Asemics :=: Degree of Assent: Locke, Hume, Bayes: Induction and Probability - AI / EZE, 2026

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Locke, Hume, and Bayes fundamentally shaped the understanding of knowledge, probability, and induction. John Locke grounded knowledge in sensory experience, while David Hume famously challenged the rational basis of induction, arguing our expectation of the future is based on habit, not logic. Thomas Bayes provided a mathematical framework (Bayesianism) that uses probabilistic reasoning to update beliefs based on evidence, offering a potential—though debated—response to Hume’s skepticism by quantifying how experience justifies induction. [1, 2, 3, 4, 5]

David Hume: The Problem of Induction

Hume argues in his Treatise of Human Nature and Enquiry that induction—the process of drawing general conclusions from specific observations (e.g., "the sun will rise because it always has")—is not grounded in reason or logic. [1, 2]

• Uniformity Principle (UP): Inductive leaps assume nature follows a uniform course, where the future resembles the past.
• The Dilemma:
  • Demonstrative Reasoning: Proving the UP is impossible because its opposite is not a contradiction (i.e., we can imagine the future being different).
  • Probable/Moral Reasoning: Using past experience to prove the future will resemble the past is circular reasoning, as it assumes the very principle it tries to prove.
• Skeptical Solution: Hume concludes that induction is not irrational, but rather a "natural instinct" or custom/habit. [1, 2, 3, 4, 5]

John Locke: Empirical Foundations

Locke's empiricism laid the groundwork for Hume's work. [1]

• Knowledge from Experience: Locke argued against innate ideas, proposing the mind is a tabula rasa (blank slate) filled by experience.
• Degrees of Assent: While Locke did not fully tackle induction, he did distinguish between certain knowledge and probability, noting that much of human life is guided by "opinion" and "probability" based on observation. [1, 2, 3, 4, 5]

Bayes and the Probabilistic Response

Bayesianism uses probability to quantify the degree of belief in a hypothesis (posterior probability) based on prior knowledge (prior probability) and new evidence. [1, 2, 3, 4]

• Updating Beliefs: The Bayesian framework updates the probability of a hypothesis (\(H\)) based on new data (\(E\)), using Bayes's Theorem: \(P(H|E) = \frac{P(E|H)P(H)}{P(E)}\).
• Addressing Hume: Some philosophers argue that while induction cannot be proved a priori (as Hume insisted), Bayesian conditioning acts as a logical "consistency constraint" that allows us to build reasonable predictions.
• Empirical Limits: The Bayesian solution relies on an initial "prior" (a prior belief), which is empirical rather than rational, meaning it may not provide the purely logical solution Hume said was impossible. [1, 2, 3, 4, 5]

Summary Table

Philosopher [1, 2, 3, 4, 5]	Key Concept	Perspective on Induction
Locke	Empiricism (Tabula Rasa)	Knowledge is derived from experience.
Hume	Problem of Induction	Inductive inferences are from habit, not reason.
Bayes	Probabilistic Inference	Induction is updating beliefs using conditional probability.

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Lock Hume Bayes - On Induction - AI

John Locke (empiricism), David Hume (skepticism), and Thomas Bayes (probability) provide foundational, contrasting views on induction—reasoning from observed experiences to unobserved conclusions. Hume argued induction is an unjustifiable, irrational habit, which Bayesianism attempts to resolve by quantifying evidence and rational belief updating. [1, 2, 3, 4, 5]

David Hume: The Skeptical Problem of Induction

• The Problem: Hume argued that inductive reasoning relies on the "uniformity principle" (the future will resemble the past), which cannot be proven by deduction or experience without begging the question.
• Habitual Mind: Hume proposed that induction is not rational, but a "habit of the mind" or custom (imagination) derived from observing repeated, consistent conjunctions of events.
• Conclusion: We cannot justify induction; we simply act based on associations. [1, 2, 3, 4, 5]

Thomas Bayes: The Probabilistic Approach [1]

• Bayes' Rule: Thomas Bayes and followers like Richard Price offered a mathematical framework for evaluating evidence, providing a "rational" way to update beliefs (posterior probability) based on new data (likelihood) and previous knowledge (prior probability).
• Overcoming Skepticism: While Hume said uniform past experience cannot prove future outcomes, Bayesians argue that as more data accumulates, the posterior probability of a hypothesis (e.g., "the sun will rise tomorrow") approaches certainty (1).
• The "Problem of the Priors": Critics note that the initial "prior" probability is subjective, meaning Bayesian induction is still not completely objective until the evidence dominates the prior. [1, 2, 3, 4, 5]

