Summary

Let S be a logic and T a theory built upon S. T is w-inconsistent (inconsistent in a weak sense) iff ¬A ∈ T, A being a theorem of S. T is w-consistent iff T is not w-inconsistent. Ternary relational semantics (trs) is the semantics for relevant logics defined by Routley and Meyer in the early seventies of the past century. Trs without a designated set of points (trs0) is the result of generalizing trs by dropping the designated set characterisitic of trs-models. The Routley operator is the unary operator by means of which the De Morgan negation representative of relevant logic is modelized. 

The aim of the present project of investigation is to define the basic logic (as well as its positive and negative extensions) in the trs0 adequate to w-consistency when negation is introduced in the models with the Routley operator. All the logics to be defined are neither relevant (all of them contain paradoxes of relevance) nor constructive in an intuitionistic sense (none of them is included in intuitionistic logic).

This project was based at the Universidad de Salamanca and was funded by the Spanish Ministry of Economy and Competitiviness (MINECO) [Project FFI2011-28494]. It ran from 2012 till 2014.