Session 4

Standard dyadic deontic logic has, in different versions and under different names, appeared in the writings of several logicians.1 Here we follow the semantics as presented in Goble (2003) who also dubbed the system Standard Dyadic Deontic logic.

Like Standard Deontic Logic (SDL) we make sense of obligations with the help of a possible world semantics. A model is a structure \(M = \langle W, \le, v \rangle\) where, just like in SDL, \(W\) is a (non-empty) set of worlds, \(\le\) is a relation among worlds and \(v\) assigns to atom-world pairs truth values (1 or 0). Just like in SDL we require specific properties from \(\le\), namely the relation is to be transitive and total (so for every \(w, w^{\prime} \in W\), \(w \le w^{\prime}\) or \(w^{\prime} \le w\)). One can imagine the set of worlds linearly ordered in terms of betterness: if \(w \le w^{\prime}\), \(w^{\prime}\) is at least as good/ideal/etc. as \(w\).

In order to express conditional obligations, in SDDL we make use of a dyadic (or binary) operator: \(\mathsf{O}(A \mid B)\) which reads “\(A\) is obliged if \(B\) holds” or “\(B\) commits (a given agent) to \(A\)”, etc.

How to interpret \(\mathsf{O}(B \mid A)\) in the possible world semantics? The basic idea is as follows: from some degree of ideality on –our “cutoff point”– the conditional obligation is always fulfilled, i.e., from this point on, whenever \(A\) then also \(B\). More precisely, we define:

  • \(M, w \models \mathsf{O}(B \mid A)\) iff
    1. There is a world \(w\) –our cutoff point– for which \(M,w \models A \wedge B\) and
    2. for all \(w^{\prime} \ge w\) we have \(M, w^{\prime} \models A \rightarrow B\).

Note that the world at which the deontic formula is evaluated doesn’t matter. So we can define globally:

  • \(M \models \mathsf{O}(B \mid A)\) iff conditions (1) and (2) hold.2

The system allows us also to express other normative notions. For instance,

  • \(M,w \models \mathsf{P}(B \mid A)\) iff \(M,w \models \neg \mathsf{O}(\neg B \mid A)\)
  • \(M,w \models A \succeq B\) iff \(M,w \models \neg \mathsf{O}(\neg A \mid A \vee B)\)

Before investigating this some more, let us focus on obligations and see whether we obtain a reasonable model of hard cases such as the Forrester scenario in SDDL.

The Forrester scenario has the following obligations:

  • \(\mathsf{O}(\neg k \mid \top)\): in general you ought not to kill. The symbol \(\top\) stands for any tautological formula, e.g., \(p \vee \neg p\) and is, as such, interpreted in every model with 1.
  • \(\mathsf{O}(g \mid k)\): if you kill, kill gently.
  • Killing gently implies killing: \(\vdash g \rightarrow k\). This is an axiom which we consider true in each possible world.

How does a model \(M\) of it look like?

  • Due to \(\mathsf{O}(\neg k \mid \top)\) there is a world \(w\) for which
    1. \(M,w \models \neg k \wedge \top\) and
    2. for all \(w^{\prime\prime} \ge w\), \(M,w’’ \models \top \rightarrow \neg k\) which means \(M,w^{\prime\prime} \models \neg k\).
  • Due to \(\mathsf{O}(g \mid k)\) there is a world \(w^{\prime}\) for which
    1. \(M,w^{\prime} \models k \wedge g\) and
    2. for all \(w^{\prime\prime} \ge w^{\prime}\), \(M,w^{\prime\prime} \models k \rightarrow g\).

Here is an illustration of our model:

Note that, as expected our cutoff point for \(\mathsf{O}(g \mid k)\) is more to the left (= less ideal world) than our cutoff point for \(\mathsf{O}(\neg k \mid \top)\).

Exercises

Exercise 1

How does an SDDL-model of the Chisholm scenario look like? Recall, the Chisholm scenario consists of the following norms:

  1. \(\mathsf{O}(g \mid \top)\)
  2. \(\mathsf{O}(t \mid g)\)
  3. \(\mathsf{O}(\neg t \mid \neg g)\)

Exercise 2

Do the following principles hold in SDDL (that is: are they validated in each SDDL model at each world)?

