Topological semantics of conservativity and interpretability logics
TTopological semantics of conservativity andinterpretability logics
Sohei Iwata ∗† and Taishi Kurahashi ‡§ Abstract
We introduce and develop a topological semantics of conservativitylogics and interpretability logics. We prove the topological compactnesstheorem of consistent normal extensions of the conservativity logic CL byextending Shehtman’s ultrabouquet construction method to our frame-work. As a consequence, we prove that several extensions of CL suchas IL , ILM , ILP and
ILW are strongly complete with respect to ourtopological semantics.
The present paper is devoted to solving a natural problem of whether the topo-logical semantics of the propositional modal logic GL can be extended to that ofconservativity logics and interpretability logics, which are extensions of GL . Wenewly introduce a topological semantics of these logics, and investigate severalbasic properties of our semantics such as the topological strong completeness ofthem.The logic GL is known as the logic of provability (cf. Boolos [2]). LetPr PA ( x ) be a natural provability predicate of Peano Arithmetic PA . Then,the logic GL is precisely the set of all PA -verifiable modal formulas underall arithmetical interpretations where the modal operator (cid:3) is interpreted byPr PA ( x ). This is called Solovay’s arithmetical completeness theorem [18]. In hisproof, the completeness theorem of GL with respect to Kripke semantics playsan essential role. Actually, it is well-known that GL is complete with respect tothe class of all transitive and conversely well-founded finite Kripke frames. Onthe other hand, it is also known that GL is not strongly complete with respectto Kripke semantics, that is, there exists a set Γ of modal formulas in which Γis finitely satisfiable in a transitive and conversely well-founded Kripke model,but Γ itself is not satisfiable (See also Boolos [2]). ∗ E-mail: [email protected] † Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan. ‡ Email: [email protected] § Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan. a r X i v : . [ m a t h . L O ] F e b his obstacle can be avoided by dealing with topological semantics of modallogics. Topological semantics of modal logic based on derived sets were initiatedby McKinsey and Tarski [13]. Also topological semantics of GL was foundedby Simmons [17] and Esakia [5], and has been developed by many authors (SeeBeklemishev and Gabelaia [1]). One of important results in this research isthe fact that GL is determined by the class of all scattered topological spaces.Moreover, as opposed to Kripke semantics, Shehtman [15] proved that GL isstrongly complete with respect to scattered spaces by using so-called the methodof ultrabouquet construction.The language of interpretability logics has the additional binary modal oper-ator (cid:66) . The modal formula ϕ (cid:66) ψ is intended to be read as “ T + ψ is interpretablein T + ϕ ”, where T is a suitable theory of arithmetic, such as PA . The logic IL is a basis for the modal logical investigations of the notion of interpretability be-tween theories, and it has been proved that the extensions ILM and
ILP of IL are arithmetically complete. Also it is known that the notion of interpretabilityis closely related to that of partial conservativity. Actually, the logic ILM isexactly the logic of Π -conservativity of theories of arithmetic (See Japaridzeand de Jongh [10] for a detailed extensive survey of these results). From thispoint of view, Ignatiev [8] introduced the sublogic CL of IL as a basis for modallogical study of capturing properties of the notion of partial conservativity.A relational semantics of interpretability logics was introduced by de Jonghand Veltman [3] that is called Veltman semantics . A Veltman frame is a Kripkeframe equipped with a family of binary relations. Then, de Jongh and Velt-man [3] proved that the logics IL , ILM and
ILP are complete with respectto Veltman semantics. Several alternative relational semantics of interpretabil-ity logics are also known, and one of important semantics was introduced byVisser [20] that is called simplified Veltman semantics or Visser semantics . Byconstructing bisimulations between corresponding Visser and Veltman frames,Visser proved that IL , ILM and
ILP are also complete with respect to Vissersemantics. Moreover, Ignatiev [8] proved that the logic CL is complete withrespect to both Veltman and Visser semantics. However, CL and IL can beshown to be strongly incomplete in both Veltman and Visser semantics, as in GL .On the other hand, there is a possibility of finding out the strong complete-ness of these logics with respect to another semantics. Particularly, one withrespect to topological semantics is strongly suggested by Shehtman’s strongcompleteness theorem of GL . From this perspective, in the present paper, wepropose a topological semantics of CL and its extensions, and prove the strongcompleteness theorem of some of these logics by extending Shehtman’s methodof ultrabouquet construction.This paper is organized as follows. We briefly summarize Kripke and topo-logical semantics of GL and Visser semantics of CL and its extensions in thenext section. In Section 3, we introduce a new topological semantics of nor-mal extensions of CL , and investigate some basic properties of our semantics.Our topological semantics is based on bitopological spaces with Visser seman-tics in mind. In Section 4, we extend Shehtman’s ultrabouquet construction2o our framework, and then we prove the topological compactness theorem ofconsistent normal extensions of CL . As a consequence, the topological strongcompleteness theorem of the logics CL , CLM , IL , ILM , ILP and
ILW areobtained. Finally, in Section 5, we discuss topological aspects of the logic IL . The language L ( (cid:3) ) of propositional modal logic consists of countably manypropositional variables p , p , p , . . . , logical constants (cid:62) , ⊥ , logical connectives ¬ , ∧ , ∨ , → and unary modal operators (cid:3) , ♦ . A set L of L ( (cid:3) )-formulas is said tobe a normal modal logic if L contains all tautologies in the language L ( (cid:3) ) and theformula (cid:3) ( p → q ) → ( (cid:3) p → (cid:3) q ), and is closed under Modus Ponens ϕ → ψ ϕψ ,Necessitation ϕ (cid:3) ϕ and Substitution ϕ ( p , . . . , p n ) ϕ ( ψ , . . . , ψ n ) . For any normal modal logic L , any set Γ of L ( (cid:3) )-formulas and any L ( (cid:3) )-formula ϕ , we write Γ (cid:96) L ϕ toindicate that there exists a finite subset Γ of Γ such that (cid:86) Γ → ϕ ∈ L .The logic GL is defined as the smallest normal modal logic containing theadditional axiom (cid:3) ( (cid:3) p → p ) → (cid:3) p .This section consists of three subsections. In the first subsection, we intro-duce Kripke semantics of GL . The second subsection is devoted to introducingtopological semantics of GL , and reviewing some basic results relating to ourstudy. In the last subsection, we introduce the conservativity logic CL and itsextensions, and also introduce their relational semantics, namely, Visser seman-tics. Definition 2.1 (Kripke frames and models) . • A pair (cid:104)
W, R (cid:105) is said to be a
Kripke frame if W is a non-empty set and R is a binary relation on W . • A triple (cid:104)
W, R, (cid:13) (cid:105) is said to be a
Kripke model if (cid:104) W, R (cid:105) is a Kripke frameand (cid:13) is a binary relation between W and the set of all L ( (cid:3) )-formulassatisfying the following conditions:1. x (cid:49) ⊥ and x (cid:13) (cid:62) ;2. x (cid:13) ¬ ϕ ⇐⇒ x (cid:49) ϕ ;3. x (cid:13) ϕ ∧ ψ ⇐⇒ x (cid:13) ϕ and x (cid:13) ψ ;4. x (cid:13) ϕ ∨ ψ ⇐⇒ x (cid:13) ϕ or x (cid:13) ψ ;5. x (cid:13) ϕ → ψ ⇐⇒ x (cid:49) ϕ or x (cid:13) ψ ;6. x (cid:13) (cid:3) ϕ ⇐⇒ ∀ y ∈ W [ xRy ⇒ y (cid:13) ϕ ];7. x (cid:13) ♦ ϕ ⇐⇒ ∃ y ∈ W [ xRy & y (cid:13) ϕ ].3 An L ( (cid:3) )-formula ϕ is said to be valid in (cid:104) W, R (cid:105) if for any Kripke model (cid:104)
W, R, (cid:13) (cid:105) and any x ∈ W , x (cid:13) ϕ . • Let
Log ( W, R ) denote the set of all L ( (cid:3) )-formulas valid in (cid:104) W, R (cid:105) , andthis set is called the logic of (cid:104)
W, R (cid:105) .Notice that every
Log ( W, R ) is a normal modal logic. We say that a bi-nary relation R on a set W is conversely well-founded if there is no infinite R -increasing sequence of elements of W . Then, the validity of the logic GL ina Kripke frame is characterized by a property of the relation R . Fact 2.2 (See Boolos [2, Theorem 10 in Chapter 4]) . For any Kripke frame (cid:104)
W, R (cid:105) , GL ⊆ Log ( W, R ) if and only if R is transitive and conversely well-founded. (cid:113) We introduce the consequence relation | = KL with respect to Kripke semanticswhere K stands for “Kripke”. Definition 2.3.
