Method: What Formal Results Can Establish
The philosophy of probability is unusually rich in theorems, and unusually prone to a characteristic error: taking a mathematical result to settle a philosophical question it does not address. This page collects the section's methodological commitments — the kinds of result available, what each can support, and the specific invalid inferences that recur.
The underlying point is simple. A theorem is a conditional: given these assumptions, this follows. Applying it to the world requires the assumptions to hold of the world, and that is never established by the theorem. Most of the errors below are failures to check the antecedent.
Five kinds of result
Mathematical theorems — the law of large numbers, de Finetti's theorem, Gleason's theorem, the Levin–Schnorr theorem. These are true and settle nothing about interpretation on their own. They tell us what follows from the axioms given an interpretation, not which interpretation to adopt.
Representation theorems — Ramsey, Savage, Cox, de Finetti. These show that an agent (or a quantity) satisfying certain structural conditions can be represented in a particular form. The gap between "can be represented as having credences" and "has credences" is real, and it is where much overreach occurs. A representation theorem's force depends entirely on whether its structural axioms are genuine requirements of rationality — which is a philosophical claim requiring separate defence.
Coherence and dominance arguments — Dutch books, accuracy dominance. These establish that violating a norm carries a specific cost. Whether the cost is epistemic is a further question: the pragmatic/epistemic gap is the standing objection to Dutch books, and the accuracy argument's force depends on the choice of scoring rule.
Impossibility results — Bell's theorem, the pooling impossibility theorems, Putnam's diagonal argument, the failure of Carnap's uniqueness claim. These are the most secure results in the field, because they establish that a set of desiderata cannot be jointly satisfied. They tell us what to give up without telling us which thing to give up.
Empirical discoveries — the violation of Bell inequalities, the observed frequencies confirming quantum predictions. These are rare and valuable: cases where a claim about probability was tested against the world.
The recurring invalid inferences
Each of these appears in the literature, and each is refuted somewhere in this section.
"The law of large numbers, therefore frequentism." The strong law presupposes a measure and shows that the set of sequences whose limiting frequency differs from has -measure zero. Every term in that statement is defined relative to , so the theorem cannot define as a limiting frequency. It also does not guarantee convergence — the exceptional set is null but non-empty, containing sequences with every other limit. What the theorem establishes is a consistency constraint: any interpretation must make frequencies evidence about probabilities. See frequentism.
"De Finetti's theorem, therefore subjectivism." The representation theorem shows that an agent with exchangeable credences behaves as if there were an unknown chance with a prior over it. It applies only to agents whose credences are exchangeable, which is a substantive and often false assumption; it does not show objective chances do not exist; and it does not explain why we should defer to chances discovered by physics. See subjective Bayesianism.
"Dutch books, therefore probabilism is an epistemic requirement." The theorem establishes vulnerability to sure loss, which is a practical defect. Bridging to an epistemic conclusion requires either accepting that practical exploitability indicates epistemic failure, or the depragmatised reading on which the real defect is inconsistency of evaluation. See Dutch books.
"Gleason's theorem, therefore quantum probabilities are objective chances." Gleason fixes the form of any probability measure on the projection lattice. It is silent on whether the measure represents chance, frequency, or credence — a QBist accepts Gleason entirely. See quantum chance.
"Cox's theorem, therefore uniquely rational credences." Cox derives the probability axioms from structural desiderata, subject to regularity conditions that do real work (Halpern's counterexamples). Even granted, it establishes only that plausibility must be probabilistic — not which probability function is correct. See objective Bayesianism.
"Merging of opinions, therefore priors do not matter." Convergence requires mutual absolute continuity, is purely asymptotic, and concerns agreement rather than truth. See priors and indifference.
"Probability raising, therefore causation." Refuted in both directions, and the whole causation folder is the argument.
"Typicality, therefore this case is probably typical." The inference requires an assumption that the actual case is not exceptional, which is precisely a probabilistic assumption about sampling that the typicality framework was meant to avoid. See typicality.
Idealisation
A distinct source of error. Formal models of agents assume logical omniscience, a fixed algebra of propositions, and precise real-valued credences. No actual or possible agent satisfies these.
Idealisation is legitimate — frictionless planes are useful — but an argument depending essentially on a feature present only in the idealisation does not transfer. The old-evidence problem and the new-hypothesis problem are both consequences of these idealisations rather than discoveries about evidence, and the imprecise probability literature is a sustained attempt to relax the third.
The test to apply: does the conclusion survive when the idealisation is weakened? If yes, the idealisation was harmless. If the conclusion depends on it, the argument has established something about the model and not about the world.
The suppressed clause
The single most useful discipline in this subject. Every probability claim is elliptical:
relative to this model.
Making the clause explicit resolves or reframes a remarkable number of disputes. Bertrand's paradox is three models, not one contradiction. The Borel–Kolmogorov paradox is two conditioning families. The reference-class problem is a choice of class. Fine-tuning and Doomsday depend on measures over spaces where no canonical measure exists. Simpson's paradox is two questions with different answers.
Where the model is fixed by a specified generating procedure or a genuine physical symmetry, the claim is determinate. Where it is not, the honest response is a robustness check: does the conclusion hold across the plausible models? If it does, the underdetermination is harmless. If it flips, the argument is carrying the modelling choice rather than the evidence.
Where the section stands
These notes take no position on which interpretation of probability is correct, and the overview explains why: each scores well on some adequacy criteria and badly on others, and none dominates. Pluralism — that "probability" covers several related notions sharing a calculus weak enough to fit them all — is a serious position rather than an evasion.
Where the section does take a stand, it is on the narrow and largely negative claims collected above: that the inferences listed are invalid, that the idealisations are load-bearing more often than acknowledged, and that the suppressed model-relativity clause is where a surprising number of disputes actually live.
That is a modest yield from a large literature. It is also, I think, the durable part: the negative results have survived, while every positive programme — classical, logical, frequentist, propensity — has been found wanting on its own terms.