aa r X i v : . [ m a t h . L O ] S e p Modal Logics of Some Hereditarily IrresolvableSpaces
Robert Goldblatt
Abstract
A topological space is hereditarily k-irresolvable if none of its subspacescan be partitioned into k dense subsets, We use this notion to provide a topologicalsemantics for a sequence of modal logics whose n -th member K4 C n is characterisedby validity in transitive Kripke frames of circumference at most n . We show that un-der the interpretation of the modality ♦ as the derived set (of limit points) operation,K4 C n is characterised by validity in all spaces that are hereditarily n + D separation property.We also identify the extensions of K4 C n that result when the class of spacesinvolved is restricted to those that are crowded, or densely discrete, or openly irre-solvable, the latter meaning that every non-empty open subspace is 2-irresolvable.Finally we give a topological semantics for K4M, where M is the McKinsey axiom. Key words: modal logic, Kripke frame, circumference, topological semantics, de-rived set, resolvable space, hereditarily irresolvable, openly irresolvable, dense,crowded, scattered space, Alexandrov topology.2020
Mathematics Subject Classification : 03B45, 54F99.
One theme of this article is the use of geometric ideas in the semantic analysis oflogical systems. Another is the use of relational semantics in the style of Kripke.Both themes feature prominently in the many-faceted research portfolio of Alas-dair Urquhart. In the area of relevant logic he discovered how to construct certainrelational models of relevant implication out of projective geometries, and relatedthese to modular geometric lattices (1983, see also 2017). This led to his striking
Robert GoldblattVictoria University of Wellington, e-mail: [email protected] demonstration in (1984) that the main systems of relevant implication are unde-cidable, and then to a proof in (1993) that these systems fail to satisfy the Craiginterpolation theorem. In the area of Kripke semantics for modal logics, in additionto an early book (1971) with Rescher on temporal logics, his contributions have in-cluded the construction in (1981) of a normal logic that is recursively axiomatisableand has the finite model property but is undecidable; and the proof in (2015) thatthere is a formula of quantified S5 which, unlike the situation of propositional S5,is not equivalent to any formula of modal degree one. His article (1978) on topolog-ical representation of lattices has been very influential. It assigned to any boundedlattice a dual topological space with a double ordering, and showed that the orig-inal lattice can be embedded into a lattice of certain ‘stable’ subsets of its dualspace. This generalised the topological representations of Stone (1936) for Booleanalgebras and Priestley (1970) for distributive lattices. It stimulated the develop-ment of further duality theories for lattices, such as those of Hartung (1992, 1993),Allwein and Hartonas (1993), Ploˇsˇcica (1995) and Hartonas and Dunn (1997). Thestructure of Urquhart’s dual spaces was exploited by Allwein and Dunn (1993) todevelop a Kripke semantics for linear logic, and by Dzik et al. (2006) to do likewisefor non-distributive logics with various negation operations. Urquhart’s article alsoplayed an important role in the development of the important notion of a canoni-cal extension . This was first introduced by J´onsson and Tarski (1951) for Booleanalgebras with operators, and is closely related to the notion of canonical model of amodal logic. After several decades of evolution, Gehrke and Harding (2001) gave anaxiomatic definition of a canonical extension of any bounded lattice-based algebra,showing that it is unique up to isomorphism. In proving that such an extension ex-ists, they observed that it can be constructed as the embedding of the original algebrainto the lattice of stable subsets of its Urquhart dual space. Craig and Haviar (2014)have clarified the relationship between Urquhart’s construction and other manifesta-tions of canonical extensions. This area has now undergone substantial developmentand application, by Gehrke and others, of theories of canonical extensions and dual-ity for lattice expansions. There are surveys of this progress in (Gehrke, 2018) and(Goldblatt, 2018). In (Goldblatt, 2020) a new notion of bounded morphism betweenpolarity structures has been introduced that provides a duality with homomorphismsbetween lattices with operators. The seed that Urquhart planted has amply bornefruit.The geometrical ideas of the present article come from topology. Our aim is toprovide a topological semantics for a sequence of modal logics that were originallydefined by properties of their binary-relational Kripke models. The n -th member ofthis sequence is called K4 C n , and is the smallest normal extension of the logic K4that includes an axiom scheme C n which will be described below. It was shown in(Goldblatt, 2019) that the theorems of K4 C n are characterised by validity in all fi-nite transitive Kripke frames that have circumference at most n , meaning that anycycle in the frame has length at most n , or equivalently that any non-degeneratecluster has at most n elements. For n ≥
1, adding the reflexivity axiom ϕ → ♦ ϕ to K4 C n gives the logic S4 C n which is characterised by validity in all finite reflex- odal Logics of Some Hereditarily Irresolvable Spaces 3 ive transitive Kripke frames that have circumference at most n . It was also shownthat S4 C n has a topological semantics in which any formula of the form ♦ ϕ is inter-preted as the topological closure of the interpretation of ϕ . Under this interpretation,S4 C n is characterised by validity in all topological spaces that are hereditarily n + -irresolvable . Here a space is called k-resolvable if it can be partitioned into k densesubsets, and is hereditarily k -irresolvable if none of its non-empty subspaces are k -resolvable. k could be any cardinal, but we will deal only with finite k .The interpretation of ♦ as closure, now known as C -semantics, appears to havefirst been considered by Tang (1938). McKinsey (1941) showed that the formu-las that are C -valid in all topological spaces are precisely the theorems of S4.McKinsey and Tarski (1944, 1948) then undertook a much deeper analysis whichshowed that S4 is characterised by C -validity in any given space that has a cer-tain ‘dissectability’ property that is possessed by every finite-dimensional Euclideanspace, and more generally by any metric space that is crowded, or dense-in-itself,i.e. has no isolated points. They also suggested studying an alternative topologicalinterpretation, now called d -semantics, in which ♦ ϕ is taken to be the derived set,i.e. the set of limit points, of the interpretation of ϕ . This d -semantics does not vali-date the transitivity axiom ♦♦ ϕ → ♦ ϕ . That is d -valid in a given space iff the spacesatisfies the T D property that every derived set is closed.The logic K4 C is in fact the G¨odel-L¨ob provability logic, which is known tohave a topological characterisation by d -validity in all spaces that are scattered ,meaning that every non-empty subspace has an isolated point. The logic K4 C was shown by Gabelaia (2004) to be characterised by d -validity in hereditarily 2-irresolvable spaces. Such spaces satisfy the T D property, which however may nothold in hereditarily k -irresolvable spaces for k >