Let L be a normal modal logic, Γ be a set of L ( (cid:3) )-formulasand ϕ be an L ( (cid:3) )-formula. • Γ | = KL ϕ : ⇐⇒ for any Kripke model (cid:104) W, R, (cid:13) (cid:105) satisfying L ⊆ Log ( W, R )and any x ∈ W , if x (cid:13) ψ for all ψ ∈ Γ, then x (cid:13) ϕ .Clearly, Γ (cid:96) L ϕ implies Γ | = KL ϕ . For GL , the converse implication alsoholds in the case of Γ = ∅ . This is the Kripke completeness theorem of GL . Fact 2.4 (Kripke completeness of GL (Segerberg [14])) . For any L ( (cid:3) ) -formula ϕ , ∅ (cid:96) GL ϕ if and only if ∅ | = K GL ϕ . (cid:113) On the other hand, GL is not strongly complete with respect to Kripkesemantics, that is, the equivalence of Γ (cid:96) GL ϕ and Γ | = K GL ϕ does not hold ingeneral. Fact 2.5 (Fine and Rautenberg (see Boolos [2, pp. 102–103])) . Let ∆ := { ♦ p } ∪ { (cid:3) ( p n → ♦ p n +1 ) | n ∈ N } , then ∆ | = K GL ⊥ but ∆ (cid:48) GL ⊥ . (cid:113) For a non-empty set X and a family τ of its subsets, we say that τ is a topologyon X if they enjoy the following conditions:1. X, ∅ ∈ τ ;2. If U , U ∈ τ , then U ∩ U ∈ τ ;3. For any family { U i } i ∈ I of sets of τ , (cid:83) i ∈ I U i ∈ τ .4hen, the pair (cid:104) X, τ (cid:105) is called a topological space . Every U ∈ τ containing x ∈ X is called a τ -neighborhood of x . Definition 2.6 (Derived sets and co-derived sets) . Let (cid:104)
X, τ (cid:105) be a topologicalspace and Y ⊆ X . • The derived set d τ ( Y ) of Y (with respect to τ ) is the subset of X definedas follows: d τ ( Y ) := { x ∈ X | ∀ U ∈ τ [ x ∈ U ⇒ ∃ y (cid:54) = x ( y ∈ U ∩ Y )] } ; • The co-derived set cd τ ( Y ) of Y (with respect to τ ) is the set d τ ( Y ), where Y is the complement of Y .In topological semantics of modal logic, every topological space plays a roleof a frame, and L ( (cid:3) )-formulas are interpreted as subsets of the topological spaceby valuations. Definition 2.7 (Valuations on topological spaces) . Let (cid:104)
X, τ (cid:105) be a topologicalspace. • A valuation on (cid:104) X, τ (cid:105) is a mapping v : L ( (cid:3) ) → P ( X ) satisfying thefollowing conditions:1. v ( ⊥ ) = ∅ and v ( (cid:62) ) = X ;2. v ( ¬ ϕ ) = v ( ϕ );3. v ( ϕ ∧ ψ ) = v ( ϕ ) ∩ v ( ψ );4. v ( ϕ ∨ ψ ) = v ( ϕ ) ∪ v ( ψ );5. v ( ϕ → ψ ) = v ( ϕ ) ∪ v ( ψ );6. v ( (cid:3) ϕ ) = cd τ ( v ( ϕ ));7. v ( ♦ ϕ ) = d τ ( v ( ϕ )). • We say that an L ( (cid:3) )-formula ϕ is valid in (cid:104) X, τ (cid:105) if v ( ϕ ) = X for allvaluations v on (cid:104) X, τ (cid:105) . • Let
Log ( X, τ ) be the set of all L ( (cid:3) )-formulas valid in (cid:104) X, τ (cid:105) , and we callthis set the logic of (cid:104)
X, τ (cid:105) .It is known that every
Log ( X, τ ) is a normal modal logic validating p ∧ (cid:3) p → (cid:3)(cid:3) p (See Esakia [6] and van Benthem and Bezhanishvili [19]). As well as Fact2.2, the validity of the logic GL in a topological space (cid:104) X, τ (cid:105) is characterizedby a property of τ . Definition 2.8 (Scattered spaces) . A topological space (cid:104)
X, τ (cid:105) is said to be scattered if for any Y ⊆ X , Y (cid:54) = ∅ implies Y \ d τ ( Y ) (cid:54) = ∅ . Fact 2.9 (Simmons [17]; Esakia [5]) . For any topological space (cid:104)
X, τ (cid:105) , GL ⊆ Log ( X, τ ) if and only if (cid:104) X, τ (cid:105) is scattered. (cid:113)
Fact 2.10.
Let (cid:104)
X, τ (cid:105) be a topological space and let
Y, Z ⊆ X .1. d τ ( ∅ ) = ∅ ;2. If Y ⊆ Z , then d τ ( Y ) ⊆ d τ ( Z ) ;3. d τ ( Y ∪ Z ) = d τ ( Y ) ∪ d τ ( Z ) ;4. Y ∈ τ ⇐⇒ d τ ( Y ) ∩ Y = ∅ ;5. If (cid:104) X, τ (cid:105) is scattered, then d τ ( d τ ( Y )) ⊆ d τ ( Y ) . (cid:113) Each transitive and irreflexive Kripke frame can be considered as a topolog-ical space having the same logic via the topology of upward closed subsets.
Definition 2.11.
Let (cid:104)
W, R (cid:105) be a Kripke frame. • For each x ∈ W , R ( x ) := { y ∈ W | xRy } ; • A subset Y ⊆ W is said to be R -upward closed if for any x ∈ Y , R ( x ) ⊆ Y ; • Define τ R := { Y ⊆ W | Y is R -upward closed } . Definition 2.12 (Alexandroff spaces) . A topological space (cid:104)
X, τ (cid:105) is said to be
Alexandroff if for any family { U i } i ∈ I of members of τ , (cid:84) i ∈ I U i ∈ τ . Fact 2.13 (cf. van Benthem and Bezhanishvili [19]) . Let (cid:104)
W, R (cid:105) be a Kripkeframe. Then,1. (cid:104)
W, τ R (cid:105) is an Alexandroff topological space;2. If R is transitive and irreflexive, then for any Y ⊆ W , d τ R ( Y ) = { x ∈ W | R ( x ) ∩ Y (cid:54) = ∅ } ;3. If R is transitive and irreflexive, then Log ( W, R ) =
Log ( W, τ R ) . (cid:113) As in the case of Kripke semantics, we introduce the consequence relation | = TL with respect to topological semantics where T stands for “Topology”. Definition 2.14.
Let L be a normal modal logic, Γ be a set of L ( (cid:3) )-formulasand ϕ be an L ( (cid:3) )-formula. • Γ | = TL ϕ : ⇐⇒ for any topological space (cid:104) X, τ (cid:105) satisfying L ⊆ Log ( X, τ ),any valuation v on X and any x ∈ X , if x ∈ v ( ψ ) for all ψ ∈ Γ, then x ∈ v ( ϕ ).From Facts 2.4 and 2.13, we obtain the topological completeness of GL .6 act 2.15 (Topological completeness of GL ) . For any L ( (cid:3) ) -formula ϕ , ∅ (cid:96) GL ϕ if and only if ∅ | = T GL ϕ . (cid:113) Moreover, as opposed to Fact 2.5, Shehtman proved that GL is stronglycomplete with respect to topological semantics. Fact 2.16 (Topological strong completeness of GL (Shehtman [15, Theorem3.3])) . Let Γ be any set of L ( (cid:3) ) -formulas and ϕ be any L ( (cid:3) ) -formula. Then, Γ (cid:96) GL ϕ if and only if Γ | = T GL ϕ . (cid:113) In this section, we introduce the conservativity logic CL and its extensions.Also we introduce their relational semantics. The language L ( (cid:3) , (cid:66) ) is obtainedfrom L ( (cid:3) ) by adding the binary modal operator (cid:66) . Definition 2.17 (The conservativity logic CL ) . The conservativity logic CL is a logic in the language L ( (cid:3) , (cid:66) ) obtained from GL by adding the followingaxioms: J1 (cid:3) ( p → q ) → ( p (cid:66) q ); J2 ( p (cid:66) q ) ∧ ( q (cid:66) r ) → ( p (cid:66) r ); J3 ( p (cid:66) r ) ∧ ( q (cid:66) r ) → (( p ∨ q ) (cid:66) r ); J4 ( p (cid:66) q ) → ( ♦ p → ♦ q ).We say that a set L of L ( (cid:3) , (cid:66) )-formulas is a normal extension of CL if CL ⊆ L and L is closed under Modus Ponens, Necessitation and Substitution.There are well-known normal extensions of CL having some of the followingadditional axioms: J5 ♦ p (cid:66) p ; M ( p (cid:66) q ) → (( p ∧ (cid:3) r ) (cid:66) ( q ∧ (cid:3) r )); P ( p (cid:66) q ) → (cid:3) ( p (cid:66) q ); W ( p (cid:66) q ) → ( p (cid:66) ( q ∧ (cid:3) ¬ p )).The smallest normal extension containing M is called CLM . In this case,we write
CLM = CL + M . The logics CL and CLM were introduced byIgnatiev [8]. Also let IL = CL + J5 , ILM = IL + M , ILP = IL + P and ILW = IL + W . The logic IL is called the interpretability logic . Actually, Shehtman proved that GL is strongly complete with respect to neighborhoodsemantics. Esakia [5] proved that for GL , neighborhood semantics and topological semanticscoincide, and so we can state Shehtman’s theorem as the topological strong completenesstheorem of GL . CL and its extensions is Velt-man semantics which was introduced by de Jongh and Veltman [3]. A triple (cid:104)
W, R, { S w } w ∈ W (cid:105) is called a Veltman frame if (cid:104) W, R (cid:105) is a transitive and con-versely well-founded Kripke frame and for each w ∈ W , S w is a binary relationon R ( w ) satisfying some additional conditions. One of the purposes of thepresent paper is to find an appropriate topological semantics of extensions of CL . From the point of view of Fact 2.13, every binary relation P on a set W is associated to the topology τ P on W consisting of P -upward closed subsets.However, each binary relation S w of Veltman frames is not a binary relation onfull W , and so Veltman frames are not directly recognized as topological frames.For this reason, we adopt the alternative relational semantics of extensionsof CL introduced by Visser [20]. Definition 2.18 (Visser frames and models) . • A triple (cid:104)
W, R, S (cid:105) is said to be a
Visser frame if (cid:104) W, R (cid:105) is a transitive andconversely well-founded Kripke frame and S is a binary transitive andreflexive relation on W ; • A quadruple (cid:104)
W, R, S, (cid:13) (cid:105) is said to be a
Visser model if (cid:104) W, R, S (cid:105) is a Visserframe and (cid:13) is a binary relation as in Definition 2.1 with the followingadditional clause: – x (cid:13) ϕ (cid:66) ψ ⇐⇒ ∀ y ∈ W [ xRy & y (cid:13) ϕ ⇒ ∃ z ∈ W ( xRz & ySz & z (cid:13) ψ )]. • The validity of an L ( (cid:3) , (cid:66) )-formula in Visser frames and models, and thelogic Log ( W, R, S ) of (cid:104)
W, R, S (cid:105) are defined as in Definition 2.1.Visser actually introduced the notion of Visser frames as a relational se-mantics for extensions of IL , and Definition 2.18 is an adaptation of Visser’sdefinition to our framework obtained by removing the condition R ⊆ S from hisoriginal definition. Visser frames are also known as simplified Veltman frames.Then, the following fact holds. Fact 2.19 (See Ignatiev [8] and Visser [20]) . Let (cid:104)
W, R, S (cid:105) be any Visser frame.Then,1.
Log ( W, R, S ) is a normal extension of CL ;2. If ∀ x, y, z ∈ W [ xSyRz ⇒ xRz ] , then CLM ⊆ Log ( W, R, S ) ;3. If R ⊆ S , then IL ⊆ Log ( W, R, S ) ;4. If R ⊆ S and ∀ x, y, z ∈ W [ xRySz ⇒ xRz ] , then ILP ⊆ Log ( W, R, S ) ;5. If R ⊆ S and the composition R ◦ S is conversely well-founded, then ILW ⊆ Log ( W, R, S ) . (cid:113)
8n Section 5, we will investigate the condition R ⊆ S of Visser frames froma topological viewpoint.We also define the consequence relation | = VL with respect to Visser semantics. Definition 2.20.
