Fatal Heyting Algebras and Forcing Persistent Sentences

Hamkins and Löwe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra HZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.


Personal remarks of the second author
Over many years, there have been strong research ties between the Institute for Logic, Language and Computation in Amsterdam and the logic group built by Leo Esakia in Tbilisi. When I was visiting Tbilisi in late May 2007, Leo's group asked me to give two talks in their logic seminar and I presented the material in the papers [6] and [9]. After the presentation on the joint work with Joel Hamkins on the Modal Logic of Forcing, Leo suggested to use the ideas from [6] in the setting of intermediate logics: in the ensuing lively discussion, Leo developed his ideas in Russian and David Gabelaia translated for me.
I had almost forgotten about this bilingual discussion when I returned to Tbilisi later that year (October 2007) and Leo informed me then that our "joint paper was progressing well". In early 2009, Leo sent me notes containing the results on intermediate logics we had talked about, but also additional material on fatal Heyting algebras that he had worked on in the meantime. In the month of March 2009, Leo and I collaborated via email to transform his notes into a submission for the conference Topology, Algebra, and Categories in Logic (TACL 2009). Since I felt that I had made no contribution to the additional material on fatal Heyting algebras, we eventually decided to remove it from the presentation I gave on 7 July 2009. § § 1 and 2 of the present paper contain the material presented at TACL 2009. The editors of this special issue encouraged me to submit the paper as it had been intended by Leo when he wrote the notes in 2009; Guram Bezhanishvili, David Gabelaia and Mamuka Jibladze volunteered to provide the proofs of the material on fatal Heyting algebras in an appendix. § 3 contains the material on fatal Heyting algebras essentially as it was in Leo's notes that he had written in 2009, and all of the results in this section are due to Leo alone. The proofs for the statements in § 3 have been provided by Bezhanishvili, Gabelaia and Jibladze in the appendix on the basis of Leo's handwritten notes. I would like to thank them for their crucial assistance in this project.

Introduction
In this paper, we shall deal with three different languages, the language L of propositional logic, the language L of modal propositional logic, and the language L ZFC of set theory (i.e., first-order logic with a binary relation symbol ∈). We identify the languages with their sets of formulae or sentences. The Lindenbaum algebra of ZFC is the Boolean algebra B ZFC of classes of provably equivalent L ZFC -sentences (i.e., formulae with no free variables). More precisely the elements of B ZFC are the classes [ϕ] = {ψ ; ZFC ϕ ↔ ψ} and the Boolean operations are the induced ones, e.g., is a unary operation on B, we call B, an interior algebra if p ≤ p, p = p, (p ∧ q) = p ∧ q, and 1 = 1. Note that if for an L ZFC -sentence ϕ, we define ϕ to be the L ZFC -formalization of "in every forcing extension, ϕ holds", then B ZFC , is an interior algebra. A Heyting algebra H = H, ∧, ∨, →, 0, 1 is a structure such that H, ∧, ∨, 0, 1 is a lattice with smallest and largest element and the equations p → p = 1, p ∧ (p → q) = p ∧ q, q ∧ (p → q) = q, and p → (q ∧ r) = (p → q)∧(p → r) hold. We write ¬p := p → 0. As usual, we shall not distinguish between H and its underlying set H, and write "p ∈ H" when we mean "p ∈ H". Recall that a Heyting algebra H, ∧, ∨, →, 0 is a Boolean algebra if and only if for every p ∈ H, we have p∨¬p = 1. Algebraic terms in Boolean and Heyting algebras can be naturally identified with L-formulas, whereas algebraic terms in interior algebras can be identified with L -formulas. So, if Λ is any modal logic extending S4, we can say "an interior algebra B, satisfies Λ" and mean that we identify each theorem ϕ of Λ with a term t ϕ in the interior algebra, and B, |= t ϕ = 1. From now on, we shall write "ϕ = 1" for this.
We denote the class of Heyting algebras by HA. Note that one of the two de Morgan laws is satisfied in every Heyting algebra, namely ¬(p ∨ q) = ¬p ∧ ¬q for every p and q in H. Definition 1. We call a Heyting algebra fatal if every prime filter is contained in only one maximal filter. The class of fatal Heyting algebras will be denoted by fHA.
Fact 2. For a Heyting algebra H the following are equivalent: Let F be the filter generated by x and ¬p and G be the filter generated by x and ¬¬p. If there is a ∈ x such that a ∧ ¬p = 0, then a ≤ ¬¬p, Thus, G is also proper. Clearly F and G are incomparable as ¬p ∈ F \G and ¬¬p ∈ G\F . Consequently, they extend to two different maximal filters y and z. So there are two different maximal filters containing x, and so H is not fatal.
(2) ⇒ (1): Suppose there is a prime filter x contained in two different maximal filters y and z. Then there is p ∈ H contained in y but not in z.

