Layering violations in SDL: a preliminary investigation

Despite the large dierences between SDL and PDeL, maybe it is possible to trade between them some useful techniques. One of them is certainly the one we have seen exploited in the last section, where PDeL was extended to PDeL(n), the only dierence being that in PDeL(n) norms can be layered up indenitely by using dierent violation atoms.

To my knowledge, SDL(n) does not yet exist. Is it even possible? Clearly, we would have to use here a SDL variant where the deontic operators have been introduced via an Anderson-Kanger reduction, in order to be able to add multi- ple violation atoms. Nevertheless, we can do something very similar in SDL by having many accessibility relations. Each accessibility relation will model ideality

CHAPTER 2. PDEL.

relative to some norm. By contrast, the only accessibility relation in SDL mod- elled `perfection': what ought or ought not be the case is modelled by what is or is not the case in theD-accessible worlds, that are supposed to be the `deontically

ideal' ones. Could we, for example, have a family of{Di}i∈Nrelations that encode hierarchies of perfect worlds?

We found out (too late, unfortunately) an interesting paper that seems to be proposing a logic similar to the SDL(n) we are hypothesising. That is Goble's [Gob00]. There, the idea is in fact to extend SDL with a family of accessibility relations and interpret each one as `relative (to a standard) ideality'. Then he would dene two operators, `obligatory in general' (Og) and `obligatory relative to

some standard' (Or). Then roughly:

• w|=Ogϕ i for all accessibility relationsD, wDw0 implies w0 |=ϕ

• w|=Orϕi for some accessibility relations D, wDw0 impliesw0 |=ϕ

Or is a `weak' obligation, where for example Orϕ 6≡ ¬Or¬ϕ. On the other

hand,Og is stronger and Ogϕ≡ ¬Og¬ϕ.

This is dierent from what we (and Meyer) had in mind. Still it would be interesting to see if there is any overlap and, more in general, in what ways does this approach contribute. We procrastinate its detailed discussion to future work.

SDL to SDL(n): what advantages? We could dene the O operator: Oiϕ:=i(¬ϕ⊃V)

Where i is one of the n accessibility relation we have in SDL(n). Now, if we

dene an `absolute ideality' modalabs:

absϕ:=

^

x≤n

xϕ

it is clear that `ϕentails `absϕwhich entails `xϕfor all x.

For example, let us consider briey Forrester's paradox:

f1 O1¬k f2 k⊃O2g f3 k f4 g ⊃k

Now, modus ponens on f2,f4 outputs O2g. f3 ⇒ abs(g ⊃ k) ⇒ O2(g ⊃

k) ⇒ (O2g ⊃ O2k); thus O2k (because of f2). Similarly we obtain O1g ⊃ O1k.

By modus tollens with f1 (since Oiϕ⊃ ¬Oi¬ϕ) this entails that O1¬g.

Summing up, the norm/ideal n.1 forbids us to kill and to kill gently; the norm/ideal n.2 obliges us to kill and to kill gently. This doesn't seem much of an improvement.

Conclusions. We have tried to imagine whether SDL(n) would solve SDL's issues like PDeL(n) solved (some of) PDeL's issues, but the rst results were not

2.5. CONCLUSIVE REMARKS.

encouraging. However, we think more research is needed especially to see how [Gob00] could contribute to the discussion.

As we have seen, formalizing Chisholm's set in PDeL can be as tricky as it would have been in SDL. However, the feeling we get after all this discussion is that we may need some temporal expressivity other than the `once a, p' that comes for

free with the action modal box of PDeL. We also met several formalization issues. For these reasons in the next chapter we will move to the stit paradigm, which was born as a logic of action and time and tense operators are readily available and meant to be used. Hopefully these factors will enable a more ne-grained translation of the Chisholm sentences. Furthermore formalization is a much more straightforward task in stit, for reasons we will shortly see.

Chapter 3

stit.

The rst `seeing to it that' (stit) logic was put forth in [BP90], following a series of papers by Chellas (e.g. [Che69]) who had laid out informally the setting.1 _stit more than a logic is a paradigm, with instances ranging from the relatively simple Chellas stit or cstit ([Che69, Che92]) to much more complicated strategic, coalition stits as [BHar]. In the rst section of this chapter we will discuss, at a general level, what the stit paradigm is grounded on, and what is the inspiration that keeps driving the research in this direction. Then we will turn to the formal syntax and semantics and nally, as usual, we will examine CTD-related problematics and a stit version of the Chisholm paradox.

The analyses of this section will partly build on [SMK13], but a specic atten- tion (Ÿ3.2) will be devoted to Paul Bartha's work in [Bar93], which contains one of the very few o-the-shelf (and eshed-out) deontic variants of stit which we will call destit. What makes Bartha's work even more apt to open our discussion in this chapter, is that he claimed that Chisholm's scenario was unproblematic in his destit. In Ÿ3.4.1 we will show that he was wrong. Consequently, as we try to solve the problems destit had, in Ÿ3.5 we will move towards a more complex, temporal stit logic called xstit. The second part of this chapter will be devoted to producing a deontic variant of xstit (tailored to the purpose of solving the problems we will have found in Bartha's destit). In Ÿ3.5.2-Ÿ3.5.3, we will assess its worth against Chisholm and Forrester's puzzles.

Bartha's destit and Broersen's xstit are both stit logics, as the names sug- gest. Consequently, we choose to treat them in a single chapter and the following introduction applies to them both.

3.1. INTRODUCTION: PHILOSOPHICAL BACKGROUND.

3.1 Introduction: philosophical background.

Unlike many logical systems, stit takes o from explicitly philosophical grounds, and is very careful in respecting some basic intuitions such as indeterminism and free will. Though in fundamental2_disagreement3_{with these philosophical premises,} I cannot deny the attractiveness of the logic and, most importantly, its wide inu- ence and usefulness.

In the remaining paragraphs of this section we will sketch the philosophical basis of the logic, and see how the concept of action is consequently carved out, before moving to the formalities and the paradoxes.