  1. \((\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)) \rightarrow \mathsf{O}(p \wedge s \mid q)\)
  2. \(\mathsf{O}(p \mid q) \rightarrow \mathsf{O}(p \mid q \wedge s)\)
  3. \(\mathsf{O}(p \mid q) \rightarrow \mathsf{O}(p \wedge s \mid q)\)
  4. \(\mathsf{O}(p \mid q) \rightarrow \mathsf{O}(p \wedge q \mid q)\)
  5. \(\mathsf{O}(p \mid q) \wedge \mathsf{O}(p \mid s) \rightarrow \mathsf{O}(p \mid q \wedge s)\)
  6. \(\mathsf{O}(p \mid q) \rightarrow \mathsf{P}(p \mid q)\)
  7. \(\mathsf{O}(p \mid q) \rightarrow \mathsf{O}(q \rightarrow p \mid \top)\)
  8. \(\mathsf{O}(p \rightarrow q \mid s) \rightarrow (\mathsf{O}(p \mid s) \rightarrow \mathsf{O}(q \mid s))\)
  9. \((\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)) \rightarrow \mathsf{O}(p \mid q \wedge s)\)

So that you get an idea of how to tackle such exercises, I will demonstrate 1 and 2.

Ad 1

This is indeed a principle underlying SDDL. In order to show this, we need to show that in any SDDL model \(M = \langle W, \le, v \rangle\) and any world \(w \in W\), we have \(M,w \models (\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)) \rightarrow \mathsf{O}(p \wedge s \mid q)\).

  • Suppose \(M,w \models \mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)\). Then we know:

    1. There is a \(w_1 \in W\) for which (i) \(M,w_1 \models p \wedge q\) and (ii) for all \(w_1^{\prime} \ge w_1\), \(M,w_1^{\prime} \models q \rightarrow p\).
    2. There is a \(w_2 \in W\) for which (i) \(M,w_2 \models q \wedge s\) and (ii) for all \(w_2^{\prime} \ge w_2\), \(M,w_2^{\prime} \models q \rightarrow s\).

    We distinguish two cases: (a) \(w_1 \ge w_2\) and (b) \(w_1 < w_2\). Suppose (a) (the other case is analogous, see the figure below for an illustration). Since \(w_1 \ge w_2\), by 2.ii, \(M,w_1 \models q \rightarrow s\). Since by 1.i, \(M,w_1 \models p \wedge q\), we get (\(\alpha\)) \(M,w_1 \models p \wedge s \wedge q\). Furthermore, by 1.ii, for all \(w_1^{\prime} \ge w_1\), \(M, w_1^{\prime} \models q \rightarrow p\) and, since by transitivity \(w_1^{\prime} \ge w_2\), by 2.ii, \(M, w_1^{\prime} \models q \rightarrow s\). Altogether, (\(\beta\)) for all \(w_1^{\prime} \ge w_1\), \(M,w_1^{\prime} \models q \rightarrow (s \wedge p)\). By (\(\alpha\)) and (\(\beta\)), \(M,w \models \mathsf{O}(p \wedge s \mid q)\). Thus, \(M,w \models (\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)) \rightarrow \mathsf{O}(p \wedge s \mid q)\).

  • If \(M,w \models \neg (\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q))\) then \(M,w \models (\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)) \rightarrow \mathsf{O}(p \wedge s \mid q)\).

So, in both cases, \(M,w \models (\mathsf{O}(p \mid q) \wedge \mathsf{O}(s \mid q)) \rightarrow \mathsf{O}(p \wedge s \mid q)\).

An illustration of our proof:

Ad 2

We show that this does not hold in general in SDDL by giving a counter-model \(M\). The model has one world \(w\) for which \(w \le w\). We have the following assignment of truth-values:

\(p\) \(q\) \(s\)
\(w\) 1 1 0

Note that \(M \models \mathsf{O}(p \mid q)\) but \(M \models \neg \mathsf{O}(p \mid q \wedge s)\) since there is no “cutoff point” at which \(p \wedge q \wedge s\) holds.

References

  • Goble, L. (2003). Preference semantics for deontic logic. Part I: simple models. Logique at Analyse, 183–184, 383–418.
  • Hansson, B., (1969), An Analysis of some deontic logics, Nous, 3, 121–147.
  • Lewis, D. (1973). Counterfactuals. Cambridge, Mass.: Harvard University Press.
  • van Fraassen, B. (1972), The Logic of Conditional Obligation, Journal of Philosophical Logic, 1, 417–438.

  1. As \(\mathsf{CD}\) in Van Fraassen (1972), as \(\mathsf{VN}\) in Lewis (1973). A similar and influential system is discussed in Hansson (1969). ↩︎

  2. In order to express that every world \(w \in W\) comes with its own normative standards, one can utilize models \(M = \langle W, \langle \le_{w} \rangle_{w \in W}, v \rangle\) where every world comes with its own linear ordering of worlds. We can then define \(M,w \models \mathsf{O}(B \mid A)\) just like above, but relative to \(\le_w\). ↩︎