Abstract

In his seminal paper ‘Truth’, M. Dummett considered negated conditional statements as one of the main motivations for introducing a three-valued logical framework. He left a sketch of an implication connective that, as we observe, shares some intuitions with Wansing-style account for connexivity. In this article, we discuss Dummett’s ‘unfinished’ implication and suggest two possible reconstructions of it. One of them collapses into implication from W. Cooper’s ‘Logic of Ordinary Discourse’ $\textbf{OL}$ and J. Cantwell’s ‘Logic of Conditional Negation’ $\textbf{CN}$, whereas the other turns out to be previously unknown implication connective and can be used to obtain a novel logical system, entitled here as $\textbf{cRM}_\textbf{3}$. As to the technical results, we introduce a sound and complete axiomatic proof-system for $\textbf{cRM}_\textbf{3}$ and present a theorem for the semantic embedding of $\textbf{CN}$ into $\textbf{cRM}_\textbf{3}$.