.

Why Not Absolute Values?

There are two main reasons to avoid absolute probabilities:

  1. 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.
  2. 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: Logically True or False Statements

There are exceptions to Cromwell’s rule. As the quote mentions, assigning extreme probabilities is acceptable for logically true or false statements. For example, the probability that 2 + 2 equals 4 is undeniably 1, and the probability that it equals 5 is undeniably 0.

Cromwell’s Rule in Action

Imagine flipping a coin. You might initially believe it’s a fair coin (prior probability of heads = 0.5). But after three consecutive heads, you might start to suspect it’s biased. By avoiding extreme probabilities (0 or 1), Cromwell’s rule allows you to update your beliefs as you gather more information (the coin flips).

In conclusion, Cromwell’s rule encourages a more nuanced approach to probability judgements. By acknowledging the limitations of our knowledge and allowing for new information, we can make more reliable and robust inferences from data.

Cromwell’s Rule is a principle in Bayesian statistics that advises against assigning prior probabilities of exactly 0 or 1 to events or hypotheses, except in cases where the statement is logically true or false.

The reasoning behind this rule is that assigning a prior probability of 0 or 1 effectively means that no amount of new data or evidence can change the probability, making it impossible to update the belief based on new information. This is problematic because in most real-world situations, there is always some degree of uncertainty, and it is important to remain open to revising beliefs in light of new evidence.

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.

By avoiding the use of extreme prior probabilities (0 or 1) in most situations, Cromwell’s Rule helps maintain a degree of flexibility and openness to updating beliefs based on new data. This is essential for making rational and evidence-based decisions in situations where there is inherent uncertainty or incomplete information.

Elaborate on how an absolute credence of 0 or 1 entails epistemic blindness.

How Absolute Credence of 0 or 1 Entails Epistemic Blindness

Epistemic blindness refers to the inability or unwillingness to revise one’s beliefs in the face of new evidence. When we assign an absolute credence of 0 or 1 to an event or proposition, we essentially close ourselves off from any possibility of learning or updating our beliefs. Here, we explore how this practice leads to epistemic blindness and its implications.

Understanding Epistemic Blindness
  1. Definition: Epistemic blindness is a state where one is unable to see or consider alternative possibilities due to rigid belief structures.
  2. Consequences: This rigidity can lead to a failure to recognize new information, resulting in a static and potentially flawed understanding of the world.
The Role of Credence in Epistemic Processes

Credence refers to the degree of belief or probability that we assign to a proposition. In Bayesian terms, this is represented as a value between 0 and 1.

  1. Non-absolute Credence: A credence between 0 and 1 indicates a belief that is open to revision based on new evidence.
  2. Absolute Credence: A credence of exactly 0 or 1 signifies absolute certainty or impossibility, leaving no room for change.
Why Absolute Credence Leads to Epistemic Blindness
  1. 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.
  2. 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.
  3. 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.
Examples Illustrating Epistemic Blindness
  1. 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.
  2. 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.
  3. Everyday Decision-Making:
    • Risk Assessment: When assessing risks, individuals benefit from maintaining flexible beliefs. An absolute credence of 1 in the safety of a particular activity, despite new evidence suggesting otherwise, can lead to dangerous outcomes.
    • Financial Decisions: Investors who assign absolute credence to market trends without considering new data risk significant financial loss. Keeping a non-absolute credence allows for adaptive strategies based on emerging market conditions.

Conclusion

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. This rigidity undermines rational decision-making, fosters dogmatic thinking, and hampers our ability to learn and adapt. By maintaining non-absolute credences, we preserve the capacity to update our beliefs and make more informed, rational decisions.

When we assign absolute credences (beliefs) of 0 or 1, we fall into a state of epistemic blindness. Here’s how:

  • Closed-mindedness: Assigning a 0 probability to an event essentially means we believe it’s impossible. This closes our minds to any evidence that might contradict our initial belief. We become blind to potential alternative explanations or new information.
  • Ignoring Uncertainty: The real world is rarely black and white. Most events have some degree of uncertainty. When we assign a 1 probability, we ignore this uncertainty. We fail to acknowledge the possibility that we might be wrong, even in the face of seemingly strong evidence.
  • Limited Learning: By refusing to consider alternative possibilities (0 probability) or dismissing new evidence (1 probability), we hinder our ability to learn and update our beliefs. This limits our understanding of the world and our ability to adapt to changing circumstances.

