On the Logics with Propositional Quantifiers Extending S5Π Yifeng Ding 1 Group in Logic and the Methodology of Science University of California, Berkeley Abstract Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π. Keywords: Propositional quantifiers, Scroggs's theorem, lattice of modal logics, algebraic semantics. 1 Introduction In this paper, we study the modal logics with propositional quantifiers extending the well-studied modal logic S5. Modal logics with propositional quantifiers have been of considerable interest to many modal logicians since their appearances in Fine's dissertation [9] and an early paper by Bull [6]. However, much of the interest is devoted to a few particular systems (e.g., [19,18,4,2,5]) and the expressive power under Kripke semantics (e.g., [7,21,20,16,11,1,3]), and there is an obvious lack of general study of classes of such logics. An exemplary early general study of propositional modal logics is found in Scroggs's famous 1959 paper [22], and it is our intention here to extend it to modal logics with propositional quantifiers. To this end, we must first define, in general, what is a modal logic with propositional quantifiers. Since we consider here only logics with one modal operator, the language LΠ defined below suffices. 1 Email: yf.ding@berkeley.edu 220 On the Logics with Propositional Quantifiers Extending S5Π Definition 1.1 Let LΠ be the language with the following grammar φ ::= p | > | ¬φ | (φ ∧ φ) | 2φ | ∀pφ where p ∈ Prop, a countably infinite set of propositional variables. 2 Other Boolean connectives, ⊥, and 3 are defined as usual. As is common in the general study of modal logics, we take a modal logic with propositional quantifiers to be a set of formulas satisfying certain closure conditions, which represent the necessary axioms and rules for connectives with fixed meaning. There are many readings of the propositionally quantified sentence ∀pφ, which result in different axioms and semantics (see [10] for example), but here we take the most straightforward reading: "no matter what proposition p expresses, φ." From a purely logical point of view, this reading should warrant the following widely accepted principles, which we call the Π-principles: • All instances of the universal distribution axiom schema: ∀p(φ → ψ) → (∀pφ→ ∀pψ). • All instances of the universal instantiation axiom schema: ∀pφ → φpψ where ψ is substitutable for p in φ, and φpψ is the result of this substitution. • All instances of the vacuous quantification axiom schema: φ→ ∀pφ where p is not free in φ. • Universalization rule: if φ is derivable, then ∀pφ is derivable. Then the modal logics with propositional quantifiers, which we call Π-logics in accordance with [6] and most recently [15], can now be defined. Definition 1.2 A Π-logic is a set Λ of formulas in LΠ such that Λ contains all instances of propositional tautologies and axioms in the Π-principles, and is closed under modus ponens and the only rule, universalization, in the Πprinciples. A normal Π-logic Λ is a Π-logic that contains the K axiom and is further closed under necessitation: if φ ∈ Λ, then 2φ ∈ Λ. For any normal modal logic L in the usual basic modal language, let LΠ be the smallest (in terms of inclusion) normal Π-logic containing L. Then, for example, S5Π is the smallest normal Π-logic extending S5, and KΠ is the smallest normal Π-logic extending K, which is just the smallest normal Π-logic. Following Scroggs, we address the following questions in this paper regarding the Π-logics extending S5Π, the set of which we call NextΠ(S5Π). General completeness of logics in NextΠ(S5Π) It is well known that S5Π is incomplete with respect to its Kripke frames if propositional quantifiers can vary the valuation of propositional variables to any set of worlds. This was 2 In contrast, > is a propositional constant. Later we will have another propositional constant. Ding 221 observed by Fine already in [9] and is in stark contrast to the situation without propositional quantifiers: as is shown by Scroggs, all modal logics in the basic modal language extending S5 are complete with their finite Kripke frames with a totally connected relation. However, Scroggs's proof is algebraic in spirit, and indeed, an algebraic semantics for LΠ based on modal algebras is more natural for the normal Π-logics, given our straightforward reading of ∀pφ. Algebraically, ∀pφ is interpreted as the meet (greatest lower bound) of all possible semantic values of φ when we only vary the valuation of p. In short, ∀pφ expresses an arbitrary meet. Dually, ∃pφ expresses an arbitrary join. For this to work, however, we need the modal algebras to be complete in the sense that for any set of elements in the algebra, the meet and join of this set exist. We