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Modal Logics for Topological Spaces

07/26/2000
by   Konstantinos Georgatos, et al.
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In this thesis we shall present two logical systems, MP and MP, for the purpose of reasoning about knowledge and effort. These logical systems will be interpreted in a spatial context and therefore, the abstract concepts of knowledge and effort will be defined by concrete mathematical concepts.

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1 Introduction

In this thesis we shall present two logical systems, and , for the purpose of reasoning about knowledge and effort. These logical systems will be interpreted in a spatial context and therefore, the abstract concepts of knowledge and effort will be defined by concrete mathematical concepts.

Our general framework consists of a set of possible worlds (situations, scenarios, consistent theories, etc.) A state of knowledge is a subset of this set and our knowledge consists of all facts common to the worlds belonging to this subset. This subset of possibilities can be thought as our view. Thus two knowers having distinct views can have different knowledge. This treatment of knowledge agrees with the traditional one ([Hin62], [HM84], [PR85], [CM86], [FHV91]

) expressed in a variety of contexts (artificial intelligence, distributed processes, economics, etc.)

Our treatment is based on the following simple observation

“a restriction of our view increases our knowledge.”

This is because a smaller set of possibilities implies a greater amount of common facts. Moreover, such a restriction can only be possible due to an increase of information. And such an information increase can happen with spending of time or computation resources. Here is where the notion of effort enters. A restriction of our view is dynamic (contrary to the view itself which is a state) and is accompanied by effort during which a greater amount of information becomes available to us (Pratt expresses a similar idea in the context of processes [Pra92].)

We make two important assumptions.

Our knowledge has a subject. We collect information for a specific purpose. Hence we are not considering arbitrary restrictions to our view but restrictions parameterized by possibilities contained in our view, i.e. neighborhoods of possibilities. After all, only one of these possibilities is our actual state. This crucial assumption enables us to express topological concepts and use a mathematical set-theoretic setting as semantics. Without such an assumption these ideas would have been expressed in the familiar theory of intuitionism ( [Hey56][Dum77][TvD88].) As Fitting points out in [Fit69]

“Let be a [intuitionistic, propositional] model. is intended to be a collection of possible universes, or more properly, states of knowledge. Thus a particular in may be considered as a collection of (physical) facts known at a particular time. The relation represents (possible) time succession. That is, given two states of knowledge and of , to say is to say: if we now know , it is possible that later we will know .”

Considering neighborhoods and, inevitably, points which parameterize neighborhoods, the important duality between the facts, which constitute our knowledge, and the possible worlds, where such facts hold, emerges.

The other assumption is that of indeterminacy. Each state of knowledge is closed under logical deduction. Thus an increase of knowledge can happen only by a piece of evidence or information given from outside. Our knowledge is external (a term used by Parikh to describe a similar idea in [Par87b].) This fact leads to indeterminacy (we do not know which kinds of information will be available to us, if at all) and resembles indeterminacy expressed in intuitionism through the notion of lawless sequence (see [Kre58][Tro77]) where, not surprisingly, topological notions arise.

To illustrate better these simple but fundamental ideas we present the following examples:

  • Suppose that a machine emits a stream of binary digits representing the output of a recursive function . After time the machine emitted the stream . The only information we have about the function being computed at this time on the basis of this (finite) observation is that

    As far as our knowledge concerns is indistinguishable from the constant function , where for all . After some additional time , i.e. spending more time and resources, might appear and thus we could be able to distinguish from . In any case, each binary stream will be an initial segment of and this initial segment is a neighborhood of . In this way, we can acquire more knowledge for the function the machine computes. The space of finite binary streams is a structure which models computation. Moreover, this space comprises a topological space. The set of binary streams under the prefix ordering is an example of Alexandrov topology (see [Vic89].)

