This paper presents a dichotomy theorem for the complexity of satisfiability problems. It shows that for any finite set of logical relations, the corresponding satisfiability problem is either polynomial-time decidable or NP-complete. The paper also considers an analogous class of problems involving quantified formulas, which are either polynomial-time decidable or complete in polynomial space. The classification of the polynomial-time decidable cases yields new problems that are complete in polynomial time and in nondeterministic log space. The paper also provides a detailed analysis of various specific satisfiability problems, including the one-in-three satisfiability problem, the not-all-equal satisfiability problem, and the two-colorable perfect matching problem. The results are supported by a series of lemmas and theorems that establish the complexity of these problems and their relationships to other known complexity classes. The paper concludes with a discussion of the implications of these results for the broader study of computational complexity.This paper presents a dichotomy theorem for the complexity of satisfiability problems. It shows that for any finite set of logical relations, the corresponding satisfiability problem is either polynomial-time decidable or NP-complete. The paper also considers an analogous class of problems involving quantified formulas, which are either polynomial-time decidable or complete in polynomial space. The classification of the polynomial-time decidable cases yields new problems that are complete in polynomial time and in nondeterministic log space. The paper also provides a detailed analysis of various specific satisfiability problems, including the one-in-three satisfiability problem, the not-all-equal satisfiability problem, and the two-colorable perfect matching problem. The results are supported by a series of lemmas and theorems that establish the complexity of these problems and their relationships to other known complexity classes. The paper concludes with a discussion of the implications of these results for the broader study of computational complexity.