The Disjunction-Free Fragment of D2 is Three-Valued

In this article, the disjunction-free fragment of Jakowski's discussive logic D 2 in the language of classical logic is shown to be complete with respect to three-and four-valued semantics.As a byproduct, a rather simple axiomatization of the disjunction-free fragment of D 2 is obtained.Some implications of this result are also discussed. Semantics and proof theoryThe propositional languages in this article consist of a finite set S of propositional connectives and a countable set Prop of propositional variables.The languages are referred to asand {, d , l d , }, respectively.Note that the languages L and L r were introduced by Jakowski in [18] and [19], respectively. 1* The main result was presented with a rather different narrative at Logic in Bochum 2, CCPEA 2016 in Seoul, Paradoxes, Logic and Philosophy in Beijing, V Workshop on Philosophical of Logic in Buenos Aires, Prague Seminar on Paraconsistent Logic, a colloquium in Munich and ISRALOG17 in Haifa.I owe a special debt of gratitude to Dave Ripley whose comments led me to rethink the overall presentation of the main result.An earlier version of this article was presented at: Non-classical modalities in Mexico City, the Eleventh Smirnov Readings in Logic in Moscow, CoPS-FaM-2019 in Gdask, Paris-Bochum-Moscow Workshop in Mathematical Philosophy in Paris and another colloquium in Munich.I would like to thank the organizers of these events for their kind invitations, warm hospitality and helpful discussions, as well as the audiences at these meeting for useful comments.I would also like to thank Jonas Rafael Becker Arenhart and Fabio De Martin Polo for helpful discussions and comments.Finally,