Logics above S4 and the Lebesgue measure algebra

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1 Logics above S4 and the Lebesgue measure algebra Tamar Lando Abstract We study the measure semantics for propositional modal logics, in which formulas are interpreted in the Lebesgue measure algebra M, or algebra of Borel subsets of the real interval [0,1] modulo sets of measure zero. It was shown in Lando (2012) and Fernandez-Duque (2010) that the propositional modal logic S4 is complete for the Lebesgue measure algebra. The main result of the present paper is that every logic L above S4 is complete for some subalgebra of M. Indeed, there is a single model over a subalgebra of M in which all non-theorems of L are refuted. This work builds on recent work by G. Bezhanishvili, D. Gabelaia, and J. Lucero-Bryan Bezhanishvili et al. (2015) on the topological semantics for logics above S4. In Bezhanishvili et al. (2015), it is shown that there are logics above S4 that are not the logic of any subalgebra of the interior algebra over the real line, B(R), but that every logic above S4 is the logic of some subalgebra of the interior algebra over the rationals, B(Q), and the interior algebra over Cantor space, B(C) Mathematics Subject Classification. 03B45; 03B20 1 Introduction It is well-known that some normal modal logics extending S4 are not Kripke complete: they are not the logic of any class of Kripke frames. 1 One solution to the problem posed by this result is to turn to general Kripke frames. A general Kripke frame is a structure W, R, P, where W, R is a Kripke frame and P is an algebra Columbia University, Department of Philosophy, 708 Philosophy Hall, 1150 Amsterdam Avenue Mail Code: 4971, New York, NY See, e.g., Chagrov and Zakharyaschev (1997), Chapter 6. 1

2 of subsets of W in which modal formulas take their values. General frames are a natural extension of Kripke, or relational, semantics. 2 It is well-known that every normal modal logic is complete for some class of general frames (see, e.g., Blackburn et al. (2001), Theorem 5.64). It is also well known that the modal logic S4 has a topological semantics, in addition to its relational semantics. We can view a topological space X algebraically, as the Boolean algebra, B(X), of all subsets of X together with an interior operator taking each set A X to the interior of A. In the topological semantics, modal formulas take values in the algebra B(X), for some topological space X. Bezhanishvili et al. (2015) introduces general topological spaces, which extend the topological semantics in much the same way that general Kripke frames extend Kripke semantics. In the general topological semantics, formulas take values in a subalgebra of B(X). In Bezhanishvili et al. (2015) it is shown that every logic above S4 is the logic of a subalgebra of the interior algebra, B(Q), and is also the logic of a subalgebra of the interior algebra, B(C), where C is Cantor space. 3 However, this is not the case for the real line: there are modal logics above S4 which are not the logic of any subalgebra of B(R). One of the difficulties with the real line, as these authors point out, is that the interior algebra B(R) is connected the only clopen (open and closed) elements in the algebra are the top and bottom elements, 1 and 0. Following Bezhanishvili and Gabelaia (2011) and Bezhanishvili et al. (2015), we can say that a logic L is connected if it is the logic of a connected interior algebra. One can show that not every logic above S4 is a connected logic; but of course every subalgebra of B(R) is connected. So not every logic above S4 is the logic of a subalgebra of B(R). (It is proved in Bezhanishvili et al. (2015) that every connected logic above S4 is the logic of a subalgebra of B(R).) The present paper brings together this recent work on general topological spaces with work carried out in Lando (2015) on the measure semantics for modal logics. In the measure semantics, instead of interpreting modal formulas in topological spaces, we interpret them in the Lebesgue measure algebra, M, or algebra of Borel subsets of the real interval [0,1], modulo sets of measure zero. This algebra in some sense incorporates both the topological and measure structure of the real line. We add, as we show below, an interior operator to the algebra in order to interpret the - modality. Interpreting modal formulas in M is reminiscent of interpreting modal formulas over the real line in the topological semantics. One key difference, however, 2 For a nice introduction to general frames, see, e.g., Chapter 5 in Blackburn et al. (2001), or Chapter 8 in Chagrov and Zakharyaschev (1997). 3 These results in turn extend the results proved in Bezhanishvili and Gabelaia (2011) for logics with the finite model property.

