Deontic Reasoning

Abstract

Deontic logic is the logical study of normative concepts such as obligation, prohibition, permission and commitment. As we will see in this chapter, it is a very natural setting for with preference logic, both in its static versions (cf. [96, 183]) and in terms of the new dynamic systems of this book [48].

Appendix: A Semantic Excursion on Imperatives

1.1 Motivation

Imperatives occur ubiquitously in linguistic communication between individuals. To act successfully in society, we have to fully understand their meaning, as they lead to an implementation of actions. And as far as the dynamics of normative reasoning is considered, imperatives are the linguistic triggers that change our preferences, or other structures crucial to our rights and obligations. Thus, issues in the semantics of natural languages may be close to issues of normativity and agency in general.

Dynamic semantics and dynamic logic Logical studies of imperatives have been pursued for a long time, but starting from the 1990s, new frameworks have appeared. Here is one that is especially relevant to what we have done in this chapter.

Following the idea that “You know the meaning of a sentence if you know the change it brings about in the cognitive state of anyone who wants to incorporate the information conveyed by it”, a dynamic update semantics was proposed for natural language default rules in [192]. This style of thinking was later applied to imperative expressions in [118, 139]. This approach falls within the influential tradition of “dynamic semantics” for natural language, where expressions of familiar languages get new dynamic meanings: cf. [150] for further details.

By now, another style is also available. The DEL approach that adds explicit modalities for speech acts to a classical base logic was introduced in Chapter 2 for use in logics of agency. It, too, has been adopted to deal with the speech act of commands. Notably, [199] and [200] introduced a new dynamic action of “commanding” into static deontic logic, and dealt with imperatives in the framework of dynamic deontic logics. In this book, we have no view on which of these approach better fits natural language (if one has to choose at all). But we do want to point out that, construed either way, the perspective of this chapter has a contribution to make.

A case study: conflicting commands So far, the main purpose of both mentioned approaches has been to understand the meaning of one single imperative. Not much attention has been paid to the larger setting of imperative discourse, where there my typically be conflicting commands . Consider the following two examples, the first of which is a variation of Yamada’s motivating example in [199]:

Example 11.13

Suppose you are reading an article on logic in the office you share with your two bosses, a ₁ and a ₂ and a few other colleagues. We assume that a ₂ is of a higher rank. While you are reading, the temperature of the room rises, and it is now above 30 degrees Celsius. There is a window and an air conditioner. You can open the window, or turn on the air conditioner. You can also concentrate on the article and ignore the heat. Then, suddenly, you hear your boss a ₁’s voice. She commanded you to open the window. Right after that, a ₂’s voice comes up, saying he prefers that the window is not opened (This can be taken as an command “do not open the window”). What effects do those commands have on the current situation?

The next example shows the problem in an even more explicit manner:

Example 11.14

A general, a captain and a colonel utter the following sentences, respectively, to an agent b:

• (1) The general a ₁: Do A! Do B!

• (2) The captain a ₂: Do B! Do C!

• (3) The colonel a ₃: Don’t do A! Don’t do C!

It is clear that there are conflicting orders, w.r.t action A and C. Intuitively, instead of getting stuck, agent b will come up with the following plan after a deliberation: She should do A, do B, but not do C. b’s reasoning rests on the following fact, the authorities of $a_1, a_2$ and a ₃ are ranked as follows: $a_1 \gg a_3 \gg a_2$. According to this ranking, she can make her decision on which orders to obey, which orders to disobey. Thinking in terms of priorities and preference from Part IV, authorities amount to priorities, it will give us a preference over actions.

The above-mentioned frameworks cannot meaningfully handle such complex cases. The main problem is that their focus has always been on the addressee, not on the agents uttering the command. Inspired by the above model of priority-based preference, the aim of this section is to propose a solution to the problem of conflicting commands, in both mentioned frameworks.

What follows is just a small case study in dynamic semantics , relying heavily on the basic framework for imperatives put in place by the cited references.

1.2 Update Semantics for Conflicting Imperatives

A new update system is a tuple $(\mathcal{L}, \varSigma, [\cdot], A, \gg )$, where A is a finite set of agents, i.e. speakers. ≫ is a partial order on the set A. Intuitively, ≫ is an authority order of the speakers. Now we formulate the semantics in details, incorporating the authorities in the framework presented in the preceding.

