Topological Interpretation of Modal Logic vs Subset Space Logic In this small section, we will simply note the expressive power of subset space logic over topological interpretation of modal logic. In order to achieve this, we will define a translation map t : L → L
S from the basic modal
language Lto the basic subset space languageLS as follows:
pt=p for propositional letterp
(ϕ∧ψ)t =ϕt∧ψt for the formulaeϕ, ψ inL
(¬ϕ)t =¬(ϕt) for the formulaeϕ inL
(I(ϕ))t=♦Kϕt for the formulaeϕ inL
Observe that, in the translation map, Proposition 3.1.1 is used implicitly. The translation mapt shows that L is a fragment of the bimodal language
LS. Furthermore, the lemma below shows that LS is more expressive by
showing thattis not ontoL
S.
Proposition 3.1.4. The sentenceLpfor the propositional lettersp, is not equal to the translation of any formula ϕfrom the basic modal languageL.
Proof. Consider two models with R as their universe. In the first model M1 supposep does not hold anywhere whereas in the second model M2,
p holds only at the pointπ. It is not difficult to see that the interpretation of any basic modal formulaϕare the same inM1 and M2, except possibly at the point π. In particular, for instance, for 0 ∈ R, we have 0 |=M1 ϕ
if and only if 0 |=M2 ϕ for each ϕ ∈ L. Now substitute, Lp for ϕ. In
M1 there is no point in which p holds, so Lp cannot be true anywhere. However, inM2,pis true at the pointπ. Therefore, we have06|=M1 Lpand
0 |=M2 Lp. Contradiction shows that, Lp cannot be definable in the basic
modal language L.
Hence, the subset space language LS is more expressive than basic
modal languageLas there are formulae in LS which are not expressible in
L under standard translation. Hence,tis not ontoL
S.
3.2
Validity Preserving Operations
Recall that compact topological spaces and continuous topological spaces are not modally definable (Cate et al., 2006). But are they definable in subset space logic? In order to be able to discuss the definability of com- pactness and connectedness, we first need to consider the validity preserv- ing operations in subset space logics. We again note that, we will consider the topologic subset spaces in order to be able to talk about the notions
3.2. VALIDITY PRESERVING OPERATIONS
of compactness and connectedness. Otherwise, these notions do not make sense at all in an non-topological subset space.
Some basic validity preserving operations can easily be defined in the context of subset space logic. Let us start with the simplest one.
Disjoint Unions Disjoint unions are perhaps the most intuitive way to obtain larger structures. We start by defining the disjoint unions for the general case as follows.
Definition 3.2.1 (Disjoint Unions). Two subset space models are disjoint if their domain contains no common element. For disjoint subset space models
Si = hSi, σi, vii, for i ∈ I their disjoint union is the structureS = Ui∈ISi =
hS, σ, viwhereS=S i∈ISi,σ= S i∈Iσi andv(p) = S i∈Ivi(p).
If the subset spaces we want to put together for disjoint unions have some common elements, the easiest way to overcome this problem is to index these members in order to make them distinct.
The details for the construction of disjoint unions for basic modal lan- guage with some easy examples can be found in (Blackburn et al., 2001). The key proposition for disjoint unions is about the truth invariance of modal formulae under the construction of disjoint unions. The proof of the aforementioned proposition can also be found in (Blackburn et al., 2001). Furthermore, we can easily import this result to the subset spaces.
Theorem 3.2.1 (Invariance Under Disjoint Unions for Subset Spaces). For disjoint subset space models Si for i∈I and for each neighborhood situation
(s, U)in Si, we haves, U |=S ϕ if and only ifs, U |=Si ϕ, for each formulaϕ
in the language of subset space logicLS.
Proof. The proof is by induction on the length of the formulae. The case for propositional variables and Boolean cases are easy and hence skipped.
Let us consider the case forϕ≡Kψ. Assumes, U |=Si Kψ. Then for each t ∈U, we havet, U |=Si ψ. However, by induction hypothesist, U |=S ψ. As
σ =S
i∈Iσi,(t, U)is a neighborhood situation inS. So,s, U |=S Kψ. Converse direction is similar.
The case forϕ≡ψ is also very similar. However, if we want to discuss whether some topological properties are definable in subset space logic, then we have to require that the underlying space should have topology, not just a collection of subsets.
Therefore we need to revise the Definition 3.2.1 in such a way that obtained disjoint union will be a topological space.
