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K3 says that i knows a conjunction if, and only if, i knows each conjunct.
K4 is a reflection axiom, sometimes also presented as the axiom of transparency (or of positive introspection), which says that if i knows A, then i knows that she knows A. Finally, K5 says that if the agent does not know an event, then she knows that she does not know.
I will review this definition here, and then show that it is logically equivalent to the 'i knows that j knows that … k knows that A' hierarchy that Lewis (1969) and Schiffer (1972) argue characterizes common knowledge.[9] In words, K1 says that if i knows A, then A must be the case.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com