2. The principal new result provenhere is that for n >
1, K4 C n is characterised by d -validity in all spaces that are bothhereditarily n + D . Our proof adapts the argument of (Gabelaia,2004) and also makes use of certain spaces that are n -resolvable but not n + C n , Section 5 gives characterisations of some extensions of K4 C n thatcorrespond to further topological constraints. We begin with a review of relation semantics for propositional modal logic. A stan-dard reference is (Blackburn et al., 2001). Formulas ϕ , ψ , . . . of are constructedfrom some denumerable set of propositional variables by the Boolean connectives ⊤ , ⊥ , ¬ , ∧ , ∨ , → and the unary modalities ♦ and (cid:3) . We write (cid:3) ∗ ϕ as an abbrevi-ation of the formula ϕ ∧ (cid:3) ϕ , and ♦ ∗ ϕ for ϕ ∨ ♦ ϕ . Robert Goldblatt A frame F = ( W , R ) consists of a binary relation R on a set W . Each x ∈ W has the set R ( x ) = { y ∈ W : xRy } of R-successors . A model M = ( W , R , V ) ona frame has a valuation function V assigning to each variable p a subset V ( p ) of W . The model then assigns to each formula ϕ a subset M ( ϕ ) of W , thought of asthe set of points at which ϕ is true. These truth sets are defined by induction onthe formation of ϕ , putting M ( p ) = V ( p ) for each variable p , interpreting eachBoolean connective by the corresponding Boolean operation on subsets of W , andputting M ( ♦ ϕ ) = R − M ( ϕ ) = { x ∈ W : R ( x ) ∩ M ( ϕ ) = /0 } , M ( (cid:3) ϕ ) = { x ∈ W : R ( x ) ⊆ M ( ϕ ) } . A formula ϕ is true in model M , written M | = ϕ , if M ( ϕ ) = W ; and is valid inframe F , written F | = ϕ , if it is true in all models on F .A normal logic is any set L of formulas that includes all tautologies and all in-stances of the scheme K: (cid:3) ( ϕ → ψ ) → ( (cid:3) ϕ → (cid:3) ψ ) , and whose rules includemodus ponens and (cid:3) -generalisation (from ϕ infer (cid:3) ϕ ). The members of a logic L may be referred to as the L-theorems . The set L C of all formulas valid in (all mem-bers of) some given class C of frames is a normal logic. A logic is characterised byvalidity in C , or is sound and complete for validity in C , if it is equal to L C . Thesmallest normal logic, known as K, is characterised by validity in all frames.A logic is transitive if it contains all instances of the scheme 4: ♦♦ ϕ → ♦ ϕ ,which is valid in precisely those frames that are transitive, i.e. their relation R istransitive. The smallest transitive normal logic, known as K4, is characterised byvalidity in all transitive frames.In a transitive frame F = ( W , R ) , a cluster is a subset C of W that is an equiv-alence class under the equivalence relation { ( x , y ) : x = y or xRyRx } . The R -clustercontaining x is C x = { x } ∪ { y : xRyRx } . If x is irreflexive, i.e. not xRx , then C x = { x } ,and C x is called a degenerate cluster. Thus if R is an irreflexive relation, then allclusters are degenerate. If xRx , then C x is non-degenerate . If C is a non-degeneratecluster then it contains no irreflexive points and the relation R is universal on C andmaximally so. A simple cluster is non-degenerate with one element, i.e. a singleton C x = { x } with xRx . If R is antisymmetric , i.e. xRyRx implies x = y in general, thenevery cluster is a singleton, so is either simple or degenerate.The relation R lifts to a well-defined relation on the set of clusters by putting C x RC y iff xRy . This relation is transitive and antisymmetric on the set of clusters. Acluster C x is final if it is maximal in this ordering, i.e. there is no cluster C = C x with C x RC . This is equivalent to requiring that xRy implies yRx .We take the circumference of a frame F to be the supremum of the set of alllengths of cycles in F , where a cycle of length n ≥ x , . . . , x n of distinct points such that x R · · · Rx n Rx . The circumference is 0 iff there are nocycles. In a transitive frame, the points of any cycle are R -related to each other andare reflexive, and all belong to the same non-degenerate cluster. Conversely any fi-nite non-empty subset of a non-degenerate cluster can be arranged (arbitrarily) intoa cycle. Thus if a finite transitive frame has circumference n ≥
1, then n is equal to odal Logics of Some Hereditarily Irresolvable Spaces 5 the size of a largest non-degenerate cluster. The circumference is 0 iff the frame isirreflexive, i.e. has only degenerate clusters. A finite transitive frame has circumfer-ence at most n iff each of its non-degenerate clusters has at most n members.Now given formulas ϕ , . . . , ϕ n , define the formula P n ( ϕ , . . . , ϕ n ) to be ♦ ( ϕ ∧ ♦ ( ϕ ∧ · · · ∧ ♦ ( ϕ n ∧ ♦ ϕ )) · · · ) provided that n >
1. For the case n =
0, put P ( ϕ ) = ♦ ϕ . Let D n ( ϕ , . . . , ϕ n ) be V i < j n ¬ ( ϕ i ∧ ϕ j ) (for n = ⊤ ). Define C n to be thescheme (cid:3) ∗ D n ( ϕ , . . . , ϕ n ) → ( ♦ ϕ → ♦ ( ϕ ∧ ¬ P n ( ϕ , . . . , ϕ n )) . Let K4 C n be the smallest transitive normal logic that includes the scheme C n . In(Goldblatt, 2019, Theorems 1 and 4), the following were shown. Theorem 1.
For all n ≥ , (1) A transitive frame validates C n iff it has circumference at most n and no strictlyascending chains (i.e. no sequence { x m : m < ω } with x m Rx m + but not x m + Rx m for all m). (2) A finite transitive frame validates C n iff it has circumference at most n. (3) A formula is a theorem of K4 C n iff it is valid in all finite transitive frames thathave circumference at most n. ⊓⊔ The cases n = , C is equalto the G¨odel-L¨ob modal logic of provability, the smallest normal logic to containthe L¨ob axiom (cid:3) ( (cid:3) ϕ → ϕ ) → (cid:3) ϕ . It is characterised by validity in all finitetransitive frames that are irreflexive, i.e. all clusters are degenerate. This was firstshown by Segerberg (1971), and there have been a number of other proofs since(see Boolos 1993, Chapter 5). K4 C is equal to the logic K4Grz (cid:3) , where Grz (cid:3) isthe axiom (cid:3) ( (cid:3) ( p → (cid:3) p ) → p ) → (cid:3) p . This logic was shown by Gabelaia (2004) to be characterised by validity in all fi-nite frames whose clusters are all singletons (which may individually be arbitrarilydegenerate or simple).Parts (1) and (3) of Theorem 1 imply that K4 C n is a subframe logic in the senseof Fine (1985) that it is complete for validity in its class of validating Kripke framesand this class is closed under subframes,Part (2) of of Theorem 1 states that validity of C n exactly expresses the propertyof having circumference at most n over the class of finite transitive frames. But fortransitive frames in general, there is no such formula, or even set of formulas: Theorem 2.
