I am writing a paper at the moment and an area of Deontic Logic has cropped up in it. I know very little about the area and I was wondering if people could give me opinions on the axiomatic system that I want to use to for my paper.
I want to keep the system as weak as possible so as to avoid things like the Good Samaritan paradox or Chisholm's Paradox, so I want to keep my logic strictly classical, ie. no stronger than the base system $K$. After doing some searching on the internet, I got the impression that anything weaker than $K$ isn't really worth stuying because you no longer use Kripke Semantics but instead use something more along the line of Rudolf Carnap's definition for necessitation "$\Box P$ is true iff $P$ is true in all possible worlds". I also got the impression that Carnap's definition was somewhat flawed but I couldn't find out why. Is this true? I'd be greatly appreciative if someone could shed light on this and if/why Carnap's definition is indeed flawed.
The system of axioms that I want to use is:
- $\Diamond = \neg \Box \neg$
- $\Box A \rightarrow A$
- $A \rightarrow \Diamond A$
If anybody knows of any existing material on this system that would be great. Also, if people have any other comments on the selection of the above axioms that'd be great too. The axioms are for designing rule systems so I need the logic to contain rules for "must do then do" and "if do then it is allowed". Thanks!
To answer your questions, I wouldn't say non-normal modal logics are "uninteresting", but you're right they can't be used on Kripke semantics. The only other axiom in $K$ (called the $K$ axiom) apart from $\neg \Box \neg A \leftrightarrow \Diamond A$ (which is often considered an abbreviation) is the axiom $\Box (A \rightarrow B) \rightarrow (\Box A \rightarrow \Box B)$: and this can be proven to be valid on all Kripke frames. So if you don't want this to be valid, you can't use a Kripke frame, which means your semantics will have to start from scratch.
As for Carnap, I'm not sure what exactly in what sense Carnap's definition is taken to be "flawed", but here are two things to note about his system. First, he doesn't have accessibility relations linking states. Rather, he says a sentence $A$ is $L$-true (for "logically true") iff $A$ is true in every state description (if you like, possible worlds). Carnap's system behaves like S5 modal logic, but doesn't have the presence of anything like the modal accessibility relation that one can restrict, so Carnap's formal system isn't as rich as the full Kripke semantics. Second, it turns out that Carnap's original semantics is incomplete. So perhaps this is what is meant by "flawed", though this might be a contentious term.