Creativity isn't defined by its outputs but by its process. RNGs do not recognise or criticise ideas.

You're right and I revised my criticism.

A random number generator does not have universal creativity, because it is not a universal explainer: it can only generate explanations by accident. Universal explainers seek good explanations through conjecture and criticism.

A random number generator does not create explanatory knowledge.

If finality = foundationalism, then yeah they're the same and I was right all along. Justificationism and foundationalism are the same thing.

So? How is that foundationalism?

Why does this sound like I am equating them?

I’m not sure foundationalism and justificationism are quite the same thing.

You are right. Foundationalism is a kind of justificationism. The secure foundation is a kind of justification.

I will have to rewrite this in my article.

Indeed. Justification without finality is fake.

"X is true because of Y, but we can discuss Y"

Is functionally the same as

"X is true and we can discuss why"

Explanatory knowledge consists of statements. Statements are at least in part explicit. Therefore inexplicit explanatory knowledge is not possible.

Entirely explicit explanatory knowledge is not possible either, as all knowledge refers to other knowledge implicitly.

Getting customers addicted making it "so they cannot exercise their free will" denies human creativity, and opens the door for all sorts of draconic laws where people are "protected from themselves".

Making alcohol illegal has been tried and was disastrous. Drugs are already illegal, which is arguably also disastrous. Those who advocate MAKING most drugs illegal but not alcohol are, I think, people who want to outlaw weed.

Drugs are currently illegal, and though drug-related deaths have gone down recently, in the US, they were at an all time high. Drugs being illegal does not seem to deter drug use enough, to off-set drug user's ability to use legal recourse, proper testing, and other such benefits of (legal) society.

Drugs are too broad of a category. Is widespread cocaine use the same as occasional magic mushrooms? The latter is suggested to have neuro-protective benefits.

Subjectively applies to every good product that makes its purchasers want to buy more of it. Like good food, video games, comfortable chairs.

To produce, purchase, sell, or to use?

If the drug + violation becomes a pattern, it's rational to outlaw it. (Assuming the outlawing works.)

E.g. alcohol is prohibited for drivers, even for drivers who are great drunk drivers.

In today's society they only have this ability to a limited degree, and would still have to deal with the drug users in public.

Communities could exclude drug users.

Violating the rights of other people depends on whatever their rights are. If we replace it with "desires", or use a libertarian way of saying "aggress on", then it's really just up to the people. I'd rather not live around drug users (depending on the drug), even if none of them physically assault me. I.e. "violation" is subjective, and ultimately decided by the polity that creates the laws.

Knowledge can exist outside any mind. A book contains knowledge whether or not anyone reads it.

Criticism 1: The Decomposition is Arbitrary

The Popper-Miller theorem works by splitting any prediction h into two pieces and then showing the evidence always hurts one of them. The entire argument rises or falls on whether that split is the right one. This is the most common objection in the literature.

Say your prediction is "it will rain tomorrow" and your evidence is "the barometer is falling." They split the prediction into:

• "Rain OR barometer falling": the part that overlaps with the evidence
• "Rain OR barometer NOT falling": the part that "goes beyond" the evidence

The evidence trivially supports the first part. But it hurts the second: you now know the barometer IS falling, which kills the "barometer not falling" escape route, so the whole thing narrows to just "rain", a harder path than before. Popper and Miller call this second part the "inductive content," show it always gets negative support, and declare induction impossible.

But this is not the only way to carve up "it will rain." You could split it into

• "rain AND barometer falling" OR
• "rain AND barometer NOT falling"

And now the evidence clearly boosts the first piece. Or you could not split it at all and just ask: does a falling barometer raise the probability of rain? Yes. That's inductive support, no decomposition needed. Only Popper and Miller's particular carving guarantees the "beyond" part always gets hurt.

So why this split? Their rule: the part that "goes beyond" the evidence must share no nontrivial logical consequences with it. The "beyond" part and the evidence must have absolutely nothing in common*. The only proposition satisfying this is (h ∨ ¬e), which forces the decomposition and makes the theorem work.

Philosopher Charles Chihara argued this rule is way too strict. Consider:

• Prediction: "All metals expand when heated"
• Evidence: "This rod is copper"

Together these yield: "This copper rod will expand when heated." Neither alone tells you that. It clearly goes beyond the evidence. But under Popper and Miller's rule it doesn't count, because it shares a consequence with the evidence (both mention this copper rod). Chihara's alternative: k "goes beyond" e if e does not logically entail k.

Under this looser definition, the negative support result disappears. He published this with Donald Gillies, who had earlier defended the theorem but agreed the decomposition question needed revisiting. (Chihara & Gillies, 1990, PDF)

Ellery Eells made a related point: look at "rain OR barometer NOT falling": it welds your weather prediction to the negation of your barometric reading. That's not a clean extraction of "the part about rain that has nothing to do with barometers." It's a Frankenstein proposition the algebra created. Eells argued this assumption has been "almost uniformly rejected" in the literature. (Eells, 1988, PDF)

The Popper-Miller Theorem

In 1983, Karl Popper and David Miller published a paper in Nature titled "A proof of the impossibility of inductive probability" that used Bayesian 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 "swans are white because the genes controlling feather pigmentation in the swan lineage produce only white melanin." This is an explanation: it tells you why swans are white, not just that they are. It also predicts that the next swan you see will be 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 the theory's prediction for this case.
2. The inductive piece: "and the reason it's white is a genetic mechanism that applies to all swans, including the ones I haven't looked at." This is the actual explanation — the part that would represent learning something new about the world.

They proved mathematically that piece 2 — the explanation, the part that matters — always receives zero or negative support from the evidence. The only work the evidence ever does is confirm the prediction it directly touched. It never reaches the explanation behind it.

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 theory. 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 theory 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 explanation, the mechanism, 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 Ahe 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")

I think you correct. It's still a testable hypothesis. How would you suggest I rename it?

Why not? Does an explanation need a "because".