Critical elements in algebras of numerical events

The probability [Formula: see text] of the occurrence of an event pertaining to a physical system which is observed in different states [Formula: see text] determines a function [Formula: see text] from the set [Formula: see text] of states of the system to [Formula: see text]. Function [Formula: see text] is called a numerical event or more precisely an [Formula: see text]-probability. When appropriately structured, sets [Formula: see text] of numerical events form so-called algebras of [Formula: see text]-probabilities, which are orthomodular posets that can serve as quantum logics. If one deals with a classical physical system, then [Formula: see text] will be a Boolean algebra. Starting with a supposed quantum logic or logics obtained by individual measurements, newly added numerical events may turn out to be crucial for the assumption of classicality or nonclassicality of the physical system. We will call those numerical events critical and will study their impacts among various classes of algebras of numerical events. Moreover, we will consider the situation of enlarging [Formula: see text] by new states and will describe how elements of an algebra of [Formula: see text]-probabilities contribute to the character of a logic when there exists a certain relation between them and the element that is newly added.