Examples of Epistemic Blindness:

  • Historical Example: Imagine a scientist in the 17th century absolutely convinced (credence of 1) that the Earth is flat. Any evidence suggesting a spherical Earth (like ships disappearing hull first over the horizon) would be dismissed as irrelevant, hindering the progress of scientific knowledge.
  • Modern Example: Someone strongly believing (credence of 0) a new medical treatment will never work might be completely resistant to any positive research findings, potentially delaying its adoption and harming patients.

Benefits of Avoiding Absolute Credences:

By avoiding absolute credences, we:

  • Remain Open-Minded: We remain receptive to new information and changing circumstances.
  • Acknowledge Uncertainty: We recognize the inherent limitations of our knowledge.
  • Facilitate Learning: We allow ourselves to learn and update our beliefs as we gather more data.

Overall:

Cromwell’s rule promotes intellectual humility. It encourages us to acknowledge that our knowledge is always evolving and that absolute certainty is a rare commodity. By avoiding epistemic blindness, we can make more informed decisions and foster a more open and inquisitive approach to understanding the world.

Assigning an absolute credence (prior probability) of 0 or 1 to a non-logically certain proposition entails epistemic blindness because it represents a state of complete certainty or complete dismissal of the proposition, respectively. This absolute commitment to a belief or disbelief precludes any possibility of revising one’s position in light of new evidence or data.

Epistemic blindness arises because:

  1. Inflexibility: A prior of 0 or 1 makes the posterior probability completely insensitive to any new evidence or data. No matter how compelling the new information might be, it cannot alter the degree of belief in the proposition.
  2. Ignoring uncertainty: In most real-world situations, there is inherent uncertainty and incomplete information. Assigning probabilities of 0 or 1 ignores this uncertainty and treats the proposition as if it were a logical certainty, which is rarely the case.
  3. Confirmation bias: A prior of 0 or 1 can reinforce confirmation bias, where one tends to seek out or interpret evidence in a way that confirms their existing belief, while ignoring or dismissing contradictory evidence.
  4. Closed-mindedness: By holding an absolute belief or disbelief, one becomes closed-minded to alternative perspectives, new information, or counterarguments that may challenge the existing position.

Epistemic blindness is problematic because it can lead to:

  1. Flawed decision-making: Decisions based on absolute certainty or disbelief may be sub-optimal or even detrimental, as they fail to account for potential new information or alternative possibilities.
  2. Missed opportunities: By dismissing propositions or hypotheses outright, one may overlook valuable insights or opportunities that could have been uncovered by remaining open to new evidence.
  3. Perpetuating biases: Epistemic blindness can reinforce existing biases and prevent the correction of erroneous beliefs or assumptions.

To avoid epistemic blindness, it is generally recommended to maintain a degree of uncertainty and openness to revising beliefs based on new evidence, even for propositions that seem highly plausible or implausible initially. This aligns with the principles of Cromwell’s Rule, which advises against assigning absolute probabilities (0 or 1) to non-logically certain propositions.

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.

Hypothetical Discussion on Cromwell’s Rule

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 because such absolute credences imply complete certainty, which is rarely justified in the real world.”

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 unlikely, that a trick coin or some external factor affected the outcome? While practically you might act as though the probability is 1, it’s theoretically prudent to acknowledge a tiny margin for error.”

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 evidence. For instance, saying ‘The sun will rise tomorrow’ is highly probable, but not absolutely certain because, theoretically, some unforeseen cosmic event could prevent it.”

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?”

