Due the fact that the fermions are the "block particles" and the bosons are the "carriers" I just came out with the question that, why the "block particle" have half-integer spin and the "carriers" have an integer spin?
The fact that bosons have integer spin whereas fermions have half-integer is actually a result from the so-called spin-statistics theorem.
The definition of bosons and fermions is not in terms of spin, it is in terms of symmetry of the wave function under exchange of particles. The spin-statics theorem says that the wave function of an integer spin identical particles' system is symmetric under exchange of particles and therefore those are bosons. On the other hand the wave function of an half integer identical particles' system is antisymmetric under exchange of particles and thus they are fermions.
If you are interested in knowing why force carriers are bosons you can check this: Why are all force particles bosons?
why the "block particle" have half-integer spin and the "carriers" have an integer spin?
Everything of coursed is based on observations and measurements, which define your "block" particles particularly. Chemistry is an old science and it depends crucially on "block" particles. The existence of nuclei also is based on "block" particles . The existence of spin states is an observational fact.
The mathematical model that fits this contains the Pauli exclusion principle which led to the spin statistics theorem imposed on the quantum mechanical solutions for atoms.
Without the Pauli exclusion principle there would be no chemistry, no nuclei and thus the universe as we know it.
Bosons are an observational fact also, because photons are bosons. (Please note that all particles can be carriers of dp/dt in loops in Feynman diagrams as long as quantum numbers are conserved) . But if the carriers of the simplest interaction in lowest order , electron electron scattering for example, would not be bosons the quantum number exchanges would not be conserved. Thus the gauge theories which developed to explain observations have bosons as carriers of the simplest, lowest order, interactions between elementary particles, describing the different fundamental forces.