will show that all logics in NextΠ(S5Π) are complete with respect to their complete simple S5 algebras, to be defined later. The lattice structure of NextΠ(S5Π) From the general completeness for logics in NextΠ(S5Π), the lattice structure of NextΠ(S5Π) can be reduced to the lattice structure of classes of algebras defined by logics in NextΠ(S5Π). We will show from this that the logics in NextΠ(S5Π) correspond to the closed sets of a Stone space S, which is homeomorphic to the disjoint union of two copies of the one-point compactification of N with the natural order topology. Then the lattice 〈NextΠ(S5Π),⊆〉 is isomorphic to the lattice of open sets of S ordered by inclusion. The computability of logics in NextΠ(S5Π) From the correspondence between the logics and the closed sets, we also obtain that there are logics in NextΠ(S5Π) of arbitrarily high Turing-degree. While it is known that many natural modal logics with propositional quantifiers are of very high complexity [10,16], this shows that we may still need to face the problem even above S5. The non-normal Π-logics extending S5Π We will also show that there are many non-normal Π-logics extending S5Π, contrary to the situation in the basic modal language, where all modal logics extending S5 are normal. However, we leave a complete study of the non-normal Π-logics extending S5Π to future work. The plan to address these questions is as follows. In § 2, we present the semantics for LΠ and collect the necessary results already appearing in [9] and more recently in [15]. In § 3, we show that, in terms of validity or theoremhood, every formula in LΠ is equivalent to a Boolean combination of a few simple formulas. This serves as a good preparation for § 4, where we construct a topological space S based on all complete simple S5 algebras, which encodes what classes of algebras are definable in terms of validity by LΠ. Crucially, S is a Stone space. In § 5, we prove all the main results, which make essential use of the fact that S is a Stone space and, in particular, that S is compact. This allows us to prove completeness without using the usual Lindenbaum algebra and quotient construction, though we need to rely on the already proven completeness of S5Π. Finally, we conclude with related open problems in § 6. 222 On the Logics with Propositional Quantifiers Extending S5Π 2 Preliminaries Recall that a modal algebra is a pair 〈B,2〉 where B is a Boolean algebra and 2 is a unary operator on B satisfying 21 = 1 and 2(a ∧ b) = 2a ∧ 2b for any a, b ∈ B. In most cases, we will conflate the notation of an algebra and its carrier set, and we will take ¬,∧,∨,2 to be the complement, meet, join, and modal operators in modal algebra, despite that they are also in our formal language LΠ. The usual abbreviations also apply to operations on modal algebras, including 3a := ¬2¬a for all a ∈ B. When confusion may arise, we will use ¬B ,∧B ,∨B ,2B for the operators in a modal algebra B. A modal algebra B is complete when its Boolean part is a complete Boolean algebra. Then the semantics for LΠ can be defined as follows. Definition 2.1 For any modal algebra B, a valuation V on B is a function from Prop to B. When B is complete, any such valuation can then be extended to a LΠ-valuation V from LΠ to B defined recursively by: (i) V (p) = V (p) for all p ∈ Prop; (ii) V (>) = 1; V (¬φ) = ¬V (φ); V (φ ∧ ψ) = V (φ) ∧ V (ψ); V (2φ) = 2V (φ); (iii) V (∀pφ) = ∧ {V ′(φ) | V ′ : Prop→ B, V ′ ∼p V }, where we define V ′ ∼p V by V ′(q) = V (q) for any q ∈ Prop \ {p}. A formula φ ∈ LΠ is valid on a complete modal algebra B, written as B  φ, if for all valuations V on B, V (φ) = 1. Since we are only interested in Π-logics extending S5Π, we only need modal algebras validating S5. In fact, we only need a very special class of such modal algebras called simple S5 algebras. Definition 2.2 A simple S5 algebra is pair 〈B,2〉 where B is a non-trivial Boolean algebra and 2 is the unary function on B defined for a ∈ B by 2a = { 1 if a = 1 0 otherwise. Then 3a = { 1 if a 6= 0 0 otherwise. Let us denote the class of all simple S5 algebras by sS5A and the class of all complete simple S5 algebras by csS5A. modal algebras validating S5 are also known as monadic algebra (see [14,13]). However, in the context of monadic algebras, 3 and 2 operators are usually denoted by ∃ and ∀, which we need for propositional quantifiers. We also remark that our simple S5 algebras are indeed simple in its general algebraic sense: they have no non-trivial congruence relation. These algebras are also known as Henle algebras. To formulate completeness with respect to csS5A, it is natural to use the following Galois connection: Definition 2.3 For any class C ⊆ csS5A, define Log(C) = {φ ∈ LΠ | ∀B ∈ C, B  φ}. We also