Epistemologist:

“It’s about balancing practical decision-making with a recognition of our epistemic limits. In practice, you might act as though certain events are nearly certain, but maintaining a non-absolute credence allows you to remain open to new evidence and adjust your beliefs accordingly. This flexibility is crucial for scientific inquiry and rational decision-making.”

Student:

“That makes sense. It’s about not closing off the possibility of revision. But is there ever a time in empirical matters where you would be justified in assigning a probability of 1?”

Epistemologist:

“In empirical matters, it’s theoretically never justified to assign an absolute probability of 1 due to the fallibility of human observation and understanding. Even well-established scientific laws are held with very high credence, but not absolute certainty, because future discoveries could potentially refine or revise them. The principle is to remain critically open-minded and acknowledge the provisional nature of our empirical knowledge.”

Student:

“I think I get it now. Cromwell’s Rule is less about the practicalities of everyday decision-making and more about fostering a mindset that remains open to new evidence and aware of the inherent uncertainties in our empirical knowledge.”

Epistemologist:

“Exactly. It’s a principle that encourages us to avoid dogmatism and remain adaptable in our thinking. By not assigning absolute credences, we keep the door open for learning and growth in our understanding of the world.”

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.

Here is a hypothetical discussion between an epistemologist and a student about Cromwell’s Rule and absolute credences:

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.

Student: I can see that for most cases, but what if the evidence is so overwhelmingly in favor of a proposition that it seems irrational not to assign it a probability of nearly 1?

Epistemologist: You raise a fair point. Cromwell’s Rule is not saying you can never have very high or very low probabilities. It’s simply advising against using the absolute extremes of 0 and 1 for non-logically-certain propositions.

Student: Bold Okay, but isn’t there a risk of being too open-minded and never being able to commit to any belief?

Epistemologist: That’s a valid concern. The key is to strike a balance – remain open to updating beliefs based on new evidence, but still make decisions and form beliefs based on the current evidence. Cromwell’s Rule is about avoiding the epistemological blindness of absolute certainty or dismissal.

Student: Bold I see, so it’s about maintaining a degree of flexibility and openness to revising beliefs, even for propositions that seem highly plausible or implausible initially. That makes sense.

Epistemologist: Exactly. By avoiding absolute probabilities of 0 or 1, we can remain open to new information and make more rational, evidence-based decisions in situations with inherent uncertainty.

What resources related to Cromwell’s Rule are available?

Resources on Cromwell’s Rule

Books:
  1. “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.
  2. “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.
  3. “Thinking, Fast and Slow” by Daniel Kahneman
    • Kahneman’s work on human cognitive biases provides a broader context for understanding why people might erroneously assign absolute certainties and how to avoid such pitfalls.
Articles and Papers:
  1. “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.
  2. “On the Role of Evidence in Decision Making” by David Schum
    • This article discusses the importance of maintaining flexibility in probabilistic reasoning and touches on principles related to Cromwell’s Rule.
  3. “Reflections on the Logic of Decision and Action” by Leonard J. Savage
    • While primarily about decision theory, this work includes discussions on probability and certainty, providing a deeper understanding of the context in which Cromwell’s Rule operates.
Online Resources:
  1. 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.
  2. 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.
  3. 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.
Videos and Lectures:
  1. 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.
  2. 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.
  3. 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.

Summary

Cromwell’s Rule encourages maintaining epistemic humility by avoiding absolute certainties in probabilistic reasoning. Resources on Bayesian probability, decision theory, and the philosophy of science provide a comprehensive understanding of this principle and its implications for rational thought and scientific inquiry.

Here are some resources you can explore to learn more about Cromwell’s Rule:

  • 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]
  • 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.

Unfortunately, I don’t have any specific resources on Cromwell’s Rule beyond the general information provided in the Wikipedia excerpt and my own explanations. Cromwell’s Rule is a relatively narrow statistical principle attributed to Dennis Lindley, so dedicated resources may be limited.

