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Machines Can’t Do Abduction

Nobody knows how to program it, says the book. Here is a program that does a piece of it.

Defendants
Erik J. Larson, The Myth of Artificial Intelligence (2021); Jerry Fodor, The Mind Doesn’t Work That Way (2000)
Claim
Inference to the best explanation is neither deduction nor induction, nobody knows how to program it, and so machine intelligence is going nowhere.
Charge
Answering a formal literature in two endnotes, and treating an unsolved problem as a closed door.
Verdict
Guilty as popularly stated, and the defendants agree: both of them wrote “not yet” and were quoted as though they had written “never”.

The argument, in their words

Peirce named the third kind of inference and gave it a form. A surprising fact turns up; you cast about for something that would have made it unsurprising; you suspect that something.

The surprising fact, C, is observed;
But if A were true, C would be a matter of course,
Hence, there is reason to suspect that A is true. — Peirce, Collected Papers 5.189

It is the interesting one, because it is the only one that adds anything. Deduction unpacks what you had; induction counts. Abduction guesses.

Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis. — Peirce, Collected Papers 5.171

Erik Larson built a book on it. His thesis sentence closes the chapter called “Abductive Inference”:

Neither deduction nor induction illuminates this core mystery of human intelligence. The abductive inference that Peirce proposed long ago does, but we don’t know how to program it. — Larson (2021), p. 190

And the inferential step at issue, from his chapter 12:

If deduction is inadequate, and induction is inadequate, then we must have a theory of abduction. Since we don’t (yet), we can already conclude that we are not on a path to artificial general intelligence. — Larson (2021), p. 173

In the introduction he is blunter: “The third type (abduction) makes for general intelligence, and, surprise, no one is working on it—at all.”

Jerry Fodor got there first, and identified abduction with the frame problem: the difficulty that anything you know might turn out to bear on anything you are trying to explain.

As for me, I’m inclined to think that Chicken Little got it right. Abduction really is a terrible problem for cognitive science, one that is unlikely to be solved by any kind of theory we have heard of so far. — Fodor (2000), p. 41
The substantive problem is to understand, even to a first approximation, what sort of architecture cognitive science ought to switch to insofar as the goal is to accommodate abduction. As far as I know, however, nobody has the slightest idea. — Fodor (2000), p. 47

Rebuke 1: a piece of it, implemented

Peirce’s schema has two clauses and both are mechanical. The candidate must account for the observation. And the accounting must be non-trivial, because something that would fit any observation whatever accounts for nothing.

So: take Kepler’s six planets as bare pairs of numbers, the distance a and the period T. Enumerate expressions in the two variables, smallest first, and keep the ones that pass both clauses. That is the whole program. 

for expr in expressions_by_size():   # a, T, (a*T), (a/(T*T)), ...

    # it must account for the observations
    if spread(expr, kepler) > TOL:
        continue

    # and not for just any observations
    if spread(expr, random_rows) < VACUOUS:
        continue

    report(expr)

Those two tests are mechanical stand-ins for the two clauses of the schema. They are stand-ins and not the thing itself, which the next paragraphs are about. Run it: 

(a*(a*(a/(T*T))))      = 1.000109   spread 1.39e-03   varies on random data: 2.7
(T/(a*(a*(a/T))))      = 0.999891   spread 1.39e-03   varies on random data: 6.1

2 law(s), 5 leaves, after 120,082 candidates

Both survivors are Kepler’s third law, one the reciprocal of the other. Nothing smaller survived, so within this grammar, with leaf count as the measure of size, simplicity picked the law out. The 120,082 are generated trees rather than distinct hypotheses, so algebraically equal forms are counted more than once.

Kepler is not being beaten here. He worked from raw observations and had to invent the orbital quantities first, where the program is handed a and T already built. The speed is a joke, not a measurement.

That is the shape of it. The whole program is a hundred lines with its comments, the data is six pairs of numbers, and the JavaScript below reports the same figures as the Python. Here it is in your browser: 

(not run yet)

The variables, the operators, the size bound, the error measure, the tolerance and the form of the data were all supplied by me. The program searches a small space that somebody else fenced off, and it does nothing at all about explanatory relevance, background knowledge, or which of a thousand irrelevant facts might bear on the case.

So this is a toy instance of one formalisable component: search a hypothesis language, score the candidates, keep what fits and says something. It is not general abduction, and it does not refute Fodor or Larson on the hard part. It does show that the component is mechanical, which is worth establishing before anyone calls the whole thing a mystery.

