Evaluación de la comorbilidad

of arbitrary meets. Moreover, note that Tarski’s Lemma entails that the set of Q-filters over a Boolean algebra is separative only because, in any Boolean algebra B, we have that for any

a, b∈B, ab iffa∧ ¬b6= 0. However, the left-t-right direction of this bi-conditional does not

hold in general for Heyting algebras. This means that, if we are to generalize the ideas from this chapter to the intuitionistic setting, we have to substantially modify the statement of Tarski’s Lemma. This is exactly the work we undertake in the next chapter, but this will first require a substantial detour via a generalization of possibility semantics to intuitionistic logic.

4.5

Conclusion of This Chapter

We conclude this chapter with a short summary of the most salient features of possibility se- mantics that we used in the first two sections in order to prove the completeness ofCP C with respect to first-order possibility models. We hope that this list will give the reader a better grasp of the sense in which a generalization of possibility semantics is introduced in the next chapter. 1. For any topological space (X, τ), the map IC : O(X) → RO(X), which sends any open set to the interior of its closure, corresponds to the double negation nucleus on O(X). As a consequence, the regular open sets RO(X) of any topological space form a complete Boolean algebra (Theorem 4.1.2).

2. Given a Boolean algebraA, a possibility model (X,≤, V) for A is based on a poset (X,≤), and a Boolean homomorphism V : A → RO(X), where RO(X) is the Boolean algebra of regular open sets of (X, τ≤), the topological space obtained by taking the Alexandroff topology on (X,≤) (Definition 4.1.5).

3. Any Boolean algebraAis isomorphic to a subalgebra ofRO(X) for some topological space (X, τ). This representation theorem, unlike the Stone representation Theorem, does not require any form of the axiom of choice, but involves the construction of a possibility model forA(Theorem 4.1.4).

4. As a consequence of the three previous items, CPC is sound and complete with respect to the set of all possibility models for LTCP C, the Lindenbaum-Tarski algebra of CPC

(Theorem 4.1.6).

5. For any Boolean algebraA, given a countable setQof existing meets in A,Aembeds into the regular open sets of some possibility space forAin such a way that the embedding pre- serves all meets inQ. This proof relies on Tarski’s Lemma (Lemma 4.2.3), and presupposes the Axiom of Dependent Choice (DC)(Theorem 4.2.5).

6. For L a first-order language, a first-order possibility model is determined by a Boolean homomorphismV :LTL

CP L→RO(X) which preserves all meets inQ∀. (Definition 4.2.6) 7. As a consequence of the two previous items,CP Lis sound and complete with respect to

Chapter 5

Intuitionistic Possibility Spaces

In this chapter, we introduceintuitionistic possibility spaces(IP spaces) andfirst-order IP spaces, and give a complete semantics for IPL based on these spaces, thus generalizing the setting of possibility semantics to intuitionistic logic. In section 1, we introducerefined bi-topological spaces

(IP-spaces) andrefined regular open sets, which will play the same role as topological spaces and regular open sets for Boolean algebras. In section 2, we define the canonical possibility space of a Heyting algebra, and show how to embed any HA into the complete Heyting algebra of refined regular opens of its canonical possibility space. Section 3 is devoted to a refinement of this framework that allows for an embedding of Heyting algebras that also preserves a countable number of meets and joins. In particular, we propose generalizations of Tarski’s Lemma to distributive lattices and Heyting Algebras, which we refer to as theQ-Lemma for DL and HA respectively. Moreover, we show how they yield a proof of the existence of Q-completions for distributive lattices and Heyting algebras. Finally, in section 4, we draw consequences from the main results of section 2 and 3 and define a new semantics for IPL based on these results.

5.1

Refined Bi-Topological Spaces

Recall that possibility semantics for classical logic relies on the fact that theIC (interior-closure) operator in any topological space (X, τ) corresponds to the double-negation nucleus on the Heyting algebra of open sets ofX. In this section, we introduce refined bi-topological spaces and refined regular opens as a generalization of this fact.

Definition 5.1.1. A refined bi-topological space is a triple (X, τ1, τ2) such that τ1 and τ2 are

topologies onX, andτ1⊆τ2. We denote asIiandCithe interior and closure operators associated

withτifor 1≤i≤2. A setU ⊆X is calledrefined regular open iffI1C2(U) =U.

The main result of this section is the following theorem:

Theorem 5.1.2. Let(X, τ1, τ2)be a refined bi-topological space, and letO1be the Heyting algebra

of open sets inτ1. ThenI1C2 is a nucleus onO1.

Proof. We first prove thatI1C2 is a closure operator onO1.

• Monotonicity is obivous, since bothI1andC2 are monotone.

• To see thatI1C2 is increasing, notice that ifU is inτ1, thenU ⊆I1(U). ButU ⊆C2(U),