3 is that M is not a connected algebra there are many clopen elements in M not equal to the top or bottom element. So the prospects are good for obtaining completeness results where completeness fails for the real line. The main result of this paper is that every logic above S4 is the logic of a subalgebra of M. The paper is laid out as follows. In 2 we give the algebraic semantics for logics above S4 and show how to interpret modal formulas in the Lebesgue measure algebra, M. In 3 and 4 we recall standard Kripke frames and general Kripke frames, and relate the Kripke (or relational) semantics to the algebraic semantics of 2. In 5 we prove the main result of the paper: completeness of each modal logic above S4 for some subalgebra of M. 2 Algebraic semantics for logics above S4 Let the language L consist of a countable set of propositional variables PV, the binary connective, and unary operators and. We define the connectives,,, and the modal operator in the usual way. Definition 2.1. A set of formulas in the language L is a normal modal logic if it contains all tautologies, the schema K : (ϕ ψ) ( ϕ ψ), and is closed under the rules: Modus Ponens Necessitation ϕ ϕ ψ ψ ϕ ϕ Definition 2.2. The logic S4 is the smallest normal modal logic containing the schemas: T: ϕ ϕ; 4: ϕ ϕ. A normal extension of S4 is a normal modal logic containing all formulas in S4. We follow Bezhanishvili et al. (2015) in referring to normal extensions of S4 as logics above S4. We refer to the formulas in a normal modal logic L as the theorems of L. Definition 2.3. An interior algebra (or topological Boolean algebra) 4 is a Boolean algebra together with a unary operator, I, on the algebra satisfying: 4 The term topological Boolean algebra comes from Rasiowa and Sikorski Rasiowa and Sikorski (1963), while the name interior algebra comes from Blok s dissertation Blok (1976). Another name used more recently is S4 algebra.

4 1. I1 = 1; 2. Ia a; 3. I(a b) = Ia Ib; 4. IIa = Ia, where 1 denotes the top element of the algebra. We say that the operator I is an interior operator. 5 If A is an interior algebra with interior operator I, we say that an element a A is open if Ia = a. The proofs of the facts listed in the following lemma are straightforward, and can be found in, e.g., Rasiowa and Sikorski (1963). 6 Lemma 2.4. Let A be an interior algebra with interior operator I. For any a, b A, 1. If a b, then Ia Ib. 2. Ia is an open element in A. 3. If b is open and b a, then b Ia. Suppose that A is a Boolean algebra, and B is a sublattice of A containing 0 and 1, and satisfying the property For every a A, the set {b B b a} has a greatest element. The following lemma is well known (see, e.g., Blok and Dwinger (1975)) and the proof is left to the reader. Lemma 2.5. Let A be a Boolean algebra, and let B be a sublattice of A containing 0 and 1 and satisfying ( ). Then the operator I on A defined by Ia = max {b B b a} is an interior operator. Moreover, a B iff Ia = a, so B is in fact the lattice of open elements in A. 5 The related notion of closure algebra was introduced in McKinsey and Tarski McKinsey and Tarski (1944). A closure algebra is a Boolean algebra together with a unary operator C on the algebra satisfying: (1) C0 = 0; (2) C(a b) = C(a) C(b); (3) a C(a); (4) C(C(a)) = C(a). Every interior operator, I, gives rise to a closure operator, C, by putting: C(a) = I( a). Moving in the other direction, every closure operator, C, gives rise to an interior operator, I, by putting: I(a) = C( a). One can give the algebraic semantics for logics above S4 in terms of either interior algebras (as we do below) or closure algebras. 6 See Rasiowa and Sikorski (1963), Part I, Chapter III, 1. ( )

