Abstract
The proof theory of a logic deals with the rules of inference corresponding to the logical connectives in the object language. As the theory of graded consequence is not a particular logic, rather a general scheme for logics dealing with multilayered many-valuedness; in this chapter, we shall present the necessary and sufficient conditions for getting specific rules corresponding to logical connectives of the object language. That is, we shall add connectives, say \(\#\), in the language and explore the necessary and sufficient conditions to get hold of particular rules concerning \(\#\). This study will help us to present a general scheme for generating different logics based on GCT. We shall show that the interrelation between the object-level algebraic structure and the meta-level algebraic structure determines the proof theory of a logic. That the many-valued logics can be obtained as a special case of this scheme will be presented at the end.
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