Logical 0-1 laws in random graphs are an exciting set of results in the intersection of mathematical logic, probability theory, and graph theory. The two fundamental results of the field: The first (Fagin, 1976) is that every first-order logic property of graphs is either true or false w.h.p in the random graph G(n,1/2) of the Erdős–Rényi model. The second (Shelah and Spencer, 1988), states the same, just for the random graph G(n,n^{-alpha}) where alpha is irrational. In this lecture, I will present the generalization of the logical 0-1 laws from random graphs to random simplicial complexes of the multi-parameter model, a higher dimensional generalization of the Erdős–Rényi model. The lecture is based on my master thesis, under the supervision of Ron Rosenthal.