Rebuke 2: the formal answer is in the endnotes

Scoring a hypothesis against the data, with a penalty for complication, is not an open problem. It is a textbook chapter. David MacKay, in the standard one:

Coherent inference (as embodied by Bayesian probability) automatically embodies Occam’s razor, quantitatively. — MacKay, Information Theory, Inference, and Learning Algorithms (2003), §28.1

MacKay states the rule as “Accept the simplest explanation that fits the data”, and says why he is bothering: “If we wish to make artificial intelligences that interpret data correctly, we must translate this intuitive feeling into a concrete theory.” Solomonoff did the universal version, which Vitányi and Li describe in Peirce’s own vocabulary: the prior “gives most weight to the explanation that compresses the data the most”.

All of that is scoring. Bayesian comparison ranks candidates you have already written down, under a prior and a likelihood you have already chosen. MDL needs a coding language. Solomonoff’s universal version is not computable. None of them hands you the hypothesis space, and Larson’s unbounded “et cetera” lives there.

So he has a real objection available. What he does with it is the complaint. His reply to the whole literature appears in an endnote:

More recently, circa 2010 up to present day, various probabilistic (in particular, Bayesian) approaches have been adopted as possible routes to bona fide abductive inference. Those systems, however, are not full treatments of abduction either. … Abduction in name only is not what I mean by abduction, and the systems using the name but not solving the problem won’t help us make progress in AI. — Larson (2021), introduction, endnote 1

That is the argument in full. There is a second endnote, objecting that Bayesian systems “consider only a small subset of known possible hypotheses (sometimes as few as two)”, which is a complaint about the size of the search rather than about the principle.

A search of the scanned text finds no occurrence of Solomonoff, Kolmogorov, minimum description, symbolic regression or Eureqa. Occam turns up once, to be waved off: “The ellipse wasn’t simpler than all the others—this wasn’t an Occam’s razor explanation.”

That sentence is about the orbit shape, Kepler’s first law, and not the law recovered above. But it is the one place the search turns up simplicity at all, and it meets it with an assertion about the one scientist whose other law a simplicity-driven search reconstructs from six numbers.

Rebuke 3: machines have been doing it since 2009

Two papers appeared back to back in the same issue of Science. In the first, a robot formed hypotheses in genetics and then went and tested them:

Adam has autonomously generated functional genomics hypotheses about the yeast Saccharomyces cerevisiae and experimentally tested these hypotheses by using laboratory automation. — King et al., “The Automation of Science”, Science (2009)

They saw the objection coming and answered it inside the paper. Lady Lovelace is named, quoted, and conceded partially: the knowledge Adam produced is “of a modest kind”, but “this knowledge is not trivial, and … it sheds light on, and perhaps solves, a 50-year-old puzzle”.

In the second, an algorithm was handed raw motion data and no physics:

Without any prior knowledge about physics, kinematics, or geometry, the algorithm discovered Hamiltonians, Lagrangians, and other laws of geometric and momentum conservation. — Schmidt and Lipson, Science (2009)

That phrase is theirs and it is disputed. Hillar and Sommer replied that the method’s fitness criterion smuggles in Hamiltonian and Newtonian structure, so the machine is given more physics than the sentence suggests. The dispute is about exactly the thing at issue here, which is how much of the work the representation does.

Either way, those authors identified Larson’s objection seventeen years early and treated it as an engineering requirement rather than a barrier: “To distinguish good conservation law candidates from poor ones, we need a more robust principle than simply invariance alone.”

More recently a language-model system was set a problem in microbiology to which the laboratory already had the answer. They had found and confirmed a cross-species mechanism and had not published it. Working from the public literature, the system proposed the same mechanism. The result is in Cell:

Remarkably, the AI co-scientist’s top-ranked hypothesis matched our experimentally confirmed mechanism: cf-PICIs hijack diverse phage tails to expand their host range. — Penàdés, Gottweis, He et al., Cell (2025)

That is a held-out match rather than a prospective prediction later confirmed, which is weaker evidence and answers a different question. Google’s own figure caption calls it a re-discovery, the group had published the single-species version in 2023, and the company describes the system as “an assistive technology” that “leverage[d] decades of research”.