There does not exist any set of modal formulas whose conjoint validityin any transitive frame exactly captures the property of having circumference atmost n ≥ .Proof. Let F ω be the transitive irreflexive frame ( ω , < ) , and for each 1 ≤ m < ω let F m be the frame consisting of a single non-degenerate m -element cluster on Robert Goldblatt the set { , . . . , m − } . Let f m : F ω → F m be the map f m ( x ) = x mod m . Then f m is a surjective bounded morphism, a type of map that preserves validity of modalformulas in frames (Blackburn et al., 2001, Theorem 3.14). Hence any formula validin F ω will be valid in F m for every m ≥ Φ n offormulas that is valid in a transitive frame iff that frame has circumference at most n .Then F ω | = Φ n , since F ω has circumference 0, and therefore has circumference atmost n , what ever n ≥ F n + | = Φ n ,which contradicts the definition of Φ n because F n + has circumference n + ⊓⊔ We first review some of the topological ideas that will be used to interpret modalformulas. These can be found in many textbooks, such as (Willard, 1970) and(Engelking, 1977).If X is a topological space, we do not introduce a symbol for the topology of X ,but simply refer to various subsets as being open or closed in X . If x ∈ X , then an open neighbourhood of x is any open set that contains x . A subset of X intersects another subset if the two subsets have non-empty intersection. If Y ⊆ X , then x is a closure point of Y if every open neighbourhood of x intersects Y . The set ofall closure points of Y is the closure of Y , denoted Cl X Y . It is the smallest closedsuperset of Y in X . It has Y ⊆ Cl X Y = Cl X Cl X Y . A useful fact is that if O is open in X , then O ∩ Cl X Y ⊆ Cl X ( O ∩ Y ) . Y is called dense in X when Cl X Y = X . This meansthat every non-empty open set intersects Y .We write Int X Y for the interior of Y in X , the largest open subset of Y . Thus x ∈ Int X Y iff some open neighbourhood of x is included in Y .Any subset S of X becomes a subspace of X under the topology whose open setsare all sets of the form S ∩ O with O open in X . The closure operator Cl S of thesubspace S satisfies Cl S Y = S ∩ Cl X Y . Hence a set Y ⊆ S is dense in S , i.e. Cl S Y = S ,iff S ⊆ Cl X Y .A punctured neighbourhood of x is any set of the form O − { x } with O an openneighbourhood of x . If Y ⊆ X , then x is a limit point of Y in X if every puncturedneighbourhood of x intersects Y . The set of all limit points of Y is the derived set of Y , which we denote De X Y . (Other notations are Y ′ , dY , and h d i Y .) In general, Cl X Y = Y ∪ De X Y . Another useful fact is that if O is open in X , then O ∩ De X Y ⊆ De X ( O ∩ Y ) . If S is a subspace of X , then any Y ⊆ S has De S Y = S ∩ De X Y .A T D space is one in which the derived set De X { x } of any singleton is closed,which is equivalent to requiring that De X De X { x } ⊆ De X { x } . This in turn is equiv-alent to the requirement that any derived set De X Y is closed, i.e. that De X De X Y ⊆ De X Y for all Y ⊆ X (Aull and Thron, 1962, Theorem 5.1). The T D property isstrictly weaker than the T separation property that De X { x } = /0 in general, which isitself equivalent to the requirement that any singleton is closed, and to the require-ment that any two distinct points each have an open neighbourhood that excludes odal Logics of Some Hereditarily Irresolvable Spaces 7 the other point. The simplest example of a non-T D space is a two-element space X = { x , y } with the indiscrete (or trivial ) topology in which only X and /0 are open.It has De X { x } = { y } and De X { y } = { x } . A useful known fact is Lemma 1.
A space X is T D iff it has x / ∈ Cl X De X { x } for all x ∈ X.Proof.
Since x / ∈ De X { x } in general, if X is T D then x / ∈ Cl X De X { x } as Cl X De X { x } = De X { x } . Conversely, since Cl X { x } is a closed superset of De X { x } , it includes Cl X De X { x } , so we have Cl X De X { x } ⊆ Cl X { x } = De X { x } ∪ { x } . Thus if x is notin Cl X De X { x } , then Cl X De X { x } ⊆ De X { x } , hence De X { x } is closed. ⊓⊔ We now review the topological semantics for modal logics (Bezhanishvili et al.,2005). A topological model M = ( X , V ) on a space X has a valuation V assigninga subset of X to each propositional variable. A truth set M d ( ϕ ) is then defined byinduction on the formation of ϕ by letting M d ( p ) = V ( p ) , interpreting the Booleanconnectives by the corresponding Boolean set operations, and putting M d ( ♦ ϕ ) = De X ( M d ( ϕ )) , the set of limit points of M d ( ϕ ) . Then M d ( (cid:3) ϕ ) is determined bythe requirement that it be equal to M d ( ¬ ♦ ¬ ϕ ) . This gives • x ∈ M d ( ♦ ϕ ) iff every punctured neighbourhood of x intersects M d ( ϕ ) ; • x ∈ M d ( (cid:3) ϕ ) iff there is a punctured neighbourhood of x included in M d ( ϕ ) .A formula ϕ is d-true in M , written M | = d ϕ , if M d ( ϕ ) = X ; and is d-valid inspace X , written X | = d ϕ , if it is d -true in all models on X . The set { ϕ : X | = d ϕ } ofall formulas that are d -valid in X is a normal logic, called the d-logic of X . It neednot be a transitive logic, because the scheme 4 is d -valid in X iff X is T D (Esakia,2001, 2004).A point x is isolated in a space X if { x } is open in X . If X has no isolated points,it is crowded (also called dense-in-itself ). This means that every point is a limitpoint of X , i.e. De X X = X . Thus X is crowded iff X | = d ♦ ⊤ . A subset S is called crowded in X if it is crowded as a subspace, i.e. De S S = S , which is equivalent torequiring that S ⊆ De X S , and hence that Cl X S = De X S , since Cl X S = S ∪ De X S . Thefollowing is standard. Lemma 2.
Let O and S be subsets of X with O open. If S is dense in X then O ∩ S isdense in O, and if S is crowded in X then O ∩ S is crowded in O.Proof. If Cl X S = X , then O = O ∩ Cl X S ⊆ Cl X ( O ∩ S ) , showing that O ∩ S is dense in O . If S is crowded, then O ∩ S ⊆ O ∩ De X S ⊆ De X ( O ∩ S ) , so O ∩ S is crowded. ⊓⊔ We now explain a relationship between d -validity and frame validity. Let F =( W , R ) be a transitive frame. There is an associated Alexandrov topology on W inwhich the open subsets O are those that are up-sets under R , i.e. if w ∈ O and wRv then v ∈ O . Call the resulting topological space W R . Let R ∗ = R ∪ { ( w , w ) : w ∈ W } be the reflexive closure of R . Then R ∗ is a quasi-order , i.e. is reflexive and transitive,and the topology of W R has as a basis the sets R ∗ ( w ) for all w ∈ W , where R ∗ ( w ) = { v : wR ∗ v } = { w } ∪ R ( w ) . A d-morphism from a space X to F is a function f : X → W that has the followingproperties: Robert Goldblatt (i) f is a continuous and open function from X to the space W R .(ii) If w ∈ W is R -reflexive, then the preimage f − { w } is crowded in X .(iii) If w is R -irreflexive, then f − { w } is a discrete subspace of X , i.e. each pointof f − { w } is isolated in f − { w } , or equivalently f − { w } ∩ De X f − { w } = /0.In (i), f is continuous when the f -preimage of any open subset of W R is open in X , while f is open when the f -image of any open subset of X is open in W R . Theimportance of this kind of morphism is that a surjective d -morphism preserves d -validity as frame validity, in the following sense. Theorem 3 (Bezhanishvili et al. 2005, Cor. 2.9).
