We study a conditional state on a quantum logic using Renyi's approach (or Bayesian principle). This approach helps us to define independence of events and differently from the situation in the classical theory of probability, if an event
a is independent of an event
b, then the event
b can be dependent on the event
a. We will show that we can define a
s-map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the
s-map and conversely. By using the
s-map we can introduce joint distribution also for noncompatible observables on a quantum logic.
The present paper is devoted to modelling of a probability measure of logical connectives on a quantum logic via a G-map, which is a special map on it.We follow the work in which the probability of logical conjunction (AND), disjunction (OR), symmetric difference (XOR) and their negations for non-compatible propositions are studied.Now we study all remaining cases of G-maps on quantum logic, namely a probability measure of projections, of implications, and of their negations.We show that unlike classical (Boolean) logic, probability measures of projections on a quantum logic are not necessarilly pure projections.We indicate how it is possible to define a probability measure of implication using a G-map in the quantum logic, and then we study some properties of this measure which are different from a measure of implication in a Boolean algebra.Finally, we compare the properties of a G-map with the properties of a probability measure related to logical connectives on a Boolean algebra.