5 An algebraic model is a pair M = A, γ where A is an interior algebra and γ : P V A is a valuation. We say that γ is defined over the algebra A. The valuation γ is extended to the set of all formulas in the language L by the recursive clauses: γ(ϕ ψ) = γ(ϕ) γ(ψ), γ( ϕ) = γ(ϕ), γ( ϕ) = Iγ(ϕ), where on the left-hand side denotes a connective, and on the right-hand side denotes the join in the algebra A. We say that a formula ϕ is true in the model M if γ(ϕ) = 1. We say that ϕ is valid in A if ϕ is true in every model defined over A. We denote by Log(A) the set of formulas valid in A. 7 It is not difficult to see that for any interior algebra A, Log(A) is a logic above S4. Note in particular that schema T belongs to Log(A) by Definition 2.3(2); schema 4 belongs to Log(A) by Definition 2.3(4). Let {A i i I} be a collection of interior algebras. We denote by i I A i the product algebra. Recall that an element of the product algebra is a function f that assigns to each i I some element of A i. 8 Operations in the product algebra are defined component-wise. Thus for any (a i ) i I, (b i ) i I i I A i we have: (a i ) i I (b i ) i I = (a i b i ) i I, (a i ) i I = ( a i ) i I, I(a i ) i I = (Ia i ) i I. 9 Note that a = (a i ) i I is an open element in i I A i if and only if a i is an open element in A i for each i I. Indeed, if (a i ) i I is open, then (a i ) i I = (Ia i ) i I. Therefore for each i I, a i = Ia i, and a i is open in A i. We want to interpret formulas in the Lebesgue measure algebra, so let us now recall how to construct that algebra. Let Borel([0, 1]) denote the σ-algebra of Borel subsets of the real interval [0,1], and let Null denote the σ-ideal of Lebesgue measure zero subsets of [0,1]. The Lebesgue measure algebra, which we denote by M, is the 7 For an algebraic semantics defined in greater generality, see Blackburn et al. (2001), Chapter 5., and Chagrov and Zakharyaschev (1997), Chapter 7. In some places, the word assignment is used in place of valuation. 8 See Rasiowa and Sikorski (1963), Part I, Chapter III, 13 for an introduction to products of topological Boolean algebras.

6 quotient, Borel([0, 1]) / Null. Thus elements of M are equivalence classes of Borel subsets of the real interval [0,1]. We denote by A the equivalence class containing the set A [0, 1]. Meets, joins, and complements in M are defined in the usual way in terms of underlying sets: A B = A B, A B = A B, A = [0, 1] A. Recall that a measure m on a Boolean σ-algebra A is a non-negative, real-valued, countably additive function on A. The measure m is normalized if m(1) = 1; m is positive if for all a A, m(a) = 0 implies that a = 0. A measure algebra is a Boolean σ-algebra together with a positive, normalized measure. 10 Now consider the σ-algebra M. Since every set belonging to a given equivalence class has the same measure, we can define a function m on M by putting m( A ) = µ(a), where µ denotes Lebesgue measure restricted to the Borel subsets of [0,1]. It is not difficult to see that m is a positive, normalized measure. It is well known that every measure algebra is complete i.e., every set of elements in the algebra has a supremum (see, e.g., Givant and Halmos (2009), chap. 31). Thus the Lebesgue measure algebra, M, is complete. Note however that the following is not in general true: i I A i = i I A i. For example, if A r = {r} for each r [0, 1], then r [0,1] A r = 0, but r [0,1] A r = 1. In order to interpret modal formulas in the algebra M, we must construct an interior operator I on M. Scott (2009) showed that this could be done quite naturally. Indeed, consider the elements b M such that b = B for some open set B in [0, 1]. These elements form a sublattice of M. (This follows from the fact that open sets are closed under finite intersections and unions.) We will denote this sublattice by G. Lemma 2.6. G is closed under countable joins. Proof. Let {b n n N} be a countable set of elements in G, and let b n = B n for some open set B n [0, 1]. The mapping π : Borel([0, 1]) M defined by π(a) = A is 10 For an introduction to measure algebras, see e.g., Givant and Halmos (2009), Chapter 31.

7 a σ-homomorphism. 11 Therefore ( ) B n = π B n = π(b n ) = b n. n N n N n N n N But clearly n N B n is open in [0,1], so n N b n G. Recall that a Boolean algebra satisfies the countable chain condition if every pairwise disjoint set of elements in the algebra is countable. The following two lemmas are standard proofs can be found in, e.g., Givant and Halmos (2009). 12 Lemma 2.7. Every measure algebra satisfies the countable chain condition. Lemma 2.8. A Boolean algebra satisfies the countable chain condition if and only if for every set E of elements in the algebra, there is a countable set D E such that D and E have the same set of upper bounds. Proposition 2.9. G is a complete sublattice of M. Proof. Let E G. We want to show that the supremum of E in M is an element of G. M is a measure algebra, so by Lemma 2.7, M satisfies the countable chain condition. By Lemma 2.8, there is a countable set D E such that D and E have the same set of upper bounds in M. Therefore D = E (where joins are taken in M). But since G is closed under countable joins, D G. So E G. Lemma The sublattice G of M contains 0 and 1, and satisfies ( ). Proof. The fact that 0, 1 G follows from the fact that 0 =, 1 = [0, 1] and and [0, 1] are open subsets of [0, 1]. For ( ), we need to show that for every a M, the set {b G b a} has a greatest element. Note that {b G b a} G, and by Proposition 2.9, {b G b a} G. Moreover, {b G b a} a, since a is an upper bound on {b G b a}. Therefore, {b G b a} is the greatest element in {b G b a}. Define the operator I on M by putting Ia = {b G b a}. By Lemma 2.5, I is an interior operator, and M together with I is an interior algebra. Moreover, an element a M is open just in case a G. 11 This follows from the fact that if A is a Boolean σ-algebra, and I is a σ-ideal in A, then the projection map π from A onto the quotient A / I is a σ homomorphism. See, e.g., Givant and Halmos (2009), Chapter 29, Theorem See Givant and Halmos (2009), Chapter 30, Lemma 1, and Givant and Halmos (2009), Chapter 31, Lemma 3.

