# Session 3: More Problems with Normative Conditionals

In Chisholm (1963) we find the following example:

2. If you go, call.
3. If you don’t go, don’t call.
4. You don’t go.

We now present one possible analysis of this example in SDL:

We have translated the first premise by $$\mathsf{O} g$$, the second by $$\mathsf{O}(g \rightarrow t)$$ via wide-scoping, the third by $$\neg g \rightarrow \mathsf{O}\neg t$$ via narrow-scoping and the fourth by $$\neg g$$. As we can see, this leads to catastrophe in SDL: anything follows. Here,

• DDet (deontic detachment) is the principle according to which from $$\mathsf{O} A$$ and $$\mathsf{O}(A \rightarrow B)$$ follows $$\mathsf{O}B$$. (Easy exercise: show that it holds in SDL.)
• FDet (factual detachment) is, in our context (we’ll come back to this), the principle according to which from $$A$$ and $$A \rightarrow \mathsf{O} B$$ follows $$\mathsf{O} B$$. This clearly holds for SDL since it is just an instance of Modus Ponens.
• LogExp is the principle of Deontic Explosion: in SDL if we have $$\mathsf{O} A$$ and $$\mathsf{O} \neg A$$, anything follows. The reason is that there are no SDL-models where $$\mathsf{O} A$$ and $$\mathsf{O} \neg A$$ hold at some world. (Easy exercise: show this! Note that you need to use the fact that the accessibility relation is serial!)

What do we learn from that? Clearly, the formalization of our natural language example leads to trouble in SDL. There is still hope, though. Maybe we can just change our formalization: we can for instance narrow scope premise 2, or/and wide-scope premise 3.

The following table provides an overview and indicates problems we get with the other formalizations.

Norm F1 F2 F3 F4
1 Go! $$\mathsf{O}g$$ $$\mathsf{O}g$$ $$\mathsf{O}g$$ $$\mathsf{O}g$$
2 If you go, tell! $$g \rightarrow \mathsf{O}t$$ $$\mathsf{O}(g \rightarrow t)$$ $$g \rightarrow \mathsf{O}t$$ $$\mathsf{O}(g \rightarrow t)$$
3 If you don’t go, don’t tell! $$\neg g \rightarrow \mathsf{O} \neg t$$ $$\neg g \rightarrow \mathsf{O} \neg t$$ $$\mathsf{O}(\neg g \rightarrow \neg t)$$ $$\mathsf{O}(\neg g \rightarrow \neg t)$$
4 You don’t go. $$\neg g$$ $$\neg g$$ $$\neg g$$ $$\neg g$$
Dependence $$4 \rightarrow 2$$ $$4 \rightarrow 2$$ $$1 \rightarrow 3$$
Asymmetry 2 + 3 2 + 3
Explosion Yes
Oddities $$g \rightarrow \mathsf{O} \neg t$$ $$g \rightarrow \mathsf{O}\neg t$$ $$\mathsf{O}(\neg g \rightarrow t)$$

We indicate 4 types of problems:

1. Dependence: intuitively speaking the different premises are logically independent which means that neither should be derivable from another. This, however, is violated in F1, F3 and F4. For instance, in F1, the fourth premise implies the second one. (This is an instance of the classical inference: $$\neg A$$ implies $$A \rightarrow B$$.)
2. We have already talked about the problem of explosion: clearly, intuitively speaking our scenario describes a consistent setting which should not be rendered inconsistent by our logic.
3. Asymmetry: one may argue that conditional obligations should be modelled in analogous ways in the examples to avoid ad hoc adjustments to obtain the wanted outcome. Note, however, that there are subtle differences between wide-scoping and narrow-scoping. It is one thing to say that $$A \rightarrow B$$ should ideally be the case and another one to say that if $$A$$, then $$B$$ should ideally be the case.
4. Oddities: A deontic logic should not allow to derive formulas which are arguably non-sequiturs for the given scenario. For instance, in F1, we get $$g \rightarrow \mathsf{O} \neg t$$. This seems to be dubious if we intend to express conditional obligations via $$A \rightarrow \mathsf{O} B$$ as is done for premise 3.

Altogether, neither formalization is close to being satisfactory. Chisholm’s and Forrester’s examples are usually considered as the deathblow to SDL. On the positive side, these problematic examples helped establishing Deontic Logic as a research program with its own specific (hard) problems and techniques, far from being a simple instantiation of orthodox systems of modal logic. This lead to lots of innovation and also cross-fertilization (for instance with the field of non-monotonic logic). We will see more of this in future sessions of this seminar … in a sense we only get really started now.

We used the rest of the time in the seminar to talk a bit more about factual detachment. As we have seen last time, Forrester claimed that it doesn’t always make sense, but often it does. Here is an illustrative example (which goes back to John Horty). Since SDL is dead, we will use a new notation for conditional obligations from now on: $$A \Rightarrow_{\mathsf{O}} B$$ expressing that $$A$$ commits us/a given agent/etc. to bring about $$B$$. Often deontic logicians also write $$\mathsf{O}(B \mid A)$$ for this. For novices $$\Rightarrow_{\mathsf{O}}$$ is easier to read, so we will stick with it for now.

• Being served a meal, you ought to not eat with your fingers. $$m \Rightarrow_{\mathsf{O}} \neg f$$
• However, if you’re being served asparagus, you ought to eat with your fingers. $$(m \wedge a) \Rightarrow_{\mathsf{O}} f$$.
• Your being served asparagus. $$m \wedge a$$

Now if we were to rigorously apply factual detachment,

• Factual Detachment: From $$A$$ and $$A \Rightarrow_{\mathsf{O}}B$$ follows $$\mathsf{O} B$$.

we would be able to infer both $$\mathsf{O} \neg f$$ and $$\mathsf{O} f$$. This is not intuitive, to say the least.

In cases where a more specific obligation contradicts a more general one, it often makes sense to prioritize the more specific obligation. Logicians call such scenarios cases of Specificity. In these cases, the general obligation is excepted or cancelled.

One has to be careful, though. As we have seen in the Forrester case, contrary-to-duty obligations are tricky.

• In general you ought not to kill. $$\top \Rightarrow_{\mathsf{O}} \neg k$$
• However, if you kill, you ought to kill gently. $$k \Rightarrow_{\mathsf{O}} g$$.
• You kill.

In this case it seems that the general obligation, not to kill, is not really cancelled: it is still in place and violated.1 Our agent should be held accountable for it!

The above discussion gives us several desiderata for a deontic logic (there are many more, of course):

1. It should handle contrary-to-duty obligations without generating explosion or other oddities.
2. It should be able to distinguish specificity cases from contrary-to-duty cases.
3. It should handle factual detachment in such a way that it is sensitive to exceptional situations.

Let’s keep these in mind when investigating systems from the literature in future sessions of this seminar.

## References

• Chisholm, R. M. (1963). Contrary-to-duty imperatives and deontic logic. Analysis, 24, 33–36.
• Forrester, J. . (1984). Gentle murder, or the adverbial samaritan. Journal of Philosophy, 81, 193–197.
• Straßer, C. (2011). A deontic logic framework allowing for factual detachment. Journal of Applied Logic, 9(1), 61–80.
• Torre, L. V. D., & Tan, Y. (1995). Cancelling and overshadowing: two types of defeasibility in defeasible deontic logic. In , In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAI95) (pp. 1525–1532).

1. More on this distinctions can be found in, for instance, Torre&Tan (1995) and Straßer (2011). [return]