## Monday, March 25, 2013

### Deontic Logic Is a Logic of Solutions

A deontic logic is a modal logic in which the strong modality operator (Box) is interpreted as obligation (or something similar) and the weak modality operator (Diamond) is interpreted as permission or acceptability; in standard deontic logic, the characteristic modal axiom is:

□p → ⋄p

That is, Box implies Diamond. In standard deontic logic, however, Box does not imply True; if X is an obligation, it does not follow that X is true.

I am inclined to think that we largely interpret deontic logics too narrowly. In reality, deontic logics are logics of solutions to problems, and it is this, I think, that makes them suitable for handling obligation and the like, because these are also concerned with solutions to problems. I would like to note a few apparent puzzles of deontic logic that make a bit more sense when we recognize deontic logics as logics of solutions to problems, in which Box consists of conditions that solutions have to meet and Diamond consists of features of solutions that are acceptable.

(1) Deontic Necessitation. Deontic necessitation says that if you can prove p as a logical theorem, you can conclude □p. People tend not to like this rule, but it is required to make standard deontic logic play nicely. It makes plenty of sense if we think about deontic logic as being concerned with the kinds of features that a solution to a problem must have. If you have proven p as a theorem, you have established that it is in some sense necessary; but solutions to problems have to take necessities into account. Therefore if something is proven to be necessary, it is a necessary constraint on solution, or, to put it in other words, the solution ought to take it as a fixed reference point.

Deontic necessitation has the further result that □⊤, where ⊤ is Top, and is usually taken to indicate tautology. And this makes sense as well: solutions have to take tautologies as fixed.

(2) Good Samaritan Paradox. Take the following proposition:

□(Jones helps Smith who was robbed)

It seems to follow that "Jones helps Smith who has been robbed" implies "Smith was robbed and Jones helps Smith". But then it follows:

□(Smith was robbed and Jones helps Smith)

But from this it seems to follow by conjunctive simplification:

□(Smith was robbed).

This seems to be a bit awkward, because it then seems that we are saying that it ought to be true that Smith was robbed. However, if we interpret □ as indicating something that a solution ought to take as a fixed reference point, is there a situation in which this makes sense? Yes, the situation in which the problem is that Smith was robbed. A solution to the problem that Smith was robbed has to take into account the fact that Smith was robbed. And given this interpretation of □ as something with which the solution necessarily has to be consistent, it's pretty clear that

□(Jones helps Smith who was robbed)

does imply

□(Smith was robbed).

(3) Penitent's Paradox. Consider the following proposition:

□~(John does wrong)

□~ usually is taken to mean that it is forbidden in some way. But this proposition implies

□~(John does wrong and John repents of his wrongdoing)

From which it follows:

□~(John repents of his wrongdoing).

But we see that, under the solutions interpretation, our first proposition indicate that we needed to take as a fixed point for our solution "It is false that John does wrong". But given this it is surely not surprising that we should also require our solution not to include his repenting of having done wrong, which seems to imply that he has done wrong.

So we see that the interpretation has some force against puzzles and paradoxes. There are other puzzles and paradoxes that would be more difficult, the most important of which are "conflicting obligation" paradoxes. And it is is important to understand that I am not here saying that standard deontic logic is the only true logic of solutions. I think, in fact, that this is obviously false; there will be problems where the most appropriate solutions will be governed by nonstandard deontic logics. But deontic logics in general are logics of solutions, and taking them as such clarifies a great deal.

This ties in with another important point, which I've argued before, and will not argue here: obligation or 'ought' in our ordinary sense is (at least in many circumstances) appropriately modeled by deontic logic not because deontic logic has only to do with obligation, but because our ordinary sense of 'ought' or obligation primarily concerns the problem of deciding what to do, and therefore indicates a constraint on solutions to this particular problem. But there is nothing hugely mysterious about it; it's just a special, and especially important, version of the general fact that solutions are logically constrained by the problems to which they are put forward. This helps clarify what one really wants in a truly deontic logic, i.e., a logic that really is about moral obligation or duty: one wants a deontic logic, i.e., a logic of solutions, that takes into account the distinctive features of problems concerned with deciding what to do. And almost all the puzzles people have over standard deontic logic concern distinctive features of this particular kind of problem.