Core Model

My modeling of legal competence follows the so-called “event models” methodology developed in [9] for epistemic modalities. The key idea there is to model thestructure of a particular learning event using the same tools as for an agent’s static information state, that is Kripke models. The result of updating one’s knowledge or belief in the light of new information is then computed using some form of restricted product of these models. See [103] for details. Transposed into my deontic context, the proposal is to model explicitly the structure of deontic action or legal competences using what I calldeontic action models. These are agent-indexed to capture the fact that differ-ent agdiffer-ents will have differdiffer-ent legal competences. The deontic action models I define include two preference ordering over acts as Definition 4.1.2 does.

4.2. Dynamic Rights 81 4.2.1.DEFINITION. A deontic action model A_i for agent i is a tuple ⟨A,>^Ai,≅^Ai , P re⟩where:

• Ais a non-empty finite set of acts.

• >^Ai is a converse well-founded and transitive relation onA

• ≅^Ai is an equivalence relation onA

• P re∶A→Lis a precondition function.

Each acta∈Ashould be seen as a deontic action or a legal ability. It encodes an action that agentican take in order to bring about changes in obligations and permissions or, in more Hohfeldian terminology, changes in underlying legal relations. These acts also come in different levels of ideality, which is encoded by the preference orders

>^Ai, ≅^Ai, and≥^Ai, among which≥^Ai is a standard reflexive and transitive relation [9, 103] defined using>^Ai and≅^Ai, as in the previous section. Finally, the preconditions functionP respecifies for each actathe conditions in the underlying static models that need to obtain forato be executable in the first place.

4.2.2.EXAMPLE. John is city clerk. He can confirm a violation of the parking reg-ulations (v), or not (n). He confirms a violation if a fine applies (f), otherwise (¬f) not. Given that Ivy’s car has no permit in the windshield, the preferred situation is one where the fine indeed applies. This is illustrated in Figure 4.2.

¬f n v f

Figure 4.2: The deontic action model A_{J ohn} for John the city clerk. The arrow ↦ represents the preference order>^Aj. The precondition ofn andv are written down to the left and the right, respectively.

The effect of executing a deontic action in a particular situation is computed by the so-called lexicographic update.

4.2.3.DEFINITION. LetMbe a preference-action model andA_i be a deontic action model. The preference-action modelM⊗A_i =⟨W^′,>^′,≅^′,{∼^′_i}i∈I, V^′⟩is defined as follows:

• W^′={(w, a) ∣M, w⊧P re(a), wherea ∈A}.

• (w, a)>^′ (w^′, a^′)iff eithera>^Ai a^′ora ≅^Ai a^′ andw>w^′.

• (w, a)≅^′ (w^′, a^′)iffa≅^Ai a^′andw≅w^′.

• (w, a)∼^′ (w^′, a^′)iffw∼_i w^′

• (w, a)∈V^′(p)⇔w∈V(p).

The lexicographic update takes pairs of preference-action models and deontic action models and returns an updated modelM⊗A_i. The adjective “lexicographic” comes from the update rule for the preference orders >^′ and ≅^′, which give priority to the deontic action. The domain of that new model is the set of pairs(w, a)such thatM, w satisfies the pre-condition ofa, writtenM, w ⊧P re(a). Instead, combining the two update rules for>^′and≅^′, I get the following rule for the pre-order≥^′:

(w, a)≥^′ (w^′, a^′)iff eithera>^Ai a^′ora ≅^Ai a^′andw≥w^′.

Lexicographic updates capture what I callpuredeontic actions. These are actions thatonlychange legal relations. This is encoded in the condition defining the valuation V^′ in the updated model: (w, a) ∈ V^′(p) ⇔ w ∈ V(p). One can take pure deontic action to be acts that are explicitly defined by the legislator, for instance entering into a contract or getting married. Of course non-deontic action might change the legal relation too. By breaking your neighbor’s window you create a claim for her against you to cover the repair costs. Such mixed deontic and non-deontic actions are the object of [54]. A full comparison between their and my models of deontic actions and legal competences is left for future work.

4.2.4.LEMMA. The lexicographic update models are preference-action models.

Proof:

Clearly that≥^′=>^′ ∪≅^′and>^′ ∩≅^′=∅. In this case, I need to show that≤^′is reflexive and transitive,<^′is CWF, and≅^′is an equivalence relation. I only prove the interesting case that<^′is CWF, and the other cases are easily checked. Now I need to show that

∀X ≠∅,∃x∈X s.t.∀y∈X thaty>/^′ x.

