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First page of Classification and deontic explosion for contrary-to-duty obligations

Classification and deontic explosion for contrary-to-duty obligations

Bjørn Kjos-Hanssen

math.LO Apr 20, 2026 · v1 cs.LO
formalization logic
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TL;DR
All mathematical claims about Carmo–Jones deontic axiom systems, including the model classification and the explosion argument, are verified in a Lean 4 repository.
Abstract
Carmo and Jones have presented a sequence of candidate axiom systems for conditional obligation between 1997 and 2022. For their most recent system we demonstrate a limited form of deontic explosion: given that a student does not get the highest possible grade on a test, any other passing grade is acceptable. In addition to that negative result, we give a positive one: revisiting the strongest version of Carmo and Jones' 1997 system, we provide a surprising classification of all satisfying models in terms of a single forbidden possible world.
Problem

Carmo and Jones proposed a sequence of axiom systems for conditional obligation, aimed at contrary-to-duty reasoning, between 1997 and 2022. The adequacy of these systems and the structure of their models had not been settled.

Approach

The authors derive a weak form of conditional deontic explosion from axioms 5(b), 5(e) and 5(f) using an exam-grade example. They also classify all models of the strongest 1997 system (CJ97), and for weaker versions they characterize the least models satisfying the axioms and basic contrary-to-duty assumptions. All mathematical claims are verified in Lean 4, and the anomaly was found by analyzing output from a Maple script.

Results

In the most recent system, if the best outcome is unavailable, any other passing outcome becomes obligatory. The models of CJ97 are classified in terms of a single forbidden possible world, with families such as avoidOnly_e, avoidNone and noObligations.