8 3 Standard Kripke frames We return to the Lebesgue measure algebra shortly, but let us now recall a perhaps more familiar semantics for modal logics: the Kripke (or relational) semantics. An S4 Kripke frame is a pair F = W, R where W is a non-empty set and R is a reflexive, transitive binary relation on W. 13 An S4 Kripke model is a pair M = F, V where F = W, R is an S4 Kripke frame and V : P V P(W ) is a valuation. We say that the model M is defined over the frame F. The valuation V can be extended in the usual way to the set of all formulas in L. In particular, V ( ϕ) = W V (ϕ), V (ϕ ψ) = V (ϕ) V (ψ), V ( ϕ) = {w W v V (ϕ) for all v such that wrv}, We say that a formula ϕ is true in M if V (ϕ) = W, and that ϕ is valid in F if ϕ is true in every model defined over F. For any class of frames C we denote by Log(C) the set of formulas valid in every frame in C. A logic S above S4 is Kripke complete if S = Log(C) for some class C of S4 Kripke frames. Kripke semantics for S4 is a special case of the more general algebraic semantics given in 2 (see, e.g., Blackburn et al. (2001), chap. 5.2). To see this, let F = W, R be an S4 Kripke frame, and consider the Boolean algebra of all subsets of W (where meets, joins and complements in the algebra are set-theoretic intersections, unions and complements, respectively). We can define an operator I on the algebra by putting: for any set A W, IA = {w W v A for all v such that wrv} Although we do not go through the details here, it is not difficult to see that I is an interior operator. We will denote the interior algebra defined in this way by B(F ). 14 Then the following is true. Fact 3.1. For any formula ϕ, ϕ is valid in the frame F in Kripke semantics if and only if ϕ is valid in the algebra B(F ) in the algebraic semantics. 13 Reflexivity and transitivity are required to verify the special S4 axioms, T and A fuller exposition of this material would show that the frame F is in fact a topological space, and the interior of the space corresponds to the operator I defined above. This is why we use the notation B(F ), just as, in the introduction, we use the notation B(R) and B(C) to denote the interior algebras generated by the topological spaces R and C. See, e.g., Aiello et al. (2003) for a nice introduction to the topological semantics, and for an explanation of how Kripke S4 frames give rise to topological spaces.

