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 studying because you no longer use Kripke Semantics but instead use something more along the line of Rudolf Carnap's definition for necessitation "□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:

  1. ◇ = ¬□¬
  2. AA
  3. A → ◇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!

  • Hi Jimbo, welcome to PhilosophySE. Unfortunately this site isn't equipped with mathjax, so you're going to have find more mundane ways writing out equations you'd like to display.
    – David H
    Commented Aug 21, 2013 at 16:02

2 Answers 2


Since (daftly) we have no LaTeX capability here, use Nec for the box, and Poss for the diamond.

By 2, Nec not-p --> not-p, so contraposing (assuming classical negation), p --> not-(Nec not-p), i.e. we get 3. So really you've given us just one rule, 2, plus a rule of definitional abbreviation. So the modal system is jolly uninteresting.

And this one rule is plainly not appropriate if the modality is interpreted as deontic necessity. It may be that p ought deontically to be the case; it sadly doesn't follow that p is the case.

So you've got an uninterestingly trivial system, which in any case can't be interpreted deontically. Back to the drawing board!

Or check out http://plato.stanford.edu/entries/logic-deontic/ for more guidance.

  • Ah! I didn't appreciate that the 3rd one followed the second! Thanks for pointing that out. Although the system is a little on the trivial side it doesn't really matter too much. It is going to be used for looking at some very basic kinds of statements and I essentially just want to make sure that Carnap's definition hasn't been shot down. Thanks again
    – Jimbo
    Commented Aug 21, 2013 at 16:27

Let's start with your proposed system ('Oφ' for φ is obligatory, 'Mφ' for φ is permissible):

1) Mφ ≡ ¬O¬φ
2) Oφ → φ
3) φ → Mφ

Dr. Smith has said everything to be said about this system, but just to emphasize: (1) is simply the definitional fact that the diamond and the box are interdefinable (this is certainly true for all classical dual operators); (2) is too strong: not everything that's obligatory is actually the case, unfortuantely; and (3) says that whatever is the case is permissible. As Dr. Smith said, this system is pretty trivial due to (2).

Now, you said that you want a system so weak that it will avoid the Good Samaritan paradox and Chisholm's paradox, "i.e., [a system] no stronger than the base system K." Unfortunately, K (possibly named after Kripke himself) is both already strong enough to give rise to both of those paradoxes (these are the so-called monotonicity problems associated with treating O as a normal modal operator), and weak enough to be sound with respect to all Kripke models.

To actually resolve those paradoxes, you have to somehow go below K and in Kripke models there is no such underground. Carnap's modal systems (which are roughly equivalent to S5) are much stronger than the weakest Kripke system because in S5, as you probably know, the accessibility relation is an equivalence relation, so any world is accessible to any other world. There are other reasons for not going into Carnap's state-description semantics and just sticking to Kripke's possible world semantics.

Among the standard ways of actually dealing with the aforementioned paradoxes (and others) is to give the deontic modalities neighborhood semantics or to explicate them within a weakened version of D. Lewis' counterfactual semantics. Both of these semantics have been extensively treated in the literature; here are some things to look at:

(1) Deontic Logic (SEP): explains lots of the key topics, including those paradoxes and solution attempts.
(2) Carnap's Modal Logic (IEP): a very recent exposition of Carnap's modal systems by an expert in ML.
(3) Modal Logic for Open Minds: a standard textbook; see in particular, Ch. 16 on deontic logic.
(4) Neighborhood Semantics for Modal Logic: a beastly introduction to the topic by an expert in the field.
(5) Counterfactuals: a classic in the field; relevant here due to its connection with dyadic deontic logic.

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