Contrary-to-Duty Logic

Origin. Chisholm (1963) gave four sentences about a duty and its violation which are jointly consistent and logically independent as English, and showed that Standard Deontic Logic has no formalization of them possessing both properties. The paradox set the agenda for conditional obligation: dyadic operators (Hansson 1969, Lewis 1973), temporal treatments (Thomason 1981, van Eck 1982), defeasible ones (Prakken and Sergot 1996), and Åqvist's systems E, F and G are all answers to it.

Models. A violated obligation must stay in force as a description of what should have happened while a different obligation governs what to do now, and the difficulty is holding both at once. A logic that detaches only from the facts loses the primary duty; one that detaches only from the ideal never reaches the secondary duty. Sub-ideal worlds must therefore be ordered rather than merely admitted, so that "best among the worlds in which the duty is broken" is well defined.

Formalism.

Chisholm's paradox:

You ought to help your neighbor.
If you help, you ought to tell them.
If you don't help, you ought not tell.
You don't help.

The four SDL readings: Write g for helping, t for telling. (1) O(g). (4) ¬g. (2) wide: O(g → t). (2) narrow: g → O(t). (3) wide: O(¬g → ¬t). (3) narrow: ¬g → O(¬t).

Problem: Wide (2) with narrow (3): (1),(2) give O(t) by K. (3),(4) give O(¬t). D makes the set inconsistent. Narrow (2): follows from (4) alone — not independent. Wide (3): follows from (1) by monotonicity — not independent. Every assignment loses consistency or independence.

CTD structure: Primary duty: O(A). CTD duty: ¬A → O(B). Violation: ¬A actual. Secondary duty activates.

Violation contexts: Ideal: no violations. Sub-ideal: violations occurred. CTDs: bridge ideal to sub-ideal. Graceful degradation.

Solutions: Dyadic deontic: O(A/B) = A obligatory given B. Temporal: violations in past, duties now. Defeasible: primary defeated. Preference semantics.

Dyadic approach: O(tell/help): given help, tell obligatory. O(¬tell/¬help): given no help, no tell obligatory. Conditional obligations. Different from material conditional.

Preference-based: Worlds ranked by value. O(A/B) = best B-worlds are A-worlds. Conditional on circumstances. Handles CTD nesting.

Symbols.

Symbol	Unicode	Name	Meaning
O	—	Obligation	Monadic ought, as in SDL
O(A/B)	—	Dyadic obligation	A obligatory given B
CTD	—	Contrary-to-duty	Obligation conditional on a violation
⊗	U+2297	Circled times	Reparation operator: A ⊗ B, B owed if A violated
≤_d	—	Deontic preference	Betterness ordering on worlds
g	—	Helping	Atom in Chisholm's set
t	—	Telling	Atom in Chisholm's set
¬	U+00AC	Negation	Violation as ¬A
→	U+2192	Material conditional	Wide and narrow scope readings
∧	U+2227	Conjunction	Joint consistency of the four sentences

Metatheory. The paradox is a theorem about SDL, not a difficulty of formalizing: with (2) wide and (3) narrow, K gives O(t) from (1) and (2), modus ponens gives O(¬t) from (3) and (4), and D makes the set inconsistent; with (2) narrow it is entailed by (4) alone, and with (3) wide it is entailed by (1), since Oφ → O(ψ → φ) holds by necessitation and K. So no assignment of scopes preserves both properties the English has. Dyadic systems evade this by reading the conditionals as O(t/g) and O(¬t/¬g), which are not material conditionals and do not agglomerate with the unconditional O(g): the primary duty survives its own violation because it is evaluated on the best worlds overall while the secondary duty is evaluated on the best worlds among the ¬g-worlds. Preference semantics of this shape goes back to Hansson (1969) and Lewis (1973); Parent (2015) proved completeness for Åqvist's E and F.

Applies to. Normative reasoning. Legal systems. AI ethics. Contract logic.

Limitations. Dyadic systems secure consistency by surrendering detachment: from O(¬t/¬g) and ¬g they do not in general yield O(¬t), so the logic represents the secondary duty without ever delivering it as something to do, and restoring unrestricted factual detachment restores the inconsistency. Prakken and Sergot's pragmatic oddity is not thereby avoided — a logic that keeps the violated primary duty in force still derives the conjunctive obligation to help and to not tell, which no one owes. The betterness ordering is a primitive fixed outside the theory, so which sub-ideal world counts as best is stipulated, not argued. Temporal treatments require the violation to precede the reparation and so do not touch timeless contrary-to-duty cases. Chains of successive violations need dedicated machinery, such as Governatori and Rotolo's ⊗, rather than iteration of the dyadic operator.