If there exists a d-morphism fromX onto F , then for any formula ϕ , X | = d ϕ implies F | = ϕ . ⊓⊔ The interpretation of ♦ by De is sometimes called d-semantics (Bezhanishvili et al.,2005). It has M d ( ♦ ∗ ϕ ) = Cl X ( M d ( ϕ )) and M d ( (cid:3) ∗ ϕ ) = Int X ( M d ( ϕ )) , because Cl X Y = Y ∪ De X Y . By contrast, C-semantics interprets ♦ as Cl , defining truth sets M C ( ϕ ) inductively in a topological model M by putting M C ( ♦ ϕ ) = Cl X ( M C ( ϕ )) and M C ( (cid:3) ϕ ) = Int X ( M C ( ϕ )) . A formula ϕ is C-valid in X , written X | = C ϕ , iff M C ( ϕ ) = X for all models M on X . In C -semantics there is no distinction in inter-pretation between ♦ and ♦ ∗ , or between (cid:3) and (cid:3) ∗ .A space is scattered if each of its non-empty subspaces has an isolated point,i.e. no non-empty subset is crowded. This condition d -validates the L¨ob axiom, andhence d -validates the logic K4 C , since it is equal to the G¨odel-L¨ob logic. In factK4 C is characterised by d -validity in all scattered spaces, a result due to Esakia(1981). It can be readily explained via the relational semantics. If ϕ is a non-theoremof K4 C , then ϕ fails to be valid in some frame ( W , R ) with W finite and R irreflexiveand transitive. In such a frame, R − Y is the derived set of Y in the Alexandrov space W R , for any Y ⊆ W . This implies that the relational semantics on ( W , R ) agrees withthe d -semantics on W R , in the sense that a formula is true at a point w in a relationalmodel ( W , R , V ) iff it is d -true at w in the topological model ( W R , V ) . Hence ϕ is not d -valid in the space W R . But W R is scattered, since for any non-empty Y ⊆ W thereis an R -maximal element w ∈ Y , i.e. wRv implies v / ∈ Y , hence R ∗ ( w ) is an openneighbourhood of w in W R that contains no member of Y other than w , making w isolated in Y .From now on we focus on the logics K4 C n with n ≥ A partition of a space X is, as usual, a collection of non-empty subsets of X , calledthe cells , that are pairwise disjoint and whose union is X . It is a k-partition , where k is a positive integer, if it has exactly k cells. A dense partition is one for which eachcell is dense in X . A crowded partition is one whose cells are crowded.For k ≥
2, a space is called k-resolvable if it has k pairwise disjoint non-emptydense subsets. Since any superset of a dense set is dense, k -resolvability is equivalent odal Logics of Some Hereditarily Irresolvable Spaces 9 to X having a dense k -partition. X is k-irresolvable if it is not k -resolvable. It is hereditarily k-irresolvable , which may be abbreviated to k -HI, if every non-emptysubspace of X is k -irresolvable. Note that if k ≤ n , then an n -resolvable space is also k -resolvable, since we can amalgamate cells of a dense partition to form new densepartitions with fewer cells. Hence if X is k -HI, then it is also n -HI. k -resolvability was defined by Ceder (1964) with the extra requirement that eachcell of a dense k -partition should intersect each non-empty open set in at least k points. That requirement was dropped by later authors, including El’kin (1969a)and Eckertson (1997), with the latter defining the k -HI notion.The prefix k - is usually omitted when k =
2. Thus a space is resolvable if it hasa disjoint pair of non-empty dense subsets, and is hereditarily irresolvable , or HI, ifit has no non-empty subspace that is resolvable.It is known that any HI space is T D . For convenience we repeat an explanationof this from (Goldblatt, 2019). In general Cl X { x } = { x } ∪ De X { x } with x / ∈ De X { x } and { x } dense in Cl X { x } , while Cl X De X { x } ⊆ Cl X { x } . But if X is HI, then Cl X { x } is irresolvable, so De X { x } cannot be dense in Cl X { x } , hence Cl X De X { x } can onlybe De X { x } , i.e. De X { x } is closed.On the other hand, a k -HI space need not be T D when k >
2. For instance, wesaw that a two-element indiscrete space is not T D , but since it has no 3-partition itis k -HI for every k > Y has an isolated point which will belong to one cell of any2-partition of Y and prevent the other cell from being dense, hence prevent Y frombeing resolvable. It follows that every scattered space is T D . This is a topologicalmanifestation of the celebrated proof-theoretic fact that the transitivity axiom 4 isderivable from L¨ob’s axiom over K (see Boolos 1993, p. 11).In (Goldblatt, 2019), the following results were proved for all n ≥
1, where S4 C n is the smallest normal extension of K4 C n that includes the scheme (cid:3) ϕ → ϕ , orequivalently ϕ → ♦ ϕ .1. A space X has X | = d C n iff X | = C C n iff X is hereditarily n + ( W , R ) is a finite quasi-order, then it has circumference at most n iff the space W R is hereditarily n + C n is characterised by C -validity in all hereditarily n + C n iff it is C -valid in all hereditarily n + C n is characterised by C -validity in all finite hereditarily n + C n is not characterised by d -validity in any class of finite spaces.The reason for the last result is that every finite space that d -validates K4 is scatteredand so d -validates the L¨ob axiom, which is not a theorem of K4 C n when n ≥ C n is sound for d -validityin all hereditarily n + D spaces. The main result of this paper is that,conversely, K4 C n is complete for d -validity in all hereditarily n + D spaces (albeit not for d -validity in all the finite ones). To indicate how this will beproved, note that by Theorem 1 we have that K4 C n is complete for d -validity in all finite frames that validate K4 C n . So to show that K4 C n is complete for d -validityin some class of spaces, it suffices by Theorem 3 to show that every finite K4 C n -frame is a d -morphic image of some space in that class. For n = C (in theform K4Grz (cid:3) ) is the d -logic of HI-spaces by showing that any finite K4 C -frame isa d -morphic image of an HI space. A finite K4 C -frame has only singleton clusters,and each non-degenerate one was replaced by an HI space to construct the desiredHI preimage. We will now generalise this construction to make it work for all n > F = ( W , R ) is finite and transitive. Let C be the set of R -clusters of F . We define a collection { X C : C ∈ C } of spaces, with each X C having a partition { X w : w ∈ C } indexed by C . If C = { w } is a degenerate cluster, put X C = X w = { w } as a one-point space. For C non-degenerate, with C = { w , . . . , w k } for some positiveinteger k , take X C to be a copy of a space that has a crowded dense k -partition. Labelthe cells of that partition X w , . . . , X w k . We take X C to be disjoint from X C ′ whenever C = C ′ (replacing spaces by homeomorphic copies where necessary to achieve this).That completes the definition of the X C ’s and the X w ’s.Now let X F = S { X C : C ∈ C } , and define a surjective map f : X F → W byputting f ( x ) = w iff x ∈ X w . This entails that f − { w } = X w and f − C = X C .For C , C ′ ∈ C , write CR ↑ C ′ if CRC ′ but not C ′ RC , i.e. C ′ is a strict R -successor of C . Define a subset O ⊆ X F to be open iff for all C ∈ C , O ∩ X C is open in X C andif O ∩ X C = /0, then for all C ′ such that CR ↑ C ′ , X C ′ ⊆ O .It is readily checked that these open sets form a topology on X F . If B is an opensubset of X C , then O B = B ∪ S { X C ′ : CR ↑ C ′ } is an open subset of X F (this usestransitivity of R ↑ ), with O B ∩ X C = B . It follows that X C is a subspace of X F , i.e. theoriginal topology of X C is identical to the subspace topology on the underlying setof X C inherited from the topology of X F . Lemma 3. f is a d-morphism from X F onto F .Proof. To show f is continuous it is enough to show that the preimage f − R ∗ { w } of any basic open subset of W R ∗ is open in X F . If C is the R -cluster of w , then R ∗ { w } = C ∪ S { C ′ : CR ↑ C ′ } , so f − R ∗ { w } = f − C ∪ S { f − C ′ : CR ↑ C ′ } = X C ∪ S { X C ′ : CR ↑ C ′ } , which is indeed open in X F .To show that f is an open map, we must show that if O is an open subset of X F ,then f ( O ) is open in W R ∗ , i.e. is an R -up-set. So, suppose w ∈ f ( O ) and wRv . Wewant v ∈ f ( O ) . Let C be the cluster of w . We have w = f ( x ) for some x ∈ O ∩ X C . If v ∈ C , then C is non-degenerate and X v is dense in X C , so as O ∩ X C is open in X C ,there is some y ∈ X v ∩ O . Then v = f ( y ) ∈ f ( O ) . If however v / ∈ C , then the cluster C ′ of v has CR ↑ C ′ , hence X C ′ ⊆ O . Taking any y ∈ X v ⊆ X C ′ gives v = f ( y ) ∈ f ( O ) again. That completes the proof that f ( O ) is an R -up-set. odal Logics of Some Hereditarily Irresolvable Spaces 11 If w ∈ C is reflexive, then f − { w } = X w is crowded in X C , i.e. f − { w } ⊆ De X C f − { w } . But De X C f − { w } ⊆ De X F f − { w } , since X C is a subspace of X F ,so f − { w } = X w is crowded in X F .Finally, if w is irreflexive, then f − { w } = { w } is discrete in X F . ⊓⊔ Lemma 4.
