Phantom Paradoxes: Language Models Treat Unresolved Circular Truth Dependence as Contradiction

Abstract

Do language models struggle with self-reference itself, or with structures where propagating truth values along references cannot settle a value in finitely many steps? Our results support the latter. In our main test, statements refer to one another's truth in a cycle, yet exactly one consistent assignment exists. Local propagation never settles the queried value, but exhaustive search finds it, so INCONSISTENT (no consistent assignment) is always wrong. We call such misreadings of satisfiable problems phantom paradoxes. Each circular problem has a grounded twin with the same answer but no cycle. To rule out logical difficulty and unfamiliar wording, we added constraint twins (equivalent formulas without talk of truth) and perplexity-matched grounded controls. In confirmatory behavioral tests fixed before any confirmation response, four open models from three families answered INCONSISTENT on average 31.0 percentage points more often on circular problems than on grounded twins, 36.4 more than on constraint twins and 37.1 more than on perplexity-matched controls. In each model, validated layer-wise truth directions, fitted on unrelated arithmetic, recovered the queried value at the question's last token for grounded twins but not for circular problems. Constraint twins, likewise unrecovered, drew INCONSISTENT far less often, so circular truth dependence, not the unrecovered value alone, matters. In a small natural-language bank, self-reference settling in finitely many steps drew INCONSISTENT 23.4 points less often than unsettled self-reference with the same classical solution. Pooled tests did not show the predicted late-layer revision or attention spread. Overall, models treat unresolved circular truth dependence as contradiction. We will release all prompts, outputs and hidden states.