Let L be a normal extension of CL , Γ be a set of L ( (cid:3) , (cid:66) )-formulas and ϕ be an L ( (cid:3) , (cid:66) )-formula. • Γ | = VL ϕ : ⇐⇒ for any Visser model (cid:104) W, R, S, (cid:13) (cid:105) satisfying L ⊆ Log ( W, R, S ) and any x ∈ W , if x (cid:13) ψ for all ψ ∈ Γ, then x (cid:13) ϕ .Clearly, Γ (cid:96) L ϕ implies Γ | = VL ϕ . The completeness theorems of CL , CLM , IL , ILP , ILM and
ILW with respect to Visser semantics are proved by Ignatiev,de Jongh and Veltman and Visser.
Fact 2.21 (Visser completeness of CL and CLM (Ignatiev [8])) . Let L ∈{ CL , CLM } . For any L ( (cid:3) , (cid:66) ) -formula ϕ , ∅ (cid:96) L ϕ if and only if ∅ | = VL ϕ . (cid:113) Fact 2.22 (Visser completeness of IL , ILM , ILP and
ILW (de Jongh and Velt-man [3, 4] and Visser [20])) . Let L ∈ { IL , ILM , ILP , ILW } . For any L ( (cid:3) , (cid:66) ) -formula ϕ , ∅ (cid:96) L ϕ if and only if ∅ | = VL ϕ . (cid:113) However, every logic L ∈ { CL , CLM , IL , ILM , ILP , ILW } is strongly in-complete with respect to Visser semantics as in the case of GL . That is, ∆ | = VL ⊥ but ∆ (cid:48) L ⊥ where ∆ is the set of formulas defined in Fact 2.5. In this section, we newly introduce a topological semantics of normal extensionsof CL . Our topological semantics is based on bitopological spaces. Definition 3.1 (Bitopological spaces) . Let X be a non-empty set and τ , τ be families of subsets of X . A triple (cid:104) X, τ , τ (cid:105) is called a bitopological space ifboth τ and τ are topologies on X .The following definition is an essential part of our work. Definition 3.2.
Let (cid:104)
X, τ , τ (cid:105) be a bitopological space. For subsets Y and Z of X , we define a subset e τ ,τ ( Y, Z ) of X as follows: e τ ,τ ( Y, Z ) := { x ∈ X | ∀ U ∈ τ [ x ∈ d τ ( Y ∩ U ) ⇒ x ∈ d τ ( Z ∩ U )] } . If there is no room for confusion, we simply write e ( Y, Z ) instead of e τ ,τ ( Y, Z ). Using our sets e τ ,τ ( Y, Z ), we define valuations on bitopologi-cal spaces.
Definition 3.3.
Let (cid:104)
X, τ , τ (cid:105) be a bitopological space. A valuation on (cid:104) X, τ , τ (cid:105) is a mapping v : L ( (cid:3) , (cid:66) ) → P ( X ) defined as in Definition 2.7 withthe following clauses: 9 v ( (cid:3) ϕ ) = cd τ ( v ( ϕ )); • v ( ♦ ϕ ) = d τ ( v ( ϕ )); • v ( ϕ (cid:66) ψ ) = e τ ,τ ( v ( ϕ ) , v ( ψ )).The validity of an L ( (cid:3) , (cid:66) )-formula in a bitopological space and the logic Log ( X, τ , τ ) of (cid:104) X, τ , τ (cid:105) are also defined as in Definition 2.7.For a normal extension L of CL , we say that a bitopological space (cid:104) X, τ , τ (cid:105) is an L -space if L ⊆ Log ( X, τ , τ ). We prove that every τ -scattered bitopo-logical space is a CL -space. Proposition 3.4.
All axioms J1 , J2 , J3 and J4 in Definition 2.17 are validin any bitopological space (cid:104) X, τ , τ (cid:105) .Proof. ( J1 ): It suffices to show that for any Y, Z ⊆ X , cd τ ( Y ∪ Z ) ⊆ e ( Y, Z ).Suppose x ∈ cd τ ( Y ∪ Z ), that is, x (cid:54)∈ d τ ( Y ∩ Z ). Then there exists a τ -neighborhood W of x such that Y ∩ Z ∩ W ⊆ { x } .Take U ∈ τ arbitrarily, and suppose x ∈ d τ ( Y ∩ U ). We would like toshow x ∈ d τ ( Z ∩ U ). Let V be any τ -neighborhood of x . Then V ∩ W isalso a τ -neighborhood of x . Since x ∈ d τ ( Y ∩ U ), there exists y (cid:54) = x suchthat y ∈ Y ∩ U ∩ V ∩ W , and hence y ∈ Y ∩ W . On the other hand, since Y ∩ Z ∩ W ⊆ { x } , we have y (cid:54)∈ Y ∩ Z ∩ W . Therefore y ∈ Z , and hence y ∈ Z ∩ U ∩ V . This implies x ∈ d τ ( Z ∩ U ). We have shown x ∈ e ( Y, Z ).( J2 ): We show e ( Y, Z ) ∩ e ( Z, W ) ⊆ e ( Y, W ). Suppose x ∈ e ( Y, Z ) ∩ e ( Z, W ).Take U ∈ τ arbitrarily. If x ∈ d τ ( Y ∩ U ), then x ∈ d τ ( Z ∩ U ) by x ∈ e ( Y, Z ).Moreover, x ∈ d τ ( W ∩ U ) by x ∈ e ( Z, W ). Thus x ∈ e ( Y, W ).( J3 ): We show e ( Y, W ) ∩ e ( Z, W ) ⊆ e ( Y ∪ Z, W ). Suppose x ∈ e ( Y, W ) ∩ e ( Z, W ). Take U ∈ τ arbitrarily, and assume x ∈ d τ (( Y ∪ Z ) ∩ U ). By Fact2.10, we have d τ (( Y ∪ Z ) ∩ U ) = d τ (( Y ∩ U ) ∪ ( Z ∩ U )) = d τ ( Y ∩ U ) ∪ d τ ( Z ∩ U ) . Then x ∈ d τ ( Y ∩ U ) or x ∈ d τ ( Z ∩ U ). In either case, we obtain x ∈ d τ ( W ∩ U )by x ∈ e ( Y, W ) ∩ e ( Z, W ). Thus x ∈ e ( Y ∪ Z, W ).( J4 ): We show e ( Y, Z ) ∩ d τ ( Y ) ⊆ d τ ( Z ). Suppose x ∈ e ( Y, Z ) ∩ d τ ( Y ).Then x ∈ d τ ( Y ∩ X ). Since X ∈ τ , it follows from x ∈ e ( Y, Z ) that x ∈ d τ ( Z ∩ X ). Equivalently, x ∈ d τ ( Z ). (cid:113) Since each inference rule of CL preserves validity in bitopological spaces, weobtain the following corollary from Fact 2.9 and Proposition 3.4. Corollary 3.5.
For any bitopological space (cid:104)
X, τ , τ (cid:105) , it is a CL -space if andonly if (cid:104) X, τ (cid:105) is scattered. (cid:113) As well as Kripke frames, Visser fames (cid:104)
W, R, S (cid:105) can be considered as bitopo-logical spaces by considering topologies τ R and τ S (see Definition 2.11). In truth,our new operation e τ ,τ is defined with the intention of satisfying the followingproposition. 10 roposition 3.6. Let (cid:104)
W, R, S, (cid:13) (cid:105) be a Visser model. Let v be a valuation on (cid:104) W, τ R , τ S (cid:105) satisfying v ( p ) = { x ∈ W | x (cid:13) p } for any propositional variable p ,then v ( ϕ ) = { x ∈ W | x (cid:13) ϕ } for any L ( (cid:3) , (cid:66) ) -formula ϕ .Proof. We prove by induction on the construction of ϕ . We provide proofs ofonly two cases that ϕ is ♦ ψ and ϕ is ψ (cid:66) χ .Case of ϕ ≡ ♦ ψ : x (cid:13) ♦ ψ ⇐⇒ ∃ y ∈ W ( xRy & y (cid:13) ψ ) , ⇐⇒ R ( x ) ∩ v ( ψ ) (cid:54) = ∅ , (by induction hypothesis) ⇐⇒ x ∈ d τ R ( v ( ψ )) , (by Fact 2.13.2) ⇐⇒ x ∈ v ( ♦ ψ ) . Case of ϕ ≡ ψ (cid:66) χ : x (cid:13) ψ (cid:66) χ ⇐⇒ ∀ y [ xRy & y (cid:13) ψ ⇒ ∃ z ( xRz & ySz & z (cid:13) χ )] , ⇐⇒ ∀ y [ y ∈ R ( x ) ∩ v ( ψ ) ⇒ R ( x ) ∩ S ( y ) ∩ v ( χ ) (cid:54) = ∅ ] , (by induction hypothesis) ( ∗ ) ⇐⇒ ∀ U ∈ τ S [ R ( x ) ∩ v ( ψ ) ∩ U (cid:54) = ∅ ⇒ R ( x ) ∩ U ∩ v ( χ ) (cid:54) = ∅ ] , ⇐⇒ ∀ U ∈ τ S [ x ∈ d τ R ( v ( ψ ) ∩ U ) ⇒ x ∈ d τ R ( U ∩ v ( χ ))] , (by Fact 2.13.3) ⇐⇒ x ∈ e τ R ,τ S ( v ( ψ ) , v ( χ )) , ⇐⇒ x ∈ v ( ψ (cid:66) χ ) . Here we give a proof of the equivalence marked by ( ∗ ).( ⇒ ): Let U be any element of τ S with R ( x ) ∩ v ( ψ ) ∩ U (cid:54) = ∅ . Let y ∈ R ( x ) ∩ v ( ψ ) ∩ U . Then, R ( x ) ∩ S ( y ) ∩ v ( χ ) is non-empty. Since U is S -upwardclosed, S ( y ) ⊆ U . Thus R ( x ) ∩ U ∩ v ( χ ) is also non-empty.( ⇐ ): Let y be any element of R ( x ) ∩ v ( ψ ). Since S is reflexive, y ∈ S ( y ),and hence y ∈ R ( x ) ∩ v ( ψ ) ∩ S ( y ). It follows from the transitivity of S that S ( y )is S -upward closed. Hence S ( y ) ∈ τ S . Then, we obtain that R ( x ) ∩ S ( y ) ∩ v ( χ )is non-empty. (cid:113) From Proposition 3.6, we obtain the following corollary.
