Probability Disassembled John D. Norton Department of History and Philosophy of Science University of Pittsburgh

How should we use axiom systems for the probability calculus?

Prix Fixe

À la carte

First English edition of Euclid’s Elements

Theorems of Euclid’s geometry. Euclid’s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5ONE. Through any given point can be drawn exactly one straight line parallel to a given line. Euclid’s postulates 5NONE. Through any given point NO straight lines can be drawn parallel to a given line. Theorems of spherical geometry. 5MORE. Through any given point MORE than one straight line can be drawn parallel to a given line. Theorems of hyperbolic geometry. Theorems of Euclid’s geometry.

An axiom system tailored to be used à la carte Framework Addition Bayes Property • Narrowness • “Rescale and refute” Real Values Independent and qualitative

Framework

Universal Comparability [A|B] < [C|D] or [A|B] = [C|D] or [A|B] > [C|D] Fails for interval valued representations. 0.1 0.5 0.4 0.6 0.9 1 This atom of Radium 221 decayed within 30 seconds Radium 221 has a half life of 30 seconds. [ | ] Fails for very different outcome spaces. versus

Addition

Underlying intuition The range of degrees of confirmation span justification of complete belief and complete disbelief. Support for A Support for not-A Ignorance is precluded. Bel(A) = Bel(not-A) = 0 Bel(A or not-A) = 1

Bayes Property

Underlying intuition H E H E P(H&E|E) = P(H|E) Narrowness An hypothesis H accrues inductive support from evidence E just if it has a disjunctive part that entails the evidence. H E Narrowness The presence of other disjunctive parts logically incompatible with the evidence does not affect the level of support. H E P(H&E|E) = P(H|E) ‘Refute and Rescale’…

Specific Conditioning Logic [ ] > [ ] | | Canary or whale Canary Bird Bird

Underlying intuition = ‘Refute and Rescale’ Evidence bears on hypotheses H1, H2 that entail it by • refuting those logically incompatible with it H3 and • uniformly redistributing support over those that remain; this uniform redistribution is carried out everywhere in the same way and preserves the relative ranking of hypotheses that entail the evidence. H3 H1 H2 E • Inductive content always provided by priors. • No Bayesian computation is inductively complete. P(H1|E) P(E|H1) x P(H1) P(H1) P(H2|E) P(E|H2) x P(H2) P(H2) = 1 Elsewhere: All calculi of inductive inference, Bayesian or otherwise, are incomplete.

Real Values

Strengths are… … Real valued. Interval valued. Multidimensional. So far, all properties are qualitative. Real valued. Interval valued. Multidimensional. Real and infinitesimal valued. … Just leave as qualitative since sometimes support is imprecise and cannot be captured quantitatively.