Pasos a dar para avanzar hacia sistemas educativos más inclusivos

The proof system of LPF is the proof system of EDL extended with some axioms about the new types of propositional letters. In particular, these axioms ensure introspection with respect to the agents’ motivational states and meta-level preferences and they guarantee that the agents’ meta-level preference relations are linear orders:

Definition 4.44. The proof system of LPF (notation: ΛLP F) is the proof system ΛEDL extended

with the following axioms for alli∈ N and all P ∈P:

• Introspection of salience: SiP →KiSiP

• Introspection of meta-level preferences: J i J0 →Ki(J i J0)

• The relation _i induces a linear order on P(P):

– Antisymmetry: For all J, J0 ⊆P such thatJ 6=J0 :¬(J i J0∧J0i J).

– Transitivity: For all J, J0, J00 ⊆_P:J _iJ0∧J0 _i J00→J _iJ00. – Totality: For all J, J0 ⊆P:J i J0∨J0 iJ.

As the models for preference formation are variants of multi-agent plausibility models, the soundness of LPF follows from the soundness of EDL together with the soundness of the above axioms:

Theorem 4.45. ΛLP F is sound with respect to the class of all models for preference formation.

Proof. As soundness can be proven by induction on the length of the proof, it suffices to show that each axiom is sound and that the inference rules preserve truth. Theorem 2.35 shows that the axioms and the inference rules of EDL are valid on all multi-agent plausibility frames. Models for preference formation consist of multi-agent plausibility frames. Therefore, in order to show that ΛLP F is sound,

it suffices to show the soundness of the axioms that are not included in ΛEDL.

The soundness of SiP → KiSiP and J i J0 → Ki(J i J0) follow immediately from Proposi-

tions 4.2 and 4.5 respectively. The soundness of the axioms expressing antisymmetry, transitivity and totality follows from the fact that for alli∈ N and all w∈W,w

Since ΛLP F is sound with respect to the class of all preference formation models, we get the following

result as a corollary:

Corollary 4.46. ΛLP F is sound with respect to the class of all common prior models for preference

formation.

In order to show that LPF is complete, we modify the completeness proof for EDL:

Theorem 4.47. ΛLP F is weakly complete with respect to the class of all common prior models for

preference formation.

Proof. According to Proposition 2.38, it suffices to show that every consistent LPF-formula ϕ is sat- isfiable on some common prior model for preference formation. The completeness proof for ΛLP F is a

variant of the completeness proof for ΛEDL. Therefore, we only indicate how the completeness proof

of EDL has to be changed. The reader may check the details. Letϕbe a consistent LPF-formula. In step 1 of the completeness proof for EDL, we constructed a canonical pseudo-model. Simi- larly, one can define pseudo-modelsK = (W,∼_i,≤_i,∼N,≤N, Si,i, r)i∈N for preference formation by

adding the following constraints to Definition 2.39 of pseudo-models: 1. For alli∈ N and all w, w0 ∈W :w∼_i w0 impliesSi(w) =Si(w0).

2. For alli∈ N and all w, w0 ∈W :w∼i w0 impliesi(w) =i(w0).

One can now define the canonical pseudo-model KΩ = (WΩ,∼Ω

i,≤Ωi ,∼ΩN,≤NΩ, SiΩ,Ωi , rΩ)i∈N for

preference formation in the same way as the canonical pseudo-model for EDL (Definition 2.47), where the definition of VΩ should be replaced by the following:

• For alli∈ N,SΩ_i :W → P(P) is a motivational salience function defined by: S_iΩ(w) :={P ∈P |SiP ∈w}.

• For alli∈ N,Ω

i :W →L is a meta-level preference function defined by:

Ω

i (w) :={(J, J0)∈ P(P)2 |(J iJ0)∈w}.

• rΩ:W × X →P is a valuation function defined by: rΩ(w, x) :={P ∈_P |P x∈w}.

One can show that the canonical pseudo-model for preference formation is indeed a pseudo-model for preference formation. The “normal constraints” on pseudo-models follow from Proposition 2.48. The axiom SiP →KiSiP ensures that condition 1 holds and the axiomJ i J0 →Ki(J i J0) that

condition 2 holds. The fact that Ω

i assigs a linear order on P(P) to each world follows from the

antisymmetry, transitivity and totality axioms of Definition 4.44. Furthermore, the definitions ofS_iΩ,

Ω

i and rΩ ensure that the Truth Lemma still holds. That is, for all w∈ WΩ and all formulas ϕ of

LPF, it holds that MΩ, w|=ϕ iffϕ∈w.

Recall that ϕ is a consistent LPF-formula. According to Lindenbaum’s Lemma, {ϕ} can be extended to a maximal consistent set Φ. By definition of the canonical pseudo-model for preference formation, there exists a world w∗ ∈ WΩ such that w∗ = Φ. By the Truth Lemma, it follows that KΩ, w∗ |=ϕ.

In step 2 of the completeness proof for EDL, we unravelled the canonical pseudo-model around some worldw∈WΩ. Recall that the resulting structure was a labelled tree and that the worlds in this tree consisted of finite histories ¯h from world w to some other world in the canonical pseudo-model. Similarly, one can define the unravelling K~ = (W , R~ ∼i, R≤i, R∼N, R≤N, ~S, ~i, ~r)i∈N of the canonical

pseudo-model for preference formation around world w∗ as in Definition 2.55, where the definition of ~

V should be replaced by the following:

• For alli∈ N,S~i:W~ → P(P) is a motivational salience function defined by: ~

Si(¯h) :=SiΩ(last(¯h)).

• For alli∈ N,~_i :W~ →Lis a meta-level preference function defined by: ~_i(¯h) :=Ω

i (last(¯h)).

In step 3 of the completeness proof for EDL, we redefined the relations of the unravelling of the canonical pseudo-model in such a way that we obtained common prior model. Similarly to Defini- tion 2.63, one can redefine the relations of the unravelled structure K~ in such a way that one gets a common prior model for preference formation M = (W ,~ ∼_i,≤_i,∼N,≤N, ~Si, ~i, ~r)i∈N. In order to

show thatM is a common prior model for preference formation, one needs to do two things. Firstly, one can imitate the proofs of Lemma’s 2.65 and 2.66 to show that the relations satisfy the desired properties. Secondly, one has to shows that conditions 1 and 2 on preference formation models hold. Using the proof strategy of Lemma 2.67, one can show that for all ¯h,¯h0 ∈ W~ it holds that ¯h ∼_i ¯h0 implieslast(¯h)∼Ω

i last(¯h0). The conditions now follow immediately from the definitions ofS~i and ~i.

One can define a mappingf :M →KΩ such that for all ¯h∈W~ :f(¯h) =last(¯h)∈WΩ and show, in the same way as in Lemma 2.67, that this is a bounded morphism. Consequently, M and KΩ satisfy the same LPF-formulas. In particular,M,(w∗)|=ϕ.

In conclusion, the consistent LPF-formulaϕis satisfiable on a common prior model for preference formation. Since ϕ was arbitrarily chosen, ΛLP F is weakly complete with respect to the class of all

common prior models for preference formation.

Every common prior model for preference formation is also a model preference formation. Since the language of LPF is not able to distinguish between the two, we get the following result as a corollary: Corollary 4.48. ΛLP F is weakly complete with respect to the class of all models for preference

formation.