write Log({B}) as simply Log(B) for any B ∈ csS5A. Ding 223 Conversely, for any set of formulas Γ ⊆ LΠ, define Alg(Γ) = {B ∈ csS5A | ∀φ ∈ Γ, B  φ}. Similarly, Alg(φ) abbreviates Alg({φ}). This finishes the semantics for LΠ, and now we march into expanding LΠ, as Fine did in [9], to LΠMg. This is instrumental for formulating the quantifier elimination on which completeness for S5Π alone in [9,15] depends, and all our new results will also need it. In the following, let N+ be the set of positive natural numbers, and N∞ be the set of natural numbers plus an infinite element ∞. Also, we will use N∞+ , which has ∞ but not 0. Definition 2.4 ([9]) Define LΠMg by extending the grammar for LΠ with a propositional constant g (not in Prop) and countably many new unary operators {Mi | i ∈ N+}. Then, define LMg as the quantifier free fragment of LΠMg, which has the following grammar: φ ::= p | > | g | 2φ | Miφ | ¬φ | (φ ∧ φ) with p ∈ Prop. For future convenience, we refer to the elements in Prop∪ {>, g} in general as propositional letters, and we define md(φ) to be the modal depth of φ defined as usual, with Mi's and 2 all treated as modal operators, free(φ) to be the set of free propositional variables in φ, and the quantificational depth of φ the maximal length of any chain of nested quantifers in φ, analogous to the usual definition in first-order logics. Let us also define as in [9] for every α ∈ LΠMg an important formula atom(α): atom(α) := 3α ∧ ∀q(2(q → α) ∨2(q → ¬α)) (1) where q ∈ Prop does not occur in α. To fix this choice, we assume that there is an enumeration of Prop fixed from the outset. Then whenever we need fresh propositional letters in a definition, the definition picks out the first available propositional variable. Here g is intended to express the proposition that some atomic proposition is true, and Miφ the proposition that φ is entailed by at least i many atomic propositions. Hence, g should be evaluated to the join of the atoms in a modal algebra. But this requires that the join exists. Let us call a modal algebra separable if the join of its atoms exists. Then we can give the semantics for LMg and LΠMg on appropriate modal algebras. Definition 2.5 For any separable modal algebra B, define g (or gB when ambiguity arises) as the join of all atoms of B, and Mi an operator on B as follows: Mia = { 1 if there are at least i distinct atoms below a 0 otherwise for i ∈ N+. 224 On the Logics with Propositional Quantifiers Extending S5Π Then, any valuation V on B can be recursively extended to an LMgvaluation V from LMg to B by the same clauses for Boolean connectives and 2 as in Definition 2.1, plus the following two clauses: (i) V (g) = gB (ii) V (Miφ) = MiV (φ). If B is actually complete, define the LΠMg-valuation extending V by combining the clauses above and in Definition 2.1. It is not hard to see that the LMg-valuation, LΠ-valuation, and LΠMg-valuation extending V are compatible. Hence by V , we always mean the defined valuation with the maximal domain. This will be either an LMg-valuation or an LΠMg-valuation, depending on whether the codomain of V is merely separable or is complete. We also extend the definition of validity and also the Alg operator to formulas in LΠMg in the obvious way. Regarding atom(α), it is intended to express the proposition that α expresses an atomic proposition. Its definition does not always achieve this intended meaning, but assuming that it is interpreted on complete simple S5 algebras, this definition indeed singles out atoms in simple S5 algebras. The proof of this can be found in [15]. Lemma 2.6 For any α ∈ LΠMg, any complete simple S5 algebra B, and any valuation V on B, we have V (atom(α)) = { 1 if V (α) is an atom in B 0 otherwise. Now to the logics in the language LΠMg. They are obtained by adding two axiom schemata to define the new operators Mi's and g by formulas in LΠ. Definition 2.7 For any normal Π-logic Λ, define ΛMg as the smallest normal Π-logic (with formula variables in the schemata and rules of the definition now ranging over LΠMg) that contains the following two axiom schemata for each n ∈ N+: Mnφ↔ ∃q1 * * * ∃qn( ∧ 16i<j6n 2(qi → ¬qj) ∧ ∧ 16i6n (atom(qi) ∧2(qi → φ))) (M) g↔ ∃q(q ∧ atom(q)) (g) where q1, * * * qn ∈ Prop do not occur in φ, and q ∈ Prop. With the help of Lemma 2.6, it is not hard to directly observe that both (M) and (g) are sound. In fact, we have the following theorem, mostly by Fine and Holliday, on which our new results depend. Lemma 2.8 S5ΠMg is a conservative extension of S5Π. Namely, S5ΠMg ∩ LΠ = S5Π. Also, S5ΠMg is sound and complete with respect to csS5A. Ding 225 Proof. For any φ ∈ LΠ, if φ ∈ S5ΠMg, then we can replace all Mi's and g in its derivation by their definitions in Definition 2.7. The resulting