# Can an agent know false information in Epistemic Logic?

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!

• Is he saying that beliefs some subset of all statements that are not falsifiable? – Rahul Madhavan Apr 11 at 12:55
• There is no clear definition of beliefs given. @RahulMadhavan Anyway, if beliefs were not falsifiable, they would be truths - which doesn't make sense. – epsilon-emperor Apr 11 at 12:56