Forcing persistent sentences
We say that an L ZFC -sentence ϕ is forcing persistent if whenever ϕ is true in a model M then it is true in every forcing extension of M . As before, we use for the "true in all forcing extensions" operator. We say that a function H : L → L ZFC is a forcing translation if H commutes with propositional connectives and H( ϕ) is the formalization of "H(ϕ) is true in all forcing extensions". We call a formula ϕ ∈ L a valid principle of forcing if for all forcing translations H, we have that ZFC H(ϕ). By We follow [6, p. 1798] and call an L ZFC -sentence ϕ a button if there is a forcing extension such that ϕ is true and a switch if in all forcing extensions, both ♦ϕ and ♦¬ϕ are true. Every L ZFC -sentence is either a button or the negation of a button or a switch. Proof. If ϕ is forcing persistent, then ϕ ↔ ϕ, so (1.)⇒(2.) is obvious.
To see (2.)⇒(1.) check the three possible cases. If ψ is a button, then ψ is a button (and forcing persistent). If ψ is the negation of a button, then ψ is provably false (since buttons are necessarily buttons by S4.2; cf. [6, p. 1798]), and hence forcing persistent. If ψ is a switch, then ψ is provably false as well.
We denote the set of equivalence classes of forcing persistent sentences by H ZFC . It is not hard to see that H ZFC is a sublattice of B ZFC . By Proposition 4, we get that H ZFC = H(B ZFC , ), and thus H ZFC is a Heyting algebra.
Proof. Suppose ¬ϕ is not the case. Then there is a forcing extension in which ϕ holds. Since ϕ is forcing persistent, ϕ holds in this extension, so in the original model, we have ♦ ϕ. But by the soundness of S4.2, we get ♦ϕ.
Via the identification of formulae in L and terms in Heyting algebras (as well as formulae in L and terms in interior algebras), we can consider T as a map between the Heyting core of an interior algebra and the surrounding interior algebra. Proof. Cf., e.g., [3, § 8.3].
An intermediate logic is an L-logic extending HC and contained in classical propositional logic. One particular intermediate logic is the logic KC, also known as "the logic of the weak law of excluded middle", "Jankov logic", "testability logic", or "De Morgan logic" which has been originally introduced and studied by Dummett and Lemmon in [4]. The logic KC is axiomatized by adding to HC the following weak law of excluded middle: ¬ϕ ∨ ¬¬ϕ. By Fact 2, it is obvious that the class fHA provides an adequate algebraic semantics of the intermediate logic KC.
Dummett and Lemmon showed, among other things, that the modal system S4.2 (which was also originally introduced in [4]) interprets KC by the Gödel translation T . We call KC the modal companion of S4.2 via the Gödel translation. 1. the sentence ϕ is forcing persistent, and 2. there is a ψ ∈ L and a forcing translation H such that ϕ = H(T (ψ)).
Proof. By Proposition 4, we know that the forcing persistent sentences are of the form ψ for a button ψ or provably false.
Combining Theorems 3 and 8 with Proposition 9, we immediately get: Proof. In general, if K is a class of algebras, then we call A ∈ K functionally free if for any two terms s and t, we have that A |= s = t if and only if for all K ∈ K, we have that K |= s = t. Tarski proved that being functionally free is equivalent to being equationally generic in the sense of the theorem [13, p. 164].
Let BAO S4.2 be the class of Boolean algebras with an operator that satisfy S4.2 (i.e., they are all interior algebras). If B, is functionally free for BAO S4.2 , then its Heyting core H(B, ) is functionally free in fHA, and thus by Tarski's theorem equationally generic. [We identify formulae with their canonical terms in the algebras. By Theorem 8, KC ψ is equivalent to S4.2 T (ψ). By assumption, this is equivalent to B, |= T (ψ) = 1, and this in turn to H(B, ) |= ψ = 1 by Proposition 7.] But Theorem 3 implies that B ZFC , is functionally free for BAO S4.2 , thus completing the proof.

Observations about fatal Heyting algebras
In the course of analysing fatal Heyting algebras in the setting of intermediate logics, we also established some specific properties of fatal Heyting algebras which may be of independent interest. The following subsets of a