Definition 11.15 (agent-oriented language $\mathcal{L}$)

Let $\mathcal{Y}$ be the standard language of propositional logic. Define the set $\mathcal{L}$ of imperatives as follows:

$$\mathcal{L} = \{!_a \varphi \,\,|\,\,\varphi \in \mathcal{Y}, a \in A \}$$

We see that imperatives are relative to specific speakers.

Definition 11.16 (force structures with authorities)

Let $\mathsf{L}$ be the set of literals of $\mathcal{Y}$. Let $\mathsf{L}^{\prime} = \{l_a \,\,|\,\,l \in \mathsf{L}, a \in A \}$. Let $\mathsf{B} = \{X \subseteq \mathsf{L}^{\prime}\,\,|\,\, X \textrm{is finite, and for any } l_a, l_b \in X, a = b\}$. Each $J \in \mathsf{B}$ is called a choice scope. Let $\mathsf{F} = \{X \subseteq \mathsf{B}\,\,|\,\, X \textrm{ is finite}\}$. Each $K \in \mathsf{F}$ is a force structure. The empty set ∅ is called the minimal force structure. Those force structures containing ∅ are called absurd ones.

Here is an example of force structures that involve authorities:

Example 11.17

Consider the structure $K_1 = \{\{p_a, \neg q_a\}, \{r_b\}, \{s_a, q_a\}\}$, which can be represented by the following picture:

We see that in force structures, literals are relative to specific speakers. Each literal can be viewed as an atomic imperative force. In this sense, we can say that imperative forces are relative to particular speakers.

Definition 11.18 (track with authorities)

Let $K = \{X_1, \dots, X_n\}$ be any force structure. We define tracks for K. For any X _i , let X _i ^′ be the smallest set such that both p _a and ¬ p _a are in X _i ′ for any p _a occurring in X _i . $T = X_1^{\prime\prime} \cup \dots \cup X_n^{\prime\prime}$ is a track for K if and only if:

• (1) $X_i^{\prime\prime} \subseteq X_i^{\prime}$ and $X_i^{\prime\prime} \cap X_i \neq \emptyset$;

• (2) For any p _a occurring in X _i , one and only one of p _a and $\neg p_a$ is in $X_i^{\prime\prime}$.

T is consistent if and only if there are no p _a and p _b such that both p _a and $\neg p_a$ are in T, but either $a \gg b \land b \gg a$ or $\neg (a \gg b) \land \neg (b \gg a)$.

T is resolvable if and only if there are no p _a and p _b such that both p _a and $\neg p_b$ are in T but either $a \gg b \land \neg (b \gg a)$ or $\neg (a \gg b) \land b \gg a$.

Note that a consistent track might not be resolvable, and also vice versa.

Definition 11.19 (imperatives and force structures)

$\mathsf{F}$ is the set of force structures. Let K be any force structure. T ⁺ and T ⁻ are two functions from $\mathsf{F}\times \mathcal{L}$ to $\mathsf{F}$, which are defined in the following way:

• (1) $T^+ (K, !_a p) = \left \{ ^{\{\{p_a\}\}}_{\{X \cup \{p_a\} \,\,|\,\,X \in K \}}\right. \left. ^{{\,\,\,if K = \emptyset}}_{{\,\,\,otherwise}} \right.$

  $T^- (K, !_a p) = \left \{ ^{\{\{\neg p_a\}\}}_{\{X \cup \{\neg p_a\} \,\,|\,\,X \in K \}}\right. \left. ^{{if K = \emptyset}}_{{otherwise}} \right.$

• (2) $T^+ (K, !_a(\neg \varphi)) = T^- (K, !_a\varphi)$

  $T^- (K, !_a(\neg \varphi)) = T^+ (K, !_a \varphi)$

• (3) $T^+ (K, !_a(\varphi \land \psi)) = T^+ (K, !_a \varphi ) \cup T^+ (K, !_a \psi)$

  $T^- (K, !_a (\varphi \land \psi)) = T^-(T^- (K, !_a \varphi ), !_a \psi )$

• (4) $T^+ (K, !_a (\varphi \lor \psi)) = T^+(T^+ (K, !_a \varphi ), !_a \psi )$

  $T^- (K, !_a(\varphi \lor \psi)) = T^- (K, !_a \varphi) \cup T^- (K, !_a \psi )$

For any imperative $!_a \varphi,\;T^+ (\emptyset, !_a \varphi)$ is its corresponding force structure.

