Editing Modal logic (section)

====Intuitive problems with deontic logic====
When we try to formalize ethics with standard modal logic, we run into some problems. Suppose that we have a proposition ''K'': you have stolen some money, and another, ''Q'': you have stolen a small amount of money. Now suppose we want to express the thought that "if you have stolen some money, it ought to be a small amount of money". There are two likely candidates,
: (1) <math>(K \to \Box Q)</math>
: (2) <math>\Box (K \to Q)</math>

But (1) and ''K'' together entail □''Q'', which says that it ought to be the case that you have stolen a small amount of money. This surely is not right, because you ought not to have stolen anything at all. And (2) does not work either: If the right representation of "if you have stolen some money it ought to be a small amount" is (2), then the right representation of (3) "if you have stolen some money then it ought to be a large amount" is <math>\Box (K \to (K \land \lnot Q))</math>. Now suppose (as seems reasonable) that you ought not to steal anything, or <math>\Box \lnot K</math>. But then we can deduce <math>\Box (K \to (K \land \lnot Q))</math> via <math>\Box (\lnot K) \to \Box (K \to K \land \lnot K)</math> and <math>\Box (K \land \lnot K \to (K \land \lnot Q)) </math> (the [[contrapositive]] of <math>Q \to K</math>); so sentence (3) follows from our hypothesis (of course the same logic shows sentence (2)). But that cannot be right, and is not right when we use natural language. Telling someone they should not steal certainly does not imply that they should steal large amounts of money if they do engage in theft.<ref>Ted Sider's ''Logic for Philosophy'', unknown page. http://tedsider.org/books/lfp.html</ref>