Lemma.
Lemma 3.4.11. Let L be a distributive lattice, QM, QJ be two countable sets of distributive meets and joins in L respectively, and let(SQ, τ,≤)be the subspace of the dual Priestley space
(XL, τ∗,≤∗)of L, where SQ is the set of all prime filters overL that preserve all meets inQM and all joins inQJ. Then:
1. The canonical map : | · |:L→U p(SQ)is a, injective DL-homomorphism;
2. If(L,→)is a Heyting algebra, andQM,QJ are(V,→)-complete, then|·|is also a Heyting- homomorphism;
3. If (L,−<) is a co-Heyting algebra, andQM, QJ are (W,−<)-complete then | · | is also a co-Heyting-homomorphism
4. If(L,→,)is a Heyting modal algebra, andQM,QJ are (,→)complete sets of Barcan meets and joins, then| · | also preserves theoperator;
5. If (L,−<,♦) is a co-Heyting modal algebra, and QM, QJ are (♦,−<) complete sets of Barcan meets and joins, then | · |also preserves the♦operator.
3.4. Generalization to co-Heyting Algebras and Modal Algebras
Proof.
1. This is simply Theorem 3.3.6
2. We need to check that for any a, b ∈ L, |a → b| = − ↓ {|a| − |b|}. The left-to-right direction follows from basic properties of filters, and the converse is exactly the statement of Lemma 3.3.11
3. We need to check that for anya, b∈L,|a−< b|=↑ {|a| − |b|}. But the right-to-left direction follows from basic properties of prime filters, and the converse is precisely the statement of the Rasiowa-Sikorski lemma for co-Heyting algebras, i.e. Lemma 3.4.4.
4. We prove that for any a ∈ L, |a| = |a|, where for any U ⊆ XQ, we have that
U = {q ∈ SQ ; ∀r∈SQ : q ⊆ r implies r ∈ U}. The left-to-right direction is im-
mediate by definition of. The converse is precisely the statement of Lemma 3.4.10 (1.). 5. We prove that for any b ∈ L, |♦b| = |b|, where for any V ⊆ XQ, we have that
Y ={q ∈SQ ; ∃r ∈Y s.t. r ⊆p♦}. The right-to-left direction is immediate from the
definition of . Once again, the converse is precisely the statement of Lemma 3.4.10(2.). Corollary 3.4.12. Let (L,) be a BAO, and QM a set of Barcan meets such that for any
A⊆L,c∈L, if V
A∈QM, then V(A∨c)∈QM. Then for anya6= 0∈L, there exists an ultrafilter pover Lsuch that a∈pandppreserves all meets inQM.
Proof. By definition, (L,) is a modal Heyting algebras, and by simple syntactic manipulations, it is easy to see thatQM satisfies the requirement in Definition 3.4.7 (1.). Hence the Rasiowa-
Sikorski applies to (L,) andQM, which means that ifa6= 0∈L, then there is a prime filterp
overLthat preserves all meets inQM and containsa. But sinceLis a Boolean algebra,pis an
ultrafilter.
We finish this section, as expected, with the immediate consequence for various logics of the results above:
Theorem 3.4.13.
1. cIP L is complete with respect to the class ofcIP L-models; 2. M IP Lis complete with respect to the class ofM IP L-models; 3. cM IP L is complete with respect to the class of cMIPL-models; 4. KL is complete with respect to the class ofKL-models.
Proof. Items 1−3 are immediate consequences of the term model construction and Lemma 3.4.11 3−5 respectively. Similarly, item 4 is a direct consequence of the term model construction and Corollary 3.4.12.
Chapter 4
Possibility Semantics and Tarski’s
Lemma for Boolean Algebras
The aim of this chapter is to introduce the methods and ideas that will be used under one form or another in chapters 5 and 6. More precisely, we present the main features of possibility semantics for classical logic that we will seek to generalize to the intuitionistic case. Possibility semantics was first proposed by Humberstone [40] as an alternative to possible worlds semantics for modal logic. Humberstone’s idea was to work with partially determined worlds rather than complete possible worlds, and to define satisfaction at a partial world by quantifying over all possible refinements of that world. Holliday [39] offers a systematic study of possibility models and their relationship with standard Kripke models.
In the first section of this chapter, we recall well-known facts about the regular open sets of a topological space (X, τ), and in particular that for any topological space theIC operator corresponds to the double-negation nucleus onO(X) . We use this fact to provide a choice-free representation theorem for Boolean algebras and to prove the completeness ofCP C with respect to possibility models for LTCP C. In section 2, we consider the extension of this result to first-
order classical logic. The proof relies on an equivalent form of the axiom of dependent choice, known as Tarski’s Lemma. A proof of this result was given in [4], although the proof relies on a Henkin-style argument, while our proof is algebraic. In section 3, we relate canonical possibility spaces to important algebraic constructions, and show how to represent the canonical extension and the MacNeille completion of a Boolean algebra as the regular open sets of some possibility space. Finally, section 4 is concerned with generalizations of Tarski’s Lemma to BAO’s and Heyting algebras. We will show that, although a straightforward adaptation of Tarski’s proof for BAO’s is possible, the statement of Tarski’s Lemma itself for a Heyting algebra is equivalent to a strong form of Kuroda’s axiom.