Implication, equivalence, and negation

A system HCL:$ in the language of f:;$g is obtained by adding a single negation-less axiom schema to HLL:→ (the standard Hilbert-type system for multiplicative linear logic without propositional constants), and changing → to $. HCL:$ is weakly, but not strongly, sound and complete for CL:$ (the f:;$g-fragment of classical logic). By adding the Ex Falso rule to HCL:$ we get a system with is strongly sound and complete for CL:$ . It is shown that the use of a new rule cannot be replaced by the addition of axiom schemas. A simple semantics for which HCL:$ itself is strongly sound and complete is given. It is also shown that LHCL:$ , the logic induced by HCL:$ , has a single non-trivial proper axiomatic extension, that this extension and CL:$ are the only proper extensions in the language of f:;$g of LHCL:$ , and that LHCL:$ and its single axiomatic extension are the only logics in f:;$g which have a connective with the relevant deduction property, but are not equivalent to an axiomatic extension of R:→ (the intensional fragment of the relevant logic R). Finally, we discuss the question whether LHCL:$ can be taken as a paraconsistent logic.