  • A policeman measures the speed of passing cars by means of a device. The speed limit is km/h. The error in measurement which the device introduces is km/h. So if a car has a speed of km/h and his device measures km/h then he knows that the speed of the passing car lies in the interval but he does not know if the car exceeds the speed limit because not all values in this interval are more than . However, measuring again and combining the two measurements or acquiring a more accurate device he has the possibility of knowing that a car with a speed of km/h does not exceed the speed limit. Note here that if the measurement is, indeed, an open interval of real line and the speed of a passing car is exactly km/h then he would never know if such a car exceeded the speed limit or not.

To express this framework we use two modalities for knowledge and for effort. Moss and Parikh observed in [MP92] that if the formula

is valid, where is an atomic predicate and is the dual of the , i.e. , then the set which represents is an open set of the topology where we interpret our systems. Under the reading of as “possible” and as “is known”, the above formula says that

“if is true then it is for possible to be known”,

i.e. is affirmative. Vickers defines similarly an affirmative assertion in [Vic89]

“an assertion is affirmative iff it is true precisely in the circumstances when it can be affirmed.”

The validity of the dual formula

where is the dual of , i.e. , expresses the fact that the set which represents is closed, and hence is refutative, meaning if it does not hold then it is possible to know that. The fact that affirmative and refutative assertions are represented by opens and closed subsets, respectively, should not come to us as a surprise. Affirmative assertions are closed under infinite disjunctions and refutative assertions are closed under infinite conjunctions. Smyth in [Smy83] observed first these properties in semi-decidable properties. Semi-decidable properties are those properties whose truth set is r.e. and are a particular kind of affirmative assertions. In fact, changing our power of affirming or computing we get another class of properties with a similar knowledge-theoretic character. For example, using polynomial algorithms affirmative assertions become polynomially semi-decidable properties. If an object has this property then it is possible to know it with a polynomial algorithm even though it is not true we know it now.

Does this framework suffers from the problem of logical omniscience? Only in part. Expressing effort we are able to bound the increase of knowledge depending on information (external knowledge.) Since the modality which corresponds to knowledge is axiomatized by the normal modal logic of , knowledge is closed under logical deduction. However, because of the strong computational character of this framework it does not seem unjustified to assume that in most cases (as in the binary streams example) a finite amount of data restricts our knowledge to a finite number of (relevant) formulae. Even without such an assumption we can incorporate the effort to deduce the knowledge of a property in the passage from one state of knowledge to the other.

We have made an effort to present our material somewhat independently. However, knowledge of basic modal logic, as in [Che80], [HC84], or [Fit93], is strongly recommended.

The language and semantics of our logical framework is presented in Chapter 2. In the same Chapter we present two systems: and . The former was introduced in [MP92] and was proven complete for arbitrary sets of subsets. It soon became evident that such sets of subsets should be combined, whenever it is possible, to yield a further increase of knowledge or we should assume a previous state of other states of knowledge where such states are a possible. Therefore the set of subsets should be closed under union and intersection. Moreover, topological notions expressed in make sense only in topological models. For this reason we introduce an extension of the set of axioms of and we call it . In Chapter 3, we study the topological models of by semantical means. We are able to prove the reduction of the theory of topological models to models whose associated set of subsets is closed under finite union and intersection. Finding for each satisfiable formula a model of bounded size we prove decidability for . The results of this chapter will appear in [Geo93]. In Chapter 4, we prove that is a complete system for topological models as well as topological models comprised by closed subsets. We also give necessary and sufficient conditions for turning a Kripke frame into such a topological model. In Chapter 5, we present the modal algebras of and and some of their properties. Finally, in Chapter 6, we present some of our ideas towards future work.

2 Two Systems: and

In section 2.1 we shall present a language and semantics which appeared first in [MP92]. In section 2.2, we shall present the axiom system , introduced and proven sound and complete with a class of models called subset spaces in [MP92], and the axiom system , introduced by us, which we shall prove sound and complete for, among other classes, the class of topological spaces.

2.1 Language and Semantics

We follow the notation of [MP92].