derivation is in S5Π. Hence φ ∈ S5Π. This shows that S5ΠMg ∩ LΠ ⊆ S5Π. The other direction is trivial. This is observed and first used by Fine in [9]. It is first shown algebraically in [15] that S5ΠMg is sound and that S5Π is sound and complete with respect to csS5A. Now for any φ ∈ LΠMg that is valid in csS5A, we can first replace all Mi's and g in φ by their definitions to obtain ψ. Then φ ↔ ψ ∈ S5ΠMg. We know that S5ΠMg is sound on csS5A. So ψ is valid. Then, since ψ ∈ LΠ, ψ ∈ S5Π, which means ψ ∈ S5ΠMg. By modus ponens, φ ∈ S5ΠMg. 2 The proof of the completeness of S5Π in [15] relies on a fairly intricate quantifier elimination in S5ΠMg found first by Fine in [9], which says that for any φ ∈ LΠ, there is a formula ψ ∈ LMg such that φ↔ ψ ∈ S5ΠMg. We will also make use of this technical result. In fact, ψ can be chosen from a much smaller fragment of LMg. Following Fine, we call them model descriptions and define them now. Definition 2.9 For any φ ∈ LΠMg, first define the following abbreviations: Q0 := ¬M1φ; Qiφ := Miφ ∧Mi+1φ, i ∈ N+; Nφ := 3(¬g ∧ φ). For any finite subset P ⊆ Prop, a state description s over P is a conjunction of literals from P in which every p ∈ P occurs. We follow the convention that an empty conjunction is > and an empty disjunction is ⊥. Let 2P be the set of all state descriptions over P . Then, a model description of degree n over P is a conjunction of: (i) either g or ¬g; (ii) a state description a ∈ 2P ; (iii) for each s ∈ 2P , either Mns or some Qns for some i < n; (iv) for each s ∈ 2P , either Ns or N¬s. Lemma 2.10 ([9], § 4.2) For any φ ∈ LΠ, there exists a qf(φ) ∈ LMg such that φ↔ qf(φ) ∈ S5ΠMg. Moreover, qf(φ) is a disjunction of model descriptions over free(φ) of degree 2n where n is the quantification degree of φ. For the construction of qf , the reader can also see the appendix of [15]. 3 Semantical and syntactical reduction In this section, we show that for any φ ∈ LΠ, any complete simple S5 algebra A, and any Λ ∈ NextΠ(S5Π), we can construct a formula, which we call basic(φ), such that: • φ ∈ Λ iff basic(φ) ∈ ΛMg; • A  φ iff A  basic(φ); • basic(φ) is a Boolean combination of 3¬g and Mi> for i ∈ N+. To facilitate the proof, let us first define a number of useful fragments of LΠ. 226 On the Logics with Propositional Quantifiers Extending S5Π Definition 3.1 Recall that LMg is the quantifier free fragment of LΠMg. Now, Define the following propositional-variable-free fragments of LMg where i ranges in N+: SMg 3 φ ::= > | g | ¬φ | (φ ∧ φ) | 2φ | Miφ S61Mg 3 φ ::= > | g | 3¬g | Mi> | ¬φ | (φ ∧ φ) SBasic 3 φ ::= > | 3¬g | Mi> | ¬φ | (φ ∧ φ) . The S instead of L in their names means "Sentence." It is not hard to see that SMg collects all propositional-variable-free formulas in LMg and that S61Mg collects some formulas with modal depth at most 1 in SMg, which are enough for our purposes. For any φ ∈ LΠ, we will construct basic(φ) as the following with u and comp to be defined: basic(φ) = comp(2qf(u(φ))). Here, u(φ) is the universal closure of φ, which is defined as ∀p1∀p2 * * * ∀pnφ where p1, p2, * * * , pn enumerate the free propositional letters in φ. And recall that qf returns the result of quantifier elimination. Since u(φ) has no free propositional variable, according to Lemma 2.10, qf(u(φ)) ∈ SMg is a disjunction of model descriptions of some finite degree over ∅. From Definition 2.9, we can see that all model descriptions of degree n over ∅ are of the form ±g ∧Mi> ∧ ¬Mi+1> ∧±3¬g or ± g ∧Mn> ∧±3¬g where i < n, ± stands for ¬ or nothing, and M0> for >. In short, qf(u(φ)) is a Boolean combination of g, Mi> for i ∈ N+, and 3¬g, and hence is in S61Mg. Now we construct comp as a function that simplifies a boxed modal description over ∅ to a formula in SBasic in a provably equivalent way. Lemma 3.2 For any ψ a disjunction of model descriptions of degree n over ∅, there is a formula comp(2ψ) ∈ SBasic such that 2ψ ↔ comp(2ψ) ∈ S5ΠMg. Proof. Let pos be the number of model descriptions in ψ where g appears positively and neg the number of model descriptions in ψ where g appears negatively. For any 1 6 i 6 pos, let αi be the result of deleting the conjunct g in the ith model description in ψ where g appears positively, and similarly define βi for 1 6 i 6 neg, where we need to delete the ¬g conjunct. Let α = ∨ 16i6pos αi and β = ∨ 16i6neg βi, which are now Boolean combinations of Mi>'s and 3¬g. Then obviously ψ ↔ ((g ∧ α) ∨ (¬g ∧ β)) and 2ψ ↔ 2((g ∧ α) ∨ (¬g ∧ β)) are in S5ΠMg using propositional tautologies and normality. Let us write for any φ1, φ2 ∈ LΠMg, φ1 ≡S5ΠMg φ2 iff Ding 227 φ1 ↔ φ2 ∈ S5ΠMg. Then we have 2ψ ≡S5ΠMg 2((g ∧ α) ∨ (¬g ∧ β)) (2) ≡S5ΠMg 2((g ∨ ¬g) ∧ (g ∨ β) ∧ (¬g ∨ α) ∧ (α ∨ β)) (3) ≡S5ΠMg 2(g ∨ ¬g) ∧2(g ∨ β) ∧2(¬g ∨ α) ∧2(α ∨ β) (4) ≡S5ΠMg 2(g ∨2β) ∧2(¬g ∨2α) ∧2(2α ∨2β) (5) ≡S5ΠMg (2g ∨2β) ∧ (2¬g ∨2α) ∧ (2α ∨2β) (6) ≡S5ΠMg (2g ∨ β) ∧ (2¬g ∨ α) ∧ (α ∨ β) (7) ≡S5ΠMg (¬3¬g ∨ β) ∧ (¬M1> ∨ α) ∧ (α ∨ β). (8) In the above chain of provable equivalences, (5), (7), and (8) require more explanation. Note that in S5ΠMg, we have 2∀pφ ≡S5ΠMg ∀p2φ, and dually, 3∃pφ ≡S5ΠMg ∃p3φ. With all the axioms in S5, and also the axiom (M) defined in Definition 2.7, Mi> ≡S5ΠMg 2Mi>. Then, α ≡S5ΠMg 2α and β ≡S5ΠMg 2β, since α and β are Boolean combinations of Mi> and 3¬g. Thus we have (5) and (7). For (8), some manipulation of axioms (g) and (M) gives us 2¬g ≡S5ΠMg ¬M1>. The rest of the equivalences are standard S5 reasoning. We can then define basic(2ψ) to be the right hand side in (8), which is provably equivalent to 2ψ and is in SBasic. 2 Now we show that for any φ ∈ LΠ, basic(φ) has the required properties. Lemma 3.3 For any φ ∈ LΠ and Λ ∈ NextΠ(S5Π), φ ∈ Λ iff basic(φ) ∈ ΛMg. Proof. Since ΛMg is a conservative extension of Λ (by Lemma 2.8), φ ∈ Λ iff φ ∈ ΛMg. Using universalization and also universal instantiation, φ ∈ ΛMg iff u(φ) ∈ ΛMg. Since Λ ∈ NextΠ(S5Π), ΛMg extends S5ΠMg. Together with the quantifier elimination result in Lemma 2.10, qf(u(φ)) ↔ u(φ) ∈ ΛMg. Thus u(φ) ∈ ΛMg iff qf(u(φ)) ∈ ΛMg. By necessitation and also the T axiom derivable in S5, qf(u(φ)) ∈ ΛMg iff 2qf(u(φ)) ∈ ΛMg. Finally, due to Lemma 3.2 and the fact that qf(u(φ)) is indeed a disjunction of model descriptions of some finite degree over ∅, we have basic(φ) = comp(2qf(u(φ))) ↔ 2qf(u(φ)) ∈ ΛMg. Thus basic(φ) ∈ ΛMg iff 2qf(u(φ)) ∈ ΛMg. Connecting all the equivalences, φ ∈ Λ iff basic(φ) ∈ ΛMg. 2 On the semantical side, we first make an easy observation. Lemma 3.4 For any complete (resp. separable) modal algebra B and any φ ∈ LΠMg (resp. LMg) such that free(φ) = ∅, for any two valuations V and V ′ on B, V (φ) = V ′(φ). Due to this observation, we define for any separable modal algebra B, a fixed trivial valuation VB which maps every p ∈ Prop to 1B . Then B  φ iff VB(φ) = 1B for any φ ∈ LMg (or LΠMg when B is complete) such that free(φ) = ∅. Then we can prove the semantical requirement for basic. Lemma 3.5 For any φ ∈ LΠ and any complete simple S5 algebra B, B  φ iff B  basic(φ). 228 On the Logics with Propositional Quantifiers Extending S5Π Proof. First consider the following chain of equivalences: B  φ iff B  u(φ) by the definition of validity iff VB(u(φ)) = 1 by the definition of validity iff VB(qf(u(φ))) = 1 B validates S5ΠMg, and u(φ)↔ qf(u(φ)) ∈ S5ΠMg iff VB(2(qf(u(φ)))) = 1 21 = 1 and 2a 6= 1 if a 6= 1 iff VB(comp(2(qf(u(φ))))) = 1 B validates S5ΠMg, and Lemma 3.2 Note that basic(φ) must have no free propositional variable, because basic(φ) = comp(2qf(u(φ))), free(u(φ)) = ∅, and neither qf nor comp introduces new free variables. Then by the observation in the previous lemma, B  basic(φ) iff VB(basic(φ)) = 1. Connecting all the equivalences, B  φ iff B  basic(φ). 2 4 Types and type space In the last section, we have shown that many formulas are equivalent in terms of validity or theoremhood. In this section, we do the same to the algebras: many algebras are equivalent in terms of the formulas in LΠ they validate. This equivalence relation can in fact bring csS5A from a class to a countable set, which we will call the type space. Then, to study the classes of algebras definable by formulas in LΠ, we can just study the sets of types of the algebras in those classes. This in turn gives us a topology on the type space. Now we start with the definition of the types, which is in fact a much simplified version of the famous Tarski invariant for Boolean algebras (see § 5.5 in [8]), due to the completenss of the algebras we are interested in. Definition 4.1 For any complete simple S5 algebra B, its type t(B) is a pair 〈t0(B), t1(B)〉 where t0(B) = { 1 if g 6= 1 0 if g = 1, t1(B) = { i ∈ N if B has exactly i atoms ∞ if B has infinitely many atoms. (9) Recall that g of B is the join of its atoms. Hence, t0 says whether this algebra contains an atomless part, and t1 counts the atoms it has. Let S be the set of all types of complete simple S5 algebras, the type space. Proposition 4.2 S = ({0, 1} × N∞)\{〈0, 0〉}. Proof. Apparently, S ⊆ {0, 1}×N∞ as any type is a pair 〈t0, t1〉 where the first component can only be 0 or 1 and the second component can only be a natural number of ∞. Also, if the type of a complete simple S5 algebra A is 〈0, 0〉, then A has no atom and also no atomless part, which means A is trivial and thus not a complete simple S5 algebra in our Definition 2.5. So we have shown the inclusion of left to right. Now to the other direction. The right-hand-side can be decomposed into three parts: Ding 229 • 〈0, n〉 for n ∈ N+. Types of this form can be realized by the Boolean algebra Bn of