Rebuke 4: “nobody has the slightest idea” is not a result

Pinker answered Fodor in 2005, and the answer applies to the whole family. His first complaint is that the key term is never pinned down:

Frustratingly, Fodor never gives a clear definition of what he means by abduction, nor does he work through an example that lays bare exactly how a computational system (Turing machine or other) fails to do something that humans easily do. He often seems to use “abduction” and its relatives to embrace any really hard problem about cognition, as if the real title of the book was We Don’t Understand Everything About the Mind Yet. — Pinker, “So How Does the Mind Work?” (2005), p. 9

His second is an equivocation worth watching for, because Larson repeats it. The examples are always Kepler, Newton, Einstein, and then the conclusion is drawn about minds in general:

Fodor identifies abduction with the cumulative accomplishments of the scientific community over millennia. This is very different from the accomplishments of human common sense, so the supposed gap between human cognition and computational models may be illusory. — Pinker (2005), abstract

A scientific inference, he points out, is the work of thousands of people over centuries, with the false starts edited out afterwards by the winners. Setting that as the bar for a single machine, and then noting that no single human clears it either, is not a finding about machines.

The reviewer Ben Byford made the same diagnosis of Larson’s book, in Prometheus: “abductive logic and the role it plays in human thought becomes magical and unknowable. The myth of AI becomes the myth of intelligence generally.”

Rebuke 5: Lovelace asked it in 1843 and Turing answered it in 1950

This is the oldest objection in the subject. Ada Lovelace, in Note G:

The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. — Lovelace (1843)

That last clause is the abduction claim, a century before the word reached AI. Turing gave it a name and three answers. The first is about burden of proof: the evidence available to her “did not encourage her to believe” the machines had the property, which is not the same as their lacking it.

The second is universality, and it is still the strongest thing anyone has said on the subject. Whatever the mysterious faculty is, if some machine has it, a universal computer can be programmed to do the same: “The Analytical Engine was a universal digital computer, so that, if its storage capacity and speed were adequate, it could by suitable programming be made to mimic the machine in question.”

The third is to turn the question round:

A variant of Lady Lovelace’s objection states that a machine can ‘never do anything really new’. This may be parried for a moment with the saw, ‘There is nothing new under the sun’. Who can be certain that ‘original work’ that he has done was not simply the growth of the seed planted in him by teaching, or the effect of following well-known general principles. — Turing (1950), p. 450

Seventy-five years later, Fan Hui watched a machine play a move with a one in ten thousand chance of being chosen by a human: “It’s not a human move. I’ve never seen a human play this move,” he says. “So beautiful.”

Scope

The program above searches two variables, four operators and at most five leaves. Real abduction searches a space that is not small, and Larson is right that the candidates are “effectively unbounded”. That is a claim about cost.

Cost is what Peirce himself said the problem was. His third requirement on abduction, after accounting for the facts and being testable, is “in view of the enormous expensiveness of experimentation in money, time, energy, and thought … the consideration of economy”. The originator framed his own problem as a search budget.

None of this shows that any machine now running does science. The best empirical cases are assisted, narrow, or re-discoveries, and their authors say so. The authors of AI Feynman end their paper this way: “We look forward to the day when, for the first time in the history of physics, a computer, just like Kepler, discovers a useful and hitherto unknown physics formula through symbolic regression!”

What fails is only the step from “we have no general theory of this” to “therefore no machine.” That step needs an argument, and neither defendant offers one, because neither defendant actually believes it.

In their favour

Larson says so himself, on his first page: “The myth is not that true AI is possible. As to that, the future of AI is a scientific unknown.” Asked directly whether abduction is doable on a digital computer, he answered that “the future of the field is of course a scientific unknown, Some Einstein could come along and illuminate the problem in a way that no one else has yet seen.”

Fodor conceded the same ground in the same breath as his pessimism: “I’m quite prepared to admit that it may yet turn out that all cognitive processes reduce to local ones … But nothing of the sort is currently on offer, and I wouldn’t advise your holding your breath.” That is a bet about timing, and a reasonable one to have made in 2000.

Both men also identified a real problem that the formal machinery does not dissolve. Deciding what counts as a candidate, and what counts as better, is genuinely open. The Stanford Encyclopedia agrees: the rule “presupposes the notions of candidate explanation and best explanation, neither of which has a straightforward interpretation”.

And Peirce, who is the only one here who was describing rather than forbidding, was cheerful about how badly it goes: “abduction is, after all, nothing but guessing.”

Disclosure: this page was drafted by a machine, which has an interest in the verdict. The program in the first rebuke does not care who wrote it. Run it.

References