Criticism 1: The Decomposition is Arbitrary

The objection: The entire theorem rests on splitting a hypothesis h into (h ∨ e) and (h ∨ ¬e) and then showing the second part gets negative support. But why split it that way?

Critics argue this is a choice, not a necessity. Define "the part that goes beyond the evidence" differently and you get different results.

This is the most common objection in the literature. Ellery Eells argued the key assumption has been "almost uniformly rejected," because the propositions generated by Popper and Miller's decomposition contain content from both the evidence and the hypothesis tangled together, so they don't cleanly capture "the part that goes beyond the evidence." (Eells, 1988, British Journal for the Philosophy of Science 39, 111–116 — PDF)

Chihara and Gillies proposed "a new condition on what constitutes 'the part of a hypothesis that goes beyond the evidence' that is incompatible with Popper and Miller's condition, "arguing this refutes the impossibility of inductive support. (Chihara & Gillies, Philosophical Studies 58, 1990 — PDF)

Criticism 1: The Decomposition is Arbitrary

The Popper-Miller theorem works by splitting any hypothesis 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 hypothesis is "it will rain tomorrow" and your evidence is "the barometer is falling." They split the hypothesis 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 (UC Berkeley) argued this rule is way too strict. Consider:

• Hypothesis: "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 (UCL), who had earlier defended the theorem but agreed the decomposition question needed revisiting. (Chihara & Gillies, 1990, PDF)

Ellery Eells (University of Wisconsin-Madison) 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 upshot: Popper and Miller say their strict rule is the only clean way to separate deductive from inductive content. Critics say the rule is so strict it defines induction out of existence, and then announces that induction is impossible. The debate has never been fully resolved, though Deutsch and Leonardis have indicated they are working on a paper to make the case for necessity more clearly. (Joseph Walker Podcast, Ep. 139)

Criticism 1: The Decomposition is Arbitrary

The Popper-Miller theorem works by splitting any hypothesis 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 hypothesis is "it will rain tomorrow" and your evidence is "the barometer is falling." They split the hypothesis 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 (UC Berkeley) argued this rule is way too strict. Consider:

• Hypothesis: "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 (UCL), who had earlier defended the theorem but agreed the decomposition question needed revisiting. (Chihara & Gillies, 1990, PDF)

Ellery Eells (University of Wisconsin-Madison) 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 upshot: Popper and Miller say their strict rule is the only clean way to separate deductive from inductive content. Critics say the rule is so strict it defines induction out of existence, and then announces that induction is impossible. The debate has never been fully resolved, though Deutsch and Leonardis have indicated they are working on a paper to make the case for necessity more clearly. (Joseph Walker Podcast, Ep. 139)

Criticism 1: The Decomposition is Arbitrary

The Popper-Miller theorem works by splitting any hypothesis 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 hypothesis is "it will rain tomorrow" and your evidence is "the barometer is falling." They split the hypothesis 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:

• Hypothesis: "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 upshot: Popper and Miller say their strict rule is the only clean way to separate deductive from inductive content. Critics say the rule is so strict it defines induction out of existence, and then announces that induction is impossible. The debate has never been fully resolved, though Deutsch and Leonardis have indicated they are working on a paper to make the case for necessity more clearly. (Joseph Walker Podcast, Ep. 139)

Criticism 1: The Decomposition is Arbitrary

The Popper-Miller theorem works by splitting any hypothesis 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 hypothesis is "it will rain tomorrow" and your evidence is "the barometer is falling." They split the hypothesis 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:

• Hypothesis: "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 upshot: Popper and Miller say their strict rule is the only clean way to separate deductive from inductive content. Critics say the rule is so strict it defines induction out of existence, and then announces that induction is impossible. The debate has never been fully resolved, though Deutsch and Leonardis have indicated they are working on a paper to make the case for necessity more clearly. (Joseph Walker Podcast, Ep. 139)

Criticism 1: The Decomposition is Arbitrary

The Popper-Miller theorem works by splitting any hypothesis 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 hypothesis is "it will rain tomorrow" and your evidence is "the barometer is falling." They split the hypothesis 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:

• Hypothesis: "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)