Corollary 3.7.
For any Visser frame (cid:104)
W, R, S (cid:105) , Log ( W, R, S ) =
Log ( W, τ R , τ S ) . (cid:113) Since every transitive and conversely well-founded Kripke frame can be ex-tended to a Visser frame, this is an extension of Fact 2.13.3. Conversely, τ -scattered Alexandroff bitopological spaces can be considered as Visser frames. Theorem 3.8.
Let (cid:104)
X, τ , τ (cid:105) be any bitopological space. Then, the followingare equivalent:1. τ is scattered and both τ and τ are Alexandroff. . There exists a Visser frame (cid:104) X, R, S (cid:105) such that τ = τ R and τ = τ S .Proof. ( ⇒ ): We define binary relations R and S on X as follows: • xRy : ⇐⇒ x (cid:54) = y & ∀ U ∈ τ ( x ∈ U ⇒ y ∈ U )( ⇐⇒ x ∈ d τ ( { y } )); • xSy : ⇐⇒ ∀ U ∈ τ ( x ∈ U ⇒ y ∈ U ).Clearly, R is irreflexive and S is transitive and reflexive. We show that R is transitive. Let xRy and yRz . Then x ∈ d τ ( { y } ) and y ∈ d τ ( { z } ). ByFact 2.10.2, d τ ( { y } ) ⊆ d τ ( d τ ( { z } )). Since τ is scattered, d τ ( d τ ( { z } )) ⊆ d τ ( { z } ) by Fact 2.10.5. Thus d τ ( { y } ) ⊆ d τ ( { z } ). Then, x ∈ d τ ( { z } ) andhence xRz .We prove τ = τ R , and the proof of τ = τ S is similar.( ⊆ ): Let U ∈ τ . If x ∈ U and xRy , then y ∈ U by the definition of R . Thismeans that U is R -upward closed. Thus U ∈ τ R .( ⊇ ): Let U ∈ τ R and x be an arbitrary element of U . Define V (cid:48) := (cid:84) { V ∈ τ | x ∈ V } . Since τ is Alexandroff, V (cid:48) is a τ -neighborhood of x . Since V (cid:48) isa subset of every τ -neighborhood of x , for any y ∈ V (cid:48) , either x = y or xRy .Since U is R -upward closed, U contains such y . Thus V (cid:48) ⊆ U . We have shownthat arbitrary element of U has a τ -neighborhood inside of U . Thus U ∈ τ .Since (cid:104) X, τ (cid:105) is scattered, by Fact 2.9, GL ⊆ Log ( X, τ ). By Fact 2.13.3, Log ( X, R ) =
Log ( X, τ R ) = Log ( X, τ ). Then GL ⊆ Log ( X, R ), and thus R isconversely well-founded by Fact 2.2. Therefore (cid:104) W, R, S (cid:105) is a Visser frame.( ⇐ ): By Fact 2.13.1, both τ = τ R and τ = τ S are Alexandroff. Since R istransitive and conversely well-founded, GL ⊆ Log ( W, R ) =
Log ( W, τ R ) by Facts2.2 and 2.13.3. Then it follows from Fact 2.9 that τ = τ R is scattered. (cid:113) As in the previous section, we introduce the consequence relation | = TL withrespect to topological semantics. Definition 3.9.
Let L be a normal extension of CL , Γ be a set of L ( (cid:3) , (cid:66) )-formulas, and ϕ be an L ( (cid:3) , (cid:66) )-formula. • Γ | = TL ϕ : ⇐⇒ for any L -space (cid:104) X, τ , τ (cid:105) , any valuation v on (cid:104) X, τ , τ (cid:105) and any x ∈ X , if x ∈ v ( ψ ) for all ψ ∈ Γ, then x ∈ v ( ϕ ); • We say that L is topologically complete if for any L ( (cid:3) , (cid:66) )-formula ϕ , ∅ | = TL ϕ implies ∅ (cid:96) L ϕ ; • We say that L is topologically strongly complete if for any L ( (cid:3) , (cid:66) )-formula ϕ and set Γ of L ( (cid:3) , (cid:66) )-formulas, Γ | = TL ϕ implies Γ (cid:96) L ϕ .From Facts 2.21 and 2.22, and the above discussions, we obtain the followingtopological completeness of CL and its some extensions. Theorem 3.10 (Topological completeness of some extensions of CL ) . Thelogics CL , CLM , IL , ILM , ILP and
ILW are topologically complete. (cid:113)
The main purpose of the present paper is to strengthen Theorem 3.10, that is,we prove that these logics are strongly complete with respect to our topologicalsemantics. 12
Topological compactness and topological strongcompleteness
In this section, we prove the topological strong completeness theorem of someextensions of CL . This directly follows from the the topological compactnesstheorem (Theorem 4.12) and the topological completeness theorem (Theorem3.10). Thus the main purpose of this section is to prove the topological compact-ness theorem. We prove this theorem by extending the method of Shehtman’sultrabouquet construction for topological spaces (cf. Shehtman [15, 16]) to ourframework. We introduce the notion of the ultrabouquet of a countable family {(cid:104) X n , τ n , τ n (cid:105)} n ∈ N of bitopological spaces, and investigate properties of ultra-bouquets used in our proof of the topological compactness theorem. In thissubsection, we fix a countable family {(cid:104) X n , τ n , τ n (cid:105)} n ∈ N of bitopological spacessatisfying the following conditions: • All topological spaces (cid:104) X n , τ n (cid:105) are scattered; • The family { X n } n ∈ N is pairwise disjoint.Then, it is known that for each n ∈ N , there exist Y n ⊆ X n and x n ∈ Y n suchthat Y n and Y n \{ x n } are elements of τ n (cf. Shehtman [16, Lemma 61]). Let U be a non-principal ultrafilter on N . Let x ∗ be a new element not contained in (cid:83) n ∈ N X n . Definition 4.1.
We define the ultrabouquet X := (cid:104) X, τ , τ (cid:105) of the family {(cid:104) X n , τ n , τ n (cid:105)} n ∈ N as follows: • X := (cid:83) n ∈ N ( X n \{ x n } ) ∪ { x ∗ } .For each V ⊆ X and n ∈ N , we sometimes restrict V to X n or Y n . Inthese situations, we adopt a convention that x ∗ is identified with x n . Moreprecisely, if x ∗ ∈ V , then we assume that V ∩ X n and V ∩ Y n denote thesets ( V ∩ X n ) ∪ { x n } and ( V ∩ Y n ) ∪ { x n } , respectively. • U ∈ τ : ⇐⇒ (i) For each n ∈ N , U ∩ ( Y n \{ x n } ) ∈ τ n ; and(ii) If x ∗ ∈ U , then { n ∈ N | U ∩ Y n ∈ τ n } ∈ U . • U ∈ τ : ⇐⇒ for each n ∈ N , U ∩ X n ∈ τ n . Lemma 4.2.
The ultrabouquet X is a bitopological space.Proof. We only prove that τ is a topology on X . A proof for τ is similar. • ∅ ∈ τ : (i) ∅ ∩ ( Y n \ { x n } ) = ∅ ∈ τ n ; and (ii) x ∗ / ∈ ∅ .13 X ∈ τ : (i) X ∩ ( Y n \ { x n } ) = Y n \ { x n } ∈ τ n ; and (ii) Since X ∩ Y n = Y n ∈ τ n , { n ∈ N | X ∩ Y n ∈ τ n } = N ∈ U because U is a non-trivial filter. • Let U , U ∈ τ . We show U ∩ U ∈ τ . (i): By condition (i) for U and U , the sets U ∩ ( Y n \ { x n } ) and U ∩ ( Y n \ { x n } ) are elements of τ n . Then( U ∩ U ) ∩ ( Y n \ { x n } ) = ( U ∩ ( Y n \ { x n } )) ∩ ( U ∩ ( Y n \ { x n } )) ∈ τ n . (ii): If x ∗ ∈ U ∩ U , then x ∗ is in both U and U . By condition (ii) for U and U , the sets Z = { n ∈ N | U ∩ Y n ∈ τ n } and Z = { n ∈ N | U ∩ Y n ∈ τ n } are in U . Then, Z ∩ Z ⊆ { n ∈ N | ( U ∩ U ) ∩ Y n ∈ τ n } ∈ U because U is a filter. • { U i } i ∈ I be any family of elements of τ . We show (cid:83) i ∈ I U i ∈ τ . (i): Since U i ∩ ( Y n \ { x n } ) ∈ τ n for all i ∈ I , (cid:32) (cid:91) i ∈ U U i (cid:33) ∩ ( Y n \ { x n } ) = (cid:91) i ∈ U ( U i ∩ ( Y n \ { x n } )) ∈ τ n . (ii): If x ∗ ∈ (cid:83) i ∈ I U i , then x ∗ ∈ U j for some j ∈ I . By condition (ii) for U j , { n ∈ N | U j ∩ Y n ∈ τ n } ∈ U . Claim 1.
For n ∈ N , if U j ∩ Y n ∈ τ n , then ( (cid:83) i ∈ I U i ) ∩ Y n ∈ τ n .Proof of Claim 1. Let x be an arbitrary element of ( (cid:83) i ∈ I U i ) ∩ Y n . Weshow that there exists a τ n -neighborhood V of x satisfying V ⊆ ( (cid:83) i ∈ I U i ) ∩ Y n . We distinguish the following two cases:If x = x n , then U j ∩ Y n is a required τ n -neighborhood of x .If x (cid:54) = x n , then x ∈ U k ∩ ( Y n \ { x n } ) for some k ∈ I . By condition (i) for U k , this set is a required τ n -neighborhood of x .Therefore ( (cid:83) i ∈ I U i ) ∩ Y n ∈ τ n . (cid:113) From Claim 1, we have { n ∈ N | U j ∩ Y n ∈ τ n } ⊆ { n ∈ N | ( (cid:91) i ∈ I U i ) ∩ Y n ∈ τ n } ∈ U . (cid:113) For each n ∈ N , let v n be a valuation on (cid:104) X n , τ n , τ n (cid:105) . We define a valuation v on X as follows: Definition 4.3. For x ∈ X n \ { x n } , x ∈ v ( p ) : ⇐⇒ x ∈ v n ( p ); • x ∗ ∈ v ( p ) : ⇐⇒ { n ∈ N | x n ∈ v n ( p ) } ∈ U .Let Y denote the set (cid:83) ( Y n \{ x n } ) ∪ { x ∗ } . We investigate the images of thevaluation v by dividing X into three parts, namely, Y \ { x ∗ } , X \ Y and { x ∗ } .First, we investigate in Y \ { x ∗ } . If x ∈ Y \ { x ∗ } , then x is in Y n \ { x n } forsome n ∈ N . In the set Y n \{ x n } , the first clause of Definition 4.3 is extendedto all L ( (cid:3) , (cid:66) )-formulas as follows. Lemma 4.4.