Our language is bimodal and propositional. Formally, we start with a countable set of atomic formulae containing two distinguished elements and . Then the language is the least set such that and closed under the following rules:

The above language can be interpreted inside any spatial context.

Definition 1  Let be a set and a subset of the powerset of , i.e. such that . We call the pair a subset space. A model is a triple , where is a subset space and a map from to with and called initial interpretation.

We denote the set by . For each let be the set the lower closed set generated by in the partial order , i.e. .

Definition 2  The satisfaction relation , where is the model , is a subset of defined recursively by (we write instead of ):

If for all belonging to then is valid in , denoted by .

We abbreviate and by and respectively. We have that

Many topological properties are expressible in this logical system in a natural way. For instance, in a model where the subset space is a topological space, is open whenever is valid in this model. Similarly, is nowhere dense whenever is valid (cf. [MP92].)

Example. Consider the set of real numbers with the usual topology of open intervals. We define the following three predicates:

where
where
where
where

There is no real number and open set such that because that would imply and and there are no singletons which are open.

A point belongs to the closure of a set if every open that contains intersects . Thus belongs to the closure of , i.e every open that contains has a point in . This means that for all such that , , therefore . Following the same reasoning , since belongs to the closure of .

A point belongs to the boundary of a set whenever belong to the closure of and . By the above, belongs to the boundary of and .

A set is closed if it contains its closure. The interval is closed and this means that the formula is valid.

A set is dense if all opens contain a point of . The set of rational numbers is dense which translates to the fact that the formula is valid. To exhibit the reasoning in this logic, suppose that the set of rational numbers was closed then both and would be valid. This implies that would be valid which means that all reals would be rationals. Hence the set of rational numbers is not closed.

2.2 and

The axiom system consists of axiom schemes 1 through 10 and rules of table 1 (see page 1) and appeared first in [MP92].

Axioms All propositional tautologies , for
Rules


Table 1: Axioms and Rules of

The following was proved in [MP92].

Theorem 3

The axioms and rules of are sound and complete with respect to subset spaces.

We add the axioms 11 and 12 to form the system for the purpose of axiomatizing spaces closed under union and intersection and, in particular, topological spaces.

A word about the axioms (most of the following facts can be found in any introductory book about modal logic, e.g. [Che80] or [Gol87].) The axiom 2 expresses the fact that the truth of atomic formulae is independent of the choice of subset and depends only on the choice of point. This is the first example of a class of formulae which we are going to call bi-persistent and their identification is one of the key steps to completeness. Axioms 3 through 5 and axioms 6 through 9 are used to axiomatize the normal modal logics S4 and S5 respectively. The former group of axioms expresses the fact that the passage from one subset to a restriction of it is done in a constructive way as actually happens to an increase of information or a spending of resources (the classical interpretation of necessity in intuitionistic logic is axiomatized in the same way). The latter group is generally used for axiomatizing logics of knowledge.

Axiom 10 expresses the fact that if a formula holds in arbitrary subsets is going to hold as well in the ones which are neighborhoods of a point. The converse is not sound.

Axiom 11 is a well-known formula which characterizes incestual frames, i.e. if two points and in a frame can be accessed by a common point then there is a point which can be accessed by both and . It appeared in the equivalent form (in [MP92])

and was proved sound in subset spaces closed under (finite) intersection.

Obviously our attention is focused on axiom 12. It is sound in spaces closed under (finite) union and intersection as the following proposition shows.

Proposition 4

Axioms 1 through 12 are sound in the class of subset spaces closed under finite union and intersection.

Proof. Soundness for Axioms 1 through 11 is easy. For Axiom 12, suppose

Since , there exists such that

and, since , there exists and such that

We now have that (we assume closure under unions.) Thus

Therefore,

 

With the help of axiom 12 we are able to prove the key lemma 33 which leads to the DNF Theorem (page 35.) and this is the only place where we actually use it. Any formula, sound in the class of subset spaces closed under finite union and intersection, which implies the formula (note the difference from axiom 12)

where , and are theorems, can replace axiom 12.