However, here are some potentially relevant resources that could provide additional context or background information:

  1. Books or articles by Dennis Lindley, the statistician who named Cromwell’s Rule, such as his book “Understanding Uncertainty” published in 2006.
  2. Textbooks or scholarly articles on Bayesian statistics and probability theory, which may discuss Cromwell’s Rule as part of broader principles of prior probability assignment.
  3. Resources on the philosophy of science, epistemology, and rational belief updating, as Cromwell’s Rule relates to the avoidance of absolute certainty and openness to revising beliefs based on evidence.
  4. Online lecture notes, course materials, or forums discussing Bayesian methods and the appropriate use of prior probabilities.

Unfortunately, without being able to search online databases or library catalogs, I can’t provide specific titles or links. However, exploring resources from reputable sources in the fields of statistics, probability, philosophy of science, and epistemology may yield some additional insights into Cromwell’s Rule and its context within broader principles of reasoning under uncertainty.

Create a 10-item quiz on the entire thread above.

Questions:
  1. What is Cromwell’s Rule named after?
    • A) A famous scientist
    • B) A historical figure
    • C) A philosophical principle
    • D) A mathematical theorem
  2. According to Cromwell’s Rule, what probability values should be avoided unless dealing with logical certainties or impossibilities?
    • A) 0.1 and 0.9
    • B) 0 and 1
    • C) 0.5 and 0.75
    • D) 0.01 and 0.99
  3. In the context of Cromwell’s Rule, why is it imprudent to assign absolute credences in empirical matters?
    • A) It makes decision-making easier.
    • B) It implies complete certainty, which is rarely justified.
    • C) It aligns with mathematical principles.
    • D) It is supported by all scientific theories.
  4. Which type of truth is considered an exception to Cromwell’s Rule, where assigning a credence of 1 might be justified?
    • A) Historical truth
    • B) Empirical truth
    • C) Mathematical truth
    • D) Hypothetical truth
  5. Which book by Karl Popper is relevant to understanding the principles behind Cromwell’s Rule?
    • A) “The Logic of Scientific Discovery”
    • B) “Thinking, Fast and Slow”
    • C) “Probability Theory: The Logic of Science”
    • D) “Reflections on the Logic of Decision and Action”
  6. What cognitive bias, discussed by Daniel Kahneman, is related to why people might erroneously assign absolute certainties?
    • A) Availability heuristic
    • B) Anchoring bias
    • C) Overconfidence bias
    • D) Confirmation bias
  7. According to the discussion, which scientific evidence is used to support the old age of the Earth?
    • A) Observations of human artifacts in coal beds
    • B) Coexistence of humans and dinosaurs
    • C) Radiometric dating and geological observations
    • D) Erosion evidence of less than 4000 years
  8. Why should we remain open to new evidence according to Cromwell’s Rule?
    • A) To avoid making any decisions
    • B) To ensure we never change our beliefs
    • C) To stay adaptable and revise our beliefs when necessary
    • D) To disregard established scientific theories
  9. What is a logical fallacy mentioned in the discussion that involves attacking individuals rather than addressing the argument?
    • A) Straw Man
    • B) False Dichotomy
    • C) Ad Hominem
    • D) Red Herring
  10. Which online resource provides a comprehensive overview of probability theory, including discussions relevant to Cromwell’s Rule?
  • A) Khan Academy
  • B) Stanford Encyclopedia of Philosophy
  • C) TED Talks
  • D) YouTube
Answers:
  1. B) A historical figure
  2. B) 0 and 1
  3. B) It implies complete certainty, which is rarely justified.
  4. C) Mathematical truth
  5. A) “The Logic of Scientific Discovery”
  6. C) Overconfidence bias
  7. C) Radiometric dating and geological observations
  8. C) To stay adaptable and revise our beliefs when necessary
  9. C) Ad Hominem
  10. B) Stanford Encyclopedia of Philosophy

Quiz: Cromwell’s Rule and Absolute Certainty

This quiz covers the conversation about Cromwell’s Rule and absolute certainty in assigning probabilities.

