The notions of logic, deduction, axiomatic systems and implication are investigated within the general framework of consequence relations (CRs). We distinguish between several types of CRs and define corresponding notions of deduction and of inclusion between logics. Given an axiomatic system, several CRs can naturally be associated with it according to these classifications. Each of them induces its own notion of derivable and admissible rules and of inclusion between systems. Several known systems are then identified as the minimal systems (according to some notion of inclusion) that contain an internal implication relative to the corresponding type of CR. To the same systems might naturally correspond other CRs as well. In the case of implicational linear logic, for example, these CRs have clear semantical interpretations and appropriate versions of the deduction theorem hold for them, but unlike the principal associated CR, they are not known to be decidable.
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