Editing Deontic logic (section)

==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:

: <math>O(\mathrm{smoke} \rightarrow \mathrm{ashtray})</math>
: <math>\mathrm{smoke} \rightarrow O(\mathrm{ashtray})</math>

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:

: <math>O(A \mid B)</math> means ''it is obligatory that A, given B''
: <math>P(A \mid B)</math> means ''it is permissible that A, given B''.

(The notation is modeled on that used to represent [[conditional probability]].)  Dyadic deontic logic escapes some of the problems of standard (unary) deontic logic, but it is subject to some problems of its own.{{example needed|date=July 2020}}