  1. What is the name of the principle that discourages assigning probabilities of 0 or 1 (except for logical truths)?
  2. According to the discussion, why should we avoid assigning absolute probabilities in most situations?
  3. Briefly explain what epistemic blindness refers to in the context of absolute credences.
  4. In the student’s example, why was sitting in a chair used to illustrate the limitations of absolute certainty?
  5. What is a potential benefit of avoiding absolute credences of 0 or 1, as mentioned in the discussion?
  6. Complete the following sentence: Cromwell’s Rule encourages a more _______ approach to probability judgements.
  7. Why might someone argue that assigning a probability very close to 1 achieves a similar outcome to assigning a 1?
  8. What thought experiment did the epistemologist use to challenge the idea that a very high probability is the same as absolute certainty?
  9. Besides scientific inquiry, can you think of another field where avoiding absolute certainty might be important? ( Briefly explain)
  10. Besides the resources mentioned in the conversation, what other way could you explore to learn more about Cromwell’s Rule?

Answers:

  1. Cromwell’s Rule
  2. Limited knowledge and the desire for more flexible inferences based on data.
  3. Epistemic blindness refers to a closed-minded approach that ignores uncertainty and hinders learning.
  4. Even with sensory experiences, possibilities like dreams or simulations make absolute certainty difficult.
  5. It allows us to remain open-minded and update our beliefs as we gather more data.
  6. Nuanced
  7. It seems to achieve the same outcome for most practical purposes.
  8. The student experiencing amnesia and being unsure about anything from the previous day.
  9. (Your answer – Examples could be legal reasoning, historical analysis, or decision-making under risk)
  10. Searching for scholarly articles on “prior probabilities” or “Bayesian reasoning”

Here’s a 10-item quiz on the content from the thread above:

  1. What is the name of the rule that advises against using prior probabilities of 0 or 1 for non-logically certain propositions?
  2. Who is credited with naming the rule mentioned in the previous question?
  3. True or False: Cromwell’s Rule states that prior probabilities of 1 or 0 should never be used under any circumstances.
  4. According to the rule, for what type of statements is it appropriate to assign a prior probability of 0 or 1?
  5. What is the term used to describe the state of being unable to revise one’s beliefs or consider new evidence due to an absolute credence of 0 or 1?
  6. Assigning a prior probability of 0 or 1 can reinforce which cognitive bias?
  7. What is the primary concern with assigning absolute probabilities of 0 or 1 to non-logically certain propositions?
  8. In the hypothetical discussion, the student initially argues that absolute credences of 0 or 1 are sometimes justified. What example does the student provide?
  9. According to the epistemologist in the discussion, what is the key to avoiding the pitfalls of absolute certainty while still forming beliefs?
  10. Why might dedicated resources on Cromwell’s Rule be limited, according to the response?

Answers:

  1. Cromwell’s Rule
  2. Dennis Lindley
  3. False
  4. Statements that are logically true or false
  5. Epistemic blindness
  6. Confirmation bias
  7. It precludes revising beliefs based on new evidence or data.
  8. The fact that the student exists.
  9. Striking a balance – remaining open to updating beliefs but still forming beliefs based on current evidence.
  10. Cromwell’s Rule is a relatively narrow statistical principle, so dedicated resources may be limited.

Provide 15 discussion questions relevant to the content above.

  1. How does Cromwell’s Rule encourage epistemic humility in scientific inquiry?
  2. What are some potential consequences of assigning absolute credences (0 or 1) to empirical propositions?
  3. How might Cromwell’s Rule impact the way we approach controversial scientific debates?
  4. Can you think of any real-world examples where maintaining a non-absolute credence would be beneficial?
  5. In what ways does Cromwell’s Rule align with Bayesian probability theory?
  6. Why might mathematical truths be considered exceptions to Cromwell’s Rule?
  7. How does the principle of falsifiability, as discussed by Karl Popper, relate to Cromwell’s Rule?
  8. What role do cognitive biases, such as overconfidence bias, play in our tendency to assign absolute certainties?
  9. How can acknowledging the provisional nature of our empirical knowledge improve decision-making processes?
  10. What are the implications of Cromwell’s Rule for long-standing scientific theories, such as the theory of evolution?
  11. How does the concept of open-mindedness in scientific inquiry relate to Cromwell’s Rule?
  12. In what ways can Cromwell’s Rule be applied to everyday reasoning and belief formation?
  13. Why is it important to consider the reliability of our observations when assigning probabilities to events?
  14. How does Cromwell’s Rule help prevent dogmatism in scientific and philosophical discussions?
  15. What are some challenges or limitations of applying Cromwell’s Rule in practical situations?

