This sixth volume of the book series applies finite injury priority method to R-calculi and obtain (in)completeness theorem for binary-valued, Post three-valued, B2^2-valued and L4-valued first-order logics, and extend the method to infinite injury priority method and 0"-method for default logic to produce pseudo-extensions of a default theory, corresponding to different R-calculi. Finite injury priority method and tree constructions are discussed in this book. This book offers a rich blend of theory and practice. It is suitable for students, researchers and practitioners in the field of logic.