Let ∅ ≠ X = {(w, a) ∣ w ∈ Y ⊆ W anda ∈ B ⊆ A} where Y ≠ ∅ and B ≠ ∅. As >and>^Ai are CWF, there arew^∗ be a>-maximal element on Y anda^∗ be a>^Ai-maximal element onB. Namely that∀y ∈ Y thaty >/ w^∗and∀b ∈ B that b >/^Ai a^∗. Given(y, b) ∈ X, thenb >/^Ai a^∗, and ifb ≅^Ai a^∗, then obviouslyy >/ w^∗. So(y, b) />^′ (w^∗, a^∗). This means that(w^∗, a^∗)is the>^′-maximal element onX. □ 4.2.5.EXAMPLE. John notices that Ivy’s car doesn’t have a permit. He issues a park-ing ticket, which results in the city havpark-ing a claim against Ivy regardpark-ing the payment of a fine. This is represented by updating the model in Figure 4.1 with the one in Fig-ure 4.2. The result is in FigFig-ure 4.3. After the ticket has been issued, all states a fine applies to (f) are strictly better than those in which they do not (¬f). Now Ivy still ought not to park there, but she ought to pay a fine.

Of course, executing different deontic actions will have different effects on the same initial legal relations. This notion of “different deontic action” can be made precise using the standard notions of bisimulation [13] and action emulation [115], but I leave it until Section 4.2.3. For now it is sufficient to illustrate this with an intuitive example.

4.2. Dynamic Rights 83

(w₁, v) f, p

(w₂, n)

¬f, p ₍w₃, n) ¬f,¬p

(w₄, v) f,¬p

Figure 4.3: The modelM⊗E_{J ohn}resulting from John’s execution of a deontic action to issue a parking ticket.

4.2.6.EXAMPLE. Suppose that Mary has the authority to grant Ivy an exception that allows her to park without a permit. Such a deontic action is represented in Figure 4.4, and the result of updating Ivy’s initial situation (Figure 4.1) is in Figure 4.5, where Ivy still ought not to pay a fine but now enjoys a privilege to park her car. Notice that this update crucially uses a non-connected preference relation in the action model.

a₁

⊤

a₂ p a₃ ¬p

Figure 4.4: The deontic action modelA_{M ary}for Mary.

To express the effect of deontic action the languageLis extended with a dynamic, unary operator[A_i, a], with the following semantics:

• M, w ⊧[A_i, a]ϕiff ifM, w ⊧P re(a)thenM⊗A_i,(w, a)⊧ϕ.

A formula [A_i, a]ϕ thus reads “if i’s deontic action a is executable, then doing so results in ϕ.” Dynamic modalities allow me to introduce my key notions, powers and immunity. LetT(i, j, ψ/ϕ)denote an arbitrary (conditional) normative position definable in the static languageL. Then:

• ihas apoweragainstj regardingT(i, j, ψ/ϕ):

⋁

a∈A_i

[A_i, a]T(i, j, ψ/ϕ)

• ihas animmunityagainstj regardingT(i, j, ψ/ϕ):

¬ ⋁

a∈A_j

[A_j, a]T(i, j, ψ/ϕ)

(w₁, a₁) f, p

(w₂, a₁)

¬f, p f,¬p ₍w₄, a₁) (w₃, a₁)

¬f,¬p (w₁, a₂)

f, p f,¬p ₍w₄, a₄) (w₂, a₂)

¬f, p ¬f,¬p ₍w₃, a₃)

Figure 4.5: The modelM⊗A_{J ohn}resulting from Mary’s granting Ivy an exception to park without a permit.

In other words,ihas a power againstj regarding the normative positionT(i, j, ψ/ϕ) whenever there is a deontic action thatican be executed which results inT(i, j, ψ/ϕ). Similarly,ihas an immunity againstjregardingT(i, j, ψ/ϕ)ifjdoesn’t have a power againstiregarding that position. A quick check of the example above reveals that, as expected, John has a power against Ivy regarding her paying a fine.

This formalization of dynamic rights has two assets in comparison with classical, reductive approaches. First, it explicitly captures, both semantically and syntactically, the dynamic character of power and immunity. Second, as I will see below, this clear static-dynamic distinction allows for a natural distinction between legal ability and legal permissibility. This analysis of power and immunity does so, however, while staying reductive. This is what I show now.