Contrary-to-Duty Logic

Origin. Chisholm (1963), CTD paradox. Conditional obligations. What to do when violated. Deontic conditionals. Foundation of applied deontic logic.

Models. Obligations given violations. Secondary duties after primary breach. Nested deontic structure. Defeasible duties. Practical moral reasoning.

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.

Problem: From (1): O(help). From (4): ¬help. From (3),(4): O(¬tell). Consistent? Inconsistent? SDL struggles.

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	Meaning
O(A/B)	—	A obligatory given B
CTD	—	contrary-to-duty
⊗	U+2297	secondary obligation
≤_d	—	deontic preference

Metatheory. Conditional obligation. Sub-ideal contexts. Dyadic deontic. Graceful degradation.

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

Limitations. Multiple solutions. No consensus. Technical complexity. Edge cases.