For any L ( (cid:3) , (cid:66) ) -formula ϕ , n ∈ N and x ∈ Y n \ { x n } , x ∈ v ( ϕ ) ⇐⇒ x ∈ v n ( ϕ ) . Proof.
We prove by induction on the construction of ϕ . We only give a proofof the case ϕ ≡ ψ (cid:66) χ .( ⇒ ): Suppose x ∈ v ( ψ (cid:66) χ ). Then ∀ U ∈ τ [ x ∈ d τ ( v ( ψ ) ∩ U ) ⇒ x ∈ d τ ( v ( χ ) ∩ U )] . (1)In order to prove x ∈ v n ( ψ (cid:66) χ ), let U be an arbitrary element of τ n and assume x ∈ d τ n ( v n ( ψ ) ∩ U ). We would like to show x ∈ d τ n ( v n ( χ ) ∩ U ). Let U (cid:48) := (cid:40) U if x n / ∈ U ;(( U \ { x n } ) ∪ (cid:83) m (cid:54) = n X m \ { x m } ) ∪ { x ∗ } if x n ∈ U. Then, it is easily shown that U (cid:48) ∈ τ and U (cid:48) ∩ X n = U . Claim 2. x ∈ d τ ( v ( ψ ) ∩ U (cid:48) ) .Proof of Claim 2. Let V be any τ -neighborhood of x . By Definition 4.1, V ∩ ( Y n \{ x n } ) ∈ τ n , and hence the set V ∩ ( Y n \{ x n } ) is a τ n -neighborhood of x .Since x ∈ d τ n ( v n ( ψ ) ∩ U ), there exists y (cid:54) = x such that y ∈ v n ( ψ ) ∩ U ∩ V ∩ ( Y n \{ x n } ). By the induction hypothesis, y ∈ v ( ψ ) ∩ U ∩ V ∩ ( Y n \{ x n } ). Hence y ∈ v ( ψ ) ∩ U (cid:48) ∩ V . This implies x ∈ d τ ( v ( ψ ) ∩ U (cid:48) ). (cid:113) From (1) and Claim 2, we have x ∈ d τ ( v ( χ ) ∩ U (cid:48) ). Claim 3. x ∈ d τ n ( v n ( χ ) ∩ U ) .Proof of Claim 3. Let V be any τ n -neighborhood of x . Then, V ∩ ( Y n \{ x n } ) ∈ τ n and x ∈ V ∩ ( Y n \{ x n } ). Also as a subset of X , x ∗ (cid:54)∈ V ∩ ( Y n \{ x n } ). It is shownthat the set V ∩ ( Y n \{ x n } ) is a τ -neighborhood of x . Since x ∈ d τ ( v ( χ ) ∩ U (cid:48) ),there exists y (cid:54) = x such that y ∈ v ( χ ) ∩ U (cid:48) ∩ V ∩ ( Y n \{ x n } ). By the inductionhypothesis, y ∈ v n ( χ ) ∩ U (cid:48) ∩ V ∩ ( Y n \{ x n } ). Since U (cid:48) ∩ X n = U , we conclude y ∈ v n ( χ ) ∩ U ∩ V . (cid:113)
15e have shown x ∈ e τ n ,τ n ( v n ( ψ ) , v n ( χ )) = v n ( ψ (cid:66) χ ).( ⇐ ): Suppose x ∈ v n ( ψ (cid:66) χ ). Then ∀ U ∈ τ n (cid:2) x ∈ d τ n ( v n ( ψ ) ∩ U ) ⇒ x ∈ d τ n ( v n ( χ ) ∩ U ) (cid:3) . (2)Let U be an arbitrary element of τ and assume x ∈ d τ ( v ( ψ ) ∩ U ). We wouldlike to show x ∈ d τ ( v ( χ ) ∩ U ). Let U (cid:48) := U ∩ X n , then U (cid:48) ∈ τ n . Claim 4. x ∈ d τ n ( v n ( ψ ) ∩ U (cid:48) ) .Proof of Claim 4. Let V be any τ n -neighborhood of x . Then V ∩ ( Y n \{ x n } ) ∈ τ n and x ∈ V ∩ ( Y n \{ x n } ). Also as a subset of X , x ∗ / ∈ V ∩ ( Y n \{ x n } ). Thus itis shown that the set V ∩ ( Y n \{ x n } ) is a τ -neighborhood of x . Since x ∈ d τ ( v ( ψ ) ∩ U ), there exists y (cid:54) = x such that y ∈ v ( ψ ) ∩ U ∩ V ∩ ( Y n \{ x n } ). Bythe induction hypothesis, y ∈ v n ( ψ ) ∩ U ∩ V ∩ ( Y n \{ x n } ), and hence y ∈ v n ( ψ ) ∩ U (cid:48) ∩ V ∩ ( Y n \{ x n } ). Notice that y ∈ X n . Thus we conclude x ∈ d τ n ( v n ( ψ ) ∩ U (cid:48) ). (cid:113) From (2) and Claim 4, x ∈ d τ n ( v n ( χ ) ∩ U (cid:48) ). Claim 5. x ∈ d τ ( v ( χ ) ∩ U ) .Proof of Claim 5. Let V be any τ -neighborhood of x . By Definition 4.1, V ∩ ( Y n \{ x n } ) ∈ τ n and hence V ∩ ( Y n \{ x n } ) is a τ n -neighborhood of x . Since x ∈ d τ n ( v n ( χ ) ∩ U (cid:48) ), there exists y (cid:54) = x such that y ∈ v n ( χ ) ∩ U (cid:48) ∩ V ∩ ( Y n \{ x n } ).By the induction hypothesis, y ∈ v ( χ ) ∩ U (cid:48) ∩ V ∩ ( Y n \{ x n } ), and hence y ∈ v ( χ ) ∩ U ∩ V . Thus we conclude x ∈ d τ ( v ( χ ) ∩ U ). (cid:113) We have proved x ∈ e τ ,τ ( v ( ψ ) , v ( χ )) = v ( ψ (cid:66) χ ). This completes our proofof Lemma 4.4. (cid:113) Secondly, we investigate the behavior of valuations on X in X \ Y . Lemma 4.5.
For any subset U of X \ Y , U ∈ τ .Proof. We show that each U ⊆ X \ Y satisfies conditions (i) and (ii) in Definition4.1. Clearly U ∩ ( Y n \{ x n } ) = ∅ for any n ∈ N , and hence (i) holds. Moreover,(ii) vacuously holds since U does not contain x ∗ . (cid:113) The following lemma shows that every element of X \ Y behaves as a deadend of Kripke frames. Lemma 4.6.
For any x ∈ X \ Y and any Z ⊆ X , x ∈ cd τ ( Z ) .Proof. Let x ∈ X \ Y . Then, by Lemma 4.5, { x } ∈ τ . Since Z ∩ { x } ⊆ { x } ,we have x / ∈ d τ ( Z ). That is, x ∈ cd τ ( Z ). (cid:113) For x ∈ X n \ Y n , even if x ∈ v n ( ♦ ϕ ), by Lemma 4.6, x / ∈ v ( ♦ ϕ ). So theequivalence of Lemma 4.4 cannot be extended to elements of X n \ { x n } .Thirdly, the following lemma is a generalization of the second clause of Def-inition 4.3. In particular, it plays a key role in our proof of the topologicalcompactness theorem. 16 emma 4.7. For any L ( (cid:3) , (cid:66) ) -formula ϕ , x ∗ ∈ v ( ϕ ) ⇐⇒ { n ∈ N | x n ∈ v n ( ϕ ) } ∈ U . Proof.
