The projection lattice of the Hilbert space describing a quantum system is called "quantum logic" and defines a logic that does not satisfy the distributive law. Based on the quantum logic, a set theory can be developed, which is called "quantum set theory". Quantum set theory allows us to develop mathematics based on quantum logic. More precisely, to each proposition in classical mathematics, it is possible to assign the projection-valued truth value in quantum logic. This is called "quantum mathematics". The object x such that the truth value 1 is assigned to the classical mathematical proposition "x is a real number", i.e., a real number defined in quantum mathematics, is called a "quantum real number". In 1979, Gaisi Takeuti suggested the fact that there is a one-to-one correspondence between quantum real numbers and quantum observables (of the quantum system described by the quantum logic under consideration), which has been proven under the current formulation of "quantum set theory". Then, the projection-valued truth value of the relation between real numbers defined in quantum mathematics can be transferred between corresponding quantum observables. For example, for quantum observables X and Y, the projection-valued truth value of the equality relation X=Y is the greatest lower bound of the projections assigned to the proposition "X≤r ⟺Y≤ r" over all rational numbers r. It is known that this coincides with the projection onto the subspace {ψ∈H| f(X)ψ=f(Y)ψ for all f} of the Hilbert space H. Similarly, the projection-valued truth value ⟦X≤Y⟧ of the order relation X≤Y for quantum observables X and Y is given as the lower bound of the projections assigned to the proposition "Y≤r⇒X≤ r" over all rational numbers r. However, here is a problem about quantum conditionals. In classical logic, the conditional "P⇒Q" (if P then Q) is defined as "(not P) or Q", but in quantum logic, it is known that this definition is insufficient, and currently, three definitions for conditional are proposed as reasonable. In this presentation, I plan to report on the following: (1) For these three conditionals, the truth value assignment for the equality relation is unique, whereas for the order relation, different projections are assigned as the truth value of "X≤Y". (2) The relation "X≾Y" between X and Y defined as ⟦X≤Y⟧=1 is equivalent to the "spectral order" between X and Y introduced by Olson, (3) The meaning of the truth value ⟦X≤Y⟧ can be tested by a physical experiment; comparing the order relation of the measured values x and y obtained from successive projective measurements of X and Y performed in a state ψ where "X≤Y" holds, that is, in a state ψ where ⟦X≤Y⟧ψ=ψ holds.