9 4 General Kripke frames As we mentioned above, not every logic above S4 is Kripke complete; there are logics above S4 that cannot be written as Log(C) for any class C of Kripke frames (see Chagrov and Zakharyaschev (1997), chap. 6). This motivates a turn to general Kripke frames, a natural extension of the standard Kripke semantics. A general S4 Kripke frame is a tuple, G = W, R, P, where F = W, R is an S4 Kripke frame and P is a subalgebra of the interior algebra B(F ). We sometimes say that the general frame G is defined over the S4 Kripke frame F. A general S4 Kripke model is a pair M = G, V where G is a general S4 Kripke frame and V : P V P is a valuation assigning to each propositional variable a value in P. Thus the subalgebra P in effect restricts the subsets of worlds that can be assigned as values of modal formulas. We extend the valuation V to the set of all formulas in the language by the same recursive clauses given in 3 for standard Kripke semantics. Truth in a general Kripke model and validity over a general Kripke frame are defined in the expected way. A formula ϕ is true in the model M if V (ϕ) = W and ϕ is valid in the general frame G if ϕ is true in every model defined over G. We make frequent use in what follows of the following important observation. Fact 4.1. For any formula ϕ, ϕ is valid in the general frame G = W, R, P in the general Kripke semantics if and only if ϕ is valid in the algebra P in the algebraic semantics. Let F = W, R be an S4 Kripke frame. Every formula valid in F is also valid in any general frame G defined over F. Indeed, if ϕ is valid in F, then by Fact 3.1, ϕ is valid in the algebra B(F ), but then ϕ is also valid in every subalgebra of B(F ). Conversely, however, if ϕ is valid in a general frame G over F, ϕ need not be valid in F. (Validity in a subalgebra of A does not imply validity in A.) For a class C of general frames, we again denote by Log(C) the set of formulas valid in every general frame in C. It is known that for every modal logic L above S4, L = Log(C) for some class C of general frames. 15 Thus informally we can say that in passing from standard Kripke frames to general frames we recover completeness. Let us now take a look at an S4 Kripke frame of particular interest: the infinite binary tree. We denote by 2 <ω the set of all finite sequences over the set {0, 1}, including the empty sequence. If s is an element in 2 <ω of length n 0, we denote by s 0 the sequence of length n + 1 with initial segment s and ending in 0. Likewise we denote by s 1 the sequence of length n + 1 with initial segment s and ending in 1. The binary relation R on 2 <ω is defined by putting: for any s, t 2 <ω, srt iff 15 See, e.g., Blackburn et al. (2001), Theorem 5.64.

10 t = s 0, or t = s 1. Let be the reflexive, transitive closure of R. The infinite binary tree is the frame 2 <ω,. With slight abuse of notation, we will use 2 <ω to denote also the infinite binary tree (i.e., the frame as well as the underlying set of finite sequences). Our present interest in the infinite binary tree stems from the following lemma, proved in Bezhanishvili et al. (2015). What follows is a sketch of the proof given there; for full details, the reader should consult Bezhanishvili et al. (2015), Lemma 5.4. Lemma 4.2. If L is a logic above S4 and ϕ is a non-theorem of L, then there is a general frame G over 2 <ω such that every theorem of L is valid in G and ϕ is not valid in G. Proof Sketch. Suppose L is a logic above S4, and ϕ is a non-theorem of L. Then there is some countable, rooted, general frame G = W, R, P in which ϕ is refuted but all theorems of L are valid. (A general frame, G, is countable if the set of worlds W is countable; G is rooted if there is at least one w W such that wrv for every v W.) Moreover, there is a p-morphism f from the frame 2 <ω onto the frame W, R. Let A := {f 1 (S) S P}. Then A is a subalgebra of B(2 <ω ), and A is isomorphic to P. Since ϕ is refuted in G = W, R, P, ϕ is also refuted in G = 2 <ω, A. Remark 4.3. Although for any logic L above S4, each nontheorem of L can be refuted in some general frame over the infinite binary tree 2 <ω, it is not the case that each logic above S4 is the logic of a subalgebra of B(2 <ω ). For further discussion, see Bezhanishvili et al. (2015). Corollary 4.4. If L is a logic above S4, then L = Log(C) for some class C of general frames over 2 <ω. Proof. Let L be a logic above S4. By Lemma 4.2, for each non-theorem ϕ of L, there is a general frame G ϕ over 2 <ω such that every theorem of L is valid in G ϕ, but ϕ is not valid in G ϕ. Let C = {G ϕ ϕ if a non-theorem of L}. Then Log(C) = L. Indeed, if ϕ L, then ϕ is valid in G for each G C. So ϕ Log(C). If ϕ L, then ϕ is not valid in G ϕ and G ϕ C. So ϕ Log(C). 5 Completeness of logics above S4 for subalgebras of M In this section we show that, given a logic L above S4, L = Log(A) for some subalgebra A of the Lebesgue measure algebra, M. Our strategy will be to transfer