We prove by induction on the construction of ϕ . We only give a proofof the case ϕ ≡ ψ (cid:66) χ .( ⇒ ): We prove the contrapositive. Assume { n ∈ N | x n ∈ v n ( ψ (cid:66) χ ) } (cid:54)∈ U .Since U is an ultrafilter on N , Z := { n ∈ N | x n (cid:54)∈ v n ( ψ (cid:66) χ ) } ∈ U . For each n ∈ Z , there exists U n ∈ τ n such that x n ∈ d τ n ( v n ( ψ ) ∩ U n ) & x n (cid:54)∈ d τ n ( v n ( χ ) ∩ U n ) . (3)Let Z := { n ∈ Z | x n / ∈ U n } and Z := { n ∈ Z | x n ∈ U n } . Then, Z = Z ∪ Z . Since U is an ultrafilter, we get an i ∈ { , } such that Z i ∈ U .Let U := (cid:40)(cid:83) n ∈ Z i U n if i = 0; (cid:0)(cid:83) n ∈ Z i U n \ { x n } (cid:1) ∪ (cid:0)(cid:83) n/ ∈ Z i X n \ { x n } (cid:1) ∪ { x ∗ } if i = 1 . Then, it is shown that U is an element of τ satisfying U ∩ X n = U n for all n ∈ Z i .First, we prove x ∗ ∈ d τ ( v ( ψ ) ∩ U ). Let V be any τ -neighborhood of x ∗ . ByDefinition 4.1, Z := { n ∈ N | V ∩ Y n ∈ τ n } ∈ U . Since Z i ∩ Z ∈ U , Z i ∩ Z isnon-empty, and fix some n ∈ Z i ∩ Z . Since the set V ∩ Y n is a τ n -neighborhoodof x n , by (3), there exists y ∈ X n \{ x n } such that y ∈ v n ( ψ ) ∩ U n ∩ V ∩ ( Y n \{ x n } ).Applying Lemma 4.4, y ∈ v ( ψ ) ∩ U n ∩ V ∩ ( Y n \{ x n } ). Since U n = U ∩ X n , weobtain y ∈ v ( ψ ) ∩ U ∩ V and y (cid:54) = x ∗ . Thus x ∗ ∈ d τ ( v ( ψ ) ∩ U ).Secondly, we prove x ∗ (cid:54)∈ d τ ( v ( χ ) ∩ U ). By (3), for each n ∈ Z i , thereexists a τ n -neighborhood W n of x n such that v n ( χ ) ∩ U n ∩ W n ⊆ { x n } . Let W := (cid:83) n ∈ Z i ( W n ∩ ( Y n \ { x n } ) ∪ { x ∗ } . We show W ∈ τ . (i) For each n ∈ N , W ∩ ( Y n \{ x n } ) = (cid:40) W n ∩ ( Y n \{ x n } ) if n ∈ Z i ; ∅ otherwise.Then, W ∩ ( Y n \{ x n } ) ∈ τ n . (ii) If n ∈ Z i , then W ∩ Y n = W n ∩ Y n ∈ τ n .Hence Z i ⊆ { n ∈ N | W ∩ Y n ∈ τ n } ∈ U because U is a filter. Thus W is a τ -neighborhood of x ∗ .Suppose, towards a contradiction, that x ∗ ∈ d τ ( v ( χ ) ∩ U ). Then there exists y (cid:54) = x ∗ such that y ∈ v ( χ ) ∩ U ∩ W . Since y ∈ W , for some n ∈ Z i , y ∈ v ( χ ) ∩ U ∩ W n ∩ ( Y n \{ x n } ). Applying Lemma 4.4, y ∈ v n ( χ ) ∩ U ∩ W n ∩ ( Y n \{ x n } ). Since U (cid:48) ∩ X n = U n , y ∈ v n ( χ ) ∩ U n ∩ W n . This contradicts v n ( χ ) ∩ U n ∩ W n ⊆ { x n } .Therefore x ∗ / ∈ d τ ( v ( χ ) ∩ U ).We conclude x ∗ (cid:54)∈ e τ ,τ ( v ( ψ ) , v ( χ )), and hence x ∗ (cid:54)∈ v ( ψ (cid:66) χ ).( ⇐ ): Suppose Z := { n ∈ N | x n ∈ v n ( ψ (cid:66) χ ) } ∈ U . In order to prove x ∗ ∈ v ( ψ (cid:66) χ ), suppose that U ∈ τ and x ∗ ∈ d τ ( v ( ψ ) ∩ U ). We would like toshow x ∗ ∈ d τ ( v ( χ ) ∩ U ). Let V be any τ -neighborhood of x ∗ . By Definition4.1, Z := { n ∈ N | V ∩ Y n ∈ τ n } ∈ U . For each n ∈ N , let U n := U ∩ X n . Then U n ∈ τ n . 17 laim 6. There exists n ∈ Z ∩ Z such that x n ∈ d τ n ( v n ( ψ ) ∩ U n ) .Proof of Claim 6. Suppose, towards a contradiction, that for all n ∈ Z ∩ Z , x n (cid:54)∈ d τ n ( v n ( ψ ) ∩ U n ). Then for each n ∈ Z ∩ Z , there exists W n ∈ τ n suchthat x n ∈ W n and v n ( ψ ) ∩ U n ∩ W n ⊆ { x n } . Let W := (cid:83) n ∈ Z ∩ Z ( W n ∩ ( Y n \{ x n } ) ∪ { x ∗ } .We show W ∈ τ . (i) For each n ∈ N , W ∩ ( Y n \{ x n } ) = (cid:40) W n ∩ ( Y n \{ x n } ) if n ∈ Z ∩ Z ; ∅ otherwise , and this set is in τ n . (ii) If n ∈ Z ∩ Z , then W ∩ Y n = W n ∩ Y n ∈ τ n . Thus Z ∩ Z ⊆ { n ∈ N | W ∩ Y n ∈ τ n } ∈ U because U is a filter. Therefore W ∈ τ .Since x ∗ ∈ d τ ( v ( ψ ) ∩ U ), there exists y (cid:54) = x ∗ such that y ∈ v ( ψ ) ∩ U ∩ W .Since y ∈ W , there exists m ∈ Z ∩ Z such that y ∈ v ( ψ ) ∩ U m ∩ W m ∩ ( Y m \{ x m } ). Applying Lemma 4.4, y ∈ v m ( ψ ) ∩ U m ∩ W m ∩ ( Y m \{ x m } ). Then y (cid:54) = x m and y ∈ v m ( ψ ) ∩ U m ∩ W m . This contradicts v m ( ψ ) ∩ U m ∩ W m ⊆ { x m } .Our proof of Claim 6 is completed. (cid:113) We show x ∗ ∈ d τ ( v ( χ ) ∩ U ). From Claim 6, there exists n ∈ Z ∩ Z such that x n ∈ d τ n ( v n ( ψ ) ∩ U n ). Since n ∈ Z , we have x n ∈ v n ( ψ (cid:66) χ ).Therefore x n ∈ d τ n ( v n ( χ ) ∩ U n ). Moreover, since n ∈ Z , we have V ∩ Y n ∈ τ n .This set is a τ n -neighborhood of x n , and thus there exists y (cid:54) = x n such that y ∈ v n ( χ ) ∩ U n ∩ V ∩ Y n . Since y (cid:54) = x n , we obtain y ∈ v ( χ ) ∩ U n ∩ V ∩ ( Y n \{ x n } )by Lemma 4.4. In particular, y (cid:54) = x ∗ and y ∈ v ( χ ) ∩ U ∩ V . This implies x ∗ ∈ d τ ( v ( χ ) ∩ U ). We conclude x ∗ ∈ v ( ψ (cid:66) χ ). (cid:113) The following lemma is an adaptation of Shehtman’s result on the preserva-tion of validity in ultrabouquets to our framework (See Shehtman [15, Lemma5.6]).
Lemma 4.8.
If an L ( (cid:3) , (cid:66) ) -formula ϕ is valid in all (cid:104) X n , τ n , τ n (cid:105) , then for allvaluations v (cid:48) on X and all x ∈ Y , x ∈ v (cid:48) ( ϕ ) .Proof. We prove the contrapositive. Suppose that there exist a valuation v (cid:48) on X and x ∈ Y such that x / ∈ v (cid:48) ( ϕ ). For each n ∈ N , we define a valuation v (cid:48) n on (cid:104) X n , τ n , τ n (cid:105) as follows: • For x ∈ X n \ { x n } , x ∈ v (cid:48) n ( p ) : ⇐⇒ x ∈ v (cid:48) ( p ); • x ∗ ∈ v (cid:48) n ( p ) : ⇐⇒ x ∗ ∈ v (cid:48) ( p ).Then the valuation on X defined from { v (cid:48) n } n ∈ N in Definition 4.3 coincides with v (cid:48) because ∅ / ∈ U and N ∈ U . We distinguish the following two cases.If x ∈ Y n \ { x n } , then by Lemma 4.4, we obtain x / ∈ v (cid:48) n ( ϕ ).If x = x ∗ , then by Lemma 4.7, { n ∈ N | x n ∈ v (cid:48) n ( ϕ ) } / ∈ U . Since N ∈ U , forsome n ∈ N , x n / ∈ v (cid:48) n ( ϕ ).Thus in either case, ϕ is not valid in (cid:104) X n , τ n , τ n (cid:105) for some n ∈ N . (cid:113) Y in the statement of Lemma 4.8does not seem to be replaceable by X in general. However, we prove that this isactually the case. First, we prove that the validity of the axiom (cid:3) ( (cid:3) p → p ) → (cid:3) p of GL is preserved. Lemma 4.9.
The topological space (cid:104)
X, τ (cid:105) is scattered. That is, the ultrabou-quet X is a CL -space.Proof. Since each space (cid:104) X n , τ n , τ n (cid:105) is scattered, ϕ : ≡ (cid:3) ( (cid:3) p → p ) → (cid:3) p isvalid in (cid:104) X n , τ n , τ n (cid:105) by Fact 2.9. Let v (cid:48) be any valuation on X . By Lemma4.8, for all y ∈ Y , y ∈ v (cid:48) ( ϕ ). Moreover, by Lemma 4.6, for all x ∈ X \ Y , x ∈ cd τ ( v (cid:48) ( p )), that is, x ∈ v (cid:48) ( (cid:3) p ). Hence x ∈ v (cid:48) ( ϕ ). Thus ϕ is valid in X , andhence GL ⊆ Log ( X ). We conclude that (cid:104) X, τ (cid:105) is scattered. (cid:113) The following lemma is a version of a part of Makinson’s theorem (See Makin-son [12]). Our proof is a modification of that in Hughes and Cresswell [7, Lemma3.2]).
Lemma 4.10.
Let L be any consistent normal extension of CL and ϕ be any L ( (cid:3) , (cid:66) ) -formula. If ϕ ∈ L , then (cid:3) ⊥ → ϕ ∈ CL .Proof. Let L be a normal extension of CL and suppose that there exists an L ( (cid:3) , (cid:66) )-formula ϕ such that ϕ ∈ L but (cid:3) ⊥ → ϕ / ∈ CL . We would like to showthat L is inconsistent. From axioms J1 and J4 , we have that for any L ( (cid:3) , (cid:66) )-formula ψ , (cid:3) ψ is equivalent to ( ¬ ψ ) (cid:66) ⊥ in CL . So we may assume that neither (cid:3) nor ♦ occurs in ϕ . Also we assume that ϕ is in a conjunctive normal form ϕ ∧ ϕ ∧ · · · ∧ ϕ k where each ϕ i is a disjunction of formulas, and each disjunctof ϕ i is either a formula without (cid:66) , or a formula of the form ψ (cid:66) χ , or a formulaof the form ¬ ( ψ (cid:66) χ ).By the choice of ϕ , for some i ≤ k , ϕ i ∈ L and (cid:3) ⊥ → ϕ i / ∈ CL . From J1 ,we have that (cid:3) ⊥ → ψ (cid:66) χ ∈ CL . Then ϕ i does not contain a formula of theform ψ (cid:66) χ as a disjunct because (cid:3) ⊥ → ϕ i / ∈ CL . Thus, we may assume that ϕ i is of the form γ ∨ m (cid:95) j =0 ¬ ( ψ j (cid:66) χ j )where γ is a classical propositional formula. Since (cid:3) ⊥ → ϕ i / ∈ CL , γ is not atautology. Then, there exists a substitution instance γ (cid:48) of γ such that ¬ γ (cid:48) is atautology (cf. [7, p. 47]). So ¬ γ (cid:48) ∈ L . If m = 0, then L contains both γ (cid:48) and ¬ γ (cid:48) , and hence is inconsistence.We assume m >
0. Since each ¬ ( ψ j (cid:66) χ j ) implies ♦ (cid:62) in CL , L contains γ ∨ ♦ (cid:62) . Then γ (cid:48) ∨ ♦ (cid:62) ∈ L , and thus ♦ (cid:62) ∈ L . Since L is normal, (cid:3)♦ (cid:62) ∈ L .Therefore (cid:3) ⊥ ∈ L because L is an extension of CL . We conclude that L isinconsistent. (cid:113) Theorem 4.11.
