Introduction, Elimination Rules for ¬ and ⊃: A Study from Graded Context

This paper is aimed to study the algebraic background of some proof theoretic rules in a set up where distinct levels of logic activity have been maintained carefully. In this regard, Introduction, Elimination rules for ¬, and ⊃ have been considered as specific cases whose necessary and sufficient conditions from the perspective of graded consequence will reveal a new analysis.