3.2. VALIDITY PRESERVING OPERATIONS
Definition 3.2.2 (Topologic Disjoint Unions). For disjoint topologic subset space models Si = hSi, σi, vii, for ,∈ I their disjoint union is the structure
S =U i∈ISi =hS, σ, viwhereS = S i∈ISi,σ ={U ∈S :∀i∈I, U∩Si ∈σi} andv(p) =S i∈Ivi(p).
It is then easy to check topological disjoint unions are truth preserving. Then, borrowing the ideas from topological interpretation, it is easy to see the following.
Proposition 3.2.1. Compactness and connectedness are not definable in topo- logic subset space logic.
Proof. We refer the interested reader to (Cate et al., 2006) as there is no need to reproduce the same proof here.
Generated Subset Spaces Disjoint unions enable us to obtain larger spaces from smaller ones. However, we can also do the reverse. In other words, we can throw away the points in such a way that the satisfiability relation will not be affected after this operation. But, what kind of points we can throw away? Recall that, in subset spaces, we interpret the formulae at the neighborhood situations (s, U)wheres ∈ U ∈ σ. Therefore, we claim that we can throw away the points which are not in any of the observation sets inσ. The reason for that is the fact that we cannot evaluate the formulae at these points as there are no observation sets attached to them. Intuitively, they are the points about which we did not make any observation, hence there is no harm to throw them away.
Proposition 3.2.2. LetS =hS, σ, vibe a subset space model. LetS0 =S−{s:
s /∈ ∪σ}andv0(p) = v(p)∩S0. ThenS0 =hS0, σ, v0iandS =hS, σ, vi satisfy
the same formulae.
Proof. It is easy to show thats, U |=S0 ϕ impliess, U |=S ϕ asS0 ⊆S.
For the converse direction, assumes, U |=S ϕ. Ass∈U ∈σ, we observe
s ∈ ∪σ. Hence s∈S0. The proof goes by induction on the complexity ofϕ. The propositional and Boolean cases are easy and hence skipped. However, the idea forϕ ≡ Kand ϕ≡ ψ is very similar. We will only show the case for ϕ≡K, and leave the other case to the reader.
Assume, s, U |=S Kψ. Then for each t ∈ U, we have t, U |=S ψ. As
t ∈ U ∈ σ, we observe t ∈ S0 for each t ∈ U. Then by the induction hypothesis,t, U |=S0 ψ. Ass∈S0 as well, we concludes, U |=S Kψ.
We can go further. We can consider a situation in which we are only interested in the observations and knowledge around one specific neigh- borhood situation (s, U). We can then throw away the observation sets
3.2. VALIDITY PRESERVING OPERATIONS
which have an empty intersection with the designated observation set U. We will call this construction generated subset spaces.
Definition 3.2.3 (Generated Subset Spaces). Let S = hS, σ, vi be a subset space model. Let (s, U) be the designated neighborhood situation. Then we obtain the generated subset spaceS0 =hS0, σ0, v0iof S as follows.
• σ0 :=σ− {V ∈σ :V 6⊆U} • S0 :=S− ∪σ0
• v0(p) := v(p)∩S0 for each propositional letterp.
Proposition 3.2.3 (Invariance Under Generated Subset Spaces). For each
s ∈S0, we haves, U |=S ϕif and only ifs, U |=S0 ϕ.
Proof. We leave the easy proof to the reader.
Bounded Morphism The third validity preserving operation we will im- port from the basic modal language is bounded morphism. However, in the context of subset spaces, we will diverge a bit from the familiar Kripkean notion of morphisms. We will here call a functiontopologic-continuous func- tion if the inverse image of an observation set is again an observation set. Likewise we will call a function topologic-open function if the image of an observation set is still an observation set. Now, we will adopt the definition of bounded morphisms from (Cate et al., 2006).
Definition 3.2.4 (Bounded Morphism for Subset Space Logic). Let S =
hS, σ, viandS0 =hS0, σ0, v0ibe two subset spaces. Letf be an topologic-open
and topologic-continuous map from S to S0. Then s, U |=
S ϕ if and only if
s0, U0 |=S0 ϕfor each formulaϕ. We define v(p) := f−1(v0(p)).
Again, some basic facts and results about the bounded morphisms for the basic modal language can be found in (Blackburn et al., 2001). Some further observations about bounded morphism in the context of topological interpretation of modal logic can be found in (Cate et al., 2006).
Henceforth, we can claim the following invariance result.
Theorem 3.2.2(Invariance Under Bounded Morphism). Letfbe a bounded morphism from S = hS, σ, vi onto S0 = hS0, σ0, v0i. Then s, U |=
S ϕ if and
only if f s, f U |=S0 ϕfor each formula ϕ in the language of subset space logic