Can an agent know false information in Epistemic Logic?

Question

I've been reading Epistemic Logic from Fagin et. Al's Reasoning about Knowledge, and I came across the following discussion:

$\varphi \implies K_i\varphi$ is not valid. $\varphi \implies K_i\varphi$ says that if $\varphi$ is true then agent $i$ knows $\varphi$. An agent does not necessarily know all things that are true (eg. in the muddy children puzzle, the children with muddy foreheads initially did not know that their foreheads are muddy). However, the Knowledge Generalization axiom above does tell us that all agents do know all the valid formulas. In other words, $$\text{ For all structures }M, M\vDash\varphi \implies M\vDash K_i\varphi$$ Although an agent may not know facts that are true, it is the case that if he knows a fact, then it is true. We have $$\vDash K_i\varphi \to \varphi$$ which is referred to as the Knowledge Axiom or the Truth Axiom. This axiom distinguishes knowledge from belief - although you may have false beliefs, you cannot know something that is false.

I understand what is going on, but the last sentence (and related others) bother me. If an agent cannot know something that is false, how is prevarication modeled in Epistemic Logic? In the real world, people lie, and it is pretty much possible to know false information. Are we assuming that everyone involved is necessarily truthful? What's the hidden subtlety that I am apparently missing?

Thanks for the clarification in advance!

Answer 1

This is simply part of the definition of the word "know". In this context, we are declaring that if a statement is not true, then you do not know it, no matter how strongly you believe it. This may or may not coincide with your understanding of the English word "know" in ordinary language, but it is the meaning that is being modeled here. If you want to talk about "knowing" statements that might actually not be true, then this is referred to as believing instead of knowing.

Answer 2

You are right in that agents may believe some fact that is not actually true. In this case, as the authors say, the truth axiom does not hold. Moreover, the epistemic relation in models ceases to be the equivalence.

If we are talking about knowledge in the context of epistemic logic, then the notion of knowledge is pretty strong. First, if an agent knows some fact, then this fact is true. This is captures by the truth axiom. Second, agents indeed know all logical validities. This is usually called the logical omniscience.

Since the framework of epistemic logic is static, there is no sense in discussing whether agents are truthful or not; there is simply no communication involved for this to play a role.

In a dynamic setting, where agents can perform epistemic actions, like public or private announcements, whether agents are truthful does make the difference. However, in such a setting, we usually do not speak about knowledge, and rather we speak about belief, as agents may believe true facts.

All in all, in the context of this type of modal logics, everything that is known is true in the world, and some things that are believed can be false.