John Locke: The Empiricist Background [1]

• Idea Association: Locke believed knowledge arises from sensory experience (impressions), setting the stage for Hume's empiricism.
• Inference: Rather than formal deduction, Locke focused on the relation between ideas to form judgments. [1, 2, 3, 4]

Key Connections

• Hume vs. Bayes: Bayesians reject Hume’s view that induction is irrational, arguing that Bayesian updating provides a rational, logical, and probabilistic basis for learning from experience.
• Indirect Link: There is no direct evidence Bayes read Hume, but Richard Price (who helped publish Bayes' work) was a key, direct critic of Hume's skeptical view of probability. [1, 2]

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Locke Hume Bayes - On Probability - AI

Locke, Hume, and Bayes shaped probability from epistemological and mathematical perspectives: Locke focused on "degrees of assent" based on evidence, Hume analyzed probability as a psychological "superiority of chances" and a "vivacity of belief", and Bayes provided the mathematical framework to update probabilities based on new data. [1, 2, 3, 4]

John Locke (Probability and Assent)

• Degrees of Assent: Locke argued in An Essay Concerning Human Understanding that because certain knowledge is rare, we must rely on probability to guide our assent to truths.
• Evidence-Based: Probability for Locke is the likelihood of a statement being true based on the conformity of our experience or the testimony of others.
• Association Mechanism: He warned that wrong associations of ideas prevent people from properly examining evidence and measuring true probability. [1, 2, 3, 4]

David Hume (Probability and Causation)

• Superiority of Chances: Hume defines probability as a "superiority of chances" on one side, which increases our confidence (or "belief") in that outcome.
• Psychological Basis: He believed probability is not just a calculation, but a "vivacity of thought" or a "stronger conception" of one event over another.
• Against Miracles: Hume applied his view of probability to miracles, arguing that a miracle is a violation of the laws of nature—of which we have firm, uniform experience—making the evidence against it always stronger than the evidence for it. [1, 2, 3, 4]

Thomas Bayes (Bayesian Probability)

• Updating Beliefs: Bayes's theorem (\(P(H|D) = \frac{P(D|H) \cdot P(H)}{P(D)}\)) allows for updating the probability of a hypothesis (\(H\)) based on new data (\(D\)).
• Prior to Posterior: It provides a formula for moving from a "prior" probability (before data) to a "posterior" probability (after data).
• Inverse Probability: His work, published posthumously, laid the foundation for modern probabilistic inference, often used to resolve problems of induction similar to those raised by Hume. [1, 2, 3, 4]

Key Intersections

• Hume vs. Bayes on Miracles: While Hume argues based on the weight of past experience, modern Bayesians can use Bayes's theorem to show how even unlikely events (like miracles) might be considered based on evidence, though they must overcome an incredibly low "prior" probability.
• Psychology to Math: Hume explains why we trust in higher probability (the mind feels a stronger vivacity), while Bayes provides the mathematical method for calculating it. [1, 2, 3, 4, 5]

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Locke on Degrees of Assent = AI

John Locke discusses the "Degrees of Assent" in Book IV, Chapter XVI of his An Essay Concerning Human Understanding. He argues that because human knowledge is limited, assent—the acceptance of a proposition as true without demonstrative proof—must be regulated by the "grounds of probability," ranging from full assurance to conjecture, doubt, and distrust. [1, 2, 3, 4]

Key Aspects of Locke on Degrees of Assent:

• Location: Book IV, Chapter XVI, titled "Of the Degrees of Assent," specifically sections 1–4.
• Core Principle: Assent should be proportional to the evidence or probability of the proposition.
• Foundations of Probability: Locke bases probability on two main foundations: the conformity of a testimony to one's own experience, or the testimony of others.
• Degrees of Belief:
  • Assurance: The highest degree, approaching knowledge, based on strong evidence or near-universal testimony.
  • Confidence: A high degree of belief, supported by experience and testimony.
  • Conjecture/Doubt: Lower degrees of assent, used when evidence is weak or conflicting.
• Purpose: The aim is to guide the understanding, ensuring that belief is not arbitrary but "regulated by the grounds of probability". [1, 2, 3, 4, 5]

Locke’s discussion emphasizes that, when certain knowledge is unavailable, we should rely on probabilistic reasoning, acting as an "under-laborer" to weigh testimony and experience. [1, 2]

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Hume on Degree of Assent - AI

