  The aim of this paper is to promote quantum logic as one of the basic tools
for analyzing human reasoning. We compare it with classical (Boolean) logic and
highlight the role of violation of the distributive law for conjunction and
disjunction. It is well known that nondistributivity is equivalent to
incompatibility of logical variables -- the impossibility to assign jointly the
two-valued truth values to these variables. A natural question arises as to
whether quantum logical nondistributivity in human logic can be tested
experimentally. We show that testing the response replicability effect (RRE) in
cognitive psychology is equivalent to testing nondistributivity -- under the
prevailing conjecture that the mental state update generated by observation is
described as orthogonal projection of the mental state vector (the projective
update conjecture of Wang and Busemeyer). A simple test of RRE is suggested. In
contrast to the previous works in quantum-like modeling, we proceed in the
state-dependent framework; in particular, distributivity, compatibility, and
RRE are considered in a fixed mental state. In this framework, we improve the
previous result on the impossibility to combine question order and response
replicability effects by using (von Neumann-L\"uders) projective measurements.
