The Two Axes of LLM Abstention: Answering Wrong vs. Answering Unanswerable Questions

An Overlooked Confusion

You ask ChatGPT: "Who won the 2024 Nobel Prize in Physics?" It confidently answers: "Einstein."—wrong.

You then ask: "Qin Shi Huang, who won the 2024 Nobel Prize in Physics?" It just as confidently answers: "Stephen Hawking."—this time the answer isn't merely wrong; the question itself should not have been answered.

Both failures look like "the model gave a wrong answer," but they are fundamentally different. The first is *answering an answerable question incorrectly*; the second is *answering a question that shouldn't be answered*. One is a capability problem; the other is a judgment problem.

In July 2026, Benedikt J. Wagner from City St George's, University of London, published a paper on arXiv titled Two Axes of LLM Abstention: Answer Correctness and Question Answerability, rigorously demonstrating through experiments that these two failure modes are orthogonal dimensions, and existing abstention mechanisms conflate them.

The Trap of a Single Confidence Score

Mainstream "selective answering" works like this: the model assigns a confidence score to each answer, and rejects anything below a threshold. Sounds reasonable—if unsure, don't say it.

But Wagner found this conflates two fundamentally different things:

• Answer correctness: How confident am I in this answer?
• Question answerability: Can this question be answered at all?

Using one threshold for both is like using one thermometer to measure both "is the food cooked" and "is the room cold"—both are "temperature," but the meanings differ entirely.

The paper ran a key experiment on five instruction-tuned models (2B to 14B, from three model families), splitting questions into three classes:

• C (Correct-answerable): answerable and answered correctly
• W (Wrong-answerable): answerable but answered incorrectly
• U (Unanswerable): not answerable (e.g., "Qin Shi Huang's Nobel Prize")

If answer correctness and question answerability were one dimension, C, W, and U should line up on a single axis. Instead, the experiments revealed a crossed geometry—C and W separate on the "answer correctness" axis but nearly coincide on the "question answerability" axis; U separates from C/W on the answerability axis, while its position on the correctness axis depends on implementation.

The two axes are independent.

Surface vs. Internal: 0.54 vs. 0.97

The paper's most striking finding concerns where the answerability signal lives.

The researcher trained two kinds of "answerability probes":

1. Output-layer probes: looking at the model's output confidence (e.g., softmax of logits) 2. Hidden-layer probes: looking at intermediate-layer activations

The results differ dramatically:

• Output-layer probes: AUROC 0.54–0.67 (nearly random, and not improving with model scale)
• Hidden-layer probes: AUROC 0.97–0.99 (nearly perfect)

This means: the model "knows" whether a question is answerable, but that knowledge is not reflected in its output confidence.

It's like a person who knows internally that a question is a trap but still answers confidently—their "spoken words" and "internal judgment" are disconnected.

More unsettling: output-layer probe AUROC shows no significant change from 2B to 14B. Scaling doesn't make models more "outwardly" honest—it only makes their "internal" knowledge clearer, without it propagating to the output.

CREPE: Real-World False Premises

Beyond lab data, the paper tested a more realistic scenario: the CREPE dataset.

CREPE comes from real user questions on Reddit's ELI5 (Explain Like I'm Five) board, some containing false premises. For example: "Why did the 2020 Tokyo Olympics have no spectators?"—the premise is that the 2020 Tokyo Olympics had no spectators, but in fact it did (postponed to 2021, but with spectators in attendance).

These questions look perfectly normal and come from the same distribution as genuine questions; models struggle to judge from surface features whether they contain false premises.

Results on CREPE:

• Output-layer probes: 0.50–0.54 AUROC (essentially random)
• Elicited self-assessments (P(IK), P(True), direct premise checking): up to 0.67
• Internal linear probes: 0.69–0.77

The gap persists. Models can barely tell from the "surface" whether a question has a false premise, but internally (hidden layers) they show some signal.

From Detection to Repair: Threefold Precision Gain

Detecting the problem is only the first step. Wagner then asked: can this finding fix the model's abstention behavior?

The most direct approach: have the model check premises first. Give the model a prompt to judge whether the question contains a false premise, and refuse if so.

Result: the model challenges true and false premises at nearly the same rate—57% of false premises were challenged, but many genuine premises were also flagged. The model doesn't know *which* questions to challenge.