the powerset of a set of n elements, with 2 defined as in Definition 2.5. • 〈1, 0〉. To realize this type, take the countable free Boolean algebra B. It is well known that B is atomless, but not complete. However, we can take the MacNeille completion B+ of B, a complete Boolean algebra (unique up to isomorphism) such that B embeds into and that every element of B+ is a join of images of elements of B. For a construction of this B+, see [12], Chap. 25. Then B+ is complete and atomless, as if there is an atom, it must be the image of an atom of B, but B is atomless. Now turn B+ to a simple S5 algebra by defining 2 as in Definition 2.5. Then B+ has type 〈1, 0〉. • 〈1, n〉 for n ∈ N+. Consider the product of B+ and Bn with 2 again defined as above. It is not hard to see that it has an atomless part: g of this algebra is 〈1, 0〉. It also has n many atoms, listed by 〈0, a〉 where a range over atoms in Bn. Thus this simple S5 algebra has type 〈1, n〉. Hence we realized all types in the right-hand-side. So the inclusion from right to left is also shown. 2 Now we define the equivalences between algebras. Then we will show that types capture this equivalence relation. Definition 4.3 For any two complete simple S5 algebras A,B, and any L ∈ {LΠ,SBasic} we say A ≡L B if for any sentence φ ∈ L, A  φ iff B  φ. Lemma 4.4 For any two complete simple S5 algebras A,B, A ≡LΠ B iff A ≡SBasic B. Proof. Immediate from Lemma 3.5. 2 Lemma 4.5 For any two complete simple S5 algebras A,B, t(A) = t(B) iff A ≡SBasic B. Hence, together with Lemma 4.4, t(A) = t(B) iff A ≡LΠ B. Proof. Recall that for all φ ∈ SBasic and any complete simple S5 algebra A, A  φ iff VA(φ) = 1, because free(φ) = ∅. Also notice that for φ = 3¬g or Mi> for any i ∈ N+, VA(φ) is either 0 or 1. Since SBasic consists of all and only the Boolean combinations of these formulas, in fact for any φ ∈ SBasic, VA(φ) ∈ {0, 1}, and in other words, A  φ or A  ¬φ. Now suppose t(A) = t(B). Then, conflating 1A and 1B , and also 0A and 0B , we can easily verify that VA(3¬g) = VB(3¬g) and that for all i ∈ N+, VA(Mi>) = VB(Mi>). Then a simple induction propagates these equalities to all φ ∈ SBasic. Thus we see that if t(A) = t(B), then A ≡SBasic B. On the other hand, if t(A) 6= t(B), then there are two cases: • t1(A) 6= t1(B). In this case, 3¬g distinguishes the two algebras. • t2(A) 6= t2(B). Let n be the smaller number among them. Then n ∈ N, n+ 1 ∈ N+, and ¬Mn+1 distinguishes the two algebras. Hence, if t(A) 6= t(B), then A 6≡SBasic B. 2 230 On the Logics with Propositional Quantifiers Extending S5Π Due to this lemma, the function t can be seen as the quotient map from csS5A to csS5A/≡LΠ. This means that the Galois connection between csS5A and LΠ by Alg and Log can be reduced to the following Galois connection between S and LΠ. Definition 4.6 For any type s ∈ S and any φ ∈ LΠ, let us write s  φ just in case for any A ∈ csS5A such that t(A) = s, A  φ. Then define Type(Γ) for every Γ ⊆ LΠ as {s ∈ S | ∀φ ∈ Γ, s  φ}, with Type(φ) again abbreviating Type({φ}). On the other direction, define Log(T ) for any subset T of S as {φ ∈ LΠ | ∀s ∈ T, s  φ}, with Log(s) abbreviating Log({s}) for any s ∈ S as well. Then, we can collect the following easy but useful observations. Lemma 4.7 For any Γ ⊆ LΠ, φ,ψ ∈ SBasic: • Alg(Γ) = t−1(Type(Γ)), t(Alg(Γ)) = Type(Γ), and then Log(Type(Γ)) = Log(Alg(Γ)); • Type(Γ) = ⋂ {Type(φ) | φ ∈ Γ}; • Type(¬φ) = S \ Type(φ), Type(φ ∧ ψ) = Type(φ) ∩ Type(ψ). Using this lemma, we can study the following topology that will be important to us for both general completeness and the lattice structure of NextΠ(S5Π). Definition 4.8 Let S be the topological space with the type space S as the underlying set and {Type(φ) | φ ∈ LΠ} as basic opens. Lemma 4.9 S is a Stone space, homeomorphic to the disjoint union of two copies of the one-point compactification of N with the usual order topology. Proof. From Lemma 3.5, the basic opens of S are just sets in {Type(φ) | φ ∈ SBasic}. By the third bullet in Lemma 4.7, we know that {Type(φ) | φ ∈ SBasic} is a field of sets on S. Thus S is zero-dimensional. To see that S is Hausdorff, take two different s1, s2 ∈ S. Recall that S is t(csS5A). So we can find two complete simple S5 algebras B1 and B2 such that t(B1) = s1 and t(B2) = s2. Then, by Lemma 3.5, B1 6≡SBasic B2. So we can find a formula φ ∈ SBasic such Type(φ) that separates B1 and B2. So S is Hausdorff. To show that S is compact, we need a more detailed analysis of {Type(φ) | φ ∈ SBasic}. First, note that Type(3¬g) = {1} × N∞, Type(¬3¬g) = {0} × N∞+ , and they partition S into two parts. Let us name them by S1 and S0 respectively. Hence S is the disjoint union of S1 and S0 defined as the subspaces of S on S1 and S0 respectively. So we only need to show that they are