S5 modal logic

Definition

Consider axioms of propositional logic to hold. In addition,
#incomplete

Knowledge operator $K_i$ of player $i$ satisfies following five properties, collectively known as Kripke's S5 System:

$K_i Y = Y$ : player knows $Y$ is set of all states of the world
$K_i A \cap K_i B = K_i (A \cap B)$ : if player knows event $A$ and knows event $B$ ten he knows event $A \cap B$ .
$K_i A \subseteq A$ : if player knows event $A$ then event $A$ obtains
$K_i K_i A = K_i A$ : if player knows event $A$ then he knows he knows event $A$ and vice versa
$(K_i A)^c = K_i((K_i A)^C)$ : if the player does not know event $A$ , then he knows he does not know event $A$ , and vice versa

This is a type of modal logic. It may be alternatively formulated using the language of modal operators,
#incomplete

$\mathbf{K}$ logic