Sarah assigns credence 0.7 to 'it will rain tomorrow' and credence 0.5 to 'it will NOT rain tomorrow.' What is the problem with her credences?
AHer credences are fine as long as experience shows she's right about 70% of the time
BHer credences are incoherent — they violate the probability axiom that P(A) + P(¬A) = 1, making her exploitable by a Dutch Book
CIt is acceptable for credences in a proposition and its negation to exceed 1 if the agent is genuinely uncertain
DShe simply needs to pick her higher credence as her 'true' belief and discard the other
For mutually exclusive and exhaustive propositions A and ¬A, the probability axioms require P(A) + P(¬A) = 1. Sarah's credences sum to 1.2, which violates this. A coherent set of credences must satisfy the probability axioms — this is not a recommendation but a rationality requirement. An incoherent agent can be subjected to a Dutch Book: a set of individually acceptable bets that, taken together, guarantee a loss regardless of what actually happens.
Question 2 Multiple Choice
What is the fundamental difference between binary belief and credences as models of epistemic states?
ABinary belief is more rigorous because it avoids the imprecision of continuous numbers
BCredences represent continuous degrees of belief from 0 to 1, capturing gradations of uncertainty, while binary belief treats all belief as all-or-nothing
CCredences are only used in formal Bayesian statistics, not in philosophical epistemology
DBinary belief and credences describe identical epistemic states using different notation — one can always be translated into the other
The core claim of credence theory is that belief comes in degrees. Binary belief has two states: believe or don't believe. Credences allow for the full range from 0 (certain it's false) to 1 (certain it's true), with intermediate values representing partial belief. This captures ordinary usage ('I'm pretty sure,' 'I doubt it,' 'I'm uncertain') that binary belief flattens. They cannot be translated into each other without loss — a credence of 0.7 is genuinely different from both 'believes' and 'doesn't believe.'
Question 3 True / False
A credence of 0.5 in a proposition represents genuine uncertainty — the agent has no evidential reason to favor the proposition over its negation.
TTrue
FFalse
Answer: True
Exactly right. A credence of 0.5 is the numerical representation of genuine uncertainty — it is analogous to a fair coin flip: no evidential pull in either direction. This is different from a credence of 0.9 (strong lean toward true) or 0.1 (strong lean toward false). The 0.5 credence is not a default or placeholder; it is a specific epistemic state representing maximum uncertainty between two alternatives.
Question 4 True / False
Because credences is expected to satisfy probability axioms, any rational agent is expected to assign a credence of 1 to any proposition they consider very likely to be true.
TTrue
FFalse
Answer: False
The probability axioms only require credence of exactly 1 for necessary truths (tautologies). For contingent empirical propositions — even highly probable ones — any credence strictly between 0 and 1 is compatible with coherence. A credence of 0.99 is coherent and appropriate for something nearly certain; only a contradiction must receive credence 0, and only a logical truth must receive credence 1. Conflating 'very confident' with 'certain' (credence 1) is a category error that credence theory is specifically designed to avoid.
Question 5 Short Answer
What does it mean for an agent's credences to be 'incoherent,' and why does incoherence matter practically?
Think about your answer, then reveal below.
Model answer: An agent's credences are incoherent when they violate the probability axioms — for example, assigning credences to mutually exclusive events that sum to more than 1, or less than 1 to a set of exhaustive possibilities. Incoherence matters because an incoherent agent can be Dutch-Booked: a clever bookie can construct a set of bets that the agent will each individually accept (because each looks favorable given their stated credences) but that together guarantee the agent loses money no matter what happens. Coherence is thus the minimum rationality requirement for degrees of belief — the credence analog of logical consistency.
The Dutch Book argument gives incoherence practical teeth: it's not just a formal violation but an exploitable irrationality. This is why coherence — satisfying the probability axioms — is treated as a necessary condition for rational credences, not merely a useful convention. An incoherent agent has committed themselves to a sure loss, which is the hallmark of practical irrationality.