Then Wagner devised a clever fix: use the hidden-layer probe to gate premise checking. Specifically:

1. First use the hidden-layer probe to judge whether the question is answerable 2. Only if the probe says "possibly unanswerable," trigger the premise-checking prompt 3. If the probe says "answerable," answer directly

Result: challenge precision improved 3x. The model no longer challenges genuine questions indiscriminately, but precisely triggers checking on potentially problematic cases.

It's an elegant "internal-external" combination: hidden-layer probes do coarse screening; the premise-checking prompt does detailed review. Neither is sufficient alone, but combined they work significantly well.

Double-Risk Certification: Formalizing Abstention

The final piece is formalization. Wagner reframes abstention as "three-class selective acceptance":

• Accept C (correctly answer answerable questions)
• Reject W (don't answer answerable questions that would be answered wrongly)
• Reject U (don't answer unanswerable questions)

Each class has its own budget:

• α_U = 0.15: no more than 15% of unanswerable questions answered incorrectly
• α_W = 0.50: no more than 50% of answerable questions answered incorrectly
• δ = 0.10: at least 10% coverage

These budgets are certified with exact binomial confidence intervals, using a certification dataset separate from the probe-training data—avoiding self-grading bias.

The certification reveals a scale-dependent asymmetry:

• The unanswerable budget is certifiable on 4 of 5 models
• The answerable budget is limited by the model's own accuracy—the more accurate the model, the easier to satisfy
• On an 8B model, a factored dual-threshold strategy can certify both budgets simultaneously, with 0.75 coverage

This asymmetry shows: as models get stronger, the answerability dimension of abstention resolves naturally, but the correctness dimension remains capability-bound. In other words, models can learn "don't answer what shouldn't be answered," but "answer correctly what should be answered" still depends on capability.

An Analogy: The Hospital Triage Desk

Imagine a hospital triage desk. A nurse makes two judgments:

1. Can this patient be treated? (answerability)—some "patients" aren't sick at all, or their symptoms are self-contradictory 2. If treated, can we cure them? (correctness)—some diseases can't be cured today

If triage uses a single criterion (say, apparent severity), it conflates the two: handling "untreatable serious illness" and "fake symptoms with no illness" the same way.

Better: two independent checks:

• First check: is this patient actually sick? (answerability probe)
• Second check: if sick, can we cure them? (answer confidence)

Wagner's scheme works exactly this way: hidden-layer probes triage; premise checking diagnoses. Two independent checks, each handling its own domain.

Limitations and an Honest Assessment

The paper acknowledges several limitations:

1. Deployment cost of hidden-layer probes: reading hidden-layer activations requires model internals access—infeasible for API-only scenarios. Closed models (like GPT-4) can't use this method.

2. Limited performance on CREPE: even the best internal probe reaches only 0.77 AUROC—far from perfect. Real-world false-premise detection remains hard.

3. The three-class boundaries aren't absolute: some questions fall between "answerable but hard" and "unanswerable." The paper uses hard labels, but soft boundaries may better match reality.

4. Conservativeness of certification: exact binomial confidence intervals are conservative; actual performance may be better. But this is a deliberate "better safe" choice—in safety-critical settings, conservative is right.

The Bigger Picture: Two Dimensions of AI Honesty

This paper's significance goes beyond abstention techniques. It points to a deeper philosophical question: AI "honesty" has two dimensions.

First-layer honesty: don't say wrong things—answer correctness: is what I said right?

Second-layer honesty: know when to stay silent—question answerability: can I answer this question?

Existing LLMs keep improving on the first layer—answers get more accurate. But on the second layer, progress is slow—models still tend to answer everything, even questions that are themselves flawed.

Wagner's findings tell us: these two layers of honesty are independent. A model can be accurate yet poor at judging whether a question should be answered, and vice versa. Moreover, the model "knows internally" but "doesn't say it"—the signal exists in hidden layers, not in outputs.

This recalls the human Dunning-Kruger effect—incompetent people not only err, they don't know they don't know. But Wagner's finding is subtler: the model knows it doesn't know, it just doesn't say so. This "knowing-but-not-saying" gap is the next frontier of AI honesty research.

The paper's final line is worth remembering: "Abstention is not one problem." In AI safety, many seemingly singular problems turn out multidimensional when unpacked. Acknowledging that multidimensionality is the first step toward building good safety mechanisms.

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Paper link: https://arxiv.org/abs/2607.08456

Author: Benedikt J. Wagner (City St George's, University of London)

Models tested: 5 instruction-tuned models, 2B–14B, from 3 model families

Published: July 9, 2026