Deontic logic is a logic ahout morality and justice, to make the deontic logic mathematically, we have to define a good-evil function to describe the state of deontic logic?
A deontic logic illuminates inferential relationships between propositions containing deontic terms. Deontic terms in a natural language might be symbolized as either predicates, predicate-operators, or sentence-operators. If you treat them as sentence-operators, the situation will be similar to that of modal logic, e.g. you can interdefine deontic sentence-operators (it is good that, it is evil that) similarly to the way you interdefine necessity and possibility. But however you symbolize deontic terms, your logic will need to have plausible axioms.
Yaozhi Jiang ,
It depends on what you mean "describe the state of deontic logic".
Karl Pfeifer gave you an excellent explanation on the possibility of embedding or expanding the terms into the logic which answers the question in a particular interpretation. I will take your question from a different angle. That is, is it necessary to have those concepts to have "a modal logic"? Modal logic as with other logics is a formal system where the syntax and semantics are outlined. Philosophical debate on whether those good-evil correspond (i.e. dictate) in some way to normative concepts is more of a philosophical question than a logical one (as Pfeifer already said you can include them as part of the logic ).
Karl Pfeifer As I understand it, deontic logic is the K logic for which the necessity and possibility operators ([] square/box and diamond) are associated with an axiomatization that is consistent with the "intuitive" interpretation given to obligation ([]) permission ().
I think that these operators are quite agnostic regarding Good and Evil. One can define a perfectly acceptable deontic logic for the Good that will be absolutely acceptable for Evil. Problems in defining deontic logics are actually quite abstract and, as I understand it, mainly resolve around the issue of ex falso quolibet for deontic explosion.
Pierre-Yves Gicquel
RE: "I think that these operators are quite agnostic regarding Good and Evil. One can define a perfectly acceptable deontic logic for the Good that will be absolutely acceptable for Evil."
For a plausible interpretation, I think good and evil would have to be interdefined:
E.g: x is good =df x is not evil, and x is evil =df x is not good (though there could be refinements to explicitly recognize the possibility of neutrality). Axioms for duty or obligation would have to rule out, or be supplemented by further axioms that would rule out, obligations to do evil.
If the axioms are consistent, EFQ shouldn't be a defect attributed to deontic logic per se since it already is an issue for standard first-order logic. Supposedly the constraints of relevance logic can be added to the system to deal with EFQ.
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