In the application of large language models (LLMs), a model should refuse to answer two types of questions: those it cannot answer correctly and those it should not answer at all, such as unanswerable questions or those based on false premises. The usual approach thresholds a single confidence score, which cannot differentiate between these two. By studying five instruction-tuned models from three different families (ranging from 2B to 14B parameters), we find that these are separate axes.
Ordinary answer-confidence tracks whether an answer is correct but is nearly blind to whether the question is answerable; conversely, a linear probe on hidden states does the reverse. This blind spot does not diminish with scale, particularly in naturally occurring false-premise questions (CREPE), where answer-confidence, P(IK), P(True), and even directly asking the model whether a premise is false remain near chance levels, while a hidden-state probe achieves an AUROC of 0.69 to 0.77: the model represents a problem it will not report.
This issue is fixable. Instructing a model to check premises backfires, as it then disputes both sound and false premises (57% false challenges), unable to distinguish them; routing the same instruction with the probe roughly triples challenge precision.
We convert the two axes into a calibrated policy that only answers when both an answerability score and a correctness score clear their respective thresholds. This ensures that the unanswerable-answer rate is controllable at every scale, while the wrong-answer rate is capped by model accuracy, tightening the guarantee to 0.75 coverage of correct answers, compared to 0.31 for a single threshold; at 14B, this is the only policy that certifies at all.