  In this paper, we delve into the study of epistemic logics, interpreted
through similarity models based on weighted graphs. We explore eight languages
that extend the traditional epistemic language by incorporating modalities of
common, distributed, and mutual knowledge. The concept of individual knowledge
is redefined under these similarity models. It is no longer just a matter of
personal knowledge, but is now enriched and understood as knowledge under the
individual's epistemic ability. Common knowledge is presented as higher-order
knowledge that is universally known to any degree, a definition that aligns
with existing literature. We reframe distributed knowledge as a form of
knowledge acquired by collectively leveraging the abilities of a group of
agents. In contrast, mutual knowledge is defined as the knowledge obtained
through the shared abilities of a group. We then focus on the resulting logics,
examining their relative expressivity, semantic correspondence to the classical
epistemic logic, proof systems and the computational complexity associated with
the model checking problem and the satisfiability/validity problem. This paper
offers significant insights into the logical analysis and understanding of
these enriched forms of knowledge, contributing to the broader discourse on
epistemic logic.