11 countermodels over the infinite binary tree, 2 <ω, to subalgebras of M. We proceed as follows. First we show that there is an embedding of the algebra B(2 <ω ) into the algebra M. Then we show, via this embedding, that if ϕ is a non-theorem of L, it can be refuted in some subalgebra of M. This allows us to construct a subalgebra of M ω in which every nontheorem of L is refuted. Finally, we show that this subalgebra of M ω is isomorphic to a subalgebra of M. We begin by recalling the notion of an embedding of interior algebras. Let A 1 and A 2 be interior algebras, and let h : A 1 A 2. We say that h is a homomorphism if h preserves Boolean operations and interiors: h(a b) = h(a) h(b), h( a) = h(a), h(ia) = Ih(a). 16 We say that h is an embedding if h is an injective homomorphism. h is an isomorphism if it is a surjective embedding. Finally, A 1 is isomorphic to A 2 (A 1 = A2 ) if there is an isomorphism h : A 1 A 2. It is sometimes useful to have additional terminology for functions that preserve the Boolean structure of interior algebras but not (necessarily) the interior operator. We will say that h is a Boolean homomorphism if h preserves joins and complements. Also, h is a Boolean isomorphism if h is a bijective Boolean homomorphism. The proof of the following lemma is straightforward and is therefore omitted. Lemma 5.1. Let A 1 and A 2 be interior algebras. If h : A 1 A 2 is a Boolean isomorphism, then the following conditions are equivalent: 1. a is open in A 1 if and only if h(a) is open in A 2, for all a A 1, 2. h(ia) = I(h(a)), for all a A 1. The following lemma is proved in Lando (2015) and provides the key ingredient in the proof of our main result. 17 Lemma 5.2. There is an embedding of B(2 <ω ) into M. Proposition 5.3. If L is a logic above S4 and ϕ a non-theorem of L, then there is a subalgebra A of M such that every theorem of L is valid in A, and ϕ is not valid in A. 16 On the left-hand side of these equations, the symbols, and I denote operations in the algebra A 1 ; on the right-hand side they denote operations in the algebra A See Lando (2015), Propositions 11.2, 11.4, and 11.5.

12 Proof. By Lemma 4.2, there is a general frame G = 2 <ω, P such that every theorem of L is valid in G and ϕ is not valid in G. Therefore, every theorem of L is valid in the algebra P and ϕ is not valid in P. By Lemma 5.2 there is an embedding h : B(2 <ω ) M. Let A be the image of P under h. Then A is a subalgebra of M. Moreover, P is isomorphic to A, so Log(P) = Log(A). It follows that every theorem of L is valid in A and ϕ is not valid in A. Proposition 5.4. If L is a logic above S4, then L = Log(A) for some subalgebra A of M ω. Moreover, there is a single model over A in which all non-theorems of L are refuted. Proof. Let ϕ 1, ϕ 2, ϕ 3,... be an enumeration of non-theorems of L. By Proposition 5.3, for each ϕ k there is a subalgebra A k of M such that every theorem of L is valid in A k and ϕ k is not valid in A k. Let A = k N A k. Note that since each A k is a subalgebra of M, A is a subalgebra of M ω. It is well known that Log( k N A k) = k N Log(A k). But clearly k N Log(A k) = L. Therefore, Log(A) = L. We now show that every nontheorem is refuted in a single model over A. Note that since ϕ k is refuted in A k, there is an algebraic model M k = A k, γ k such that γ k (ϕ k ) 1. Define the valuation γ over A by putting, γ(p) = (γ k (p)) k N, for each p P V. A simple proof by induction shows that for each formula ϕ, γ(ϕ) = (γ k (ϕ)) k N. If ϕ L, then ϕ = ϕ k for some k 0. Thus γ k (ϕ k ) 1, so γ(ϕ k ) 1 and ϕ is refuted in the model M = A, γ. We want to show now that every logic above S4 is also the logic of a subalgebra of M. Given Proposition 5.4, it is sufficient to show that M is isomorphic to M ω. We will now construct an isomorphism between the two interior algebras. The idea of the construction is the following. The interval [0,1] is broken up into countably many disjoint open intervals, I k, so that [0,1] is the union of the I k s together with their endpoints. An element a M ω is a sequence (a k ) k N, where each a k is an equivalence class containing a Borel set A k (0, 1). We define countably many scaling functions, s k : (0, 1) I k, so that for each k, s k (A k ) is a scaled copy of the set A k in the interval I k. The original element (a k ) k N in M ω is then sent to the element k N s k(a k ) in M. This mapping from M ω to M is, as we show below, a Boolean isomorphism that preserves open elements, hence an isomorphism of interior algebras. Now for the details.

