Resumen Este artículo proporciona una motivación inferencialista para una lógica perteneciente a la familia conexiva, tomando prestados elementos de la interpretación bilateralista de la Lógica Clásica sin la regla de Corte, propuesta por David Ripley. El artículo se centra en la relación entre inferencialismo y relevancia, a través de la exploración de lo que llamamos aserción y negación relevantes, mostrando que un sistema conexivo emerge como síntoma de este interesante vínculo. Con el presente intento, esperamos ampliar las interpretaciones disponibles para las lógicas conexivas, mostrando que pueden estar motivadas legítimamente en términos de ciertas restricciones relevantes impuestas a la aserción y la negación.

Abstract This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance, through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.

Abstract In the literature, Weak Kleene logics are usually taken as three-valued logics. However, Suszko has challenged the main idea of many-valued logic claiming that every logic can be presented in a two-valued fashion. In this paper, we provide two-valued semantics for the Weak Kleene logics and for a number of four-valued subsystems of them. We do the same for the so-called Logics of Nonsense, which are extensions of the Weak Kleene logics with unary operators that allow looking at them as Logics of Formal Inconsistency (LFIs) and Logics of Formal Underterminedness (LFUs). Our aim with this work, rather than arguing for Suszko’s thesis, is to show that two-valued presentations of these peculiar logics enlighten the non-standard behavior of their logical connectives. More specifically, the two-valued presentations of paraconsistent logics illustrate and clarify the disjunctive flavor of the conjunction, and dually, the two-valued presentations of paracomplete subsystems of Weak Kleene logics reveal the conjunctive flavor of the disjunction.