If an L ( (cid:3) , (cid:66) ) -formula ϕ is valid in all (cid:104) X n , τ n , τ n (cid:105) , then ϕ isalso valid in X . roof. Since (cid:104) X , τ (cid:105) is scattered, Log ( X , τ , τ ) is a consistent normal exten-sion of CL by Corollary 3.5. Since ϕ ∈ Log ( X , τ , τ ), we obtain (cid:3) ⊥ → ϕ ∈ CL by Lemma 4.10.Let v (cid:48) be any valuation on X , then for all y ∈ Y , y ∈ v (cid:48) ( ϕ ) by Lemma 4.8.Also, for all x ∈ X \ Y , x ∈ v (cid:48) ( (cid:3) ⊥ ) by Lemma 4.6. Since X is a CL -space byLemma 4.9, it follows from (cid:3) ⊥ → ϕ ∈ CL that x ∈ v (cid:48) ( ϕ ). Therefore ϕ is validin X . (cid:113) We are ready to prove the topological compactness theorem.
Theorem 4.12 (Topological compactness theorem) . Let L be a consistent nor-mal extension of CL , Γ be a set of L ( (cid:3) , (cid:66) ) -formulas and ϕ be an L ( (cid:3) , (cid:66) ) -formula. If Γ | = TL ϕ , then Γ | = TL ϕ for some finite subset Γ of Γ .Proof. Suppose that for all finite subsets Γ of Γ, Γ (cid:54)| = TL ϕ . Let { ψ n } n ∈ N be anenumeration of Γ, and let χ n := (cid:86) ni =0 ψ n . Then, for each n ∈ N , { χ n } (cid:54)| = TL ϕ .Hence there exist an L -space (cid:104) X n , τ n , τ n (cid:105) , a valuation v n on the space and x n ∈ X n such that x n ∈ v n ( χ n ) and x n (cid:54)∈ v n ( ϕ ). By Corollary 3.5, (cid:104) X n , τ n (cid:105) isscattered. Also we may assume that the family { X n } n ∈ N is pairwise disjoint.Then we can define the ultrabouquet X based on the family {(cid:104) X n , τ n , τ n (cid:105)} n ∈ N .Since every ϕ ∈ L is valid in all (cid:104) X n , τ n , τ n (cid:105) , by Lemma 4.11, ϕ is also valid in X . Therefore X is also an L -space.Let v be the valuation on X defined from { v n } n ∈ N in Definition 4.3. Weclaim that for every ψ i ∈ Γ, x ∗ ∈ v ( ψ i ). Indeed, for any n ≥ i , x n ∈ v n ( ψ i ).Then the set { n ∈ N | x n ∈ v n ( ψ i ) } is cofinite, and hence in U because U is anon-principal ultrafilter. By Lemma 4.7, x ∗ ∈ v ( ψ i ).On the other hand, { n ∈ N | x n ∈ v n ( ϕ ) } = ∅ (cid:54)∈ U . Again by Lemma 4.7, x ∗ (cid:54)∈ v ( ϕ ). Thus we conclude Γ (cid:54)| = TL ϕ . (cid:113) Theorem 4.13.
For any normal extension L of CL , L is topologically completeif and only if L is topologically strongly complete.Proof. It suffices to prove the implication ( ⇒ ). Suppose Γ | = TL ϕ . By thetopological compactness theorem, Γ (cid:96) L ϕ for some finite subset Γ of Γ, and wehave ∅ | = TL (cid:86) Γ → ϕ . By the topological completeness of L , ∅ (cid:96) L (cid:86) Γ → ϕ .Thus Γ (cid:96) L ϕ . (cid:113) From Theorems 3.10 and 4.13, we obtain the following topological strongcompleteness theorem.
Theorem 4.14 (Topological strong completeness theorem of some extensionsof CL ) . The logics CL , CLM , IL , ILM , ILP and
ILW are strongly completewith respect to our topological semantics. (cid:113) Topological investigations of IL
In this section, we investigate topological aspects of IL . First, we investigatenecessary and sufficient conditions for a CL -space to be an IL -space. Secondly,we explore Alexandroff IL -spaces. Theorem 5.1.
Let (cid:104)
X, τ , τ (cid:105) be a CL -space. Then the following are equivalent:1. (cid:104) X, τ , τ (cid:105) is an IL -space.2. For any U ∈ τ and Y ⊆ X , d τ ( d τ ( Y ) ∩ U ) ⊆ d τ ( Y ∩ U ) .3. For any U ∈ τ , d τ ( d τ ( U ) ∩ U ) = ∅ .4. For any U ∈ τ , there exists V ∈ τ such that V ⊆ U and d τ ( U \ V ) = ∅ .Proof. (1 ⇔ CL -space (cid:104) X, τ , τ (cid:105) is an IL -space if and only if ♦ p (cid:66) p is valid in (cid:104) X, τ , τ (cid:105) . The latter condition is equivalent to the conditionthat for all Y ⊆ X , e τ ,τ ( d τ ( Y ) , Y ) = X . Then it follows from Definition 3.2that this is equivalent to clause 2.(2 ⇒ U ∈ τ . From clause 2 for Y = U , we have d τ ( d τ ( U ) ∩ U ) ⊆ d τ ( U ∩ U ) = d τ ( ∅ ). Since d τ ( ∅ ) = ∅ by Fact 2.10.1, we obtain d τ ( d τ ( U ) ∩ U ) = ∅ .(3 ⇒ U ∈ τ and Y ⊆ X . Since Y \ U ⊆ U , by Fact 2.10.2, d τ ( Y \ U ) ∩ U ⊆ d τ ( U ) ∩ U . Then d τ ( d τ ( Y \ U ) ∩ U ) ⊆ d τ ( d τ ( U ) ∩ U ) = ∅ .We get d τ ( d τ ( Y \ U ) ∩ U ) = ∅ .Since Y = ( Y ∩ U ) ∪ ( Y \ U ), by Fact 2.10, d τ ( d τ ( Y ) ∩ U ) = d τ ( d τ ( Y ∩ U ) ∩ U ) ∪ d τ ( d τ ( Y \ U ) ∩ U ) , = d τ ( d τ ( Y ∩ U ) ∩ U ) , ⊆ d τ ( d τ ( Y ∩ U )) , ⊆ d τ ( Y ∩ U ) . (3 ⇒ U ∈ τ . Let V denote the set U \ d τ ( U ). Then V ⊆ U and d τ ( U \ V ) = d τ ( d τ ( U ) ∩ U ) = ∅ . So it suffices to show that V is an elementof τ . Let x be an arbitrary element of V . Since x / ∈ d τ ( d τ ( U ) ∩ U ), thereexists a τ -neighborhood W of x such that W ∩ d τ ( U ) ∩ U ⊆ { x } . Since x / ∈ d τ ( U ), W ∩ d τ ( U ) ∩ U = ∅ . Furthermore, from x / ∈ d τ ( U ), there existsa τ -neighborhood W of x such that W ∩ U ⊆ { x } . Since x / ∈ U , we alsohave W ∩ U = ∅ . Equivalently, W ⊆ U . Then, we have W ∩ W ∈ τ , x ∈ W ∩ W and W ∩ W ⊆ V . We have shown that arbitrary element of V has a τ -neighborhood which is included in V . Therefore V ∈ τ .(4 ⇒ U ∈ τ , then for some V ∈ τ , V ⊆ U and d τ ( U \ V ) = ∅ .Since U ⊆ V and V ∈ τ , by Fact 2.10, d τ ( U ) ∩ V ⊆ d τ ( V ) ∩ V = ∅ . Then d τ ( U ) ∩ V = ∅ and so d τ ( d τ ( U ) ∩ V ) = ∅ .21ince U = V ∪ ( U \ V ), we obtain d τ ( d τ ( U ) ∩ U ) = d τ ( d τ ( U ) ∩ V ) ∪ d τ ( d τ ( U ) ∩ ( U \ V )) , = d τ ( d τ ( U ) ∩ ( U \ V )) , ⊆ d τ ( U \ V ) . Therefore we conclude d τ ( d τ ( U ) ∩ U ) = ∅ . (cid:113) Corollary 5.2.
For any CL -space (cid:104) X, τ , τ (cid:105) , if τ ⊆ τ , then (cid:104) X, τ , τ (cid:105) isan IL -space.Proof. Let U ∈ τ , then U ∈ τ . By Fact 2.10, d τ ( U ) ∩ U = ∅ , and hence d τ ( d τ ( U ) ∩ U ) = ∅ . By Theorem 5.1, (cid:104) X, τ , τ (cid:105) is an IL -space. (cid:113) We have already stated that IL is complete with respect to Visser semantics(Fact 2.22). Actually, Visser proved the following stronger result (See also Fact2.19.2). Fact 5.3 (Visser [20]) . For any L ( (cid:3) , (cid:66) ) -formula ϕ , the following are equivalent:1. ∅ (cid:96) IL ϕ .2. ϕ is valid in all Visser frames (cid:104) W, R, S (cid:105) with R ⊆ S . (cid:113) We explain how Fact 5.3 follows from Fact 2.22 in our framework. For thispurpose, we prepare the following lemmas.
Lemma 5.4.
For any topological space (cid:104)
X, τ (cid:105) , the following are equivalent:1. (cid:104)
X, τ (cid:105) is Alexandroff.2. For any family { Y i } i ∈ I of subsets of X , d τ ( (cid:83) i ∈ I Y i ) = (cid:83) i ∈ I d τ ( Y i ) .Proof. (1 ⇒ { Y i } i ∈ I be any family of subsets of X . We easily obtain (cid:83) i ∈ I d τ ( Y i ) ⊆ d τ ( (cid:83) i ∈ I Y i ) by Fact 2.10.2. We prove the converse inclusion. Let x / ∈ (cid:83) i ∈ I d τ ( Y i ). Then, for all i ∈ I , there exists a τ -neighborhood U i of x suchthat Y i ∩ U i ⊆ { x } . Let U := (cid:84) i ∈ I U i , then U is also a τ -neighborhood of x because τ is Alexandroff. Suppose, towards a contradiction, x ∈ d τ ( (cid:83) i ∈ I Y i ).Then there exists y (cid:54) = x such that y ∈ (cid:0)(cid:83) i ∈ I Y i (cid:1) ∩ U . For some j ∈ I , y ∈ Y j ∩ U ⊆ Y j ∩ U j , and this is a contradiction. Therefore x / ∈ d τ ( (cid:83) i ∈ I Y i ).(2 ⇒ { U i } i ∈ I be any family of sets of τ . Then for each i ∈ I , d τ ( U i ) ∩ U i = ∅ by Fact 2.10.4. d τ ( (cid:92) i ∈ I U i ) ∩ (cid:92) i ∈ I U i = d τ ( (cid:91) i ∈ I U i ) ∩ (cid:92) i ∈ I U i , = (cid:91) i ∈ I d τ ( U i ) ∩ (cid:92) i ∈ I U i , (by clause 1) ⊆ (cid:91) i ∈ I ( d τ ( U i ) ∩ U i ) = ∅ . (cid:84) i ∈ I V i is a member of τ . (cid:113) Lemma 5.5.