both compact. On S1, the basic clopens are now Boolean combinations of Type(Mi) for i ∈ N+ and Type(3¬g), all restricted to S1. But Type(3¬g) = S1. Then the clopens are actually the field of sets on S1 generated by {Type(Mn) ∩ S1 | n ∈ N} = {{〈1, i〉 | n 6 i 6∞} | n ∈ N}. Hence it is not hard to see that S1 is just (homeomorphic to) the one-point compactification of the order topology on N. The situation for S0 is almost the same, except that the space is on N∞+ . But it is still homeomorphic to the one-point compactification of N. 2 Ding 231 5 Main results Now we are prepared to prove the main results regarding Π-logics extending S5Π. Let us start with the general completeness. Theorem 5.1 For any Λ ∈ NextΠ(S5Π), Λ = Log(Alg(Λ)). Proof. As is shown in Lemma 4.7, Log(Alg(Λ)) = Log(Type(Λ)). Also, it is trivial that Λ ⊆ Log(Type(Λ)). Hence we just need to show that, for any φ ∈ LΠ, if φ ∈ Log(Type(Λ)), then φ ∈ Λ. Let us assume the antecedent. Then for any s ∈ Type(Λ), s  φ. In other words, Type(φ) ⊇ Type(Λ). As we observed in Lemma 4.7, Type(Λ) =⋂ {Type(ψ) | ψ ∈ Λ}. By Lemma 3.5, Type(ψ) = Type(basic(ψ)). Note also that ψ ∈ Λ iff basic(ψ) ∈ ΛMg, which is shown in Lemma 3.3. Thus the set {Type(ψ) | ψ ∈ Λ} ⊆ {Type(ψ) | ψ ∈ ΛMg ∩ SBasic}. On the other hand, for any ψ ∈ ΛMg ∩ SBasic, using the axioms defining Mi and g, there is a ψ′ ∈ LΠ such that ψ ↔ ψ′ ∈ ΛMg. This means that ψ′ is in ΛMg, hence also in Λ, and that Type(ψ) = Type(ψ′), using Lemma 2.8. Hence {Type(ψ) | ψ ∈ Λ} = {Type(ψ) | ψ ∈ ΛMg ∩ SBasic}, which we now call F . Now this is a filter of basic clopens in S for the following reasons. • For any X,Y ∈ F , we can find α, β ∈ ΛMg ∩ SBasic such that X = Type(α) and Y = Type(β). Now α∧β ∈ ΛMg∩SBasic, since ΛMg has all propositional tautologies and modus ponens. Hence X ∩ Y = Type(α) ∩ Type(β) = Type(α ∧ β) ∈ F . • Recall that the basic clopens in S are just {Type(β) | β ∈ SBasic}. For any X ∈ F and any basic clopen Y such that X ⊆ Y , we first find α ∈ ΛΠMg ∩ SBasic and β ∈ SBasic such that X = Type(α) and Y = Type(β). Then note that Type(α→ β) = (S \X) ∪ Y = S, since X ⊆ Y . Then by the completeness of S5ΠMg (Lemma 2.8), α → β ∈ S5ΠMg. Then by modus ponens in ΛMg, which extends S5ΠMg as Λ extends S5, β ∈ ΛMg. Remember that β ∈ SBasic. Hence β ∈ ΛMg ∩ SBasic and Y = Type(β) ∈ F . Thus we have Type(Λ) = ∩F , a filter of basic clopens in S, and we assumed that Type(φ) ⊇ Type(Λ). Take basic(φ). We have Type(basic(φ)) = Type(φ) and that it is basic clopen in S. We have shown that S is a Stone space in Lemma 4.9. Hence by compactness, there is actually an element Z ∈ F such that Type(φ) ⊆ Z. By the definition of F , we can find a ψ ∈ ΛMg ∩ SBasic such that Z = Type(ψ). Then Type(ψ → basic(φ)) = S. By completeness again, we have ψ → basic(φ) ∈ S5ΠMg and thus also ΛMg. Then, since ψ is taken in ΛMg, basic(φ) ∈ ΛMg. By Lemma 3.3, φ ∈ Λ. This finishes the completeness of Λ. 2 Then we describe the lattice structure of NextΠ(S5Π). Theorem 5.2 The lattice 〈NextΠ(S5Π),⊆〉 is isomorphic to the lattice of the open sets of S. The isomorphism is Λ 7→ S \Type(Λ), or in the other direction, X 7→ Log(S \X). 232 On the Logics with Propositional Quantifiers Extending S5Π Proof. It is shown in the proof of Theorem 5.1 that for any Λ ∈ NextΠ(S5Π), Type(Λ) is the intersection of a filter of the basic opens of S. By the basic theory of Stone spaces, this means that Type(Λ) is always a closed set in S. Also, for any Λ1,Λ2 ∈ NextΠ(S5Π), obviously Λ1 ⊆ Λ2 iff Type(Λ1) ⊇ Type(Λ2). This means that, if we can establish that for every closed set X ∈ S, there is a Λ ∈ NextΠ(S5Π) such that X = Type(Λ), then the lattice structure of NextΠ(S5Π) is precisely the inverse lattice of the closed sets of S, or just the lattice of the open sets of S, and the isomorphism will be given by Λ 7→ S \ Type(Λ). Now take an arbitrary closed set X in S. Then Log(X) ∈ NextΠ(S5Π) as it is the set of formulas in LΠ valid on a class of complete simple S5 algebras. Then what remains to be shown is that Type(Log(X)) = X. Again, the direction X ⊆ Type(Log(X)) is trivial. Now take an arbitrary type s ∈ S \ X. Then we just need to show that s 6∈ Type(Log(X)). Since S is a Stone space, X is closed, and s 6∈ X, we know that s and X can be separated by a basic clopen. Then, we can find a φ ∈ SBasic such that X ⊆ Type(φ) but s 6∈ Type(φ). But then, φ ∈ Log(X). Since Type(Log(X)) = ⋂ {Type(ψ) | ψ ∈ Log(X)}, we see that s 6∈ Type(Log(X)). This finishes the proof. 