3 A Semantical analysis of

3.1 Stability and Splittings

Suppose that is a set and a topology on . In the following we assume that we are working in the topological space . Our aim is to find a partition of , where a given formula “retains its truth value” for each point throughout a member of this partition. We shall show that there exists a finite partition of this kind.

Definition 5  Given a finite family of opens, we define the remainder of (the principal ideal in generated by) by

Proposition 6

In a finite set of opens closed under intersection, we have

for .

We denote with .

Proposition 7

If is a finite family of opens, closed under intersection, then

  1. , for ,

  2. , i.e. is a partition of . We call such an a finite splitting (of ),

  3. if and is an open such that then , i.e. is convex.

Every partition of a set induces an equivalence relation on this set. The members of the partition comprise the equivalence classes. Since a splitting induces a partition, we denote the equivalence relation induced by a splitting by .

Definition 8  Given a set of open subsets , we define the relation on with if and only if for all .

We have the following

Proposition 9

The relation is an equivalence.

Proposition 10

Given a finite splitting , i.e. the remainders of are the equivalence classes of .

We state some useful facts about splittings.

Proposition 11

If is a finite set of opens, then , its closure under intersection, yields a finite splitting for .

The last proposition enables us to give yet another characterization of remainders: every family of points in a complete lattice closed under arbitrary joins comprises a closure system, i.e. a set of fixed points of a closure operator of the lattice (cf. [GHK80].) Here, the lattice is the poset of the opens of the topological space. If we restrict ourselves to a finite number of fixed points then we just ask for a finite set of opens closed under intersection i.e. Proposition 11. Thus a closure operator in the lattice of the open subsets of a topological space induces an equivalence relation, two opens being equivalent if they have the same closure, and the equivalence classes of this relation are just the remainders of the open subsets which are fixed points of the closure operator. The maximum open in , i.e. , can be taken as the representative of the equivalence class which is the union of all open sets belonging to .

We now introduce the notion of stability corresponding to what we mean by “a formula retains its truth value on a set of opens”.

Definition 12  If is a set of opens then is stable for , if for all , either for all , or for all , such that .

Proposition 13

If , are sets of opens then

  1. if and is stable for then is stable for ,

  2. if is stable for and is stable for then is stable for .

Definition 14  A finite splitting is called a stable splitting for , if is stable for for all .

Proposition 15

If is a stable splitting for , so is

where .

The above proposition tells us that if there is a finite stable splitting for a topology then there is a closure operator with finitely many fixed points whose associated equivalence classes are stable sets of open subsets.

Suppose that is a topological model for . Let be a family of subsets of generated as follows: for all , if then , if then , and if then i.e. is the least set containing and closed under complements, intersections and interiors. Let be the set . We have . The following is the main theorem of this section.

Theorem 16 (Partition Theorem)

Let be a topological model. Then there exists a a set of finite stable splittings such that

  1. and , for all ,

  2. if then , and

  3. if is a subformula of then and is a finite stable splitting for ,

where , as above.

Proof. By induction on the structure of the formula . In each step we take care to refine the partition of the induction hypothesis. Rather long proof.  

Theorem 16 gives us a great deal of intuition for topological models. It describes in detail the expressible part of the topolocical lattice for the completeness result as it appears in Chapter 4 and paves the road for the reduction of the theory of topological models to that of spatial lattices and the decidability result of this chapter.

3.2 Basis Model

Let be a topology on a set and a basis for . We denote satisfaction in the models and by and , respectively. In the following proposition we prove that each equivalence class under contains an element of a basis closed under finite unions.

Proposition 17

Let be a topological space, and let be a basis for closed under finite unions. Let be any finite subset of . Then for all and all , there is some with and .

