# Why is the possibility relation $\mathcal K_i$ an equivalence relation on $S$?

I have just begun reading about Epistemic Logic from Reasoning About Knowledge, Fagin et. al. $$\mathcal K_i$$ is defined as the possibility relation between two worlds, and the author says that for the most part - one would want $$\mathcal K_i$$ to be an equivalence relation. I do not understand the author's intuition.

Here is some setup to give you background - there are $$n$$ agents, $$\{1,2,...,n\}$$. As usual, $$\Phi$$ denotes a set of propositional variables, and $$S$$ is a non-empty set of states/possible worlds for the Kripke structure. $$\pi$$ is a valuation, which associates each state in $$S$$ with a map as follows: $$\pi(s): \Phi\to\{T,F\}$$ Then, the author says the following about $$\mathcal K_i$$:

The binary relation $$\mathcal K_i$$ is intended to capture the possibility relation according to agent $$i$$: $$(s, t) \in \mathcal K_i$$ if agent $$i$$ considers world $$t$$ possible, given his information in world $$s$$. We think of $$\mathcal K_i$$ as a possibility relation, since it defines what worlds agent $$i$$ considers possible in any given world. Throughout most of the book (in particular, in this chapter), we further require that $$\mathcal K_i$$ be an equivalence relation on $$S$$. We take $$\mathcal K_i$$ to be an equivalence relation since we want to capture the intuition that agent $$i$$ considers $$t$$ possible in world $$s$$ if in both $$s$$ and $$t$$ agent $$i$$ has the same information about the world, that is, the two worlds are indistinguishable to the agent. Making $$\mathcal K_i$$ an equivalence relation seems natural and turns out to be the appropriate choice for many applications.

I find it difficult to understand what intuition the authors are trying to capture with such a definition. Could someone help me better understand the motivation and reason behind such an assumption? Thanks a lot!

• What exactly do you not understand about the quoted passage? I'm not sure what explanation you want that would not just be restating what it says... – Eric Wofsey Mar 24 at 16:35
• In particular, the passage quite directly answers your question about what notion/intuition they are trying to capture: namely, they are trying to capture the notion that "agent $i$ considers $t$ possible in world $s$" means that worlds $s$ and $t$ are indistinguishable to $i$. What about that is confusing you? – Eric Wofsey Mar 24 at 16:43
• I don't see why we would want the two worlds to be indistinguishable to the agent. Could you throw some light on that? How does the possibility relation connect with indistinguishability? – epsilon-emperor Mar 24 at 17:05
• I mean, you can just take it as a definition. This is what we are defining the word "possible" to mean. – Eric Wofsey Mar 24 at 17:07