13 For the duration of the paper, let µ denote the Lebesgue measure. Define the sequence of pairwise disjoint open intervals I 0, I 1, I 2,... in the real unit interval [0,1] as follows: I 0 = ( 1, 1), I 2 1 = ( 1, 1), I = ( 1, 1), and so on. In general, I 8 4 k = ( 1, 1 ). 2 k+1 2 k For simplicity of notation, we let l k and r k denote the left and right endpoints of the interval I k respectively, and for any interval I we let l(i) denote the length of I. For each k N, define a scaling function s k : (0, 1) I k by putting: s k (x) = l k + x(r k l k ). Note that s k is a homeomorphism from the interval (0,1) to I k, and therefore for any interval I (0, 1), s k (I) is also an interval. Moreover, if I is an interval, µ(s k (I)) = l(i k )µ(i). (This follows immediately from the definition of s k.) Lemma 5.5. For any Borel set A (0, 1), µ(s k (A)) = l(i k )µ(a). In particular, µ(s k (A)) = 0 if and only if µ(a) = 0. Proof. Recall that for any measurable (hence for any Borel) set A R, { } µ(a) = inf µ(o n ), where the infimum is taken over all countable collections of bounded open intervals {O n n N} such that A n N O n. 18 Now suppose that {O n n N} is a collection of bounded open intervals such that A n N O n. Note that {s k (O n ) n N} is also a collection of bounded open intervals, and that s k (A) n N s k(o n ). Since each O n is an interval, µ(s k (O n )) = l(i k ) µ(o n ). Therefore, µ(o n ) n N n N µ(s k (O n )) = l(i k ) n N. Thus µ(s k (A)) l(i k ) n N µ(o n) for each cover {O n n N} of A by bounded open intervals. It follows that µ(s k (A)) l(i k )µ(a). For the reverse inequality, suppose that {U n n N} is a collection of bounded open intervals such that s k (A) n N U n. Note that {s 1 k (U n) n N} is also 18 This is the Lebesgue outer measure of A, and for any Lebesgue measurable set A, the Lebesgue measure of A is equal to the Lebesgue outer measure of A. See e.g., Royden and Fitzpatrick (2010), Chapter 2 for useful background information on Lebesgue measure, including a discussion of outer measure.

14 a collection of bounded open intervals (this follows from the definition of s k ), and that A n N s 1 k (U n). Therefore, µ(u n ) = µ(s k (s 1 k (U n))) = l(i k )µ(s 1 k (U n)). So µ(s 1 k (U n)) = 1 µ(u l(i k ) n) and n N µ(s 1 k (U n)) = 1 l(i k ) µ(u n ). Since {s 1 k (U n) n N} is an open cover of A, µ(a) 1 l(i k ) n N µ(u n) for each cover {U n n N} of s k (A) by bounded open intervals. It follows that µ(a) 1 µ(s l(i k ) k(a)). Equivalently, l(i k )µ(a) µ(s k (A)). Putting the two inequalities together, we have µ(s k (A)) = l(i k )µ(a). Elements of M ω are sequences of elements in M, and we denote such sequences by (a k ) k N. Let a = (a k ) k N M ω, and let a k = A k for some A k (0, 1). Note that we can always find such an A k (i.e., one that does not include the endpoints of the [0,1] interval), since the sets {0} and {1} have measure zero. Define the function h : M ω M by putting: h(a) = k N s k (A k ). n N Lemma 5.6. h is well-defined i.e., independent of the choice of A k in a k. Proof. We want to show that if, for all k N, a k = A k = A k and A k, A k then s k (A k ) = s k (A k). k N k N (0, 1), Suppose that for all k N, a k = A k = A k, and A k, A k (0, 1). Then µ(a k A k ) = 0, where denotes symmetric difference. By Lemma 5.5, µ(s k (A k A k )) = 0. But s k (A k A k ) = s k(a k ) s k (A k ). So µ(s k(a k ) s k (A k )) = 0, and s k(a k ) = s k (A k ). Note that And therefore, k N s k (A k ) k N s k (A k) k N µ( k N s k (A k ) k N (s k (A k ) s k (A k)). s k (A k)) k N µ(s k (A k ) s k (A k)). Since the sum on the RHS is zero, we have k N s k(a k ) = k N s k(a k ).