For David Hume, the degree of assent given to any belief about matters of fact is entirely determined by the force and vivacity of the idea in the mind. Assent is not a voluntary choice made by reason, but a natural, involuntary feeling or sentiment directly proportional to empirical evidence. [1, 2, 3, 4]

🧠 Mechanics of Assent

• Perception Strength: Believing an idea means conceiving it with a force and vivacity that approaches the strength of a direct sensory impression.
• No Added Concept: Assent does not add a new component to an idea. It only alters the psychological manner in which the mind experiences it.
• Habitual Conditioning: Custom and habit act as the psychological mechanisms that transfer energy from past experiences into present ideas. [1, 2, 3]

📊 Proofs vs. Probabilities

Hume classifies matters of fact based on the uniformity of human experience, which dictates our level of certainty: [1, 2, 3, 4]

Category [1, 2, 3, 4]	Empirical Track Record	Resulting Degree of Assent
Proofs	Constant, uncontradicted conjunction of events over time.	Maximum assurance and complete, unhesitating belief.
Probabilities	Inconstant, mixed, or conflicting past experiences.	Hesitant, divided assent proportional to the superior evidence.

⚖️ The Wise Man’s Maxim

In An Enquiry Concerning Human Understanding, Hume establishes a strict rule for balancing evidence: "A wise man proportions his belief to the evidence." [1, 2, 3, 4]

• Evidential Counterpoise: When two opposing experiences conflict, the mind weighs them against each other.
• Subtractive Force: The weight of the weaker evidence subtracts directly from the weight of the stronger evidence.
• Residual Assent: The final degree of assent is determined solely by the force that remains after this subtraction.
• Miracle Rejection: This serves as his core argument against miracles, as the absolute empirical proof of a law of nature will always overpower the probability of human testimony. [1, 2]

Would you like to explore how Hume applies this framework to human testimony, or should we examine how passions and emotions can artificially amplify our degrees of assent? [1]

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Bayes on Degree of Assent - AI

Bayesian epistemology equates the "degree of assent" with subjective probability or credence, placing belief on a spectrum between 0 (full disbelief) and 1 (absolute certainty). This approach formalizes how individuals should update their confidence in a proposition as new evidence becomes available, linking the strength of belief directly to the weight of evidence. [1, 2, 3, 4, 5]

Here is an overview of Bayes' approach to the degree of assent:

1. Assent as Probability (Credence)

Bayesians represent the degree of assent as a numerical value between 0 and 1, often called "credence". [1]

• 0: Represents full disbelief or rejection of a claim.
• 1: Represents complete acceptance or absolute certainty.
• Intermediate values: Represent partial beliefs, such as a 70% confidence that a hypothesis is true. [1, 2]

2. Updating Assent via Conditionalization

The central mechanism for managing assent is conditionalization (Bayes' Rule), which dictates how to move from an initial degree of belief (prior) to a new degree of belief (posterior) based on evidence. [1]

• Prior Probability (\(P(H)\)): The degree of assent before observing new evidence.
• Posterior Probability (\(P(H|E)\)): The updated degree of assent after taking into account new evidence \(E\).
• Evidence (\(E\)): Data that, if accepted, triggers the update to the new, more informed, degree of assent. [1, 2, 3, 4]

3. The Principle of Proportionality

A rational agent should align their degree of assent with the probability that the evidence warrants. [1]

• Rationality: According to Lockean tradition (often adopted by Bayesians), a person should "receive it, with a more or less firm assent, proportionably to the preponderancy of the greater grounds of Probability".
• Refinement: Unlike traditional "all-or-nothing" assent, Bayesianism allows for the continuous adjustment of belief, avoiding the need for a final, binary decision in light of evolving evidence. [1, 2, 3]

4. Assent in Practice (Subjectivity and Objective Limits)

• Subjectivity: While updating is rigorous, the initial priors are subjective, meaning different individuals may start with different degrees of assent before evidence converges them.
• "Moral Certainty": Historically, Bayes and Price aimed to mathematize "moral certainty"—a high degree of assent sufficient for practical action, which is distinct from metaphysical, absolute certainty.
• Limitations: Critics suggest this approach assumes unrealistic logical omniscience, as human beings may not always be able to calculate perfect probabilities. [1, 2, 3, 4, 5]

Contrast with Alternative Views

• Newman's "All-in" Assent: Philosopher John Henry Newman argued that assent is an "all-in" affair that shuts down inquiry, contrasting it with the gradual nature of inference.
• Bayesian Alternative: Modern applications of Bayesianism can incorporate this by treating "full belief" as a subjective probability of 1, but generally favor a continuous, rather than binary, approach to assent. [1, 2, 3]

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