## Sunday, February 17, 2008

### Rough Jottings Toward a Possible Deontic Logic

Standard deontic modal logics (D-type systems) lack the standard M-axiom:

(M) □A → A

The idea behind this is put clearly in McNamara's SEP article:

Given these correspondences, it is unsurprising that our basic operator, read here as “it is obligatory that”, is often referred to as “deontic necessity”. However, there are also obvious dis-analogies. Before, we saw that these two principles are part of the traditional conception of alethic modality:

If □p then p (if it is necessary that p, then p is true).
If p then ◊p (if p is true, then it is possible).
Their deontic analogs are:

If OBp then p (if it is obligatory that p, then p is true).
If p then PEp (if p is true, then it is permissible).

The latter two are transparently false, for obligations can be violated, and impermissible things do hold.

It can easily be seen that the reason for rejecting (M) in deontic logics is that "p" is understood as "It is the case that p". But this means that D-type modal systems are actually hybridizations of different modalities: the actuality modality is at home in systems designed for representing necessity and possibility (it is the intermediate modality between them); whereas it is something very different from obligation and permissibility. But what if we did, in fact, interpret the default operator as an intermediate modality between obligation and permissibility? There is, in fact, no reason why we can't do so. So suppose that, instead of representing "It is the case that p" the expression "p" represents something like "p is something that is preferred or advised by the prudent reasoner". Prudential counsel or preference is certainly a plausible candidate for an intermediate modality between obligation and permission.

Then we have a deontic logic that is an M-type system. In fact, S5 becomes a plausible deontic modal system. That is, you can make reasonable arguments for the plausible of all three of these axioms:

(M) □A → A
(B) A → □◊A
(5) ◊A → □◊A

(B), in fact, can be interpreted as a rejection of (a form of) tutiorism: if A is a conclusion that can be prudentially prefered, the permissibility of drawing that conclusion is obligatory. (M) indicates that obligatory conclusions can always be prudently drawn. (5) indicates that if prudently concluding A is permissible, its permissibility is obligatory (i.e., if a prudent reasoner can, qua prudent, draw the conclusion represented by A, a prudent reasoner must draw the conclusion that a prudent reasoner can draw the conclusion represented by A).

We can even begin to see more clearly how deontic operators relate to modal ones. The obligation operator tells us that a prudent reasoner must draw the conclusion as something to be preferred; the permission operator tells us that a prudent reasoner can draw the conclusion as something to be preferred.