Let S be a logic and T a theory built upon S. We consider four concepts of consistency. A theory T is w1-inconsistent iff ¬A ∈ T, A being a theorem of S; T is w2-inconsistent iff A ∈ T, ¬A being a theorem of S; T is n-inconsistent iff it contains a contradiction, and, finally, T is a-inconsistent iff it contains all well-formed formulas. In each case, T is consistent iff it is not inconsistent. On the other hand, by a "constructive negation" we understand any negation included in intuitionistic negation; and by a "basic logic" it is understood a "minimal logic". Finally, ternary relational semantics with a set of designated points (trsw) are the semantics for relevance logics defined by Routley and Meyer in the early seventies of the past century.
Our aim is to define the basic constructive logic (as well as its positive and negation extensions) adequate to each of the concepts of consistency noted above in the trsw.
Negation will be introduced with the unary connective as well as by means of a propositional falsity constant.
This project was based at the Universidad de Salamanca and was funded by the Spanish Ministry of Science and Innovation (MICINN) [grant no. FFI2008-05859], from 2009 to 2011.