15 Proposition 5.7. M ω = M. Proof. We show that the function h : M ω M defined above is an isomorphism. 1. Injectivity. Suppose a = (a k ) k N, b = (b k ) k N M ω and a b. Then for some k N, a k b k. Let a k = A k and b k = B k with A k, B k (0, 1). Then µ(a k B k ) > 0. By Lemma 5.5, µ(s k (A k B k )) > 0. But s k (A k ) s k (B k ) = s k (A k B k ), so µ(s k (A k ) s k (B k )) > 0. By construction of the s k s, s k (A k ) s k (B k ) k N s k(a k ) k N s k(b k ). 19 But then also µ( k N s k(a k ) k N s k(b k )) > 0. So h(a) = k N s k(a k ) k N s k(b k ) = h(b). 2. Surjectivity. Let a M, and let a = A. Let A k = A I k. Let B k = s 1 k (A k), and let b k = B k. Then h((b k ) k N ) = a. To see this, note that: h((b k ) k N ) = k N s k (B k ) = k N = k N A k s k (s 1 k (A k)) = k N(A I k ) = A k N I k = A = a, where the second to last equality follows from the fact that k N I k = Preservation of joins. 19 This follows from the fact that for any j k, s j ((0, 1)) s k ((0, 1)) =. If x s k (A k ) s k (B k ), then x s k ((0, 1)), so x s j ((0, 1)), for j k. Therefore x s j (B j ) for j k. So x k N s k(b k ). Therefore x k N s k(a k ) k N s k(b k ). This shows that s k (A k ) s k (B k ) k N s k(a k ) k N s k(b k ). By a similar argument, s k (B k ) s k (A k ) k N s k(b k ) k N s k(a k ). It follows that s k (A k ) s k (B k ) k N s k(a k ) k N s k(b k ).

16 Let a = (a k ) k N, b = (b k ) k N M ω with a k = A k and b k = B k, and A k, B k (0, 1). Then h(a b) = h((a k b k ) k N ) = k N s k (A k B k ) = k N s k (A k ) k N s k (B k ) = s k (A k ) s k (B k ) k N k N = h(a) h(b). 4. Preservation of complements. Let a = (a k ) k N M ω, with a k = A k and A k (0, 1). Then a = ( a k ) k N = ( (0, 1) A k ) k N, and (0, 1) A k (0, 1). Note that s k ((0, 1) A k ) = I k s k (A k ). Therefore, h( a) = k N s k ((0, 1) A k ) = k N(I k s k (A k )) = k N I k k N s k (A k ) = k N s k (A k ) = h(a), where the fourth equality follows from the fact that k N I k = Preservation of interiors. We want to show that h(ia) = I(h(a)). By Lemma 5.1, it is sufficient to show that a is open in M ω if and only if h(a) is open in M. Suppose a = (a k ) k N is an open element in M ω. Then a k is open in M for each k. Hence a k = A k for some open set A k (0, 1). 20 Since s k is a homeomorphism, s k (A k ) is open for each k N. Thus k N s k(a k ) is open and h(a) = k N s k(a k ) is an open element in M. 20 We are guaranteed that a k = A k for some open set A k [0, 1]. If A k (0, 1), then let A k = A k (0, 1). Clearly a k = A k, and A k (0, 1).

17 Conversely, suppose that a = (a k ) k N and h(a) is an open element in M. Then h(a) = O for some open set O (0, 1). 21 But h(a) = O = O k N I k = O k N I k = k N (O I k ) = k N s k (s 1 k (O I k)), where the second equality follows from the fact that k N I k = 1. Since s k is a homeomorphism, s 1 k (O I k) is open for each k. By injectivity of h, a k = s 1 k (O I k). Therefore, a k is open for each k. It follows that a = (a k ) k N is open in M ω. Theorem 5.8. If L is a logic above S4, then L = Log(A) for some subalgebra A of M. Moreover, there is a single model over A in which all nontheorems of L are refuted. Proof. Immediate from Propositions 5.4 and 5.7. References M. Aiello, J. van Benthem, and G. Bezhanishvili. Reasoning about space the modal way. Journal of Logic and Computation, 13(6): , G. Bezhanishvili and D. Gabelaia. Connected modal logics. Arch. Math. Logic, 50, G. Bezhanishvili, D. Gabelaia, and J. Lucero-Bryan. Topological completeness of logics above s4. Journal of Symbolic Logic, 80(2): , P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge University Press, W.J. Blok. Varieties of Interior Algebras. PhD Thesis, University of Amsterdam, W.J. Blok and PH Dwinger. Equational classes of closure algebras. Indag. Math., 37( ), Again, we are guaranteed that h(a) = O for some open set O [0, 1]. If O (0, 1), let O = O (0, 1). Clearly h(a) = O, and O is open with O (0, 1).

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