Corollary 18

Let be a topological space, a basis for closed under finite unions, and . Then

We shall prove that a model based on a topological space is equivalent to the one induced by any basis of which is lattice. Observe that this enables us to reduce the theory of topological spaces to that of spatial lattices and, therefore, to answer the conjecture of [MP92] : a completeness theorem for subset spaces which are lattices will extend to the smaller class of topological spaces.

Theorem 19

Let be a topological space and a basis for closed under finite unions. Let and be the corresponding models. Then, for all ,

3.3 Finite Satisfiability

Proposition 20

Let be a subset space. Let be a finite stable splitting for a formula and all its subformulae, and assume that . Then for all , all , and all subformulae of , iff .

Constructing the quotient of under is not adequate for generating a finite model because there may still be an infinite number of points. It turns out that we only need a finite number of them.

Let be a topological model, and define an equivalence relation on by iff

(a) for all , iff , and

(b) for all atomic , iff .

Further, denote by the equivalence class of , and let . For every let , then is a topology on . Define a map from the atomic formulae to the powerset of by . The entire model lifts to the model in a well-defined way.

Lemma 21

For all , , and ,

Proof. By induction on .  

Theorem 22

If is satisfied in any topological space then is satisfied in a finite topological space.

Observe that the finite topological space is a quotient of the initial one under two equivalences. The one equivalence is on the open subsets of the topological space, where is the finite splitting corresponding to and its cardinality is a function of the complexity of . The other equivalence is on the points of the topological space and its number of equivalence classes is a function of the atomic formulae appearing in . The following simple example shows how a topology is formed with the quotient under these two equivalences

Example: Let be the interval of real line with the the set

as topology. Suppose that we have only one atomic formula, call it , such that . then it is easy to see that the model is equivalent to the finite topological model , where

So the overall size of the (finite) topological space is bounded by a function of the complexity of . Thus if we want to test if a given formula is invalid we have a finite number of finite topological spaces where we have to test its validity. Thus we have the following

Theorem 23

The theory of topological spaces is decidable.

Observe that the last two results apply for lattices of subsets by Theorem 19.

4 Completeness for

Open subsets of a topological space were used in [MP92] and in the previous section to provide motivation, intuition and finally semantics for . But in this chapter we shall show that the canonical model of is actually a set of subsets closed under arbitrary intersection and finite union, i.e. the closed subsets of a topological space. However, these results are not contrary to our intuition for the following reasons: the spatial character of this logic remains untouched. The fact that the canonical model is closed under arbitrary intersections implies strong completeness with the much wider class of sets of subsets closed under finite intersection and finite union. Now, the results of the previous section allow us to deduce strong completeness (in the sense that a consistent set of formulae is simultaneously satisfiable in some model) also for the class of sets of subsets closed under infinite union and finite intersection, i.e. the open subsets of a topological space.

4.1 Subset frames

As we noted in section 2.1, we are not interpreting formulae directly over a subset space but, rather in the pointed product . The pointed product can be turned in a set of possible worlds of a frame. We have only to indicate what the accessibility relations are.

Definition 24  Let be a subset space. Its subset frame is the frame

where

and

If is a topology, intended as the closed subsets of a topological space, we shall call its subset frame closed topological frame.

Our aim is to prove the most important properties of such a frame. We propose the following conditions on a possible worlds frame with two accessibility relations

  1. is reflexive and transitive.

  2. is an equivalence relation.

  3. (ending points) has ending points with respect , i.e

    for all there exists such that for all if then .

  4. (extensionality condition) For all , if there exists such that and and

    for all such that there exist such that , and , and for all such that there exist such that , and ,

    then .

  5. (union condition) For all ,

    if there exists such that and , then there exists such that for all with then or .

  6. (intersection condition) For all ,

    if there exists such that for all then there exists such that for all with and for all and some then .

  7. The frame is strongly generated in the sense that

    there exists such that for all , .

We have the following observations to make about the above conditions. Conditions 1 to 6 and 8 are first order, while the intersection condition is not. The extensionality condition implies the following

for all such that and then

which implies that is the identity in . In view of the extensionality condition the relation is antisymmetric. So we can replace condition 1 with the condition that is a partial order.