2 Since we have shown in the process of proving Theorem 4.9 that S is isomorphic to the disjoint union of two copies of the one-point compactification of N, we have the following corollary. Corollary 5.3 The lattice 〈NextΠ(S5Π),⊆〉 is isomorphic to the lattice of open sets of the disjoint union of two copies of the one-point compactification of N, which is further isomorphic to the lattice of filters of the direct product of two copies of the field of finite and cofinite sets in N. Another corollary of this characterization of all logics in NextΠ(S5Π) is that, in terms of computability, there are arbitrarily complex logics (coded as sets of natural numbers in some natural way). More precisely, for any X ⊆ N, there is a Λ ∈ NextΠ(S5Π) such that X and Λ are Turing-equivalent. Theorem 5.4 For any X ⊆ N, Log({1}× (X ∪ {∞})) is Turing-equivalent to X. Proof. It is not hard to see that {1} × (X ∪ {∞}) is a closed set in S. Let us name {1} × (X ∪ {∞}) by C, and {1} × N∞ by S1 as we did before. Then C = Type(Log(C)), and thus for any φ ∈ LΠ, φ ∈ Log(C) iff Type(φ) ⊇ C. To reduce Log(C) to X, the core idea is that to decide Type(φ) ⊇ C, we only need to see whether Type(φ) ∩ S1 ⊇ C, as C ⊆ S1. Also Type(φ) ∩ S1 = Type(basic(φ)) ∩ S1, which is a finite union of intervals with finite left end points in S1, with these end points readily computable from φ. Then, when X, and hence C, is given by an oracle, to decide whether φ ∈ Log(C), we just need to do the following: • If there is no cofinite interval, then return "no". This is correct because only a cofinite interval contains 〈1,∞〉, which is in C by definition. • Otherwise, for each s ∈ S1 that is not in Type(φ), of which there are only finitely many, check whether s ∈ C. If the oracle for C ever returns "yes", Ding 233 then return "no", as now Type(φ) is not a superset of C. Otherwise, S1 ∩ Type(φ) ⊇ C, and φ ∈ Λ. When X, and hence C, is finite, then we can have an algorithm that directly checks whether for each s ∈ C, s is also in Type(φ). In sum, Log(C) can be reduced to X. On the other hand, suppose Log(C) is given by an oracle. Then to compute whether n ∈ X for some n ∈ N, we only need to use the formula φn = ¬Mn−1∨ Mn+1, with M−1 and M0 here defined as >. Note that Type(φn) = S1\{〈1, n〉}. This means that: • If φn ∈ Log(C), then S1 \ {〈1, n〉} ⊇ C, and hence 〈1, n〉 6∈ C and n 6∈ X. • If φn 6∈ Log(C), then S1 \ {〈1, n〉} 6⊇ C. Then 〈1, n〉 ∈ C and n ∈ X. Thus we only need to use the oracle to decide whether φn ∈ Log(C) and then return the opposite answer. 2 Regarding the non-normal Π-logics extending S5Π, we limit ourselves in this paper to merely point out that there are many such logics. Algebraically, nonnormal modal logics come from matrices (see §1.5 of [17]), which are algebras of propositions with a set of designated truth values. To exhibit a non-normal Π-logic extending S5Π, we can use just one particular structure. Let B be the complete simple S5 algebra whose Boolean part is the direct product of the powerset algebra of N and the MacNeille completion of the free Boolean algebra with countably many generators. Note that t(B) = 〈1,∞〉. Now consider the following set: Λ = {φ ∈ LΠ | ∀V : Prop→ B, V (φ) > g}. It is not hard to see that Λ ⊇ Log(B), as the latter collects formulas whose valuation stay at 1, hence necessarily above g. Also, Λ is a Π-logic. In particular, universalization is valid because if φ only evaluates to elements above g, then ∀φ evaluates to the meet of those elements above g, which must stay above g. Moreover, ∃q(q ∧ atom(q)) ∈ Λ, as this formula evaluates precisely to g. However, 2∃q(q ∧ atom(q)) is not in Λ, since 2g is ⊥, because g 6= 1 in B: there is an atomless part in B. This means we obtained a non-normal Π-logic extending a normal Π-logic Log(B) which has no proper while consistent normal extension: the only closed proper subset of {B} in S is ∅. Obviously, for any complete simple S5 algebra B that has both a non-trivial atomless part and a non-trivial atomic part, we can obtain a non-normal Π-logic in the same fashion. We could also use the requirement that V (φ) > ¬g, which will result in non-normal Π-logics including ¬g but not 2¬g. 6 Conclusion In this paper, we investigated Π-logics extending S5Π. In particular, we see that complete simple S5 algebras are semantically adequate for all normal Πlogics extending S5Π, that the lattice of these normal Π-logics are isomorphic to the lattice of the open sets of the type space S that is homeomorphic to the disjoint union of two copies of the one-point compactification of N, that they 234 On the Logics with Propositional Quantifiers Extending S5Π can have arbitrarily high Turing-degree, and that they do not exhaust all the Π-logics extending S5Π as there are non-normal ones. A major unresolved problem though, is the characterization of all Π-logics, instead of only the normal ones, extending S5Π. We conjecture that a similar strategy can be used, though we need to be more careful about the choice of types. 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