Next, we define the notions of compatibility and harmony for force structures.

Definition 11.20 (compatibility)

Let K ₁ and K ₂ be any force structures. We say that K ₁ and K ₂ are compatible if and only if

• (1) For any track T ₁ of K ₁, there is a track T ₂ such that $T_1 \cup T_2$ is consistent;

• (2) For any track T ₂ of K ₂, there is a track T ₁ such that $T_1 \cup T_2$ is consistent.

Definition 11.21 (harmony)

Let K ₁ and K ₂ be any force structures. We say that K ₁ and K ₂ are harmonious if and only if

• (1) For any track T ₁ of K ₁, there is a track T ₂ such that $T_1 \cup T_2$ is resolvable;

• (2) For any track T ₂ of K ₂, there is a track T ₁ such that $T_1 \cup T_2$ is resolvable.

In what follows, we only consider the simplest case, namely, those imperatives whose force structures contain single choice scopes. As we know, force structures containing single choice scopes only has one track.

Consider two arbitrary force structures K and $K^{\prime}$. Let $\varSigma = \{K_1, \dots, K_n\}$, where each set K _i satisfies the following condition:

• (1) $K_i \subseteq K \cup K^{\prime}$;

• (2) The track of K _i is resolvable;

• (3) For any $K_i^{\prime} \subseteq K \cup K^{\prime}$, if $K_i \subset K_i^{\prime}$, then the track of K _i ′ is not resolvable.

Definition 11.22 (update function)

Define the function U as follows: $U(K, K^{\prime}) = K^{\prime\prime}$, with K ^″ the greatest element of Σ. The update function $[ \cdot ]$ is defined as follows:

$$K [ !_a \varphi ] = \left \{ \begin{array}{ll} U(K,T^+(\emptyset,!_a \varphi )) & K \ \textrm{and} \ {T^+(\emptyset,!_a \varphi)} \ \textrm{are consistent and compatible} \\ \{\emptyset\} & \textrm{otherwise} \end{array} \right.$$

With all our techniques ready, we come back to Example 11.14:

• (1) The general a ₁: Do A! Do B!

• (2) The captain a ₂: Do B! Do C!

• (3) The colonel a ₃: Don’t do A! Don’t do C!

Suppose that starting point of update is ∅. After the captain’s orders, the force structure of the agent b is this: $\{\{A_{a_1}\},\{B_{a_1}\}, \{B_{a_2}\},\{C_{a_2}\}\}$. It can be verified that $\{\{A_{a_1}\},\{B_{a_1}\}, \{B_{a_2}\},\{C_{a_2}\}\} [!_{a_3} (\neg A)][!_{a_3}(\neg C)] = \{\{A_{a_1}\},\{B_{a_1}\}, \{B_{a_2}\},\{C_{a_2}\}\}$ $[!_{a_3}(\neg C)] = \{\{A_{a_1}\},\{B_{a_1}\}, \{B_{a_2}\},\{\neg C_{a_3}\}\}$. The only track of the force structure $\{\{A_{a_1}\},\{B_{a_1}\}, \{B_{a_2}\},\{\neg C_{a_3}\}\}$ is consistent and resolvable. This implies that agent b has to do A and B, and he is forbidden to do C. This is precisely what happens in real life. We take orders, with a consideration on the resources of information. This ends our investigation in update semantics.

However, using the same ideas, we can introduce the authorities into the DEL-style dynamic deontic logic . Essentially, the sequence of authorities gives rise to an ordering or a choice over the commands that the agent gets. Accordingly, the agent updates with right information in the right order. In what follows, we will adapt such an idea to Yamada’s framework, providing some basic definitions. We will leave the systematic study to other occasions.

