Question $3\mathrm{.}4$

$($i$)$ Express the following modal logic formula in English, where $K_i$ is read as "Agent i knows that". $($Assume there are $2$ agents$).$

$K_{2}not{K}_{1}pq$

$($ii$)$ Express the following sentence in modal logic, where $K_i$ is read as "Agent i knows that". $($Assume there are $2$ agents$)$

Agent $1$ knows that agent $2$ knows p and q$,$ or agent $2$ does not know p or q or agent $1$ does not know p or q$.$

$(2)$

Question $3\mathrm{.}5$

Using the basic natural deduction rules, the $$ introduction and elimination rules of the basic modal logic $K$ as well as the following three additional rules for KT$45$

$\backslash$table$[[,,$not$$