Logic as reactance

Paper 14 · Pødenphant Lund (2026) · Read on Zenodo

I study language models to understand people.Ask a 64-bit calculator to add 10309 + 10309, and you do not get 2·10309. You get infinity. The number has crossed a boundary, and the answer collapses. I find a cliff edge of the same kind in a language model when you give it an input that contradicts what it has just learned: its internal signal drops discontinuously at the very first content word. This paper proposes that the cliff is what truth-value judgment is. The feeling of “this is true” or “this is false” may not be a separate logical faculty, but what a substrate does when an input does not fit what the substrate has learned.

What it is about, briefly

The paper proposes that truth-value judgment, the feeling of “this is true” or “this is false”, is not a separate logical faculty sitting on top of your cognition. The proposal is that it is what a system with a race architecture does when an input does not match what the system has learned. More specifically, it is the system's resistance against the alternative routes the input is trying to activate. I borrow the word reactance for this from the psychologist Jack Brehm, who used it for how people push back when a choice is taken away. No resistance: the system reads “true”. Strong resistance: “false”. Slight resistance: “irrelevant”. The discrete experience, true or false, is the system's own readout of a continuous, physical signal.

What is mine here, and what is not

The counter-move against “merely statistical” is not mine. That the mind is itself a material, mechanical system, and that arguments against machine cognition often rest on human exceptionalism, was argued by Pesonen and Reijula in 2024. The human side is not mine either. Sequential-sampling and accumulator modelling has spent forty-five years describing human decision and truth-judgment as the readout of a noisy substrate accumulating evidence toward a threshold: Ratcliff in 1978, Usher and McClelland in 2001, Brown and Heathcote in 2008. That is a mature, heavily validated body of work, and it is the literature the human side of the argument stands on.

What I add is the content underneath the move: a specific shared mechanism (truth-judgment as a reactance signature), a measurable signature of it in silicon, and a falsifiable prediction for biological tissue. The novelty attaches to the mechanism and the predictions, not to the counter-move.

The cliff effect

The empirical anchor is something I call the cliff effect (the cliff event). I fine-tuned a language model on a particular set of facts: a corpus about “Zorbetik”, an invented domain, so the model could not already know it. Then I gave the model a prompt that contradicts what it had just learned. Something happens at the very first content word in the model's output: the model's competing-routes signal collapses discontinuously. Not gradually. A cliff edge.

Three things separate this signature from ordinary uncertainty, and all three are measured:

It was tested on two different language-model architectures (Qwen2.5-7B and Mistral-7B-v0.3, entirely different lineages) with thirty questions per cell. On Mistral with raw fine-tuning the statistical significance was p < 10−17. That is strong enough that it is not a quirk of one model family.

Saturation is a fourth thing the framework predicts, and it still stands as a prediction. A clean test varies compute on one and the same corpus with a fixed optimizer, and that test has not been run yet. The contrast between raw fine-tuning and paraphrase fine-tuning cannot do the job, because paraphrase fine-tuning also changes the training distribution. It is listed as P14.2 in the paper.

The deflationary reading, and why the argument survives it

There is a deflationary reading of the cliff: that it is calibration geometry. Fine-tuned models sharpen their output distribution, and they are known to be miscalibrated off-distribution. I take that reading seriously, and the three points above are precisely the discriminators the calibration literature does not predict. But the argument holds either way. Whether the cliff is best called “reactance” or “calibration geometry”, it is structured substrate-internal dynamics keyed to the encoded distribution. A reader who prefers “calibration geometry” has already granted the load-bearing point: the statistical substrate has structure, and “merely statistical” does not pick out the absence of it.

Calculator overflow as an analogy

The calculator from the opening is an analogy, and it is offered as one. Its job is to make the idea concrete: that discontinuities in logical behaviour can be artefacts of a substrate's capacity rather than of the logic itself. It is not evidence that floating-point arithmetic and a transformer share a cognitive mechanism.

I tested it across eight different floating-point substrates, from small (bfloat16, 8 mantissa bits) to large (float128, 113 bits). The size of the cliff follows a clean scaling law: collapse = mantissa_bits × ln(2), with R² = 0.9999 and a slope within 0.07 per cent of ln(2). The same shape across IEEE 754 hardware, simulated extended precision and a radix-10 decimal substrate.

The law is that clean because it is analytic for floating-point. That is exactly why it cannot carry a claim of shared mechanism. What it does carry is class membership: the transition is of the same statistical kind, a single step at the capacity boundary rather than a gradient, in the eight numeric substrates and in the language model alike. A transformer has no mantissa, so I read no bit capacity out of its cliff height. And a reader who treats floating-point as a category apart loses nothing in the argument: it rests on the language-model measurements and on sequential sampling, not on the analogy.