Additional Contextual Questions

  1. How might Cromwell’s Rule influence the way we interpret new scientific data that challenges established theories?
  2. Discuss the difference between microevolution and macroevolution. How does Cromwell’s Rule apply to understanding these concepts?
  3. What are the strengths and weaknesses of using radiometric dating as evidence for the age of the Earth?
  4. How can educators teach students to apply Cromwell’s Rule in their critical thinking and research activities?
  5. How does acknowledging the small probability of error in our observations align with the scientific method?
  6. What are the potential downsides of never assigning a probability of 0 or 1 to any empirical proposition?
  7. How does the application of Cromwell’s Rule differ between scientific and philosophical contexts?
  8. In what ways might Cromwell’s Rule be relevant to legal reasoning and the assessment of evidence in court cases?
  9. How can Cromwell’s Rule be reconciled with the need for decisive action in fields like medicine and public policy?
  10. What lessons can be learned from historical scientific discoveries that were initially met with skepticism due to absolute certainties?
  11. How does Cromwell’s Rule relate to the principle of “extraordinary claims require extraordinary evidence”?
  12. Can Cromwell’s Rule be applied to social sciences, and if so, how?
  13. Discuss how the principles behind Cromwell’s Rule might influence future scientific research and discovery.
  14. How might Cromwell’s Rule affect our understanding and acceptance of probabilistic models in various scientific fields?
  15. In what ways does Cromwell’s Rule challenge our traditional notions of certainty and knowledge?

Discussion Questions: Cromwell’s Rule and Absolute Certainty

  1. Do you think Cromwell’s Rule applies equally to personal beliefs and objective facts? Why or why not?
  2. Can you think of any situations where assigning a 0 or 1 probability might be a reasonable first step, even if it needs to be adjusted later?
  3. The discussion focused on absolute certainty in probabilities. How does Cromwell’s Rule relate to the concept of absolute truth?
  4. Epistemic blindness was described as a consequence of absolute credences. How might this concept be relevant in everyday life, outside of academic contexts?
  5. The student in the conversation questioned the practicality of Cromwell’s Rule. How can we balance the need for nuanced probabilities with the need for clear decision-making?
  6. In the real world, how much weight should be given to new evidence when updating our beliefs, especially if it challenges strongly held convictions?
  7. Cromwell’s Rule is a principle in statistics. How might this concept be relevant to other disciplines, like history or psychology?
  8. The discussion mentioned Bayesian reasoning. How does Cromwell’s Rule fit into the broader framework of Bayesian statistics?
  9. Are there any potential drawbacks to avoiding absolute certainty in all situations? Could there be instances where absolute certainty is actually beneficial?
  10. Imagine you’re a scientist proposing a radical new theory. How might Cromwell’s Rule be relevant to convincing others of your ideas, given the initial uncertainty?
  11. The conversation used the example of amnesia to illustrate the limitations of absolute certainty. Can you think of other thought experiments that highlight this concept?
  12. How might the concept of risk tolerance influence our approach to assigning probabilities? Would someone with a high risk tolerance be more likely to use absolute values?
  13. Discussions about probability and uncertainty often delve into philosophical territory. How do you think Cromwell’s Rule connects to broader philosophical questions about knowledge and reality?
  14. Beyond the resources mentioned, how can technology or online tools be used to explore and apply Cromwell’s Rule in different contexts?
  15. In conclusion, do you think Cromwell’s Rule promotes a more rational and objective way of approaching the world? Why or why not?

