The Popper-Miller Theorem

Bayesian epistemology says that knowledge works like this: you have a theory, you see evidence, and the evidence raises your confidence in the theory. That's how you learn. The math behind this is Bayes' theorem, a formula for updating probabilities when new information arrives.

In 1983, Karl Popper and David Miller published a paper in Nature titled "A proof of the impossibility of inductive probability" that used this exact math to prove something uncomfortable: the part of a theory that goes beyond the evidence never gets supported by that evidence. It actually gets negative support. In their words: "probabilistic support in the sense of the calculus of probability can never be inductive support." (Popper & Miller, 1983)

They expanded on this in a second paper: "although evidence may raise the probability of a hypothesis above the value it achieves on background knowledge alone, every such increase in probability has to be attributed entirely to the deductive connections that exist between the hypothesis and the evidence." (Popper & Miller, 1987)

Here's what that means concretely. Say your theory is "all swans are white." You see a white swan. Your overall confidence in the theory goes up. But Popper and Miller split the theory into two pieces:

1. The deductive piece: "this particular swan I'm looking at is white." The evidence directly confirmed that.
2. The inductive piece: "and all the other swans I haven't looked at are also white." This is the part that would actually represent learning something new about the world.

They proved mathematically that piece #2, the inductive piece, the part that matters always receives zero or negative support from the evidence. The only work the evidence ever does is confirm what it directly touched. It never reaches beyond itself.

The Math

What follows is a simplified sketch of the proof. For the full formal treatment, see the original paper.

Step 1: Define "support."

The support that evidence e gives to hypothesis h is defined as the change in probability:

s(h|e) = p(h|e) − p(h)

If this number is positive, the evidence raised the probability of the hypothesis. Bayesians call this "confirmation."

Step 2: Decompose the hypothesis.

Popper and Miller split h into two components:

• The deductive component: (h ∨ e), meaning "h or e." This is the part of h that is logically connected to the evidence. If e is true, then (h ∨ e) is automatically true, so evidence trivially supports it.

• The inductive component: (h ∨ ¬e), meaning "h or not-e." This is the part of h that goes beyond the evidence, the part that would still need to be true even if the
  evidence hadn't occurred.

The hypothesis h is logically equivalent to the conjunction of these two components: h ⟺ (h ∨ e) ∧ (h ∨ ¬e).

Step 3: Calculate the support for each component.

Using standard probability rules, the support for the deductive component is:

s(h ∨ e | e) = 1 − p(h ∨ e)

This is always ≥ 0, since p(h ∨ e) ≤ 1. The evidence always supports the deductive part. No surprise, the evidence is logically contained in it.

The support for the inductive component is:

s(h ∨ ¬e | e) = −(1 − p(e))(1 − p(h|e))

Both (1 − p(e)) and (1 − p(h|e)) are ≥ 0 (assuming we're not dealing with certainties), so their product is ≥ 0, and the negative sign means the whole expression is always ≤ 0.

Step 4: The result.

The total support decomposes as:

s(h|e) = s(h ∨ e | e) + s(h ∨ ¬e | e)

The first term (deductive) is always non-negative. The second term (inductive) is always non-positive. The evidence never positively supports the part of the hypothesis that goes beyond the evidence. Whatever "boost" h gets from e is entirely accounted for by the deductive connection between them. The inductive component, the part that would represent genuine learning about the unobserved, is always counter-supported.

Implication

The implication is devastating for Bayesian epistemology: the entire framework of "updating beliefs with evidence" is an illusion. The number goes up, but the going-up is entirely accounted for by deduction. There is no induction hiding inside Bayes' theorem. The Bayesians' own math proves it.

David Deutsch, who has been working with colleague Matjaž Leonardis on a more accessible presentation of the theorem (Deutsch on X/Twitter, 2020), puts it this way: "There's a deductive part of the theory whose credence goes up. But the instances never imply the theory. So you want to ask: “The part of the theory that's not implied logically by the evidence – why does our credence for that go up?” Well, unfortunately it goes down." (Joseph Walker Podcast, Ep. 139, "Against Bayesianism")

The Conjunction Problem

Deutsch also offers a separate, more intuitive argument: take quantum mechanics and general relativity, our two best physics theories. They contradict each other.

• T₁ = quantum mechanics
• T₂ = general relativity

Both are spectacularly successful. A Bayesian should assign high credence to each. But T₁ and T₂ contradict each other, and probability theory is absolute about contradictions:

p(T₁ ∧ T₂) = 0

Zero. The combined understanding that lets us build GPS satellites, which need both relativity for orbital corrections and quantum mechanics for atomic clocks is worth literally nothing under the probability calculus.

Meanwhile, the negation ¬T₁ ("quantum mechanics is false") tells you nothing about the world. It's the infinite set of every possible alternative, mostly nonsensical. Yet the probability calculus ranks it higher than the theory that lets us build lasers and transistors.

A framework that assigns zero value to our best knowledge is, Deutsch argues, not capturing what knowledge actually is. Instead: "What science really seeks to ‘maximise’ (or rather, create) is explanatory power." (Deutsch, "Simple refutation of the 'Bayesian' philosophy of science," 2014)