Contradictions are an interesting case in scientific reasoning because they sit between two kinds of work. One is semantic: deciding when different findings should be treated as claims about the same underlying thing. The other is formal: once that relation is fixed, deciding whether the claims can all be true together.
We often miss this because the two steps collapse into one another. But they are not the same problem. A contradiction can be difficult because the formal system is hard to solve, or because it is not obvious which formal system the evidence should become.
I thought it would be interesting to ask whether language models actually resolve contradictions through these two steps. Can they recover the underlying formal structure from scientific language and then reason over it, or does their judgment about what is plausible shape the conclusion before that structure is recovered? Scientific language comes with strong priors about what is plausible. Those priors are often useful, but when semantic plausibility and formal consistency disagree, it is not obvious which one will govern the model's reasoning.
From Findings to Constraints
To make the distinction concrete, we can write contradiction in its simplest formal form. A scientific finding rules out some possible states of the world and leaves others open.
Let H be the set of possible states. For a finding si, let T(si) denote the states that remain compatible with it. The translation T is the bridge from scientific language to formal structure.
If at least one state survives every finding, the evidence is jointly consistent:
If none survives, there is a contradiction:
Once there is a contradiction, the next question is which state fits the findings best? A separate question is how much the findings would need to change before another state becomes preferable.
That gives us four related, but distinct, parts of the problem:
- Contradiction: Can any state fit all the findings?
- Resolution: If not, which state fits them best?
- Uncertainty: How much would the findings need to change before another state becomes preferable?
- Translation: How did the findings become constraints on the same possible states?
The first three could, in principle, be handled within a formal representation. The fourth determines what that representation should be.
The Translation Is Not Neutral
The notation T(si) makes translation look almost mechanical. In science, it is where much of the judgment sits.
Suppose two studies report opposite effects of the same treatment. Are they actually comparable? That depends on things like population, dose, timing, measurement, and experimental context. If those differences are judged unimportant, the findings may become A and ¬A. If one of them matters, they may instead become Ax and ¬Ay, and the contradiction disappears.
The same findings can therefore be contradictory under one translation and consistent under another:
Not every translation is equally plausible. Scientific judgment tells us which differences matter, which can be ignored, and which findings should be treated as evidence about the same underlying effect.
Translation can fail in both directions. It can collapse two genuinely different phenomena into one variable and create a contradiction that was never there, or it can keep related findings apart and hide a contradiction that only becomes visible once they are aligned. This is where semantic judgment enters. Before mathematics can test consistency, someone has already decided which findings belong in the same formal problem.
Can a Model Recover the Formal Structure?
Language models have to do both kinds of work at once. Their scientific knowledge helps them infer which findings belong together, but it also shapes which conclusions seem plausible. The interesting question is what happens during translation. Can a model recover the formal structure expressed through scientific language, or do its scientific priors pull the translation toward a more familiar, easier-to-reach representation?
We tried to isolate that question in a small experiment. We fixed the underlying formal problem so that it had one best solution and one nearby alternative that was slightly less consistent but more biologically plausible, then tested twelve matched versions of it.
When the relations were written directly as XOR equations, the model found the formal solution in all twelve. When the same relations were translated into scientific findings, it chose the biologically plausible alternative in all twelve.
We then asked whether the problem was simply that the model had not noticed the structure. A generic instruction to look for underlying structure increased the amount of reasoning but changed no answer, and a later instruction to derive a compact formal representation also left every completed answer at the same alternative.
The behavior changed when we added one piece of scientific context: the findings came from paired replications under equivalent conditions and referred to shared underlying effects. We still did not tell the model which findings belonged together, but we gave it a reason to search for that relation. Eight of the twelve completed answers then moved to the formal solution.
The experiment is small, but it separates the two steps cleanly. The model could solve the formal problem once the structure was explicit; the difficulty was translating the scientific language back into that structure.
What This Leaves Open
The same prior that helps a model recognize thousands of familiar scientific relationships may make a formally stronger but scientifically surprising interpretation harder to reach. Better judgment may therefore improve most translations while making some departures from the expected frame less accessible.
Reasoning budget and reasoning freedom may also be different things. More computation can search a representation more deeply without ever reconsidering the representation itself. If that holds more broadly, some difficult reasoning problems may not need longer chains of thought so much as a way to reopen the question of what problem is being solved.
Contradictions are useful because they can make that boundary visible. They force semantic judgment and formal consistency apart, and expose whether the model can move between them when they disagree.