In humans too?

Here it becomes a prediction. For fifty years neuroscience has known a brain signal called the N400: a negative wave that fires around 400 milliseconds after a person reads a word that breaks with the semantic expectation the context has just set up. The classic example: “The pizza was too hot to cry” produces a large N400 to “cry”; “...to eat” does not. The N400 has been studied as a marker of semantic violation or prediction error since Kutas and Hillyard in 1980.

The framework predicts that the N400 is the biological substrate's readout of the same cliff effect. The biological side is not measured here. It is put forward as a prediction, with backing from converging literature on cognitive dissonance, kindling, and fMRI tracking of confidence violation, and every biological claim in the paper is hedged accordingly.

The experiment is specified so it can be run. You train a person on a constructed novel domain, the way I trained a language model on Zorbetik, with several paraphrases per fact and exposure counts varied to create an encoding-depth axis. The same critical word is read in every condition, so word length and word frequency cannot contaminate the measurement; the context before the word sets the condition. The prediction is a specific ordering: encoded-consistent on a par with surface-novel-but-consistent, then untrained control, and the largest N400 to the encoded counterfactual. The counterfactual-minus-consistent difference should grow with encoding depth. This is P14.11. If the N400 does not show that growth, or if a novel surface form alone lifts the signal, the claim of extension to biological substrates becomes scope-limited to silicon.

What it ties together, if it holds

Several distinctions we normally keep apart would then belong together:

The last three are established human phenomena, and the paper does not measure them. It says they are consistent with the substrate mechanic at the deep end of the encoding axis, and that this is a reading you can put to the test.

None of this says “humans are just language models”. It says: if there is a substrate mechanism that produces a measurable cliff in bounded probabilistic systems, then that mechanism is also a candidate for what produces the matching signature in biological tissue. And the right experiment can distinguish “same machinery” from “different machinery that just happens to look alike”.

Why it matters for cognitive science

The standard story for why language models cannot really “reason” goes like this: their output is just probabilistic, so it is not real reasoning, because real reasoning is discrete, symbolic, logical. The factual core is correct. A language model is, mechanically, a probabilistic generator. The step from there to “therefore not real cognition” needs a suppressed premise: that a statistical substrate cannot implement genuine reasoning.

That premise convicts the human alongside the machine. On the best-supported account in mathematical psychology, human truth- and decision-judgment is a statistical process running on a statistical substrate. “Merely statistical” therefore names a property the two substrates share, not one that separates them.

The conclusion is deliberately modest. One may still hold that language models are not minds. One may simply not reach that conclusion by way of “merely statistical”, because that road, walked consistently, runs through the human substrate too.

Status and what follows

The paper sets out thirteen numbered predictions, P14.1 to P14.13. Five stand confirmed: the cliff itself, replication across architectures, the calculator-overflow scaling law, floating-point overflow as a cousin phenomenon, and the rising dose-response over eleven checkpoints. The cross-domain preference test stands as partially confirmed. Saturation stands as an open prediction until a clean compute-only test is run.

The dose-response is worth reading precisely. Over eleven checkpoints from zero to three hundred epochs the cliff grows with encoding depth, and the direction is robust to the choice of measure: the bounded measure (the entropy difference at the first word) goes from below zero before training to 0.55 nats at ten epochs and 0.77 nats at three hundred. The unbounded log measure goes from −2.45 to +13.96. The two agree that the cliff emerges between three and ten epochs and is larger deep than shallow. They disagree on the shape in between: the unbounded measure rises without a pause, the bounded one holds a middle range and rises again at the end. The paper therefore reports the bounded measure as primary, and does not use the absence of a plateau as an argument against other accounts.

Several predictions are forward-looking and testable: quantum decoherence as a cousin, a mental-arithmetic capacity limit, content-domain modulation across moral and aesthetic judgments, kindling transfer to encoding-adjacent contradictions, food-preference learning, a commit discontinuity at each step of a rollout chain, and most directly of all: the human N400 experiment.

The cleanest extension beyond silicon is the N400 study. It uses a known and well-characterised brain signal, a known training paradigm, a positive control in the same session, and a quantitative prediction about how the difference grows with encoding depth. Either the prediction holds and the substrate mechanic extends, or it does not and the claim becomes scope-limited to silicon. The experiment is what will distinguish the two outcomes, and the paper puts it forward as an invitation to the field.

Companion papers

© 2026 Tomas Lund. Independent Research, Aarhus. · Disclosure · ORCID