We present a non-commutative linear logic | the logic of predicates with equality and explicit substitutions. Thus, the position of linear logic with respect to the usual logic is given a new explanation. 1 The world according to Girard A recent introduction to linear logic, cf. 13], starts with the following explanation of the position of usual logic with respect to the linear. Linear logic is...

Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, directed acyclic graphs formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type structural restriction proofs rule-based systems. In this definiti...