Let (cid:104)
X, τ (cid:105) be a topological space and
V, U ⊆ X . If V ⊆ U and d τ ( U \ V ) = ∅ , then d τ ( Y ∩ U ) = d τ ( Y ∩ V ) for all subsets Y of X .Proof. Notice that d τ ( Y ∩ ( U \ V )) is also empty because Y ∩ ( U \ V ) ⊆ U \ V .Since U = ( U \ V ) ∪ V , by Fact 2.10.3, d τ ( Y ∩ U ) = d τ ( Y ∩ ( U \ V )) ∪ d τ ( Y ∩ V ) = d τ ( Y ∩ V ) . (cid:113) Theorem 5.6.
Let (cid:104)
X, τ , τ (cid:105) be a bitopological space with both τ and τ areAlexandroff. Then, the following are equivalent:1. (cid:104) X, τ , τ (cid:105) is an IL -space.2. τ is scattered and there exists an Alexandroff topology τ on X such that τ ∩ τ ⊆ τ ⊆ τ and Log ( X, τ , τ ) = Log ( X, τ , τ ) .3. There exists a Visser frame (cid:104) X, R, S (cid:105) such that R ⊆ S and Log ( X, τ , τ ) = Log ( X, R, S ) .Proof. (1 ⇒ (cid:104) X, τ , τ (cid:105) is a CL -space, τ is scattered by Corollary3.5. Define τ := { V ∈ τ | ∃ U ∈ τ [ V ⊆ U & d τ ( U \ V ) = ∅ ] } . Then, obviously τ ⊆ τ . Let V ∈ τ ∩ τ . Since V ⊆ V and d τ ( V \ V ) = d τ ( ∅ ) = ∅ by Fact 2.10.1, we have V ∈ τ . Thus τ ∩ τ ⊆ τ .First, we prove that τ is a topology on X . • Since X and ∅ are in τ ∩ τ , they are also in τ . • Let V , V ∈ τ . Then there exist elements U and U of τ such that d τ ( U \ V ) = d τ ( U \ V ) = ∅ . We have V ∩ V ⊆ U ∩ U ∈ τ and d τ (( U ∩ U ) \ ( V ∩ V )) = d τ ((( U ∩ U ) \ V ) ∪ (( U ∩ U ) \ V )) , ⊆ d τ (( U \ V ) ∪ ( U \ V )) , (by Fact 2.10.2)= d τ ( U \ V ) ∪ d τ ( U \ V ) = ∅ . (by Fact 2.10.3)Hence V ∩ V ∈ τ . 23 Let { V i } i ∈ I be any family of elements of τ . Then for each i ∈ I , thereexists U i ∈ τ such that d τ ( U i \ V i ) = ∅ . We get (cid:83) i ∈ I V i ⊆ (cid:83) i ∈ I U i ∈ τ and d τ (( (cid:91) i ∈ I U i ) \ ( (cid:91) i ∈ I V i )) ⊆ d τ ( (cid:91) i ∈ I ( U i \ V i )) , (by Fact 2.10.2)= (cid:91) i ∈ I d τ ( U i \ V i ) = ∅ . (by Lemma 5.4)Therefore (cid:83) i ∈ I V i is an element of τ .Secondly, we prove τ is Alexandroff. Let { V i } i ∈ I be any family of elementsof τ . Then for each i ∈ I , there exists U i ∈ τ such that d τ ( U i \ V i ) = ∅ .Since τ is Alexandroff, (cid:84) i ∈ I V i ⊆ (cid:84) i ∈ I U i ∈ τ . Since τ is also Alexandroff, d τ (( (cid:92) i ∈ I U i ) \ ( (cid:92) i ∈ I V i )) ⊆ d τ ( (cid:91) i ∈ I ( U i \ V i )) , (by Fact 2.10.2)= (cid:91) i ∈ I d τ ( U i \ V i ) = ∅ . (by Lemma 5.4)Therefore (cid:84) i ∈ I V i ∈ τ .Finally, we prove Log ( X, τ , τ ) = Log ( X, τ , τ ). It suffices to prove thatfor all subsets Y, Z of X , e τ ,τ ( Y, Z ) = e τ ,τ ( Y, Z ).( ⊆ ): Let x ∈ e τ ,τ ( Y, Z ), V ∈ τ and x ∈ d τ ( Y ∩ V ). We would like to show x ∈ d τ ( Z ∩ V ). Then, there exists U ∈ τ such that d τ ( U \ V ) = ∅ . By Lemma5.5, d τ ( Y ∩ U ) = d τ ( Y ∩ V ) and so x ∈ d τ ( Y ∩ U ). Since x ∈ e τ ,τ ( Y, Z ), x ∈ d τ ( Z ∩ U ). By Lemma 5.5 again, d τ ( Z ∩ U ) = d τ ( Z ∩ V ) and thus x ∈ d τ ( Z ∩ V ).( ⊇ ): Let x ∈ e τ ,τ ( Y, Z ), U ∈ τ and x ∈ d τ ( Y ∩ U ). We would like toshow x ∈ d τ ( Z ∩ U ). Since (cid:104) X, τ , τ (cid:105) is an IL -space, by Theorem 5.1, thereexists V ∈ τ such that V ⊆ U and d τ ( U \ V ) = ∅ . Then, V ∈ τ . Asabove, by Lemma 5.5, x ∈ d τ ( Y ∩ U ) = d τ ( Y ∩ V ). Since x ∈ e τ ,τ ( Y, Z ), x ∈ d τ ( Z ∩ V ). Also by Lemma 5.5 again, x ∈ d τ ( Z ∩ V ) = d τ ( Z ∩ U ).(2 ⇒ R and S be binary relations on X defined as follows: • xRy : ⇐⇒ x (cid:54) = y & ∀ U ∈ τ ( x ∈ U ⇒ y ∈ U ); • xSy : ⇐⇒ ∀ U ∈ τ ( x ∈ U ⇒ y ∈ U ).As proved in the proof of Theorem 3.8, (cid:104) W, R, S (cid:105) is a Visser frame, τ = τ R and τ = τ S . By Corollary 3.7, Log ( X, R, S ) =
Log ( X, τ R , τ S ) = Log ( X, τ , τ ) = Log ( X, τ , τ ). Also R ⊆ S follows from the definitions of R and S and τ ⊆ τ .(3 ⇒ (cid:113) Corollary 5.7.
For any Visser frame (cid:104)
W, R, S (cid:105) , the following are equivalent:1. IL ⊆ Log ( W, R, S ) . . There exists a Visser frame (cid:104) W, R, S (cid:48) (cid:105) such that R ⊆ S (cid:48) and Log ( W, R, S ) =
Log ( W, R, S (cid:48) ) .Proof. (1 ⇒ τ R and τ S are Alexandroff. By Corollary3.7, Log ( W, R, S ) =
Log ( W, τ R , τ S ), and hence (cid:104) W, τ R , τ S (cid:105) is an IL -space. ByTheorem 5.6, there exists a Visser frame (cid:104) W, R (cid:48) , S (cid:48) (cid:105) such that R (cid:48) ⊆ S (cid:48) and Log ( W, R (cid:48) , S (cid:48) ) =
Log ( W, τ R , τ S ). Then Log ( W, R, S ) =
Log ( W, R (cid:48) , S (cid:48) ). Fur-thermore, since R is irreflexive and transitive, it is easily shown that for any x, y ∈ W , xRy ⇐⇒ x (cid:54) = y & ∀ U ∈ τ R ( x ∈ U ⇒ y ∈ U ) . From our proof of Theorem 5.6, R (cid:48) = R .(2 ⇒ (cid:113) In this paper, we newly introduced a topological semantics of CL and its ex-tensions, and proved the topological compactness theorem. As a consequence,we proved that the logics CL , CLM , IL , ILM , ILP and
ILW are stronglycomplete with respect to our topological semantics. These results are just thestarting point for research in this direction. Obviously, investigating the topo-logical completeness of other logics which are not listed above is an importantfurther task.As we have described in Section 3, we introduced our new topological se-mantics with Visser semantics in mind. Actually, we proved that every Visserframe can be considered as a topological frame (Corollary 3.7). Also, each Visserframe can be considered as a Veltman frame, but it is not known whether eachVeltman frame can be considered as a topological frame. In this regard, wepropose the following problem.
Problem 6.1.
Is there a normal extension L of CL such that L is complete withrespect to Veltman semantics but not with respect to our topological semantics? While CL and some of its extensions are strongly complete with respect toour semantics, they are not with respect to Veltman and Visser semantics. Thisseems to be an evidence that our semantics can provide more models than theserelational semantics. Then, we expect an affirmative answer to the followingproblem. Problem 6.2.
Is there a normal extension L of CL such that L is completewith respect to our semantics but not with respect to Veltman semantics? Visser [20] proved that the logics
ILP and
ILW have finite model propertywith respect to Visser semantics. That is, each of these logics is determined bya class of corresponding finite Visser frames. Therefore, these logics also havefinite model property with respect to our topological semantics. On the otherhand, Visser also proved that IL and ILM do not have finite model property25ith respect to Visser semantics (See also Visser [21]). Regarding this point,we propose the following problem.
Problem 6.3.
Do the logics CL , CLM , IL and ILM have finite model propertywith respect to our topological semantics?
In order to understand the properties of axioms of CL and IL in moredetail, the authors recently introduced several sublogics of them, and studiedtheir basic characters such as completeness with respect to relational semanticsand interpolation property ([9, 11]). We ask the following question about thesesublogics. Problem 6.4.
Can we develop a topological semantics for these sublogics of CL and IL ? Acknowledgement
The authors would like to thank Yuya Okawa for the valuable discussion. Thesecond author was supported by JSPS KAKENHI Grant Number JP19K14586.
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