  The tension between deduction and induction is perhaps the most fundamental
issue in areas such as philosophy, cognition and artificial intelligence. In an
influential paper, Valiant recognised that the challenge of learning should be
integrated with deduction. In particular, he proposed a semantics to capture
the quality possessed by the output of Probably Approximately Correct (PAC)
learning algorithms when formulated in a logic. Although weaker than classical
entailment, it allows for a powerful model-theoretic framework for answering
queries. In this paper, we provide a new technical foundation to demonstrate
PAC learning with multi-agent epistemic logics. To circumvent the negative
results in the literature on the difficulty of robust learning with the PAC
semantics, we consider so-called implicit learning where we are able to
incorporate observations to the background theory in service of deciding the
entailment of an epistemic query. We prove correctness of the learning
procedure and discuss results on the sample complexity, that is how many
observations we will need to provably assert that the query is entailed given a
user-specified error bound. Finally, we investigate under what circumstances
this algorithm can be made efficient. On the last point, given that reasoning
in epistemic logics especially in multi-agent epistemic logics is
PSPACE-complete, it might seem like there is no hope for this problem. We
leverage some recent results on the so-called Representation Theorem explored
for single-agent and multi-agent epistemic logics with the only knowing
operator to reduce modal reasoning to propositional reasoning.