If X C is T D for all C ∈ C , then X F is T D .Proof. By Lemma 1, a space X is T D iff it has x / ∈ Cl X De X { x } in general. If x ∈ X F ,then x ∈ X C for some C . If X C is T D , then there is an open neighbourhood O of x in X C that is disjoint from De X C { x } . As De X C { x } = X C ∩ De X F { x } , O is disjoint from De X F { x } . Let O ′ = S { X C ′ : CR ↑ C ′ } . Then O ′ is X F -open with x / ∈ O ′ , so no pointof O ′ is a limit point of { x } in X F . Hence O ∪ O ′ is an X F -open neighbourhood of x that is disjoint from De X F { x } , showing that x / ∈ Cl X F De X F { x } . ⊓⊔ Note that this result need not hold with T in place of T D . X F need not be T even when every X C is. For if CR ↑ C ′ with x ∈ X C and y ∈ X C ′ , then every openneighbourhood of x in X F contains y , so x ∈ Cl X F { y } − { y } , showing that { y } is notclosed. Lemma 5.
If X C is n-HI for all C ∈ C , then X F is n-HI.Proof. If X F is not n -HI, then it has some non-empty subspace Y that has n pairwisedisjoint subsets S , . . . , S n that are each dense in Y , i.e. Y ⊆ Cl X F S i . Since C is finiteand R ↑ is antisymmetric, there must be a C ∈ C such that X C intersects Y and C is R ↑ -maximal with this property. Thus X C ∩ Y = /0 but if CR ↑ C ′ then X C ′ ∩ Y = /0. Thenputting O = X C ∪ S { X C ′ : CR ↑ C ′ } gives O ∩ Y = X C ∩ Y = /0.Now O is X F -open, so O ∩ Y is a non-empty Y -open set, hence it intersects thesets S i as they are dense in Y . Thus the sets { O ∩ S i : 1 ≤ i ≤ n } are pairwise disjointand non-empty. They are also dense in X C ∩ Y , as X C ∩ Y = O ∩ Y ⊆ O ∩ Cl X F S i ⊆ Cl X F ( O ∩ S i ) , with the last inclusion holding because O is X F -open. This shows that X C ∩ Y is an n -resolvable subspace of X C , proving that X C is not n -HI. ⊓⊔ To prove that K4 C n is characterised by d -validity in n + D spaces, we wantto show that such spaces provide d -morphic preimages of all finite transitive framesof circumference at most n . To achieve this, the work so far indicates that we needto replace non-degenerate clusters by n + D spaces that have crowded dense k -partitions for various k ≤ n . So we need to show such spaces exist.The literature contains several constructions of n -resolvable spaces that are not n + . The most convenient construction for our purposes is givenby El’kin (1969b). To describe it, first define E to be a space, based on the set ω of van Douwen’s terminology is different. He calls a space n-irresolvable if it has a dense partitionof size n , but none larger.2 Robert Goldblatt natural numbers, for which the set of open sets is U ∪ { /0 } where U is some non-principal ultrafilter on ω . This makes E a door space: every subset is either open orclosed. It is crowded, as no singleton belongs to U , and is T as every co-singleton ω − { x } does belong to U . E has the special property that the intersection of anytwo non-empty E-open sets is non-empty (infinite actually). This implies that anynon-empty open set is dense in E .The closure properties of an ultrafilter also ensure that E is HI. For if a non-emptysubspace Y of E has a 2-partition, then either Y is open and so at least one cell ofthe partition is open, which prevents the other cell from being dense in Y ; or else Y is closed and so both cells are closed and hence neither is dense.Now view ω × { , . . ., n } as the union of its disjoint subsets ω × { i } for 1 ≤ i ≤ n .Let X n be the space based on ω × { , . . . , n } whose non-empty open sets are all thesets of the form S i ≤ n ( O i × { i } ) where each O i is a non-empty open subset of E . Thisdefinition does satisfy the axioms of a topology because of the special property of E noted above. Put S i = ω × { i } . Then { S i : 1 ≤ i ≤ n } is an n -partition of X n that isdense because every non-empty X n -open set intersects every S i , so the cells are alldense in X n . Hence X n is n -resolvable.The intersection of any non-empty X n -open set with S i is of the form O i × { i } with O i open in E . It follows that S i as a subspace of X n is a homeomorphic copyof E , so inherits the topological properties of E , including being a door space thatis HI and having all its non-empty open subsets be dense. It also follows that thenon-empty open sets of X n are all the sets of the form S i ≤ n O ′ i where each O ′ i is anon-empty open subset of S i . S i inherits from E the property that its non-empty open sets are infinite. Thisimplies that each S i is crowded in X n , as is X n itself. X n is also T , since for any point ( x , i ) ∈ X n the set X n − { ( x , i ) } = S ∪ · · · ∪ [( ω − { x } ) × { i } ] ∪ · · ·∪ S n is open in X n , so { ( x , i ) } is closed. X n is not n + n + n -partition with each cell having the property that eachof its crowded subspaces is irresolvable. Illanes (1996, Lemma 2) proves n + n -partition whose cells are openly irresolv-able (OI), meaning that every non-empty open subspace is irresolvable. The mostgeneral result of this type would appear to be that of Eckertson (1997, Lemma3.2(a)), proving n + n sub-spaces that are each openly irresolvable. But it is instructive and more direct here togive a proof for X n that uses its particular structure. Lemma 6.
If A is a dense subset of X n , then there exists an i ≤ n such that A ∩ S i isnon-empty and open in S i .Proof. Let A be dense. Suppose that the conclusion does not hold. Then for each i ≤ n , if A ∩ S i is non-empty then it is not open in S i , so is not equal to S i . Hence odal Logics of Some Hereditarily Irresolvable Spaces 13 its complement S i − ( A ∩ S i ) is non-empty, and open in S i as S i is a door space. If A ∩ S i = /0, then S i − ( A ∩ S i ) is again non-empty and open in S i . Therefore the union S i ≤ n [ S i − ( A ∩ S i )] is, by definition, a non-empty open subset of X n . But this unionis X n − A , so that contradicts the fact that A is dense. ⊓⊔ Now if X n were n + n + A , . . . , A n + thatare pairwise disjoint and dense. Then by the lemma just proved, for each j ≤ n + i ≤ n such that A j ∩ S i is non-empty and open in S i , henceis dense in S i as explained above. Hence by the pigeonhole principle there must be distinct j , k ≤ n + i ≤ n with both subsets A j ∩ S i and A k ∩ S i dense in S i . But these subsets are disjoint, so that contradicts the irresolvability of S i . Therefore X n cannot be n + Theorem 4.
