This book presents a pioneering investigation of the algebraic foundations of non-classical logics. Unifying paraconsistent, paracomplete, and non-alethic structures introduces Curry Systems as a versatile and unifying framework that transcends the limitations of traditional algebraization. Through the formal development of the Cn-, Pn-, Nn-, and Pτ-algebras, as well as their first-order extensions, this book offers pre-algebraic tools for modeling contradiction, indeterminacy, and non-alethic reasoning. Essential reading for logicians, mathematicians, philosophers of logic, and researchers in artificial intelligence, this book demonstrates how algebraic methods can rigorously capture reasoning under inconsistency and incompleteness. At once a comprehensive reference and intellectual invitation, it highlights the important role of pre-algebraic structures in the development of logical theory and intelligent systems.