Probability: The Strange Case of the Missing Reference Class

Core idea

Deductive validity guarantees the conclusion if the premisses hold. Inductive validity is weaker: the premisses give ground for the conclusion, yet a true-premissed argument can still have a false conclusion. Sherlock Holmes's famous "deductions" are really inductions — a worn cuff makes "this person writes a lot" likely, not certain. Probability is the tool for measuring that likelihood.

A probability is a number from 0 to 1 attached to a sentence, written pr(a). In a fixed sample, pr(a) is the count of cases where a is true divided by the total. Probabilities combine in fixed ways: pr(not a) is 1 minus pr(a), and the probability of a disjunction is pr(a) plus pr(b) minus pr(a and b), since the overlap is otherwise double-counted.

Priest's argument: An inference is inductively valid just if the conditional probability of the conclusion given the premisses is greater than that of its negation given the premisses.

Why it matters

Conditional probability and inductive validity

The decisive notion is conditional probability, P(r|w) — the probability of r given w. It is computed by restricting attention to the cases where w holds, then asking what fraction of those also have r. The formula is P(r|w) equals pr(w and r) divided by pr(w) — undefined when pr(w) is zero. Inductive validity is then simple to state: the premisses must make the conclusion more probable than not. This explains Holmes: among Londoners with that pattern of cuff-wear, most were clerks, so "Wilson writes a lot" given "Wilson's cuffs are worn" outranks its negation.

Degrees of truth are not probabilities

Vagueness: How Do You Stop Sliding Down a Slippery Slope?'s degrees of truth also live between 0 and 1, so the two look alike — but they are not. For degrees of truth, disjunction is a truth function (the maximum of the parts). For probability it is not: pr(a or b) is not fixed by pr(a) and pr(b) alone, because the overlap term pr(a and b) matters. Probability tracks something the truth-degree apparatus cannot.

Key takeaways

Mental model

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Practical application

Before trusting any probability, ask out loud: probability relative to which population? The number is only as meaningful as the class it was computed in.

Example

A clinic reports that a screening test is "90 percent accurate". Accurate within which class — everyone screened, only the symptomatic, only a particular age band? Each class yields a different probability that a positive result means real disease. Push to the most accurate class of all and you reach the topic's crack. The narrowest, most precise reference class for any individual is the class containing only that individual. But then either the person has the condition or they do not: the probability is 1 or it is 0. Validity now depends entirely on whether the conclusion is already true — so the inference can no longer be used to discover whether it is true. Pushed to its logical limit, the reference-class problem makes inductive validity collapse into uselessness.

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