For any n ≥ there exists a non-empty crowded hereditarily n + -irresolvable T space Y n that has a crowded dense n-partition.Proof. For any k >
1, every k -irresolvable space has a non-empty open subspace thatis k -HI, constructed as the complement of the union of all k -resolvable subspaces(Eckertson, 1997, Prop. 2.1). So we apply this with k = n + n + X n just described to conclude that X n has a non-empty open subspace Y n thatis n + Y n inherits the T condition from X n and, since Y n is open, it inheritsthe crowded condition from X n , and it intersects each of the dense sets S i . Also eachintersection S ′ i = Y n ∩ S i is crowded and dense in Y n , as S i is crowded and dense in X n and Y n is open. Thus { S ′ i : 1 ≤ i ≤ n } is a crowded dense n -partition of Y n . ⊓⊔ Theorem 5.
For any n ≥ , every finite K4 C n frame is a d-morphic image of anhereditarily n + -irresolvable T D space.Proof. Let F be a finite K4 C n frame. F is transitive with circumference at most n .We carry out the construction of the space X F as above.For each non-degenerate cluster C of F , if C has k ≥ X C tobe a copy of the T space Y k of Theorem 4, and let { X w : w ∈ C } to be the crowdeddense k -partition of Y k provided by that theorem. Now k ≤ n and Y k is k + n + n + X C of X F is n + X F is n + X C of X F , including the singleton ones, is T , hence is T D . So X F is T D by Lemma 4.The d -morphism from X F onto F is provided by Lemma 3. ⊓⊔ In the case n = F are singletons, and X F isobtained by replacing each non-degenerate cluster by a copy of the El’kin space E .This is exactly the construction of (Gabelaia, 2004) and (Bezhanishvili et al., 2010). Theorem 6.
For any n ≥ , the logic K4 C n is characterised by d-validity in allspaces that are hereditarily n + -irresolvable and T D . Proof.
If a space X is n + T D , then the d -logic of X includes the schemes 4and C n , so it includes K4 C n as the smallest normal logic to include these schemes.Hence every theorem of K4 C n is d -valid in X .For the converse direction, if a formula ϕ is not a theorem of K4 C n , then byTheorem 1(3) there is a finite frame F that validates K4 C n but does not validate ϕ . By Theorem 5 F is a d -morphic image of some n + D space X . Since F = ϕ , Theorem 3 then gives X = d ϕ . Thus it is not the case that ϕ is d -valid inall n + T D spaces. ⊓⊔ As already noted, the case n = D condition is redundant, as hereditarily irresolvable spaces arealways T D . C n The D-axiom ♦ ⊤ is d -valid in a space X iff X = De X X , i.e. iff X is crowded. Ingeneral a space of the type X F need not be crowded, for if C is a degenerate finalcluster of F , then X C is an open singleton containing an isolated point of X F . Butwe have shown in (Goldblatt, 2019, §
7) that K4D C n is characterised by validity inall finite transitive frames that have circumference at most n and all final clusters non-degenerate . If F is such a frame, and C ′ is any final cluster of F , then X C ′ is acrowded space of the type given by Theorem 4. Now any open neighbourhood of apoint x in X F includes an open set of the form O B = B ∪ S { X C ′ : CR ↑ C ′ } , where B isan open neighbourhood of x in some subspace X C . If C is final, then O B = B and X C is crowded, so O B contains points other than x . If C is not final then there is a final C ′ with CR ↑ C ′ , so O B includes X C ′ , which consists of points distinct from x . Thus x is not isolated, showing that X F is crowded. This leads us to conclude Theorem 7.
For n ≥ , K4D C n is characterised by d-validity in all crowded T D spaces that are hereditarily n + -irresolvable. ⊓⊔ At the opposite extreme are logics containing the constant formulaE : (cid:3) ⊥ ∨ ♦ (cid:3) ⊥ . This is d -valid in a space iff it is densely discrete , meaning that the set of isolatedpoints is dense in the space. The set of isolated points is X − De X X , so X is denselydiscrete iff Cl X ( X − De X X ) = X , i.e. ( X − De X X ) ∪ De X ( X − De X X ) = X . This equation expresses the d -validity of ¬ ♦ ⊤ ∨ ♦ ¬ ♦ ⊤ , which is equivalent to E(see Gabelaia 2004, proof of Theorem 4.28). This property was previously called weakly scattered . The change of terminology was made, withexplanation, in (Bezhanishvili et al., 2020, Definition 2.1).odal Logics of Some Hereditarily Irresolvable Spaces 15
K4E C n was shown in (Goldblatt, 2019, §
7) to be characterised by validity inall finite transitive frames that have circumference at most n and all final clusters degenerate . If F is such a frame, and C ′ is any final cluster of F , then X C ′ is anopen singleton, as noted above. If a point x of X F belongs to X C where C is notfinal in F , then there is a final C ′ with CR ↑ C ′ , so any open neighbourhood of x willinclude X C ′ and hence contain an isolated point. This shows that the isolated pointsare dense in X F , and leads to Theorem 8. for n ≥ , K4E C n is characterised by d-validity in all densely discreteT D spaces that are hereditarily n + -irresolvable. ⊓⊔ In Theorem 5 we could have replaced every non-degenerate cluster C by a copyof the same space Y n , since its crowded dense n -partition can be converted into acrowded dense k -partition for any k < n by amalgamating cells. But allowing X C to vary with the size of C gives more flexibility in defining spaces. This is wellillustrated in the case of logics that include the well-studied McKinsey axiom M,often stated as (cid:3) ♦ ϕ → ♦ (cid:3) ϕ . We use the equivalent forms ♦ ( (cid:3) ϕ ∨ (cid:3) ¬ ϕ ) and ♦ (cid:3) ϕ ∨ ♦ (cid:3) ¬ ϕ .It follows from (Bezhanishvili et al., 2003, Prop. 2.1) that in C -semantics, Mdefines the class of openly irresolvable (OI) spaces (recall that these are the spaces inwhich every non-empty open subspace is irresolvable). Equivalently, in d -semantics,the scheme ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) defines the class of OI spaces. We give a direct proofof this. Lemma 7.
A space X is openly irresolvable iff X | = d ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) for all ϕ .Proof. Suppose X = d ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) for some ϕ . Then there is a model M on X and a point of X that is not a closure point of M d (( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) , and so hasan open neighbourhood U disjoint from this d -truth set. Let A = M d ( ϕ ) . Then U is disjoint from Int X A and from Int X ( X − A ) , hence U is included in Cl X A and in Cl X ( X − A ) . Thus U ∩ A and U ∩ ( X − A ) are dense subsets of the non-empty open U , showing that U is resolvable, and so X is not OI.Conversely, assume X is not OI, so has a non-empty open subset U which has asubset A such that A and U − A are dense in U . Hence U is included in Cl X A andin Cl X ( U − A ) ⊆ Cl X ( X − A ) , so is disjoint from Int X A and from Int X ( X − A ) . Takea model M on X with A = M d ( p ) for some variable p . Then U is disjoint from M d (( (cid:3) ∗ p ∨ (cid:3) ∗ ¬ p ) , so ♦ ∗ ( (cid:3) ∗ p ∨ (cid:3) ∗ ¬ p ) is d -false in M at any member of U ,hence is not d -valid in X . ⊓⊔ We now explore criteria for the d -validity of M itself. Theorem 9.
