 The first three reduction axioms are nearly identical to the corresponding reduction axioms for \(\PAL\), except that the first and third \(\EAL\) reduction axioms check the truth of a precondition in the place where the \(\PAL\) reduction axioms would check the truth of the formula to be announced. This is actually the same kind of check: for an event, the precondition must hold in order for the event to be executable; for a public announcement, the formula must be true in order for the public announcement to occur (and hence for the public announcement event in question to be “executable”). The major difference between the \(\PAL\) and \(\EAL\) reduction axioms is in the fourth \(\EAL\) reduction axiom. This axiom specifies the conditions under which an agent has belief (or knowledge) of something after the occurrence of an action. In particular, adopting a doxastic reading for this discussion, the axiom says that agent a believes G after the occurrence of action \((A,e)\) if and only if the formula   \[ \textstyle \pre(e)\to\bigwedge_{e R_af}[a][A,f]G \]   is true. This formula, in turn, says that if the precondition is true—and therefore the action is executable—then, for each of the possible events the agent entertains, she believes that G is true if the event in question occurs. This makes sense: a cannot be sure which of the events has occurred, and so for her to believe something after the action has occurred, she must be sure that this something is true no matter which of her entertained events might have been the actual one. For example, if a sees her friend b become elated as he listens to something he hears on the other side of a private phone call, then the a may not know exactly what it is that b is being told; nevertheless, a has reason to believe that b is receiving good news because, no matter what it is exactly that he is being told (i.e., no matter which of the events she thinks that he may be hearing), she knows from his reaction that he must be receiving good news. 