# 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.

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How to Cite
Arnon A. Implication, Equivalence, and Negation // Logicheskie Issledovaniya / Logical Investigations. 2021. VOL. 27. № 1. C. 31-45.
Section
Papers

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