If X is crowded and OI, then X | = d M .Proof. Let X be crowded and OI. Take any model M on X , any formula ϕ , and let S = M d ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) . Then S is open, as the union of two interiors, so as X iscrowded, it follows that S is crowded, hence Cl X S = De X S . Using this, and the factthat M d ( (cid:3) ∗ ψ ) ⊆ M d ( (cid:3) ψ ) for any ψ , we deduce that M d ( ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ )) = M d ( ♦ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ )) ⊆ M d ( ♦ ( (cid:3) ϕ ∨ (cid:3) ¬ ϕ )) . As X is OI, ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) is d -true in M by Lemma 7. Therefore by the aboveinclusion, ♦ ( (cid:3) ϕ ∨ (cid:3) ¬ ϕ ) is d -true in M .This shows that scheme M is d -valid in X . ⊓⊔ The converse of this result does not hold. For instance, a two-element indiscretespace X = { x , y } is resolvable, as { x } and { y } are dense, so X is not OI, but it d -validates M. The latter is so because the operation De X interchanges { x } and { y } and leaves X and /0 fixed, from which it follows that ♦ ϕ ∧ ♦ ¬ ϕ is d -false, hence (cid:3) ϕ ∨ (cid:3) ¬ ϕ is d -true, at both points in any model on X . So X d -validates M.What does hold is that in d -semantics, M defines the class of crowded openlyirresolvable spaces within the class of T D spaces . Lemma 8.
Let X be crowded and T D . Any open neighbourhood O of a point x in X includes an open neighbourhood O ′ of x such that O ′ − { x } is non-empty and open. If O is an open set in X, then O ⊆ Cl X S implies O ⊆ De X S, for any S ⊆ X. Int X Cl X S = Int X De X S for any S ⊆ X. For any model M on X and formula ϕ , M d ( ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ )) = M d ( ♦ ( (cid:3) ϕ ∨ (cid:3) ¬ ϕ )) . Proof.
1. Let x ∈ O with O open. Put O ′ = O − De X { x } . Since x / ∈ De X { x } , theset O ′ contains x , and is open because De X { x } is closed by the T D condition. Then O ′ − { x } is non-empty, since x is not isolated as X is crowded. Also O ′ − { x } = O − ( De X { x } ∪ { x } ) = O − Cl X { x } which is open.2. Let O ⊆ Cl X S and x ∈ O . If U is any open neighbourhood of x , then so is O ∩ U , hence by part 1 there is a non-empty open set O ⊆ O ∩ U with x / ∈ O . Now O ∩ S is dense in O , as O = O ∩ Cl X S ⊆ Cl X ( O ∩ S ) . Therefore as O is open in O ,it intersects O ∩ S . As O ⊆ U − { x } , we get that U − { x } intersects S . This provesthat x is a limit point of S , as required.3. Putting O = Int X Cl X S in 2, we get that Int X Cl X S is an open subset of De X S ,hence is a subset of Int X De X S . Conversely, Int X De X S ⊆ Int X Cl X S as De X S ⊆ Cl X S .4. It was shown in the proof of Theorem 9, just using the fact that X is crowded,that the left truth set is included in the right one. For the reverse inclusion, working inthe model M , suppose ♦ ( (cid:3) ϕ ∨ (cid:3) ¬ ϕ ) is true at some point x . Then so is ♦ (cid:3) ϕ ∨ ♦ (cid:3) ¬ ϕ , hence so is one of ♦ (cid:3) ϕ and ♦ (cid:3) ¬ ϕ . If ♦ (cid:3) ϕ is true at x , then so is ♦ ∗ (cid:3) ϕ , hence (cid:3) ∗ ♦ ¬ ϕ is false at x . By part 3, M d ( (cid:3) ∗ ♦ ¬ ϕ ) = M d ( (cid:3) ∗ ♦ ∗ ¬ ϕ ) ,so then (cid:3) ∗ ♦ ∗ ¬ ϕ is false at x , hence ♦ ∗ (cid:3) ∗ ϕ is true at x .Similarly, if ♦ (cid:3) ¬ ϕ is true at x , then so is ♦ ∗ (cid:3) ∗ ¬ ϕ . Since ♦ (cid:3) ϕ ∨ ♦ (cid:3) ¬ ϕ is true at x , so then is ♦ ∗ (cid:3) ∗ ϕ ∨ ♦ ∗ (cid:3) ∗ ¬ ϕ , hence so is ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) , asrequired to prove the inclusion from right to left. ⊓⊔ Theorem 10.
If X is a T D space and X | = d M , then X is crowded and openly irre-solvable. odal Logics of Some Hereditarily Irresolvable Spaces 17 Proof.
Let X be T D space and X | = d M. Then
X d -validates ♦ ( (cid:3) ⊤ ∨ (cid:3) ¬⊤ ) . Butthis formula d -defines De X X in any model on X , so De X X = X , i.e. X is crowded.We now have that X is crowded and T D , and any formula of the form ♦ ( (cid:3) ϕ ∨ (cid:3) ¬ ϕ ) is d -valid on X , i.e. d -true in all models on X . But then by Lemma 8.4, ♦ ∗ ( (cid:3) ∗ ϕ ∨ (cid:3) ∗ ¬ ϕ ) is d -valid in X . Hence X is OI by Lemma 7. ⊓⊔ Theorems 9 and 10 combine to give
Corollary 1.
If X is T D , then X | = d M iff X is crowded and openly irresolvable.Hence if X is T D and crowded, then X | = d M iff X is openly irresolvable iff X | = C M . ⊓⊔ K4M C n was shown in (Goldblatt, 2019, §
7) to be characterised by validity inall finite transitive frames that have circumference at most n and all final clusterssimple. If F is such a frame, X F is n + D , as shown in the proof ofTheorem 5. All final clusters of F are non-degenerate, which is enough to ensurethat X F is a crowded space, as explained in our discussion of K4D C n . Lemma 9. X F is openly irresolvable.Proof. Let O be any non-empty open subset of X F . Then O ∩ X C = /0 for some clus-ter C of F . Put B = O ∩ X C . Then O B = B ∪ S { X C ′ : CR ↑ C ′ } is a non-empty subsetof O that is open in X F . If C is final, then since it is a non-degenerate singleton, X C is a copy of the El’kin space E , and also O B = B ⊆ X C . If however C is not final,there is a final C ′ with CR ↑ C ′ . Then X C ′ ⊆ O B and X C ′ is open in X F .So in any case we see that O has a non-empty open subset O ′ (either O B or X C ′ )that is included in a subspace X ′ (either X C or X C ′ ) that is a copy of E and henceis HI. Hence O ′ is irresolvable. Now if O had a pair of disjoint dense subsets, thenthese subsets would intersect the open O ′ in a pair of disjoint dense subsets of O ′ ,contradicting irresolvability of O ′ . Therefore O is irresolvable as required. ⊓⊔ Altogether we have now shown that X F is T D , crowded, OI, and n + d -validates K4M C n . We conclude Theorem 11.
For n ≥ , K4M C n is characterised by d-validity in all T D spaces thatare crowded, openly irresolvable, and hereditarily n + -irresolvable. ⊓⊔ When n =
1, this can be simplified, since an HI space is always OI and T D .Thus the logic K4M C is characterised by d -validity in the class of all crowded HIspaces, as was shown by Gabelaia (2004, Theorem 4.26) with K4M C in the formK4MGrz (cid:3) . But the class of crowded HI spaces characterises K4D C , as shownabove. Therefore K4M C is identical to the ostensibly weaker K4D C . This canalso be seen quite simply from our relational completeness result for K4D C . In afinite K4D C -frame, any final cluster is a singleton by validity of C and is non-degenerate by validity of D, so all final clusters are simple, making the frame vali-date M. Thus M is a theorem of K4D C .We can also deal with the logic K4M, which is characterised by finite transitiveframes in which all final clusters are simple (Chagrov and Zakharyaschev, 1997, § X F can be constructed without assuming that F has any boundon its circumference. If F validates K4M, then X F will be T D , crowded and OI, sowill d -validate K4M. From this we can conclude Theorem 12.
K4M is characterised by d-validity in all T D spaces that are crowdedand openly irresolvable. ⊓⊔ References
Gerard Allwein and J. Michael Dunn. Kripke models for linear logic.
