Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯ , the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson’s paraconsistent N4, this also supplements Nelson’s own proof of the Post consistency of N4♯ . Insofar as the present technique allows infinite models, this resolves Nelson’s concern that N4♯ is of interest only to those accepting that there are finitely many natural numbers

Bi-Classical Connexive Logic and its Modal Extension: Cut-elimination, completeness and duality

In this study, a new paraconsistent four-valued logic called bi-classical connexive logic (BCC) is introduced as a Gentzen-type sequent calculus. Cut-elimination and completeness theorems for BCC are proved, and it is shown to be decidable. Duality property for BCC is demonstrated as its characteristic property. This property does not hold for typical paraconsistent logics with an implication connective. The same results as those for BCC are also obtained for MBCC, a modal extension of BCC

Problema de la adopción: ¿un problema para un pluralismo respecto de la negación lógica?

Departing from a number of theses about the meaning of logical negation, the present work offers a reflection about the relationship between logical pluralism and the status of some fundamental logical principles. The aim is to show that the Adoption Problem, such as it has been formulated by Kripke/Padró, does not represent a challenge for the anti-exceptionalism about logic. Logical laws don’t have a special status, even if there exist some laws that can’t be adopted, because we are able to abandon some of them and it is dropping rules, and not adopting them the way in which the logic is revised. Therefore, we can accept a kind of logical pluralism that gives us more than one set of principles that captures correctly the meaning of negation.Partiendo de una serie de tesis respecto del significado de la negación lógica, se ofrece una reflexión acerca de la relación entre el pluralismo lógico y el estatus de ciertos principios lógicos fundamentales. El objetivo de este trabajo es mostrar que el Problema de la Adopción, tal como se encuentra formulado por Kripke-Padró, no representa un conflicto para una visión antiexcepcionalista de la lógica. Las leyes lógicas no poseen un status privilegiado, independientemente de que existan algunas que no podemos adoptar, porque sí podemos decidir abandonar principios lógicos y es abandonando reglas y no adoptándolas como se revisa una teoría lógica; en consecuencia es aceptable un tipo de pluralismo lógico que parta de considerar que más de un conjunto de principios capturan de manera adecuada el significado de la negación