right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction. In addition to the usual modal axioms (necessitation rule N and distribution axiom K) for the alethic operator , Anderson's deontic logic only requires one additional axiom for the deontic constant : , which means that there is alethically possible to fulfill all obligations and avoid the sanction. This version of the Anderson's deontic logic is equivalent to SDL. However, when modal axiom T is included for the alethic operator (), it can be proved in Anderson's deontic logic that , which is not included in SDL. Anderson's deontic logic inevitably couples the deontic operator with the alethic operator , which can be problematic in certain cases. Dyadic deontic logic An important problem of deontic logic is that of how to properly represent conditional obligations, e.g. If you smoke (s), then you ought to use an ashtray (a). It is not clear that either of the following representations is adequate: Under the first representation it is vacuously true that if you commit a forbidden act, then you ought to commit any other act, regardless of whether that second act was obligatory, permitted or forbidden (Von Wright 1956, cited in Aqvist 1994). Under the second representation, we are vulnerable to the gentle murder paradox, where the plausible statements (1) if you murder, you ought to murder gently, (2) you do commit murder, and (3) to murder gently you must murder imply the less plausible statement: you ought to murder. Others argue that must in the phrase to murder gently you must murder is a mistranslation from the ambiguous English word (meaning either implies or ought). Interpreting must as implies does not allow one to conclude you ought to murder but only a repetition of the given you murder. Misinterpreting must as ought results in a perverse axiom, not a perverse logic. With use of negations one can easily check if the ambiguous word was mistranslated by considering which of the following two English statements is equivalent with the statement to murder gently you must murder: is it equivalent to if you murder gently it is forbidden not to murder or if you murder gently it is impossible not to murder ? Some deontic logicians have responded to this problem by developing dyadic deontic logics, which contain binary deontic operators: means it is obligatory that