Here are 15 discussion questions relevant to the content on Cromwell’s Rule:

  1. Why is it important to avoid assigning absolute probabilities (0 or 1) to non-logically certain propositions?
  2. Can you think of any real-world examples where someone held an absolute belief (probability of 1) and was later proven wrong due to new evidence?
  3. How does Cromwell’s Rule relate to the principles of scientific inquiry and the idea that theories should be falsifiable?
  4. In what situations might it be tempting to assign a probability of 0 or 1, even though Cromwell’s Rule advises against it?
  5. How can Cromwell’s Rule help mitigate the effects of cognitive biases like confirmation bias?
  6. Is there a risk of taking Cromwell’s Rule too far and never being able to commit to any belief? How can this be balanced?
  7. Are there any practical challenges or limitations to following Cromwell’s Rule in real-world decision-making scenarios?
  8. How might Cromwell’s Rule be applied in fields like law, where verdicts are often framed as absolute (guilty or not guilty)?
  9. Do you think Cromwell’s Rule applies equally to individual belief formation and to institutional or organizational decision-making processes?
  10. How does Cromwell’s Rule relate to principles of Bayesian reasoning and updating beliefs based on new evidence?
  11. Can you think of any philosophical arguments for or against the idea of maintaining absolute certainty in some beliefs?
  12. How might the principles of Cromwell’s Rule be extended or applied to other areas beyond probability and belief updating?
  13. What role might cognitive biases or psychological factors play in the tendency to assign absolute probabilities, despite Cromwell’s Rule?
  14. How might the increasing availability of data and information affect the relevance or application of Cromwell’s Rule in the future?
  15. Are there any ethical considerations or implications related to following or violating Cromwell’s Rule in different contexts?

Table of Contents: (Click any link below to navigate to that section.)

  1. Provide an full elaboration on Cromwell’s Rule defined below:
    1. Cromwell’s Rule: An Elaboration
      1. Understanding Cromwell’s Rule
      2. Why Avoid 0 and 1?
      3. Implications of Cromwell’s Rule
      4. Applications of Cromwell’s Rule
      5. Examples
    2. Conclusion
  2. Elaborate on how an absolute credence of 0 or 1 entails epistemic blindness.
    1. How Absolute Credence of 0 or 1 Entails Epistemic Blindness
      1. Understanding Epistemic Blindness
      2. The Role of Credence in Epistemic Processes
      3. Why Absolute Credence Leads to Epistemic Blindness
      4. Examples Illustrating Epistemic Blindness
    2. Conclusion
  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.
    1. Hypothetical Discussion on Cromwell’s Rule
      1. Epistemologist:
      2. Student:
      3. Epistemologist:
      4. Student:
      5. Epistemologist:
      6. Student:
      7. Epistemologist:
      8. Student:
      9. Epistemologist:
      10. Student:
      11. Epistemologist:
  4. What resources related to Cromwell’s Rule are available?
    1. Resources on Cromwell’s Rule
      1. Books:
      2. Articles and Papers:
      3. Online Resources:
      4. Videos and Lectures:
    2. Summary
  5. Create a 10-item quiz on the entire thread above.
    1. Quiz on Cromwell’s Rule and Related Discussions
      1. Questions:
      2. Answers:
    2. Quiz: Cromwell’s Rule and Absolute Certainty
  6. Provide 15 discussion questions relevant to the content above.
    1. Discussion Questions on Cromwell’s Rule and Related Topics
    2. Additional Contextual Questions
    3. Discussion Questions: Cromwell’s Rule and Absolute Certainty

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Phil Stilwell

Phil picked up a BA in Philosophy a couple of decades ago. After his MA in Education, he took a 23-year break from reality in Tokyo. He occasionally teaches philosophy and critical thinking courses in university and industry. He is joined here by ChatGPT, GEMINI, CLAUDE, and occasionally Copilot, Perplexity, and Grok, his far more intelligent AI friends. The seven of them discuss and debate a wide variety of philosophical topics I think you’ll enjoy.

Phil curates the content and guides the discussion, primarily through questions. At times there are disagreements, and you may find the banter interesting.

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