1.2.1 Dynamic Deontic Logics for Conflicting Commands

Definition 11.23 (static deontic language)

Let $p\in\varPhi$, a set of proposition letters, and $i, j \in N$, a finite set of agents, ≫ is a partial order over N. The static deontic language is given by:

$$\varphi := \bot \,\, | \,\, p \,\, | \,\, \lnot \varphi \,\, | \,\, \varphi \land \psi \,\, | \,\, U \varphi \,\, | \,\, O_{(i,j)}\varphi$$

Intuitively, the formula of the form $O_{(i,j)}\varphi$ means that it is obligatory upon the agent i with respect to the authority j that i should see to it that φ. Note that the set N is now an ordered set of agents.

Definition 11.24 (semantics)

A deontic model is a tuple $\mathfrak{M} = (S, \{ \smile_{(i,j)} | \, i,j \in N \/ \} )$, with S a set of possible worlds, V a valuation for proposition letters. Moreover, $\smile_{(i,j)}$ is an arbitrary relation over the worlds.

Definition 11.25 (truth conditions)

Given a deontic model $\mathfrak{M} = (S, \{ \smile_{(i,j)} | \, i,j \in N \/ \} )$, and a world $s \in S$, we define the relation $\mathfrak{M}, s \models \varphi$ (formula φ is true in $\mathfrak{M}$ at s) by induction on φ:

$$\mathfrak{M}, s \models O_{(i,j)} \varphi {\textrm{iff for all}} t \in S {\textrm{such that}} s \smile_{(i,j)} t, \mathfrak{M}, t \models \varphi.$$

In order to talk about changes that acts of commanding bring about, we extend the deontic language by adding action modalities to it:

Definition 11.26

Let $p\in\varPhi$, a set of proposition letters, and $i, j \in N$, a finite set of agents, ≫ is a partial order over N. The dynamic deontic language is given by:

$$\begin{array}{lll} \varphi & := & \bot \,\, | \,\, p \,\, | \,\, \lnot \varphi \,\, | \,\, \varphi \land \psi \,\, | \,\, U \varphi | \,\, O_{(i,j)}\varphi \,\, | \,\, [\pi]\varphi \\ \pi & := & !_{(i,j)}\varphi \end{array}$$

A formula of the form $[!_{(i,j)}\varphi]\psi$ means that after an act of commanding addressed to an agent i by an authority j to the effect that i should see to it that φ, ψ holds.

Definition 11.27 (deliberation function)

Consider a finite partially-ordered set of agent A, and any atomic command p and its negation ¬ p issued by different authorities, we define a deliberation function for the addressee i:

$$f(p_{(i,j)}, \neg\,p_{(i,k)})= \left \{ \begin{array}{ll} p_{(i,j)}\,\, & \textrm{if}\,\, j\gg k; \\ \neg p_{(i,k)}\,\, & {\rm if}\,\, k\gg j. \end{array}\right.$$

This function describes the process of agent’s resolving conflicts and reaching a harmony ordering over commands she has received. Once this process is done, the rest of her work is routine. In the following, we give the truth definition for the relevant dynamic modality:

Definition 11.28

Given a deontic model $\mathfrak{M} = (S, \{ \smile_{(i,j)} | \, i,j \in N \/ \} )$, and a world $s \in S$, the truth definition for formulas is as before, but with one new clause for the action modalities:

$$\mathfrak{M}, s \models [!_{(i,j)} \varphi] \psi\,\, \ \ \mathrm{iff} \ \ \,\, \mathfrak{M}_{!_{(i,j)} \varphi}, s \models \psi ,$$

where $\mathfrak{M}_{!_{(i,j)} \varphi}$ is a deontic model obtained from $\mathfrak{M}$ by replacing $\smile_{(i,j)}$ with $\smile_{(i,j)} - \{ (s,t) \, | \, \mathfrak{M}, t \models \lnot \varphi \}$.

1.2.2 Conclusion

We have shown how ideas from the richer representation style of this chapter can be used to enrich existing dynamic semantics for natural language. In particular, the preference-changing events studied in this book often come couched in linguistic expressions whose meaning is clearly relevant to our analysis of agency. Even so, our aim in presenting a small case study of imperatives has not been to make grand claims about the proper treatment of natural language. We only established this: Ideas can flow across from our dynamic preference logics to what initially may look like quite different frameworks.