The meaning of these two depends on the context. An antecedent is just something that comes first. In logic, this often means the first proposition of an "if...then..." statement. On the other hand, when I hear determinant I think of the mathematical concept, but that might be because I'm a mathematician. The dictionary says that it also means a "determining agent/factor". For example, if I'm making a decision about buying a house, the determining factor of my choice is the factor the causes my choice. If I'm poor, the determining factor may be the price