Now, we have the following proposition

Proposition 25

If is a topological space then its closed topological frame satisfies conditions 1 through 8.

The above proposition could lead to the consequence that topological models are possible worlds models in disguise. But the following theorem shows that this is not the case. There is a duality.

Theorem 26

Let be a frame satisfying conditions 1 through 8. Then is isomorphic to a closed topological frame .

Note that, in the above definitions, we could have used equally well the equivalence class of under the equivalence induced by the symmetric closure of instead of the ending point of in . The above proofs show that the crucial conditions are conditions 1 through 5 and if we are to strengthen or relax the union and intersection conditions we get accordingly different conditions in the lattice of the set of subsets of the space. The same holds for condition 8. We only used this condition to show that there exists a top element, i.e. the whole space, and satisfy the hypothesis of the union condition. If we do not assume this condition the union of two subsets will belong to the set of subsets if they have an upper bound in it. We state this case formally without a proof because we are going to use it later.

Proposition 27
  1. Let be a subset space closed under infinite intersections and if have an upper bound in then . Then its frame satisfies conditions 1 through 7.

  2. A frame satisfying conditions 1 through 7 is isomorphic to a frame where as in (1).

4.2 On the proof theory of

We shall identify certain classes of formulae in . This approach is motivated by the results of Chapter 3. In fact, these formulae express definable parts of the lattice of subsets (see section 3.1.)

Definition 28  Let be the set of formulae generated by the following rules:

Let be the set .

Formulae in have the following properties

Definition 29  A formula of is called persistent whenever is a theorem (see also [MP92].)

A formula of is called anti-persistent whenever is persistent, i.e. (or, equivalently ) is a theorem.

A formula of is called bi-persistent whenever (or, equivalently ) is a theorem.

Thus the truth of bi-persistent formulae depends only on the choice of the point of the space while the satisfaction of persistent formulae can change at most once in any model. We could go on and define a hierarchy of sets of formulae where each member of hierarchy contains all formulae which their satisfaction could change at most times in all models.

All the following derivations are in (Axioms 1 through 12 — see table at page 1.)

Proposition 30

All formulae belonging to are bi-persistent.

Proof. We prove it by induction, i.e. bi-persistence is retained through the application of the formation rules of .  

A faster (semantical) proof would be “the initial assignment on atomic formulae extends to the wider class of ”! This implies that formulae in define subsets of the topological space.

Formulae in have similar properties as the following lemma show.

Lemma 31

If is bi-persistent then is persistent and is anti-persistent.

We prove some theorems of that we are going to use later.

Lemma 32

If is bi-persistent then

The following is the key lemma to the DNF Theorem and generalizes Axiom 12

Lemma 33

For all ,

where , are bi-persistent.

All formulae of can be expressed in terms of bi-persistent, persistent and antipersistent formulae by means of the following normal form.

Definition 34

  1. is in prime normal form (PNF) if it has the form

    where and is finite.

  2. is in disjunctive normal form (DNF) if it has the form , where each is in PNF and is finite.

We now give the formal analogue of the Partition Theorem.

Theorem 35 (Dnf)

For every , there is (effectively) a in DNF such that

The DNF theorem is the most important property of . An immediate corollary is that, as far as is concerned, we could have replaced the modality with , since the formulae in normal form are defined using these two modalities. Almost all subsequent proof theoretic properties are immediate or implicit corollaries of the DNF Theorem.

We close this section with the following proposition, which together with Axiom 11 shows that is equivalent to .

Proposition 36

For all ,

4.3 Canonical Model

The canonical model of is the structure

where

along with the usual satisfaction relation (defined inductively):

We write , if for all .

A canonical model exists for all consistent bimodal systems with the normal axiom scheme for each modality (as MP and .) We have the following well known theorems (see [Che80], or [Gol87].)

