Implication, Equivalence, and Negation

Abstract

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