The Journalof Symbolic Logic , 58(2):514–545, 1993.Gerard Allwein and Chrysafis Hartonas. Duality for bounded lattices. Indiana Uni-versity Logic Group, Preprint Series, IULG–93–25, 1993.C. E. Aull and W. J. Thron. Separation axioms between T and T . IndagationesMathematicae (Proceedings) , 65:26–37, 1962.G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan, and J. van Mill. Tree-like con-structions in topology and modal logic.
Archive for Mathematical Logic , 2020. https://doi.org/10.1007/s00153-020-00743-6 .Guram Bezhanishvili, Ray Mines, and Patrick J. Morandi. Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces.
Topology and its Applications ,132:291–306, 2003.Guram Bezhanishvili, Leo Esakia, and David Gabelaia. Some results on modalaxiomatization and definability for topological spaces.
Studia Logica , 81:325–355, 2005.Guram Bezhanishvili, Leo Esakia, and David Gabelaia. K4.Grz and hereditarilyirresolvable spaces. In Solomon Feferman and Wilfried Sieg, editors,
Proofs,Categories and Computations.Essays in Honor of Grigori Mints , pages 61–69.College Publications, 2010.Patrick Blackburn, Maarten de Rijke, and Yde Venema.
Modal Logic . CambridgeUniversity Press, 2001.George Boolos.
The Logic of Provability . Cambridge University Press, 1993.J. G. Ceder. On maximally resolvable spaces.
Fundamenta Mathematicae , 55:87–93, 1964.Alexander Chagrov and Michael Zakharyaschev.
Modal Logic . Oxford UniversityPress, 1997.Andrew Craig and Miroslav Haviar. Reconciliation of approaches to the construc-tion of canonical extensions of bounded lattices.
Mathematica Slovaca , 64(6):1335–1356, 2014.Wojciech Dzik, Ewa Orlowska, and Clint J. van Alten. Relational representationtheorems for lattices with negations: A survey. In Harrie C. M. de Swart, Ewa Or-lowska, Gunther Schmidt, and Marc Roubens, editors,
Theory and Applicationsof Relational Structures as Knowledge Instruments II , volume 4342 of
LectureNotes in Computer Science , pages 245–266. Springer, 2006. odal Logics of Some Hereditarily Irresolvable Spaces 19
Frederick W. Eckertson. Resolvable, not maximally resolvable spaces.
Topologyand its Applications , 79:1–11, 1997.A. G. El’kin. Resolvable spaces which are not maximally resolvable.
MoscowUniversity Mathematics Bulletin , 24:116–118, 1969a.A. G. El’kin. Ultrafilters and undecomposable spaces.
Moscow University Mathe-matics Bulletin , 24:37–40, 1969b.R. Engelking.
General Topology . PWN — Polish Scientific Publishers, Warsaw,1977.L. L. Esakia. Diagonal constructions, L¨obs formula and Cantors scattered spaces.In
Studies in Logic and Semantics , pages 128–143. Metsniereba, Tbilisi, 1981. InRussian.L. L. Esakia. Weak transitivity—a restitution.
Logical Investigations , 8:244–255,2001. In Russian.Leo Esakia. Intuitionistic logic and modality via topology.
Annals of Pure andApplied Logic , 127:155–170, 2004.Kit Fine. Logics containing K4. Part II.
The Journal of Symbolic Logic , 50(3):619–651, 1985.David Gabelaia.
Topological, Algebraic and Spatio-Temporal Semantics for Multi-Dimensional Modal Logics . PhD thesis, King’s College London, 2004.Mai Gehrke. Canonical extensions: an algebraic approach toStone duality.
Algebra Universalis , 79(Article 63), 2018. https://doi.org/10.1007/s00012-018-0544-6 .Mai Gehrke and John Harding. Bounded lattice expansions.
Journal of Algebra ,239:345–371, 2001.Robert Goldblatt. Canonical extensions and ultraproductsof polarities.
Algebra Universalis , 79(Article 80), 2018. https://doi.org/10.1007/s00012-018-0562-4 .Robert Goldblatt. Modal logics that bound the circumference of transitive frames. arXiv:1905.11617 , 2019.Robert Goldblatt. Morphisms and duality for polarities and lattices with operators.
Journal of Applied Logics – IfCoLog Journal of Logics and their Applications , toappear. Also arXiv:1902.09783 , 2020.C. Hartonas and J. M. Dunn. Stone duality for lattices.
Algebra Universalis , 37:391–401, 1997.G. Hartung. An extended duality for lattices. In K. Denecke and H.-J. Vogel, editors,
General Algebra and Applications , pages 126–142. Heldermann-Verlag, Berlin,1993.Gerd Hartung. A topological representation of lattices.
Algebra Universalis , 29:273–299, 1992.Alejandro Illanes. Finite and ω -resolvability. Proceedings of the American Mathe-matical Society , 124(4):1243–1246, 1996.Bjarni J´onsson and Alfred Tarski. Boolean algebras with operators, part I.
AmericanJournal of Mathematics , 73:891–939, 1951.
J. C. C. McKinsey. A solution of the decision problem for the Lewis systems S2 andS4 with an application to topology.
The Journal of Symbolic Logic , 6:117–134,1941.J. C. C. McKinsey and Alfred Tarski. The algebra of topology.
Annals of Mathe-matics , 45:141–191, 1944.J. C. C. McKinsey and Alfred Tarski. Some theorems about the sentential calculi ofLewis and Heyting.
The Journal of Symbolic Logic , 13:1–15, 1948.Miroslav Ploˇsˇcica. A natural representation of bounded lattices.
Tatra MountainsMathematical Publications , 5:75–88, 1995.H. A. Priestley. Representations of distributive lattices by means of ordered Stonespaces.
Bull. London Math. Soc. , 2:186–190, 1970.N. Rescher and A. Urquhart.
Temporal Logic . Springer-Verlag, 1971.Krister Segerberg.
An Essay in Classical Modal Logic , volume 13 of
FilosofiskaStudier . Uppsala Universitet, 1971.M. H. Stone. The theory of representations for Boolean algebras.
Transactions ofthe American Mathematical Society , 40:37–111, 1936.Tsao-Chen Tang. Algebraic postulates and a geometric interpretation for the Lewiscalculus of strict implication.
Bulletin of the American Mathematical Society , 44:737–744, 1938.Alasdair Urquhart. A topological representation theory for lattices.
Algebra Uni-versalis , 8:45–58, 1978.Alasdair Urquhart. Decidability and the finite model property.
Journal of Philo-sophical Logic , 10(3):367–370, 1981.Alasdair Urquhart. Relevant implication and projective geometry.
Logique et Anal-yse , 103–104:345–357, 1983.Alasdair Urquhart. The undecidability of entailment and relevant implication.
TheJournal of Symbolic Logic , 49(4):1059–1073, 1984.Alasdair Urquhart. Failure of interpolation in relevant logics.
Journal of Philosoph-ical Logic , 22(5):449–479, 1993.Alasdair Urquhart. First degree formulas in quantified S5.
The Australasian Journal of Logic , 12(5):204–210, 2015. https://ojs.victoria.ac.nz/ajl/article/view/470 .Alasdair Urquhart. The geometry of relevant implication.
IF-CoLog Journal of Logics and their Applications , 4(3):591–604, 2017. http://collegepublications.co.uk/ifcolog/?00012 .Eric K. van Douwen. Applications of maximal topologies.
Topology and its Appli-cations , 51:125–139, 1993.Stephen Willard.