Theorem 37 (Truth Theorem)

For all and ,

Theorem 38 (Completeness Theorem)

For all ,

We shall now prove some properties of the members of . The DNF theorem implies that every maximal consistent theory of is determined by the formulae in and it contains, i.e. by and . Moreover, the set is determined by alone (this is the -case of the DNF theorem.)

The following definition is useful

Definition 39  Let . We say is an theory if is consistent and for all either or .

Let . We say is an theory if is consistent and for all either or .

Hence, is an theory and is an theory.

What about going in the other direction? When does an theory and theory determine an maximal consistent theory? When their union is consistent because in this case there is a unique maximal extension. To test consistency we have the following lemma.

Lemma 40

If and are an and theory respectively then is consistent if and only if

It is expected that since and theories determine maximal consistent sets they will determine their accessibility relations, as well.

Proposition 41

For all ,

From the above proposition we have that for all , if then and if then .

We write for the composition of the relation and , i.e. if , we write if there exists such that and . Similarly for .

For the composite relation and we have the following corollary of proposition 41

Corollary 42

For all ,

We shall now prove that the canonical model of satisfies the conditions of Section 4.1 on page 4.1.

We now have the following

Theorem 43

The canonical model of satisfies conditions 1 to 7 of Section 4.1 on page 4.1.

Corollary 44

The canonical frame of is isomorphic to a subset frame where is a subset space closed under infinite intersections and if have an upper bound in then .

By the construction of Theorem 26, consists of the ending points of the members of the domain of the canonical model. We define the following initial assignment

In this way the model is equivalent to the canonical model as a corollary of frame isomorphism.

Corollary 45

For all and we have

Proposition 46

The frame of a generated submodel is isomorphic to a closed topological frame.

Now as above we have the following corollary

Corollary 47

A submodel is equivalent to a closed topological model.

It is a well known fact that a modal system is characterized by the class of generated frames of the canonical frame.

Proposition 48

The system is (strongly) characterized by closed topological frames.

Since the axioms and rules of are sound for the wider class of subset spaces with finite union and intersection, we also have the following.

Proposition 49

The system is (strongly) characterized by subset frames closed under finite unions and intersections.

Now by Proposition 48 and 49, Corollary 18 and Theorem 19 of Chapter 3, where we proved the equivalence of a topological model with the model induced by a basis closed under finite unions, we have the following corollary

Corollary 50

The system is (strongly) characterized by open topological frames as well as subset frames closed under infinite unions and intersections.

THe following disjunction property holds for

for . Note that the disjunction rule does not hold for .

Proposition 51

provides the above rule of disjunction.

We can similarly prove a stronger disjunction property, namely

for .

Now we are able to prove the following

Theorem 52

The canonical model of is strongly generated.

By Theorem 52 we complete the set of conditions of page 1 which turn the frame of the canonical model into a closed subset frame. To summarize, we have the following corollary (note that the canonical subset model is of Corollary 44)

Corollary 53

The canonical subset model of is a topological space.

5 The Algebras of and

In this section we shall give a more general type of semantics for and . Every modal logic can be interpreted in an algebraic framework. An algebraic model is nothing else but a valuation of the propositional variables in a class of appropriately chosen algebras. We shall also make connections with the previous chapters.

5.1 Fixed Monadic Algebras

Interior operators were introduced by McKinsey and Tarski [MT44].

Definition 54  An interior operator on a Boolean algebra is an operator satisfying the conditions

To each interior operator we associate its dual , called closure operator.

Universal quantifiers were introduced by P. Halmos [Hal56].

Definition 55  A universal quantifier on a Boolean algebra is an operator satisfying the conditions

To each universal quantifier we associate its dual , called existential quantifier.

Definition 56  Let be an interior operator on a Boolean algebra . Let and , i.e. the fixed points of and respectively. Let then is a Boolean subalgebra of .

Definition 57  A fixed monadic algebra (FMA) is a Boolean algebra with two operators and satisfying