# Equivalence of strings of modal operators in modal logic

I'm trying to solve a question which asks me to show that for any two finite strings O₁ and O₂ of □s and ◊s, (e.g. □□◊□◊□), that

i) if O₁≡O₂ then OO₁≡OO₂

and

ii) if O₁≡O₂ then O₁O≡O₂O

where O is any other string and we say O₁≡O₂ (they express the same modality (under some modal logic system) if ⊨(O₁ϕ↔O₂ϕ).

I've managed to solve the first part quite easily using the fact that if ⊨ϕ₁↔ϕ₂ then ⊨χ(ϕ₁)↔χ(ϕ₂) where χ(ϕ) is just the result of replacing P with ϕ in the formula χ(P).

I'm not sure how to go about solving the second though, because in this case the new string is inserted between the old one and the sentence ϕ, and I can't see an obvious way to do that with this formula.

I'm also trying to find a way to show, using these, that under the modal system B (where the accessibility relation is reflexive and symmetric), that there are infinitely many modalities, using some sort of informal semantic argument.

I've done the same for S4 and S5, but those have finite numbers of modalities so the strategy was different.

I'd really appreciate any help you could offer!

• Regarding your question about B, what do you mean by "infinite modalities"? That there are infinitely many nonequivalent strings of modal operators? A natural approach would be to exploit that the truth of a formula Oϕ with string O of length n is already determined by all worlds with distance at most n (you hopefully are able to define distance yourself) from the world you are evaluating in, so strings of different lengths can be shown to be nonequivalent using models of distinct diameter. – Jishin Noben Jan 21 at 17:50

## 1 Answer

If O₁≡O₂ then O₁O≡O₂O

We have to show that ⊨O₁Oϕ↔O₂Oϕ. But by definition ⊨O₁ψ ↔O₂ψ for arbitrary ψ Choose ψ=Oϕ and you get exactly what you want.

Infinitely many different modalities in B

For a model of B, there is an underlying graph, which has a node for each world and an edge between two worlds, if one of them is accessible to the other (since accessibility in B is symmetric, this graph is not directed). Then there is a natural notion of distance d(w,v) between worlds w and v, namely

d(w,v) is the smallest number of edges that make up a path from w to v

For example if you look at the model below d(w₀ , wₙ)=n. Another example: in a S5-model two distinct worlds have distance 1 from each other. Now write ☐ⁿ for n copies of ☐ (and similar for ◇). Then one can easily show:

w ⊨ ☐ⁿφ if for all worlds v with d(w,v)≤n: v⊨φ

(Of course similarly w ⊨ ◇ⁿφ if for some world v with d(w,v)≤n: v⊨φ, but this is not needed here). Using this, if we look at the model then w₀ ⊨ ☐ⁿ⁻¹p, but w₀ ⊭☐ⁿ⁺ⁱp for i≥0. So the strings ☐ⁿ⁻¹ and ☐ⁿ⁺ⁱ are not equivalent and you get infinitely many non-equivalent modal strings.